Christoph Schiller
MOTION MOUNTAIN
www.motionmountain.net
Christoph Schiller
M
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e Adventure of Physics
available free of charge at www.motionmountain.net
Editio undevicesima.
Proprietas scriptoris © Christophori Schiller secundo anno Olympiadis vicesimae sextae tertio anno Olympiadis vicesimae octavae.
Omnia proprietatis iura reservantur et vindicantur. Imitatio prohibita sine auctoris permissione. Non licet pecuniam expetere pro aliquo, quod partem horum verborum continet; liber pro omnibus semper gratuitus erat et manet.
Nineteenth revision.
Copyright © – by Christoph Schiller, between the second year of the th olympiad and the third year of the th olympiad.
All rights reserved. Commercial reproduction, distribution or use, in whole or in part, is not allowed without the written consent of the copyright owner. You are not allowed to charge money for anything containing any part of this text; it was and remains free to read for everybody. Details of the cover photographs are on page .
To Esther τῷ ἐµοὶ δαὶµονι
Die Menschen stärken, die Sachen klären.
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Preface
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A request
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An appetizer
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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1. Why should we care about motion?
2. Galilean physics – motion in everyday life
3. Global descriptions of motion – the simplicity of complexity 4. From the limitations of physics to the limits of motion
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5. Maximum speed, observers at rest, and motion of light
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6. Maximum force – general relativity in one statement
7. e new ideas on space, time and gravity
8. Motion in general relativity – bent light and wobbling vacuum
9. Why can we see the stars? – Motion in the universe
10. Black holes – falling forever
11. Does space di er from time?
12. General relativity in ten points – a summary for the layman
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13. Liquid electricity, invisible elds and maximum speed
14. What is light?
15. Charges are discrete – the limits of classical electrodynamics
16. Electromagnetic e ects and challenges
17. Classical physics in a nutshell – one and a half steps out of three
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18. Minimum action – quantum theory for poets and lawyers
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19. Light – the strange consequences of the quantum of action
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20. Motion of matter – beyond classical physics
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21. Colours and other interactions between light and matter
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22. Are particles like gloves?
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23. Rotations and statistics – visualizing spin
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24. Superpositions and probabilities – quantum theory without ideology
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25. Applied quantum mechanics – life, pleasure and the means to achieve them
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
26. Quantum electrodynamics – the origin of virtual reality 27. Quantum mechanics with gravitation – the rst approach
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28. e structure of the nucleus – the densest clouds
29. e strong nuclear interaction and the birth of matter
30. e weak nuclear interaction and the handedness of nature
31. e standard model of elementary particle physics – as seen on television
32. Grand uni cation – a simple dream
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33. Does matter di er from vacuum?
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34. Nature at large scales – is the universe something or nothing?
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35. e physics of love – a summary of the rst two and a half parts
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36. Maximum force and minimum distance – physics in limit statements
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37. e shape of points – extension in nature
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38. String theory – a web of dualities
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20 Preface 21 A request 23 An appetizer
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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28 1. Why should we care about motion?
Does motion exist? 30 • How should we talk about motion? 32 • What are the
types of motion? 33 • Perception, permanence and change 36 • Does the world
need states? 39 • Curiosities and fun challenges about motion 40
42 2. Galilean physics – motion in everyday life
What is velocity? 43 • What is time? 45 • Why do clocks go clockwise? 48 • Does
time ow? 49 • What is space? 49 • Are space and time absolute or relative? 52 •
Size – why area exists, but volume does not 53 • What is straight? 56 • A hollow
Earth? 56 • Curiosities and fun challenges about everyday space and time 57
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How to describe motion – kinematics
rowing and shooting 66 • What is rest? 68 • Objects and point particles 71 •
Legs and wheels 73
74
Objects and images
Motion and contact 75 • What is mass? 77 • Is motion eternal? 82 • More on
conservation – energy 84 • Is velocity absolute? – e theory of everyday relativ-
ity 86 • Rotation 88 • Rolling wheels 91 • How do we walk? 92 • Is the Earth
rotating? 94 • How does the Earth rotate? 98 • Does the Earth move? 100 • Is
rotation relative? 104 • Curiosities and fun challenges about everyday motion 104
• Legs or wheels? – Again 114
116 Dynamics due to gravitation
Properties of gravitation 119 • Dynamics – how do things move in various dimen-
sions? 123 • Gravitation in the sky 123 • e Moon 124 • Orbits 127 • Tides 129
• Can light fall? 132 • What is mass? – Again 133 • Curiosities and fun challenges
about gravitation 135
146 What is classical mechanics?
Should one use force? 147 • Complete states – initial conditions 152 • Do surprises
exist? Is the future determined? 154 • A strange summary about motion 157
158 Bibliography
173 3. Global descriptions of motion – the simplicity of complexity
176 Measuring change with action
e principle of least action 179 • Why is motion so o en bounded? 183 • Curios-
ities and fun challenges about Lagrangians 186
189 Motion and symmetry
Why can we think and talk? 189 • Viewpoints 190 • Symmetries and groups 192
• Representations 192 • Symmetries, motion and Galilean physics 195 • Reprodu-
cibility, conservation and Noether’s theorem 198 • Curiosities and fun challenges
about motion symmetry 203
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203 Simple motions of extended bodies – oscillations and waves Waves and their motion 205 • Why can we talk to each other? – Huygens’ principle 210 • Signals 210 • Solitary waves and solitons 212 • Curiosities and fun challenges about waves and extended bodies 214
218 Do extended bodies exist? Mountains and fractals 219 • Can a chocolate bar last forever? 219 • How high can animals jump? 221 • Felling trees 221 • e sound of silence 222 • Little hard balls 223 • Curiosities and fun challenges about uids and solids 225
233 What can move in nature?
234 How do objects get warm?
235 Temperature Entropy 237 • Flow of entropy 239 • Do isolated systems exist? 240 • Why do balloons take up space? – e end of continuity 240 • Brownian motion 242 • Entropy and particles 244 • e minimum entropy of nature – the quantum of information 245 • Why can’t we remember the future? 246 • Is everything made of particles? 247 • Why stones can be neither smooth nor fractal, nor made of little hard balls 248 • Curiosities and fun challenges about heat 249
255 Self-organization and chaos Curiosities and fun challenges about self-organization 261
263 4. From the limitations of physics to the limits of motion Research topics in classical dynamics 263 • What is contact? 264 • Precision and accuracy 264 • Can all of nature be described in a book? 265 • Why is measurement possible? 265 • Is motion unlimited? 266
267 Bibliography
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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275 5. Maximum speed, observers at rest, and motion of light Can one play tennis using a laser pulse as the ball and mirrors as rackets? 280 • Special relativity in a few lines 283 • Acceleration of light and the Doppler e ect 284 • e di erence between light and sound 287 • Can one shoot faster than one’s shadow? 287 • e composition of velocities 289 • Observers and the principle of special relativity 290 • What is space-time? 294 • Can we travel to the past? –
Time and causality 295
297 Curiosities of special relativity Faster than light: how far can we travel? 297 • Synchronization and time travel – can a mother stay younger than her own daughter? 297 • Length contraction 300 • Relativistic lms – aberration and Doppler e ect 302 • Which is the best seat in a bus? 305 • How fast can one walk? 306 • Is the speed of shadow greater than the speed of light? 306 • Parallel to parallel is not parallel – omas rotation 309 • A
never-ending story – temperature and relativity 310
310 Relativistic mechanics Mass in relativity 310 • Why relativistic snooker is more di cult 312 • Mass is concentrated energy 313 • Collisions, virtual objects and tachyons 315 • Systems of particles – no centre of mass 317 • Why is most motion so slow? 317 • e history of the mass–energy equivalence formula of de Pretto and Einstein 318 • 4vectors 319 • 4-momentum 322 • 4-force 323 • Rotation in relativity 324 • Wave motion 325 • e action of a free particle – how do things move? 326 • Conformal
transformations – why is the speed of light constant? 327
329 Accelerating observers
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Acceleration for inertial observers 330 • Accelerating frames of reference 331 • Event horizons 335 • Acceleration changes colours 336 • Can light move faster than c? 337 • What is the speed of light? 338 • Limits on the length of solid bodies 339
340 Special relativity in four sentences Could the speed of light vary? 340 • What happens near the speed of light? 341
342 Bibliography
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349 6. Maximum force – general relativity in one statement e maximum force and power limits 350 • e experimental evidence 352 • Dedu-
cing general relativity 353 • Space-time is curved 357 • Conditions of validity of the force and power limits 359 • Gedanken experiments and paradoxes about the force limit 359 • Gedanken experiments with the power limit and the mass ow limit 364 • Hide and seek 367 • An intuitive understanding of general relativity 368 • An intuitive understanding of cosmology 370 • Experimental challenges for the third millennium 371 • A summary of general relativity 372 • Acknowledgement 373
374 Bibliography
377 7. e new ideas on space, time and gravity Rest and free fall 377 • What is gravity? – A second answer 378 • What tides tell us about gravity 381 • Bent space and mattresses 382 • Curved space-time 384 • e speed of light and the gravitational constant 386 • Why does a stone thrown into the air fall back to Earth? – Geodesics 388 • Can light fall? 390 • Curiosities and fun challenges about gravitation 391 • What is weight? 395 • Why do apples fall? 396
397 8. Motion in general relativity – bent light and wobbling vacuum
397 Weak elds e irring e ects 397 • Gravitomagnetism 399 • Gravitational waves 402 •
Bending of light and radio waves 409 • Time delay 410 • E ects on orbits 411 • e geodesic e ect 414 • Curiosities and fun challenges about weak elds 414
415 How is curvature measured? Curvature and space-time 418 • Curvature and motion in general relativity 420 • Universal gravity 421 • e Schwarzschild metric 421 • Curiosities and fun challenges about curvature 422
422 All observers – heavier mathematics e curvature of space-time 422 • e description of momentum, mass and en-
ergy 424 • Hilbert’s action – how things fall? 425 • e symmetries of general relativity 426 • Einstein’s eld equations 427 • More on the force limit 430 • Deducing universal gravity 430 • Deducing linearized general relativity 431 • How to calculate the shape of geodesics 431 • Mass in general relativity 433 • Is gravity an interaction? 433 • e essence of general relativity 434 • Riemann gymnastics 435 • Curiosities and fun challenges about general relativity 437
437 9. Why can we see the stars? – Motion in the universe Which stars do we see? 438 • What do we see at night? 439 • What is the universe? 445 • e colour and the motion of the stars 447 • Do stars shine every night? 449 • A short history of the universe 450 • e history of space-time 454 • Why is the sky dark at night? 458 • Is the universe open, closed or marginal? 460 • Why is the universe transparent? 461 • e big bang and its consequences 462 • Was the big bang a big bang? 463 • Was the big bang an event? 463 • Was the big bang a beginning? 463 • Does the big bang imply creation? 464 • Why can we see the Sun? 465 • Why are the colours of the stars di erent? 466 • Are there dark
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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stars? 467 • Are all stars di erent? – Gravitational lenses 467 • What is the shape of the universe? 469 • What is behind the horizon? 470 • Why are there stars all over the place? – In ation 471 • Why are there so few stars? – e energy and entropy content of the universe 471 • Why is matter lumped? 473 • Why are stars so small compared with the universe? 473 • Are stars and galaxies moving apart or is the universe expanding? 473 • Is there more than one universe? 473 • Why are the stars xed? – Arms, stars and Mach’s principle 474 • At rest in the universe 475 • Does light attract light? 475 • Does light decay? 476
476 10. Black holes – falling forever Why study black holes? 476 • Horizons 477 • Orbits 479 • Hair and entropy 482 • Black holes as energy sources 484 • Curiosities and fun challenges about black holes 485 • Formation of and search for black holes 488 • Singularities 489 • A quiz – is the universe a black hole? 490
491 11. Does space di er from time? Can space and time be measured? 492 • Are space and time necessary? 493 • Do closed timelike curves exist? 494 • Is general relativity local? – e hole argument 494 • Is the Earth hollow? 495 • Are space, time and mass independent? 496
497 12. General relativity in ten points – a summary for the layman e accuracy of the description 498 • Research in general relativity and cosmo-
logy 499 • Could general relativity be di erent? 501 • e limits of general relativity 502
503 Bibliography
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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517 13. Liquid electricity, invisible elds and maximum speed
517 Amber, lodestone and mobile phones How can one make lightning? 520 • Electric charge and electric elds 522 • Can we detect the inertia of electricity? 526 • Feeling electric elds 528 • Magnets 529 • Can humans feel magnetic elds? 530 • How can one make a motor? 531 • Magnetic elds 532 • How motors prove relativity to be right 536 • Curiosities and fun
challenges about things electric and magnetic 537
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e description of electromagnetic eld evolution
Colliding charged particles 547 • e gauge eld – the electromagnetic vector poten-
tial 548 • Energy and linear and angular momentum of the electromagnetic eld 552
• e Lagrangian of electromagnetism 552 • Symmetries – the energy–momentum
tensor 554 • What is a mirror? 554 • What is the di erence between electric and
magnetic elds? 556
557 Electrodynamic challenges and curiosities Could electrodynamics be di erent? 557 • e toughest challenge for electrodyna-
mics 558
558 14. What is light? e slowness of progress in physics 566 • How does the world look when riding on
a light beam? 567 • Does light travel in a straight line? 567 • e concentration of light 571 • Can one touch light? 572 • War, light and lies 574 • What is colour? 574 • What is the speed of light? – Again 577 • 200 years too late – negative refraction indices 579 • Signals and predictions 580 • Does the aether exist? 581
582 Curiosities and fun challenges about light How to prove you’re holy 582 • Do we see what exists? 583 • How does one make pictures of the inside of the eye? 585 • How does one make holograms and other
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
three-dimensional images? 586 • Imaging 588 • Light as weapon? 589
589 15. Charges are discrete – the limits of classical electrodynamics How fast do charges move? 590 • Challenges and curiosities about charge discreteness 591
592 16. Electromagnetic e ects and challenges Is lightning a discharge? – Electricity in the atmosphere 597 • Does gravity make charges radiate? 600 • Research questions 600 • Levitation 602 • Matter, levitation and electromagnetic e ects 604 • Why can we see each other? 612 • A summary of classical electrodynamics and of its limits 614
614 17. Classical physics in a nutshell – one and a half steps out of three e future of planet Earth 616 • e essence of classical physics – the in nitely small
implies the lack of surprises 618 • Why have we not yet reached the top of the mountain? 618 620 Bibliography
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Evolution 632
Children and physics Why a brain? 635 • What is information? 636 • What is memory? 637 • e capa-
city of the brain 639
What is language? What is a concept? 644 • What are sets? What are relations? 646 • In nity 648 • Functions and structures 650 • Numbers 651 • Why use mathematics? 656 • Is mathematics a language? 657 • Curiosities and fun challenges about mathemat-
ics 657
Physical concepts, lies and patterns of nature Are physical concepts discovered or created? 660 • How do we nd physical patterns and rules? 661 • What is a lie? 662 • Is this statement true? 667 • Curiosities and
fun challenges about lies 668
Observations Have enough observations been recorded? 670 • Are all physical observables known? 671 • Do observations take time? 673 • Is induction a problem in phys-
ics? 673
e quest for precision and its implications What are interactions? – No emergence 676 • What is existence? 677 • Do things exist? 678 • Does the void exist? 679 • Is nature in nite? 680 • Is the universe a set? 681 • Does the universe exist? 682 • What is creation? 683 • Is nature designed? 685 • What is a description? 686 • Reason, purpose and explanation 687 • Uni cation and demarcation 688 • Pigs, apes and the anthropic principle 689 • Does one need cause and e ect in explanations? 691 • Is consciousness required? 691 • Curiosity 692 • Courage 694
Bibliography
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704 18. Minimum action – quantum theory for poets and lawyers
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e e ects of the quantum of action on motion 705 • e consequences of the quantum of action for objects 706 • What does ‘quantum’ mean? 708 • Quantum surprises 710 • Waves 714 • Information 715 • Curiosities and fun challenges
about the quantum of action 716
717 19. Light – the strange consequences of the quantum of action What is colour? 717 • What is light? – Again 720 • e size of photons 721 • Are photons countable? – Squeezed light 722 • e position of photons 724 • Are photons necessary? 726 • How can a wave be made up of particles? 727 • Can light move faster than light? – Virtual photons 733 • Indeterminacy of electric elds 734 • Curiosities and fun challenges about photons 734
735 20. Motion of matter – beyond classical physics Wine glasses and pencils 735 • Cool gas 736 • No rest 736 • Flows and the quantization of matter 737 • Quantons 737 • e motion of quantons – matter as waves 738 • Rotation and the lack of North Poles 740 • Silver, Stern and Gerlach 742 • e language of quantum theory and its description of motion 743 • e state – or wave function – and its evolution 745 • Why are atoms not at? Why do shapes exist? 747 • Rest – spread and the quantum Zeno e ect 747 • Tunnelling, hills and limits on memory 748 • Spin and motion 749 • Relativistic wave equations 750 • Maximum acceleration 751 • Curiosities and fun challenges about
quantum theory 751
753 21. Colours and other interactions between light and matter What are stars made of? 753 • What determines the colour of atoms? 754 • Relativistic hydrogen 756 • Relativistic wave equations – again 757 • Antimatter 759 • Virtual particles and QED diagrams 759 • Compositeness 760 • Curiosities and fun challenges about colour 761 • e strength of electromagnetism 762
764 Bibliography
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771 22. Are particles like gloves? Why does indistinguishability appear in nature? 773 • Can particles be counted? 773 • What is permutation symmetry? 774 • Indistinguishability and symmetry 775 • e behaviour of photons 776 • e energy dependence of permutation symmetry 776 • Indistinguishability in quantum eld theory 777 • How accurately is permutation symmetry veri ed? 778 • Copies, clones and gloves 778
780 23. Rotations and statistics – visualizing spin e belt trick 783 • e Pauli exclusion principle and the hardness of matter 785 •
Integer spin 786 • Is spin a rotation about an axis? 787 • Why is fencing with laser beams impossible? 787 • Rotation requires antiparticles 788 • Limits and open
questions of quantum statistics 789
791 Bibliography
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793 24. Superpositions and probabilities – quantum theory without ideology 794 Why are people either dead or alive?
Conclusions on decoherence, life and death 799 800 What is a system? What is an object?
Is quantum theory non-local? – A bit about the Einstein-Podolsky-Rosen paradox 801 • Curiosities and fun challenges about superpositions 803 805 What is all the fuss about measurements in quantum theory? Hidden variables 809
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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811 Conclusions on probabilities and determinism What is the di erence between space and time? 813 • Are we good observers? 814 • What connects information theory, cryptology and quantum theory? 814 • Does
the universe have a wave function? And initial conditions? 815
816 25. Applied quantum mechanics – life, pleasure and the means to achieve them
816 Biology Reproduction 817 • Quantum machines 818 • How do we move? – Molecular motors 820 • Curiosities and fun challenges about biology 822
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e nerves and the brain 827 • Clocks in quantum mechanics 827 • Do clocks
exist? 828 • Living clocks 830 • Metre sticks 831 • Why are predictions so di cult,
especially of the future? 831 • Decay and the golden rule 832 • Zeno and the present
in quantum theory 833 • What is motion? 834 • Consciousness – a result of the
quantum of action 834 • Why can we observe motion? 835 • Curiosities and fun
challenges about quantum experience 835
837 Chemistry – from atoms to DNA Ribonucleic acid and Deoxyribonucleic acid 838 • Curiosities and fun challenges
about chemistry 839
840 Materials science Why does the oor not fall? 840 • Rocks and stones 841 • How can one look through matter? 842 • What is necessary to make matter invisible? 842 • How does matter behave at the lowest temperatures? 844 • Curiosities and fun challenges about
materials science 844
846 Quantum technology Motion without friction – superconductivity and super uidity 847 • Quantized conductivity 848 • e fractional quantum Hall e ect 849 • Lasers and other spin-one vector boson launchers 849 • Can two photons interfere? 851 • Can two electron beams interfere? 852 • Challenges and dreams about quantum technology 853
854 26. Quantum electrodynamics – the origin of virtual reality Ships, mirrors and the Casimir e ect 854 • e Banach–Tarski paradox for vacuum 857 • e Lamb shi 857 • e QED Lagrangian 857 • Interactions and virtual particles 857 • Vacuum energy 858 • Moving mirrors 858 • Photons hitting photons 858 • Is the vacuum a bath? 859 • Renormalization – why is an electron
so light? 860
860 Curiosities and fun challenges of quantum electrodynamics
How can one move on perfect ice? – e ultimate physics test 864
865 Summary of quantum electrodynamics
Open questions in QED 867
868 27. Quantum mechanics with gravitation – the rst approach Corrections to the Schrödinger equation 869 • A rephrased large number hypothesis 869 • Is quantum gravity necessary? 870 • Limits to disorder 870 • Measur-
ing acceleration with a thermometer – Fulling–Davies–Unruh radiation 871
872 Black holes aren’t black Gamma ray bursts 875 • Material properties of black holes 877 • How do black holes evaporate? 878 • e information paradox of black holes 878 • More para-
doxes 879
880 Quantum mechanics of gravitation e gravitational Bohr atom 880 • Decoherence of space-time 881 • Do gravitons
exist? 881 • Space-time foam 882 • No particles 883 • No science ction 883 •
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Not cheating any longer 883 884 Bibliography
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897 28. e structure of the nucleus – the densest clouds A physical wonder – magnetic resonance imaging 897 • e size of nuclei 898 • Nuclei are composed 902 • Nuclei can move alone – cosmic rays 904 • Nuclei decay 908 • Why is hell hot? 910 • Nuclei can form composites 911 • Nuclei have colours and shapes 911 • Motion in the nuclear domain – four types of motion 913 • Nuclei react 913 • Bombs and nuclear reactors 914 • e Sun 915 • Curiosities and fun challenges on radioactivity 917
921 29. e strong nuclear interaction and the birth of matter Why do the stars shine? 921 • Why are fusion reactors not common yet? 923 • Where do our atoms come from? 926 • e weak side of the strong interaction 926 • Bound motion, the particle zoo and the quark model 927 • e mass, shape and colour of protons 928 • Experimental consequences of the quark model 929 • e Lagrangian of quantum chromodynamics 930 • e sizes and masses of quarks 933 • Con nement and the future of the strong interaction 933 • Curiosities about the strong interactions 934
935 30. e weak nuclear interaction and the handedness of nature Curiosities about the weak interactions 936 • Mass, the Higgs boson and a ten thousand million dollar lie 937 • Neutrinium and other curiosities on the electroweak interaction 938
940 31. e standard model of elementary particle physics – as seen on television Conclusion and open questions about the standard model 940
941 32. Grand uni cation – a simple dream Experimental consequences 941 • e state of grand uni cation 942
945 Bibliography
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951 Quantum theory’s essence – the lack of the in nitely small
Achievements in precision 951 • Physical results of quantum theory 953 • Results
of quantum eld theory 955 • Is quantum theory magic? 956 • e dangers of
buying a can of beans 957
959
e essence and the limits of quantum theory
What is unexplained by quantum theory and general relativity? 959 • How to delude
oneself that one has reached the top of Motion Mountain 961 • What awaits us? 964
967 Bibliography
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Bumblebees and other miniature ying systems 969 • Swimming 972 • Falling cats and the theory of shape change 976 • Turning a sphere inside out 976 • Knots, links and braids 978 • Knots in nature and on paper 980 • Clouds 983 • A one-dimensional classical analogue of the Schrödinger equation 983 • Fluid spacetime 987 • Solid space-time 987 • Swimming in curved space 989 • Curiosities and fun challenges on wobbly entities 990 • Outlook 991
992 Bibliography
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e contradictions 996
998 33. Does matter di er from vacuum? Planck scales 999 • Farewell to instants of time 1001 • Farewell to points in space 1003 • Farewell to the space-time manifold 1005 • Farewell to observables and measurements 1008 • Can space-time be a lattice? – A glimpse of quantum geometry 1009 • Farewell to particles 1010 • Farewell to mass 1013 • Curiosities and fun challenges on Planck scales 1016 • Farewell to the big bang 1020 • e baggage
le behind 1020
1021 Some experimental predictions
1026 Bibliography
1032 34. Nature at large scales – is the universe something or nothing? Cosmological scales 1032 • Maximum time 1033 • Does the universe have a de nite age? 1033 • How precise can age measurements be? 1034 • Does time exist? 1035 • What is the error in the measurement of the age of the universe? 1036 • Maximum length 1039 • Is the universe really a big place? 1040 • e boundary of space-time – is the sky a surface? 1041 • Does the universe have initial conditions? 1042 • Does the universe contain particles and stars? 1042 • Does the universe contain masses and objects? 1043 • Do symmetries exist in nature? 1045 • Does the universe have a boundary? 1045 • Is the universe a set? – Again 1046 • Curiosities and fun challenges about the universe 1048 • Hilbert’s sixth problem settled 1049 • Does the universe make sense? 1049 • A concept without a set eliminates contradictions 1051 • Extremal scales and open questions in physics 1051 • Is extremal identity a prin-
ciple of nature? 1051
1054 Bibliography
1056 35. e physics of love – a summary of the rst two and a half parts 1067 Bibliography
1068 36. Maximum force and minimum distance – physics in limit statements
1068 Fundamental limits to all observables Special relativity in one statement 1068 • Quantum theory in one statement 1069 • General relativity in one statement 1071 • Deducing general relativity 1072 • Deducing universal gravitation 1075 • e size of physical systems in general relativity 1075 • A mechanical analogy for the maximum force 1075
1076 Units and limit values for all physical observables Limits to space and time 1078 • Mass and energy limits 1078 • Virtual particles – a new de nition 1079 • Limits in thermodynamics 1079 • Electromagnetic limits and units 1080 • Vacuum and mass–energy – two sides of the same coin 1081 •
Curiosities and fun challenges about Planck limits 1082
1085 Upper and lower limits to observables Size and energy dependence 1085 • Angular momentum, action and speed 1085 • Force, power and luminosity 1086 • Acceleration 1087 • Momentum 1087 • Lifetime, distance and curvature 1088 • Mass change 1088 • Mass and density 1088 • e strange charm of the entropy bound 1089 • Temperature 1090 • Electromagnetic observables 1091 • Curiosities and fun challenges about limits to observ-
ables 1092
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
1093 Limits to measurement precision and their challenge to thought Measurement precision and the existence of sets 1093 • Why are observers needed? 1094 • A solution to Hilbert’s sixth problem 1095 • Outlook 1095
1097 Bibliography
1101 37. e shape of points – extension in nature Introduction – vacuum and particles 1102 • How else can we show that matter and vacuum cannot be distinguished? 1103
1104 Argument 1: e size and shape of elementary particles Do boxes exist? 1105 • Can the Greeks help? – e limits of knives 1105 • Are cross-sections nite? 1106 • Can one take a photograph of a point? 1106 • What is the shape of an electron? 1108 • Is the shape of an electron xed? 1109
1110 Argument 2: e shape of points in vacuum Measuring the void 1111 • What is the maximum number of particles that ts inside a piece of vacuum? 1111
1112 Argument 3: e large, the small and their connection Small is large? 1113 • Uni cation and total symmetry 1113
1115 Argument 4: Does nature have parts? Does the universe contain anything? 1117 • An amoeba 1117
1118 Argument 5: e entropy of black holes 1120 Argument 6: Exchanging space points or particles at Planck scales 1121 Argument 7: e meaning of spin 1122 Present research
Conceptual checks of extension 1123 • Experimental falsi cation of models based on extended entities 1124 • Possibilities for con rmation of extension models 1124 • Curiosities and fun challenges about extension 1125 • An intermediate status report 1127 • Sexual preferences in physics 1127 • A physical aphorism 1127
1128 38. String theory – a web of dualities 1129 Strings and membranes – why string theory is so di cult 1129 Matrix models and M-theory
Masses and couplings 1130 • Outlook 1130 1132 Bibliography
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1137 C
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e Latin alphabet 1142 • e Greek alphabet 1143 • e Hebrew alphabet and other scripts 1146 • Digits and numbers 1146 • e symbols used in the text 1147 • Calendars 1149 • Abbreviations and eponyms or concepts? 1151
1152 Bibliography
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Planck’s natural units 1157 • Other unit systems 1158 • Curiosities and fun challenges about units 1160 • Precision and accuracy of measurements 1164 • Basic physical constants 1165 • Useful numbers 1170
1171 Bibliography
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1194 A
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1194 Numbers as mathematical structures Complex numbers 1196 • Quaternions 1197 • Octonions 1202 • Other types of numbers 1204 • Grassmann numbers 1204
1204 Vector spaces
1207 Algebras Lie algebras 1209 • Classi cation of Lie algebras 1210 • Lie superalgebras 1211 • e Virasoro algebra 1212 • Kac–Moody algebras 1212
1213 Topology – what shapes exist? Topological spaces 1213 • Manifolds 1214 • Holes, homotopy and homology 1216
1217 Types and classi cation of groups Lie groups 1218 • Connectedness 1219 • Compactness 1220
1224 Mathematical curiosities and fun challenges
1226 Bibliography
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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P
“Primum movere, deinde docere.
” Antiquity
e intensity with which small children explore their environment suggests that there is
a drive to grasp the way the world works, a ‘physics instinct’, built into each of us. What
would happen if this drive, instead of being sti ed during school education, as it usually
is, were allowed to thrive in an environment without bounds, reaching from the atoms
to the stars? Probably most adolescents would then know more about nature than most
senior physics teachers today. is text tries to provide this possibility to the reader. It
acts as a guide in an exploration, free of all limitations, of physics, the science of motion.
e project is the result of a threefold aim I have pursued since : to present the basics
of motion in a way that is simple, up to date and vivid.
In order to be simple, the text focuses on concepts, while keeping mathematics to the
necessary minimum. Understanding the concepts of physics is given precedence over
using formulae in calculations. e whole text is within the reach of an undergraduate. It
presents simple summaries of the main domains of physics.
ere are three main stages in the physical description of motion. First, there is every-
day physics, or classical continuum physics. It is based on the existence of the in nitely
small and the in nitely large. In the second stage, each domain of physics is centred
around a basic inequality for the main observable. us, statistical thermodynamics lim-
its entropy by S k ; special relativity limits speeds by v c; general relativity limits
force by F c G; quantum theory limits action by L ħ ; and quantum electrody-
namics limits change of charge by ∆q e. ese results, though not so well known, are
proved rigorously. It is shown that within each domain, the principal equations follow
from the relevant limit. Basing the domains of physics on limit principles allows them to
be introduced in a simple, rapid and intuitive way. e third and nal stage is the uni c-
ation of all these limits in a single description of motion. is unusual way of learning
physics should reward the curiosity of every reader – whether student or professional.
In order to be up to date, the text includes discussions of quantum gravity, string theory
and M theory. Meanwhile, the standard topics – mechanics, electricity, light, quantum
theory, particle physics and general relativity – are enriched by the many gems – both
theoretical and empirical – that are scattered throughout the scienti c literature.
In order to be vivid, a text must be challenging, questioning and daring. is text tries
to startle the reader as much as possible. Reading a book on general physics should be like
going to a magic show. We watch, we are astonished, we do not believe our eyes, we think,
and nally – maybe – we understand the trick. When we look at nature, we o en have
the same experience. e text tries to intensify this by following a simple rule: on each
page, there should be at least one surprise or provocation for the reader to think about.
Numerous interesting challenges are proposed. Hints or answers to these are given in an
appendix.
e strongest surprises are those that seem to contradict everyday experience. Most
of the surprises in this text are taken from daily life: in particular, from the the things
one experiences when climbing a mountain. Observations about trees, stones, the Moon,
the sky and people are used wherever possible; complex laboratory experiments are men-
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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tioned only where necessary. ese surprises are organized so as to lead in a natural way to the most extreme conclusion of all, namely that continuous space and time do not exist.
e concepts of space and time, useful as they may be in everyday life, are only approximations. Indeed, they turn out to be mental crutches that hinder the complete exploration of the world.
Giving full rein to one’s curiosity and thought leads to the development of a strong and dependable character. e motto of the text, a famous statement by Harmut von Hentig on pedagogy, translates as: ‘To clarify things, to fortify people.’ Exploring any limit requires courage; and courage is also needed to abandon space and time as tools for the description of the world. Changing habits of thought produces fear, o en hidden by anger; but we grow by overcoming our fears. Achieving a description of the world without the use of space and time may be the most beautiful of all adventures of the mind.
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Eindhoven and other places, May
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
A e text is and remains free for everybody. In exchange for getting the le for free, please
send me a short email on the following issues: — What was unclear? — What did you miss? Challenge 1 ny — What should be improved or corrected? Feedback on the speci c points listed on the http://www.motionmountain.net/project. html web page is most welcome of all. On behalf of myself and all other readers, thank you in advance for your input. For a particularly useful contribution you will be mentioned – if you want – in the acknowledgements, receive a reward, or both. But above all, enjoy the reading.
C. Schiller fb@motionmountain.net
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
A
Page 1295
Many people who have kept their gi of curiosity alive have helped to make this project come true. Most of all, Saverio Pascazio has been – present or not – a constant reference for this project. Fernand Mayné, Anna Koolen, Ata Masafumi, Roberto Crespi, Luca Bombelli, Herman Elswijk, Marcel Krijn, Marc de Jong, Martin van der Mark, Kim Jalink, my parents Peter and Isabella Schiller, Mike van Wijk, Renate Georgi, Paul Tegelaar, Barbara and Edgar Augel, M. Jamil, Ron Murdock, Carol Pritchard and, most of all, my wife Britta have all provided valuable advice and encouragement.
Many people have helped with the project and the collection of material. Most useful was the help of Mikael Johansson, Bruno Barberi Gnecco, Lothar Beyer, the numerous improvements by Bert Sierra, the detailed suggestions by Claudio Farinati, the many improvements by Eric Sheldon, the continuous help and advice of Jonatan Kelu, and in particular the extensive, passionate and conscientious help of Adrian Kubala.
Important material was provided by Bert Peeters, Anna Wierzbicka, William Beaty, Jim Carr, John Merrit, John Baez, Frank DiFilippo, Jonathan Scott, Jon aler, Luca Bombelli, Douglas Singleton, George McQuarry, Tilman Hausherr, Brian Oberquell, Peer Zalm, Martin van der Mark, Vladimir Surdin, Julia Simon, Antonio Fermani, Don Page, Stephen Haley, Peter Mayr, Allan Hayes, Igor Ivanov, Doug Renselle, Wim de Muynck, Steve Carlip, Tom Bruce, Ryan Budney, Gary Ruben, Chris Hillman, Olivier Glassey, Jochen Greiner, squark, Martin Hardcastle, Mark Biggar, Pavel Kuzin, Douglas Brebner, Luciano Lombardi, Franco Bagnoli, Lukas Fabian Moser, Dejan Corovic, Paul Vannoni, John Haber, Saverio Pascazio, Klaus Finkenzeller, Leo Volin, Je Aronson, Roggie Boone, Lawrence Tuppen, Quentin David Jones, Arnaldo Uguzzoni, Frans van Nieuwpoort, Alan Mahoney, Britta Schiller, Petr Danecek, Ingo ies, Vitaliy Solomatin, Carl O ner, Nuno Proença, Elena Colazingari, Paula Henderson, Daniel Darre, Wolfgang Rankl, John Heumann, Joseph Kiss, Martha Weiss, Antonio González, Antonio Martos, John Heumann, André Slabber, Ferdinand Bautista, Zoltán Gácsi, Pat Furrie, Michael Reppisch, Enrico Pasi, omas Köppe, Martin Rivas, Herman Beeksma, Tom Helmond, John Brandes, Vlad Tarko, Nadia Murillo, Ciprian Dobra, Romano Perini, Harald van Lintel, Andrea Conti, François Belfort, Dirk Van de Moortel, Heinrich Neumaier, Jarosław Królikowski, John Dahlman, Fathi Namouni, Elmar Bartel plus a number of people who wanted to remain unnamed.
e so ware tools were re ned with extensive help on fonts and typesetting by Michael Zedler and Achim Blumensath and with the repeated and valuable support of Donald Arseneau; help came also from Ulrike Fischer, Piet van Oostrum, Gerben Wierda, Klaus Böhncke, Craig Upright, Herbert Voss, Andrew Trevorrow, Danie Els, Heiko Oberdiek, Sebastian Rahtz, Don Story, Vincent Darley, Johan Linde, Joseph Hertzlinger, Rick Zaccone and John Warkentin.
Many illustrations in the text were made available by the copyright holders. A warm thank you to all of them; they are mentioned in Appendix G. In particular, Luca Gastaldi and Antonio Martos produced images speci cally for this text. e improvement in the typesetting is due to the professional typographic consulting of Ulrich Dirr. e design of the book and the website owe much to the suggestions and support of my wife Britta.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
A
Die Lösung des Rätsels des Lebens in Raum und Zeit liegt außerhalb von Raum und Zeit.*
“ Ludwig Wittgenstein, Tractatus, . ” is the most daring and amazing journey we can make in a lifetime?
e can travel to remote places, like adventurers, explorers or cosmonauts;
We can look at even more distant places, like astronomers; we can visit the past, like
historians, archaeologists, evolutionary biologists or geologists; or we can delve deeply into the human soul, like artists or psychologists. All these voyages lead either to other places or to other times. However, we can do better.
e most daring trip of all is not the one leading to the most inaccessible place, but the one leading to where there is no place at all. Such a journey implies leaving the prison of space and time and venturing beyond it, into a domain where there is no position, no present, no future and no past, where we are free of the restrictions imposed by space and time, but also of the mental reassurance that these concepts provide. In this domain, many new discoveries and new adventures await us. Almost nobody has ever been there; humanity’s journey there has so far taken at least years, and is still not complete.
To venture into this domain, we need to be curious about the essence of travel itself. e essence of travel is motion. By exploring motion we will be led to the most fascinating adventures in the universe.
e quest to understand motion in all its details and limitations can be pursued behind a desk, with a book, some paper and a pen. But to make the adventure more vivid, this text uses the metaphor of a mountain ascent. Every step towards the top corresponds to a step towards higher precision in the description of motion. In addition, with each step the scenery will become more delightful. At the top of the mountain we shall arrive in a domain where ‘space’ and ‘time’ are words that have lost all meaning and where the sight of the world’s beauty is overwhelming and unforgettable.
inking without time or space is di cult but fascinating. In order to get a taste of the issues involved, try to respond to the following questions without referring to either Challenge 2 n space or time:**
— Can you prove that two points extremely close to each other always leave room for a third point in between?
— Can you describe the shape of a knot over the telephone? — Can you explain on the telephone what ‘right’ and ‘le ’ mean, or what a mirror is? — Can you make a telephone appointment with a friend without using any terms of time
or position, such as ‘clock’, ‘hour’, ‘place’, ‘where’, ‘when’, ‘at’, ‘near’, ‘before’, ‘a er’, ‘near’, ‘upon’, ‘under’, ‘above’, ‘below’? — Can you describe the fall of a stone without using the language of space or time? — Do you know of any observation at all that you can describe without concepts from the domains of ‘space’, ‘time’ or ‘object’?
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* ‘ e solution of the riddle of life in space and time lies outside space and time.’ ** Solutions to, and comments on, challenges are either given on page 1233 or later on in the text. Challenges are classi ed as research level (r), di cult (d), normal student level (n) and easy (e). Challenges for which no solution has yet been included in the book are marked (ny).
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
— Can you explain what time is? And what clocks are? — Can you imagine a nite history of the universe, but without a ‘ rst instant of time’? — Can you imagine a domain of nature where matter and vacuum are indistinguishable? — Have you ever tried to understand why motion exists?
is book explains how to achieve these and other feats, bringing to completion an ancient dream of the human spirit, namely the quest to describe every possible aspect of motion.
Why do your shoelaces remain tied? ey do so because space has three dimensions. Why not another number? e question has taxed researchers for thousands of years. e answer was only found by studying motion down to its smallest details, and by exploring its limits.
Why do the colours of objects di er? Why does the Sun shine? Why does the Moon not fall out of the sky? Why is the sky dark at night? Why is water liquid but re not? Why is the universe so big? Why is it that birds can y but men can’t? Why is lightning not straight? Why are atoms neither square, nor the size of cherries? ese questions seem to have little in common – but they are related. ey are all about motion – about its details and its limitations. Indeed, they all appear, and are answered, in this text. Studying the limits of motion, we discover that when a mirror changes its speed it emits light. We also discover that gravity can be measured with a thermometer. We nd that there are more cells in the brain than stars in the galaxy, giving substance to the idea that people have a whole universe in their head. Exploring any detail of motion is already an adventure in itself.
By exploring the properties of motion we will nd that, despite appearance, motion never stops. We will nd out why the oor cannot fall. We will understand why computers cannot be made arbitrarily fast. We will see that perfect memory cannot exist. We will understand that nothing can be perfectly black. We will learn that every clock has a certain probability of going backwards. We will discover that time does not exist. We will
nd that all objects in the world are connected. We will learn that matter cannot be distinguished precisely from empty space. We will learn that we are literally made of nothing. We will learn quite a few things about our destiny. And we will understand why the world is the way it is.
e quest to understand motion, together with all its details and all its limits, involves asking and answering three speci c questions.
How do things move? Motion usually de ned as is an object changing position over time. is seemingly mundane de nition actually encompasses general relativity, one of the most amazing descriptions of nature ever imagined. We will nd that space is warped, that light does not usually travel in a straight line, and that time is not the same for everybody. We will discover that there is a maximum force of gravity, and that gravity is not an interaction, but rather the change of time with position. We will see how the blackness of the sky at night proves that the universe has a nite age. We will also discover that there is a smallest entropy in nature, which prevents us from knowing everything about a physical system. In addition, we will discover the smallest electrical charge. ese and other strange properties and phenomena of motion are summarized in the rst part of this text, whose topic is classical physics. It leads directly to the next question.
What are things? ings are composites of particles. Not only tangible things, but all interactions and forces – those of the muscles, those that make the Sun burn, those that
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Page 957
make the Earth turn, those that determine the di erences between attraction, repulsion, friction, creation and annihilation – are made of particles as well. e growth of trees, the colours of the sky, the burning of re, the warmth of a human body, the waves of the sea and the mood changes of people are all composed of particles in motion. is story is told in more detail in the second part of the text, which deals with quantum mechanics. Here we will learn that there is a smallest change in nature. is minimum value forces everything to keep changing. In particular, we will learn that it is impossible to completely
ll a glass of wine, that eternal life is impossible, and that light can be transformed into matter. If you nd this boring, you can read about the substantial dangers involved in buying a can of beans.
e rst two parts of this text can be summarized with the help of a few limit principles:
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statistical thermodynamics limits entropy: special relativity limits speed:
general relativity limits force: quantum theory limits action: quantum electrodynamics limits change of charge:
Sk vc
F cG Lħ ∆q e .
In other words, each of the constants of nature k , c, c G, ħ and e that appear above is a limit value. We will discover in each case that the equations of the corresponding domain of physics follow from this limit property. A er these results, the path is prepared for the nal part of our mountain ascent.
What are particles, position and time? e recent results of an age-long search are making it possible to start answering this question. One just needs to nd a description that explains all limit principles at the same time. is third part is not yet complete, because the necessary research results are not yet available. Nevertheless, some of the intermediate results are striking:
Page 998
— It is known already that space and time are not continuous; that – to be precise – neither points nor particles exist; and that there is no way to distinguish space from time, nor vacuum from matter, nor matter from radiation.
— It is known already that nature is not simply made of particles and vacuum. — It seems that position, time and particles are aspects of a complex, extended entity that
is incessantly varying in shape. — Among the mysteries that should be cleared up in the coming years are the origin of
the three dimensions of space, the origin of time and the details of the big bang. — Current research indicates that motion is an intrinsic property of matter and radiation
and that, as soon as we introduce these two concepts in our description of nature, motion appears automatically. Indeed, it is impossible not to introduce these concepts, because they necessarily appear when we divide nature into parts, an act we cannot avoid because of the mechanisms of our senses and therefore of our thinking. — Current research also indicates that the nal, completely precise, description of nature does not use any form of in nity. We nd, step by step, that all in nities appearing in the human description of nature – both the in nitely large and the in nitely small – result from approximations. ‘In nity’ turns out to be merely a conceptual convenience
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that has no place in nature. However, we nd that the precise description does not include any nite quantities either! ese and many other astonishing results of modern physics appear in the third part of this text.
is third and nal part of the text thus describes the present state of the search for a uni ed theory encompassing general relativity and quantum mechanics. To achieve such a description, the secrets of space, time, matter and forces have to be unravelled. It is a fascinating story, assembled piece by piece by thousands of researchers. At the end of the ascent, at the top of the mountain, the idea of motion will have undergone a complete transformation. Without space and time, the world will look magical, incredibly simple and fascinating: pure beauty.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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First Part
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Where the experience of hiking and other motion leads us to introduce, for its description, the concepts of velocity, time, length, mass and charge, as well as action, eld and manifold, allowing us to discover limits to speed, entropy, force and charge, and thus to understand – among other things – why we have legs instead of wheels, how empty space can bend, wobble and move, what love has to do with magnets and amber, and why we can see the stars.
C
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GALILEAN MOTION
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W ! e lightning striking the tree nearby violently disrupts our quiet forest alk and causes our hearts to suddenly beat faster. In the top of the tree e see the re start and fade again. e gentle wind moving the leaves around us helps to restore the calmness of the place. Nearby, the water in a small river follows its complicated way down the valley, re ecting on its surface the ever-changing shapes of the clouds.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
.
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“All motion is an illusion.
Motion is everywhere: friendly and threatening, terrible and
” Zeno of Elea*
beautiful. It is fundamental to our human existence. We need
motion for growing, for learning, for thinking and for enjoying
life. We use motion for walking through a forest, for listening
to its noises and for talking about all this. Like all animals, we
rely on motion to get food and to survive dangers. Plants by
contrast cannot move (much); for their self-defence, they de-
veloped poisons. Examples of such plants are the stinging nettle,
the tobacco plant, digitalis, belladonna and poppy; poisons in-
clude ca eine, nicotine, curare and many others. Poisons such
as these are at the basis of most medicines. erefore, most
medicines exist essentially because plants have no legs. Like all
living beings, we need motion to reproduce, to breathe and to
digest; like all objects, motion keeps us warm.
Motion is the most fundamental observation about nature F I G URE 1 An example of at large. It turns out that everything that happens in the world is motion observed in nature
some type of motion. ere are no exceptions. Motion is such
a basic part of our observations that even the origin of the word is lost in the darkness
of Indo-European linguistic history. e fascination of motion has always made it a fa-
vourite object of curiosity. By the h century in ancient Greece, its study had been
Ref. 1 given a name: physics.
* Zeno of Elea (c. 450 ), one of the main exponents of the Eleatic school of philosophy.
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BIOLOGY
ASTRONOMY
tom MATERIAL SCIENCES
CHEMISTRY
MEDICINE
part II: qt
part III: mt
Motion Mountain
part I: clm, gr & em
PHYSICS
GEOSCIENCES
emotion bay THE HUMANITIES
MATHEMATICS social sea
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F I G U R E 2 Experience Island, with Motion Mountain and the trail to be followed (clm: classical mechanics, gr: general relativity, em: electromagnetism, qt: quantum theory, mt: M-theory, tom: the theory of motion)
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Motion is also important to the human condition. Who are we? Where do we come from? What will we do? What should we do? What will the future bring? Where do people come from? Where do they go? What is death? Where does the world come from? Where does life lead? All these questions are about motion. e study of motion provides answers that are both deep and surprising. Ref. 3 Motion is mysterious. ough found everywhere – in the stars, in the tides, in our eyelids – neither the ancient thinkers nor myriads of others in the centuries since then have been able to shed light on the central mystery: what is motion? We shall discover that the standard reply, ‘motion is the change of place in time’, is inadequate. Just recently an answer has nally been found. is is the story of the way to nd it.
Motion is a part of human experience. If we imagine human experience as an island, then destiny, symbolized by the waves of the sea, carried us to its shore. Near the centre of the island an especially high mountain stands out. From its top we can see over the whole landscape and get an impression of the relationships between all human experiences, in particular between the various examples of motion. is is a guide to the top of what I have called Motion Mountain. e hike is one of the most beautiful adventures of the human mind. Clearly, the rst question to ask is:
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•.
F I G U R E 3 Illusions of motion: look at the figure on the left and slightly move the page, or look at the white dot at the centre of the figure on the right and move your head back and forward
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
D
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Ref. 2 Challenge 3 n
Ref. 4 Ref. 5
Challenge 4 n
Ref. 6
“Das Rätsel gibt es nicht. Wenn sich eine Frage überhaupt stellen läßt, so kann sie beantwortet werden.*
” Ludwig Wittgenstein, Tractatus, .
To sharpen the mind for the issue of motion’s existence, have a look at Figure and follow
the instructions. In both cases the gures seem to rotate. One can experience similar
e ects if one walks over Italian cobblestone in wave patterns or if one looks at the illusions
on the webpage www.ritsumei.ac.jp/~akitaoka/. How can one make sure that real motion
is di erent from these or other similar illusions?**
Many scholars simply argued that motion does not exist at all. eir arguments deeply
in uenced the investigation of motion. For example, the Greek philosopher Parmenides
(born c.
in Elea, a small town near Naples, in southern Italy) argued that since
nothing comes from nothing, change cannot exist. He underscored the permanence of
nature and thus consistently maintained that all change and thus all motion is an illusion.
Heraclitus (c. to c.
) held the opposite view. He expressed it in his famous
statement πάντα ῥεῖ ‘panta rhei’ or ‘everything ows’.*** He saw change as the essence
of nature, in contrast to Parmenides. ese two equally famous opinions induced many
scholars to investigate in more detail whether in nature there are conserved quantities or
whether creation is possible. We will uncover the answer later on; until then, you might
ponder which option you prefer.
Parmenides’ collaborator Zeno of Elea (born c.
) argued so intensely against
motion that some people still worry about it today. In one of his arguments he claims –
in simple language – that it is impossible to slap somebody, since the hand rst has to
travel halfway to the face, then travel through half the distance that remains, then again
so, and so on; the hand therefore should never reach the face. Zeno’s argument focuses
on the relation between in nity and its opposite, nitude, in the description of motion.
In modern quantum theory, a similar issue troubles many scientists up to this day.
Zeno also maintained that by looking at a moving object at a single instant of time,
* e riddle does not exist. If a question can be put at all, it can be answered. ** Solutions to challenges are given either on page 1233 or later on in the text. Challenges are classi ed as research level (r), di cult (d), normal student level (n) and easy (e). Challenges with no solution as yet are
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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F I G U R E 4 How much water is required to make a bucket hang vertically? At what angle does the pulled reel change direction of motion? (© Luca Gastaldi)
one cannot maintain that it moves. Zeno argued that at a single instant of time, there is no di erence between a moving and a resting body. He then deduced that if there is no di erence at a single time, there cannot be a di erence for longer times. Zeno therefore questioned whether motion can clearly be distinguished from its opposite, rest. Indeed, in the history of physics, thinkers switched back and forward between a positive and a negative answer. It was this very question that led Albert Einstein to the development of general relativity, one of the high points of our journey. We will follow the main answers given in the past. Later on, we will be even more daring: we will ask whether single instants of time do exist at all. is far-reaching question is central to the last part of our adventure.
When we explore quantum theory, we will discover that motion is indeed – to a certain extent – an illusion, as Parmenides claimed. More precisely, we will show that motion is observed only due to the limitations of the human condition. We will nd that we experience motion only because we evolved on Earth, with a nite size, made of a large but
nite number of atoms, with a nite but moderate temperature, electrically neutral, large compared with a black hole of our same mass, large compared with our quantum mechanical wavelength, small compared with the universe, with a limited memory, forced by our brain to approximate space and time as continuous entities, and forced by our brain to describe nature as made of di erent parts. If any one of these conditions were not ful-
lled, we would not observe motion; motion, then, would not exist. Each of these results can be uncovered most e ciently if we start with the following question:
marked (ny). *** Appendix A explains how to read Greek text.
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Anaximander
Empedocles Eudoxus
Ctesibius
Anaximenes
Aristotle
Archimedes
Pythagoras
Heraclides
Konon
Almaeon Philolaos
Theophrastus Chrysippos
Heraclitus Zeno
Autolycus Eratosthenes
Xenophanes
Anthistenes
Euclid Dositheus
Thales Parmenides
Archytas Epicure
Biton
Strabo
Frontinus Cleomedes
Varro Athenaius
Maria the Jew
Artemidor
Josephus Sextus Empiricus
Eudoxus of Kyz.
Pomponius Mela
Sosigenes
Dionysius Athenaios Diogenes Periegetes of Nauc. Laertius
Marinus
Virgilius
Menelaos
Philostratus
Polybios
Horace
Nicomachos Apuleius
Alexander Ptolemaios II
Ptolemaios VIII
Caesar
Nero Trajan
600 BCE
500
400
300
200
100
1
100
200
Socrates Plato Ptolemaios I
Cicero
Seneca
Anaxagoras
Aristarchus
Asclepiades Livius Dioscorides Ptolemy
Leucippus
Pytheas Archimedes
Seleukos
Vitruvius Geminos Epictetus
Protagoras
Erasistratus Diocles
Manilius
Demonax
Diophantus
Oenopides Hippocrates
Aristoxenus Aratos Berossos
Philo Dionysius
of Byz.
Thrax
Diodorus Valerius
Theon
Siculus Maximus of Smyrna
Alexander of Aphr.
Herodotus
Herophilus
Apollonius
Democritus
Straton Hipparchus
Theodosius Lucretius
Plinius Senior
Rufus
Galen
Aetius Arrian
Hippasos Speusippos
Dikaiarchus
Poseidonius
Heron Plutarch Lucian
F I G U R E 5 A time line of scientific and political personalities in antiquity (the last letter of the name is aligned with the year of death)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 5 ny
Je hais le mouvement, qui déplace les lignes, Et jamais je ne pleure et jamais je ne ris.
“ Charles Baudelaire, La Beauté.* ” Like any science, the approach of physics is twofold: we advance with precision and with
curiosity. Precision makes meaningful communication possible, and curiosity makes it worthwhile.** Whenever one talks about motion and aims for increased precision or for more detailed knowledge, one is engaged, whether knowingly or not, in the ascent of Motion Mountain. With every increase in the precision of the description, one gains some height. e examples of Figure make the point. When you ll a bucket with a small amount of water, it does not hang vertically. (Why?) If you continue adding water, it starts to hang vertically at a certain moment. How much water is necessary? When you pull a thread from a reel in the way shown, the reel will move either forwards or backwards, depending on the angle at which you pull. What is the limiting angle between the two possibilities?
High precision means going into ne details. is method actually increases the pleasure of the adventure.*** e higher we get on Motion Mountain, the further we can see
Ref. 7 Ref. 8 Challenge 6 n
* Charles Baudelaire (b. 1821 Paris, d. 1867 Paris) Beauty: ‘I hate movement, which changes shapes, and never do I weep and never do I laugh.’ Beauty. ** For a collection of interesting examples of motion in everyday life, see the excellent book by Walker. *** Distrust anybody who wants to talk you out of investigating details. He is trying to deceive you. Details are important. Also, be vigilant also during this walk.
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TA B L E 1 Content of books about motion found in a public library
M
M
motion pictures
motion as therapy for cancer, diabetes, acne and depression
motion perception Ref. 21
motion sickness
motion for tness and wellness
motion for meditation
motion control in sport
motion ability as health check
perpetual motion
motion in dance, music and other arts
motion as proof of various gods Ref. 10
motion of stars and angels Ref. 11
economic e ciency of motion
the connection between motional and emotional habits
motion as help to overcome trauma
motion in psychotherapy Ref. 12
locomotion of insects, horses and robots commotion
motions in parliament
movements in art, sciences and politics
movements in watches
movements in the stock market
movement teaching and learning
movement development in children
musical movements
troop movements Ref. 13
religious movements
bowel movements
moves in chess
cheating moves in casinos Ref. 14
connection between gross national product and citizen mobility
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
and the more our curiosity is rewarded. e views o ered are breathtaking, especially from the very top. e path we will follow – one of the many possible routes – starts from Ref. 9 the side of biology and directly enters the forest that lies at the foot of the mountain.
Intense curiosity drives us to go straight to the limits: understanding motion requires exploration of the largest distances, the highest velocities, the smallest particles, the strongest forces and the strangest concepts. Let us begin.
W
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Every movement is born of a desire for change.
“ ” Antiquity
e best place to obtain a general overview on the types of motion is a large library; this is shown in Table . e domains in which motion, movements and moves play a role are indeed varied. Already in ancient Greece people had the suspicion that all types of motion, as well as many other types of change, are related. It is usual to distinguish at least three categories.
e rst category of change is that of material transport, such as a person walking or a leaf falling from a tree. Transport is the change of position or orientation of objects. To a large extent, the behaviour of people also falls into this category.
A second category of change groups observations such as the dissolution of salt in water, the formation of ice by freezing, the putrefaction of wood, the cooking of food, the
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 6 An example of transport
Ref. 15 Ref. 16
coagulation of blood, and the melting and alloying of metals. ese changes of colour, brightness, hardness, temperature and other material properties are all transformations. Transformations are changes not visibly connected with transport. To this category, a few ancient thinkers added the emission and absorption of light. In the twentieth century, these two e ects were proven to be special cases of transformations, as were the newly discovered appearance and disappearance of matter, as observed in the Sun and in radioactivity. Mind change, such as change of mood, of health, of education and of character, is also (mostly) a type of transformation.
e third and especially important category of change is growth; it is observed for animals, plants, bacteria, crystals, mountains, stars and even galaxies. In the nineteenth century, changes in the population of systems, biological evolution, and in the twentieth century, changes in the size of the universe, cosmic evolution, were added to this category. Traditionally, these phenomena were studied by separate sciences. Independently they all arrived at the conclusion that growth is a combination of transport and transformation.
e di erence is one of complexity and of time scale. At the beginnings of modern science during the Renaissance, only the study of transport was seen as the topic of physics. Motion was equated to transport. e other two domains were neglected by physicists. Despite this restriction, the eld of enquiry remains large, covering a large part of Experience Island. e obvious temptation is to structure the eld by distinguishing types of transport by their origin. Movements such as those of the legs when walking are volitional, because they are controlled by one’s will, whereas movements of external objects, such as the fall of a snow ake, which one cannot in uence by will-power, are called passive. Children are able to make this distinction by about the age of six, and this marks a central step in the development of every human towards a precise description of the environment.* From this distinction stems the historical but
* Failure to pass this stage completely can result in a person having various strange beliefs, such as believing
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 7 Transport, growth and transformation (© Philip Plisson)
Ref. 17
now outdated de nition of physics as the science of the motion of non-living things. en, one day, machines appeared. From that moment, the distinction between vo-
litional and passive motion was put into question. Like living beings, machines are selfmoving and thus mimic volitional motion. However, careful observation shows that every part in a machine is moved by another, so that their motion is in fact passive. Are living beings also machines? Are human actions examples of passive motion as well? e accumulation of observations in the last years made it clear that volitional movement* indeed has the same physical properties as passive motion in non-living systems. (Of course, from the emotional viewpoint, the di erences are important; for example, grace can only be ascribed to volitional movements.) e distinction between the two types is thus not necessary and is dropped in the following. Since passive and volitional motion have the same properties, through the study of motion of non-living objects we can learn something about the human condition. is is most evident when touching the topics of determinism, causality, probability, in nity, time and sex, to name but a few of the themes we will encounter on the way.
With the accumulation of observations in the nineteenth and twentieth centuries, even more of the historical restrictions on the study of motion were put into question. Extensive observations showed that all transformations and all growth phenomena, including behaviour change and evolution, are also examples of transport. In other words, over
years of studies have shown that the ancient classi cation of observations was use-
in the ability to in uence roulette balls, as found in compulsive players, or in the ability to move other bod-
ies by thought, as found in numerous otherwise healthy-looking people. An entertaining and informative
account of all the deception and self-deception involved in creating and maintaining these beliefs is given
by J
R , e Faith Healers, Prometheus Books, 1989. A professional magician, he presents many
similar topics in several of his other books. See also his http://www.randi.org website for more details.
* e word ‘movement’ is rather modern; it was imported into English from the old French and became
popular only at the end of the eighteenth century. It is never used by Shakespeare.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 7 n Ref. 18
less: all change is transport. In the middle of the twentieth century this culminated in the con rmation of an even more speci c idea already formulated in ancient Greece: every type of change is due to the motion of particles. It takes time and work to reach this conclusion, which appears only when one relentlessly pursues higher and higher precision in the description of nature. e rst two parts of this adventure retrace the path to this result. (Do you agree with it?)
e last decade of the twentieth century changed this view completely. e particle idea turns out to be wrong. is new result, already suggested by advanced quantum theory, is reached in the third part of our adventure through a combination of careful observation and deduction. But we still have some way to go before we reach there.
At present, at the beginning of our walk, we simply note that history has shown that classifying the various types of motion is not productive. Only by trying to achieve maximum precision can we hope to arrive at the fundamental properties of motion. Precision, not classi cation is the path to follow. As Ernest Rutherford said: ‘All science is either physics or stamp collecting.’
To achieve precision in our description of motion, we need to select speci c examples of motion and study them fully in detail. It is intuitively obvious that the most precise description is achievable for the simplest possible examples. In everyday life, this is the case for the motion of any non-living, solid and rigid body in our environment, such as a stone thrown through the air. Indeed, like all humans, we learned to throw objects long before we learned to walk. rowing is one of the rst physical experiment we performed by ourselves.* During our early childhood, by throwing stones, toys and other objects until our parents feared for every piece of the household, we explored the perception and the properties of motion. We do the same.
Die Welt ist unabhängig von meinem Willen.**
“ ” Ludwig Wittgenstein, Tractatus, .
P
,
Only wimps study only the general case; real scientists pursue examples.
“ Beresford Parlett ” Human beings enjoy perceiving. Perception starts before birth, and we continue enjoying
it for as long as we can. at is why television, even when devoid of content, is so successful. During our walk through the forest at the foot of Motion Mountain we cannot avoid perceiving. Perception is rst of all the ability to distinguish. We use the basic mental act of distinguishing in almost every instant of life; for example, during childhood we rst learned to distinguish familiar from unfamiliar observations. is is possible in combination with another basic ability, namely the capacity to memorize experiences. Memory gives us the ability to experience, to talk and thus to explore nature. Perceiving, classify-
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* e importance of throwing is also seen from the terms derived from it: in Latin, words like subject or ‘thrown below’, object or ‘thrown in front’, and interjection or ‘thrown in between’; in Greek, it led to terms like symbol or ‘thrown together’, problem or ‘thrown forward’, emblem or ‘thrown into’, and – last but not least – devil or ‘thrown through’. ** e world is independent of my will.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 19 Ref. 20
ing and memorizing together form learning. Without any one of these three abilities, we could not study motion.
Children rapidly learn to distinguish permanence from variability. ey learn to recognize human faces, even though a face never looks exactly the same each time it is seen. From recognition of faces, children extend recognition to all other observations. Recognition works pretty well in everyday life; it is nice to recognize friends, even at night, and even a er many beers (not a challenge). e act of recognition thus always uses a form of generalization. When we observe, we always have a general idea in our mind. We specify the main ones.
All forests can remind us of the essence of perception. Sitting on the grass in a clearing of the forest at the foot of Motion Mountain, surrounded by the trees and the silence typical of such places, a feeling of calmness and tranquillity envelops us. Suddenly, something moves in the bushes; immediately our eyes turn and our attention focuses. e nerve cells that detect motion are part of the most ancient part of our brain, shared with birds and reptiles: the brain stem. en the cortex, or modern brain, takes over to analyse the type of motion and to identify its origin. Watching the motion across our eld of vision, we observe two invariant entities: the xed landscape and the moving animal. A er we recognize the animal as a deer, we relax again.
How did we distinguish between landscape and deer? Several steps in the eye and in the brain are involved. Motion plays an essential part in them, as is best deduced from the ip movie shown in the lower le corners of these pages. Each image shows only a rectangle lled with a mathematically-random pattern. But when the pages are scanned, one discerns a shape moving against a xed background. At any given instant, the shape cannot be distinguished from the background; there is no visible object at any given instant of time. Nevertheless it is easy to perceive its motion.* Perception experiments such as this one have been performed in many variations. In one, it was found that detecting such a window is nothing special to humans; ies have the same ability, as do, in fact, all animals that have eyes.
e ip movie in the lower le corner, like many similar experiments, shows two central connections. First, motion is perceived only if an object can be distinguished from a background or environment. Many motion illusions focus on this point.** Second, motion is required to de ne both the object and the environment, and to distinguish them from each other. In fact, the concept of space is – among others – an abstraction of the idea of background. e background is extended; the moving entity is localized. Does this seem boring? It is not; just wait a second.
We call the set of localized aspects that remain invariant or permanent during motion, such as size, shape, colour etc., taken together, a (physical) object or a (physical) body. We will tighten the de nition shortly, since otherwise images would be objects as well. In other words, right from the start we experience motion as a relative process; it is perceived
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Ref. 21
* e human eye is rather good at detecting motion. For example, the eye can detect motion of a point of
light even if the change of angle is smaller than that which can be distinguished in a xed image. Details of
this and similar topics for the other senses are the domain of perception research.
** e topic of motion perception is full of interesting aspects. An excellent introduction is chapter 6 of the
beautiful text by D
D. H
, Visual Intelligence – How We Create What We See, W.W. Norton
& Co., 1998. His collection of basic motion illusions can be experienced and explored on the associated
http://aris.ss.uci.edu/cogsci/personnel/ho man/ho man.html website.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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TA B L E 2 Family tree of the basic physical concepts
motion the basic type of change
parts
permanent bounded shaped
relations
variable unbounded unshaped
background
permanent extended measurable
objects
images
impenetrable penetrable
states global
e corresponding aspects:
mass size charge spin etc.
intensity
instant
colour
position
appearance momentum
disappearance energy
etc.
etc.
interactions local
source domain strength direction etc.
phase space composed
dimension distance volume subspaces etc.
space-time simple
curvature topology distance area etc.
world – nature – universe – cosmos the collection of all parts, relations and backgrounds
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Challenge 8 n Page 1018
in relation and in opposition to the environment. e concept of an object is therefore also a relative concept. But the basic conceptual distinction between localized, isolable objects and the extended environment is not trivial or unimportant. First, it has the appearance of a circular de nition. (Do you agree?) is issue will keep us very busy later on. Second, we are so used to our ability of isolating local systems from the environment that we take it for granted. However, as we will see in the third part of our walk, this distinction turns out to be logically and experimentally impossible!* Our walk will lead us to discover the reason for this impossibility and its important consequences. Finally, apart from moving entities and the permanent background, we need a third concept, as shown in Table .
Ref. 22
Wisdom is one thing: to understand the thought which steers all things through all things.
“ ” Heraclitus of Ephesus
* Contrary to what is o en read in popular literature, the distinction is possible in quantum theory. It becomes impossible only when quantum theory is uni ed with general relativity.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 10 n
“Das Feste, das Bestehende und der Gegenstand sind Eins. Der Gegenstand ist das Feste, Bestehende; die Kon guration ist das Wechselnde, Unbeständige.* ” Ludwig Wittgenstein, Tractatus, . – .
What distinguishes the various patterns in the lower le corners of this text? In everyday life we would say: the situation or con guration of the involved entities. e situation somehow describes all those aspects that can di er from case to case. It is customary to call the list of all variable aspects of a set of objects their (physical) state of motion, or simply their state.
e situations in the lower le corners di er rst of all in time. Time is what makes opposites possible: a child is in a house and the same child is outside the house. Time describes and resolves this type of contradiction. But the state not only distinguishes situations in time: the state contains all those aspects of a system (i.e., of a group of objects) that set it apart from all similar systems. Two objects can have the same mass, shape, colour, composition and be indistinguishable in all other intrinsic properties; but at least they will di er in their position, or their velocity, or their orientation. e state pinpoints the individuality of a physical system,** and allows us to distinguish it from exact copies of itself. erefore, the state also describes the relation of an object or a system with respect to its environment. Or in short: the state describes all aspects of a system that depend on the observer. ese properties are not boring – just ponder this: does the universe have a state?
Describing nature as a collection of permanent entities and changing states is the starting point of the study of motion. e various aspects of objects and of their states are called observables. All these rough, preliminary de nitions will be re ned step by step in the following. Using the terms just introduced, we can say that motion is the change of state of objects.***
States are required for the description of motion. In order to proceed and to achieve a complete description of motion, we thus need a complete description of objects and a complete description of their possible states. e rst approach, called Galilean physics, consists in specifying our everyday environment as precisely as possible.
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Challenge 9 n
* Objects, the unalterable, and the subsistent are one and the same. Objects are what is unalterable and subsistent; their con guration is what is changing and unstable. ** A physical system is a localized entity of investigation. In the classi cation of Table 2, the term ‘physical system’ is (almost) the same as ‘object’ or ‘physical body’. Images are usually not counted as physical systems (though radiation is one). Are holes physical systems? *** e exact separation between those aspects belonging to the object and those belonging to the state depends on the precision of observation. For example, the length of a piece of wood is not permanent; wood shrinks and bends with time, due to processes at the molecular level. To be precise, the length of a piece of wood is not an aspect of the object, but an aspect of its state. Precise observations thus shi the distinction between the object and its state; the distinction itself does not disappear – at least not for quite while.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 8 A block and tackle and a differential pulley
C Motion is not always a simple topic.*
** Challenge 11 n Is the motion of a ghost an example of motion?
Challenge 12 n
**
A man climbs a mountain from 9 a.m. to 1 p.m. He sleeps on the top and comes down the next day, taking again from 9 am to 1 pm for the descent. Is there a place on the path that he passes at the same time on the two days?
**
Challenge 13 n Can something stop moving? If yes: how would you show it? If not: does this mean that nature is in nite?
** Challenge 14 n Can the universe move?
** Challenge 15 n To talk about precision with precision, we need to measure it. How would you do that?
** Challenge 16 n Would we observe motion if we had no memory?
** Challenge 17 n What is the lowest speed you have observed? Is there a lowest speed in nature?
* Sections entitled ‘curiosities’ are collections of topics and problems that allow one to check and to expand the usage of concepts already introduced.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 18 n
**
According to legend, Sessa ben Zahir, the Indian inventor of the game of chess, demanded from King Shirham the following reward for his invention: he wanted one grain of rice for the rst square, two for the second, four for the third, eight for the fourth, and so on. How much time would all the rice elds of the world take to produce the necessary rice?
**
When a burning candle is moved, the ame lags behind the candle. How does the ame Challenge 19 n behave if the candle is inside a glass, still burning, and the glass is accelerated?
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Challenge 20 d
**
A good way to make money is to build motion detectors. A motion detector is a small box with a few wires. e box produces an electrical signal whenever the box moves. What types of motion detectors can you imagine? How cheap can you make such a box? How precise?
**
A perfectly frictionless and spherical ball lies near the edge of a perfectly at and horiChallenge 21 d zontal table. What happens? In what time scale?
**
You step into a closed box without windows. e box is moved by outside forces unknown Challenge 22 n to you. Can you determine how you move from inside the box?
**
What is the length of rope one has to pull in order to li a mass by a height h with a block Challenge 23 n and tackle with four wheels, as shown in Figure 8?
**
When a block is rolled over the oor over a set of cylinders, how are the speed of the Challenge 24 n block and that of the cylinders related?
Ref. 15 Challenge 25 n
**
Do you dislike formulae? If you do, use the following three-minute method to change the situation. It is worth trying it, as it will make you enjoy this book much more. Life is short; as much of it as possible, like reading this text, should be a pleasure.
1 - Close your eyes and recall an experience that was absolutely marvellous, a situation when you felt excited, curious and positive.
2 - Open your eyes for a second or two and look at page 321 – or any other page that contains many formulae.
3 - en close your eyes again and return to your marvellous experience. 4 - Repeat the observation of the formulae and the visualization of your memory – steps 2 and 3 – three more times. en leave the memory, look around yourself to get back into the here and now, and test yourself. Look again at page 321. How do you feel about formulae now?
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Challenge 26 n
**
In the sixteenth century, Niccolò Tartaglia* proposed the following problem. ree young couples want to cross a river. Only a small boat that can carry two people is available. e men are extremely jealous, and would never leave their brides alone with another man. How many journeys across the river are necessary?
.
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Challenge 27 n
Physic ist wahrlich das eigentliche Studium des Menschen.**
“ Georg Christoph Lichtenberg ” e simplest description of motion is the one we all, like cats or monkeys, use uncon-
sciously in everyday life: only one thing can be at a given spot at a given time. is general description can be separated into three assumptions: matter is impenetrable and moves, time is made of instants, and space is made of points. Without these three assumptions (do you agree with them?) it is not possible to de ne velocity in everyday life. is description of nature is called Galilean or Newtonian physics.
Galileo Galilei ( – ), Tuscan professor of mathematics, was a founder of modern physics and is famous for advocating the importance of observations as checks of statements about nature. By requiring and performing these checks throughout his life, he was led to continuously increase the accuracy in the description of motion. For example, Galileo studied motion by measuring change of position with a self-constructed stopwatch. His approach changed the speculative description of ancient Greece into the experimental physics of Renaissance Italy.***
e English alchemist, occultist, theologian, physicist and politi- Galileo Galilei cian Isaac Newton ( – ) was one of the rst to pursue with vigour the idea that di erent types of motion have the same properties, and he made important steps in constructing the concepts necessary to demonstrate this idea.****
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 23 Ref. 24
* Niccolò Fontana Tartaglia (1499–1557), important Venetian mathematician. ** ‘Physics truly is the proper study of man.’ Georg Christoph Lichtenberg (1742–1799) was an important physicist and essayist. *** e best and most informative book on the life of Galileo and his times is by Pietro Redondi (see the footnote on page 220). Galileo was born in the year the pencil was invented. Before his time, it was impossible to do paper and pencil calculations. For the curious, the http://www.mpiwg-berlin.mpg.de website allows you to read an original manuscript by Galileo. **** Newton was born a year a er Galileo died. Newton’s other hobby, as master of the Mint, was to supervise personally the hanging of counterfeiters. About Newton’s infatuation with alchemy, see the books by Dobbs. Among others, Newton believed himself to be chosen by god; he took his Latin name, Isaacus Neuutonus, and formed the anagram Jeova sanctus unus. About Newton and his importance for classical mechanics, see the text by Cli ord Truesdell.
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TA B L E 3 Properties of everyday – or Galilean – velocity
V
P
M
D
Be distinguished Change gradually Point somewhere Be compared Be added Have de ned angles Exceed any limit
distinguishability continuum direction measurability additivity direction in nity
element of set real vector space vector space, dimensionality metricity vector space Euclidean vector space unboundedness
Page 646
Page 69, Page 1214
Page 69 Page 1205 Page 69 Page 69 Page 647
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Page 69 Challenge 28 d
“ ere is nothing else like it.
” Jochen Rindt*
Velocity fascinates. To physicists, not only car races are interesting, but any moving entity
is. erefore they rst measure as many examples as possible. A selection is given in
Table .
Everyday life teaches us a lot about motion: objects can overtake each other, and they
can move in di erent directions. We also observe that velocities can be added or changed
smoothly. e precise list of these properties, as given in Table , is summarized by math-
ematicians in a special term; they say that velocities form a Euclidean vector space.** More
details about this strange term will be given shortly. For now we just note that in describ-
ing nature, mathematical concepts o er the most accurate vehicle.
When velocity is assumed to be an Euclidean vector, it is called Galilean velocity. Ve-
locity is a profound concept. For example, velocity does not need space and time meas-
urements to be de ned. Are you able to nd a means of measuring velocities without
measuring space and time? If so, you probably want to skip to page , jumping
years of enquiries. If you cannot do so, consider this: whenever we measure a quantity we
assume that everybody is able to do so, and that everybody will get the same result. In
other words, we de ne measurement as a comparison with a standard. We thus implicitly
assume that such a standard exists, i.e. that an example of a ‘perfect’ velocity can be found.
Historically, the study of motion did not investigate this question rst, because for many
centuries nobody could nd such a standard velocity. You are thus in good company.
Some researchers have specialized in the study of the lowest velocities found in nature:
* Jochen Rindt (1942–1970), famous Austrian Formula One racing car driver, speaking about speed. ** It is named a er Euclid, or Eukleides, the great Greek mathematician who lived in Alexandria around 300 . Euclid wrote a monumental treatise of geometry, the Στοιχεῖα or Elements, which is one of the milestones of human thought. e text presents the whole knowledge on geometry of that time. For the rst time, Euclid introduces two approaches that are now in common use: all statements are deduced from a small number of basic ‘axioms’ and for every statement a ‘proof ’ is given. e book, still in print today, has been the reference geometry text for over 2000 years. On the web, it can be found at http://aleph0.clarku. edu/~djoyce/java/elements/elements.html.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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TA B L E 4 Some measured velocity values
O
Stalagmite growth Can you nd something slower? Growth of deep sea manganese crust Lichen growth Typical motion of continents Human growth during childhood, hair growth Tree growth Electron dri in metal wire Sperm motion Speed of light at Sun’s centre Ketchup motion Slowest speed of light measured in matter on Earth Speed of snow akes Signal speed in human nerve cells Wind speed at 1 Beaufort (light air) Speed of rain drops, depending on radius Fastest swimming sh, sail sh (Istiophorus platypterus) Fastest running animal, cheetah (Acinonyx jubatus) Wind speed at 12 Beaufort (hurricane) Speed of air in throat when sneezing Fastest measured throw: cricket ball Freely falling human Fastest bird, diving Falco peregrinus Fastest badminton serve Average speed of oxygen molecule in air at room temperature Speed of sound in dry air at sea level and standard temperature Cracking whip’s end Speed of a ri e bullet Speed of crack propagation in breaking silicon Highest macroscopic speed achieved by man – the Voyager satellite Average (and peak) speed of lightning tip Speed of Earth through universe Highest macroscopic speed measured in our galaxy Speed of electrons inside a colour TV Speed of radio messages in space Highest ever measured group velocity of light Speed of light spot from a light tower when passing over the Moon Highest proper velocity ever achieved for electrons by man Highest possible velocity for a light spot or shadow
V
. pm s
Challenge 29 n
am s
down to pm s
mm a = . nm s
nm s
up to nm s
µm s
60 to µm s
. mm s
mm s
. m s Ref. 25
. m s to . m s
. m s to m s Ref. 26
below . m s
m s to m s
ms
ms
above m s
ms
ms
50 to m s
ms
ms
ms
ms
ms
km s
km s
km s
km s (
km s)
km s
. ë m s Ref. 27
ë ms
ms
ë ms
ë ms
ë ms
in nite
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 28
they are called geologists. Do not miss the opportunity to walk across a landscape while listening to one of them.
Velocity is a profound subject for a second reason: we will discover that all properties of Table are only approximate; none is actually correct. Improved experiments will uncover limits in every property of Galilean velocity. e failure of the last three properties will lead us to special and general relativity, the failure of the middle two to quantum theory and the failure of the rst two properties to the uni ed description of nature. But for now, we’ll stick with Galilean velocity, and continue with another Galilean concept derived from it: time.
“Without the concepts place, void and time, change cannot be. [...] It is therefore clear [...] that their investigation has to be carried out, by studying each of them separately. ” Aristotle* Physics, Book III, part .
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Ref. 18 Challenge 30 n
“Time does not exist in itself, but only through the perceived objects, from which the concepts of past, of present and of future ensue. ” Lucrece,** De rerum natura, lib. , v. ss.
In their rst years of life, children spend a lot of time throwing objects around. e term ‘object’ is a Latin word meaning ‘that which has been thrown in front.’ Developmental psychology has shown experimentally that from this very experience children extract the concepts of time and space. Adult physicists do the same when studying motion at university.
When we throw a stone through the air, we can de ne a sequence of observations. Our memory and our senses give us this ability. e sense of hearing registers the various sounds during the rise, the fall and the landing of the stone. Our eyes track the location of the stone from one point to the next. All observations have their place in a sequence, with some observations preceding them, some observations simultaneous to them, and still others succeeding them. We say that observations are perceived to happen at various instants and we call the sequence of all instants time.
An observation that is considered the smallest part of a se- F I G URE 9 A typical path quence, i.e. not itself a sequence, is called an event. Events are followed by a stone central to the de nition of time; in particular, starting or stop- thrown through the air ping a stopwatch are events. (But do events really exist? Keep this question in the back of your head as we move on.)
Sequential phenomena have an additional property known as stretch, extension or duration. Some measured values are given in Table .*** Duration expresses the idea that
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* Aristotle (384/3–322), Greek philosopher and scientist. ** Lucretius Carus (c. 95 to c. 55 ), Roman scholar and poet. *** A year is abbreviated a (Latin ‘annus’).
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TA B L E 5 Selected time measurements
O
Shortest measurable time Shortest time ever measured Time for light to cross a typical atom Period of caesium ground state hyper ne transition Beat of wings of fruit y Period of pulsar (rotating neutron star) PSR 1913+16 Human ‘instant’ Shortest lifetime of living being Average length of day 400 million years ago Average length of day today From birth to your 1000 million seconds anniversary Age of oldest living tree Use of human language Age of Himalayas Age of Earth Age of oldest stars Age of most protons in your body Lifetime of tantalum nucleus Ta Lifetime of bismuth Bi nucleus
T
−s
−s
−s
.
ps
ms
.
( )s
ms
.d
s
. ( )s
.a
a
ëa
35 to ë a
.ë a
. Ga
. Ga
a
. ( )ë a
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 18
Page 1161 Challenge 32 n
sequences take time. We say that a sequence takes time to express that other sequences
can take place in parallel with it.
How exactly is the concept of time, including sequence and duration, deduced from ob-
servations? Many people have looked into this question: astronomers, physicists, watch-
makers, psychologists and philosophers. All nd that time is deduced by comparing mo-
tions. Children, beginning at a very young age, develop the concept of ‘time’ from the
comparison of motions in their surroundings. Grown-ups take as a standard the motion
of the Sun and call the resulting type of time local time. From the Moon they deduce a
lunar calendar. If they take a particular village clock on a European island they call it the
universal time coordinate (UTC), once known as ‘Greenwich mean time.’*Astronomers use
the movements of the stars and call the result ephemeris time. An observer who uses his
personal watch calls the reading his proper time; it is o en used in the theory of relativity.
Not every movement is a good standard for time. In the year an Earth rotation
did not take
seconds any more, as it did in the year , but
. seconds.
Can you deduce in which year your birthday will have shi ed by a whole day from the
time predicted with
seconds?
Challenge 31 n
* O cial UTC time is used to determine power grid phase, phone companies’ bit streams and the signal to the GPS system used by many navigation systems around the world, especially in ships, aeroplanes and lorries. For more information, see the http://www.gpsworld.com website. e time-keeping infrastructure is also important for other parts of the modern economy. Can you spot the most important ones?
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Challenge 33 n Ref. 30
Page 655
All methods for the de nition of time are thus based on comparisons of motions. In order to make the concept as precise and as useful as possible, a standard reference motion is chosen, and with it a standard sequence and a standard duration is de ned. e device that performs this task is called a clock. We can thus answer the question of the section title: time is what we read from a clock. Note that all de nitions of time used in the various branches of physics are equivalent to this one; no ‘deeper’ or more fundamental de nition is possible.* Note that the word ‘moment’ is indeed derived from the word ‘movement’. Language follows physics in this case. Astonishingly, the de nition of time just given is
nal; it will never be changed, not even at the top of Motion Mountain. is is surprising at rst sight, because many books have been written on the nature of time. Instead, they should investigate the nature of motion! But this is the aim of our walk anyhow. We are thus set to discover all the secrets of time as a side result of our adventure. Every clock reminds us that in order to understand time, we need to understand motion.
A clock is a moving system whose position can be read. Of course, a precise clock is a system moving as regularly as possible, with as little outside disturbance as possible. Is there a perfect clock in nature? Do clocks exist at all? We will continue to study these questions throughout this work and eventually reach a surprising conclusion. At this point, however, we state a simple intermediate result: since clocks do exist, somehow there is in nature an intrinsic, natural and ideal way to measure time. Can you see it?
Time is not only an aspect of observations, it is also a facet of personal experience. Even in our innermost private life, in our thoughts, feelings and dreams, we experience sequences and durations. Children learn to relate this internal experience of time with external observations, and to make use of the sequential property of events in their actions. Studies of the origin of psychological time show that it coincides – apart from its lack of accuracy – with clock time.** Every living human necessarily uses in his daily life the concept of time as a combination of sequence and duration; this fact has been checked in numerous investigations. For example, the term ‘when’ exists in all human languages.
Time is a concept necessary to distinguish between observations. In any sequence, we observe that events succeed each other smoothly, apparently without end. In this context, ‘smoothly’ means that observations that are not too distant tend to be not too di erent. Yet between two instants, as close as we can observe them, there is always room for other events. Durations, or time intervals, measured by di erent people with di erent clocks agree in everyday life; moreover, all observers agree on the order of a sequence of events. Time is thus unique.
e mentioned properties of everyday time, listed in Table , correspond to the precise version of our everyday experience of time. It is called Galilean time; all the properties can be expressed simultaneously by describing time with real numbers. In fact, real numbers have been constructed to have exactly the same properties as Galilean time, as explained in the Intermezzo. Every instant of time can be described by a real number, o en abbreviated t, and the duration of a sequence of events is given by the di erence between the values for the nal and the starting event.
Page 830 Ref. 29
* e oldest clocks are sundials. e science of making them is called gnomonics. An excellent and complete introduction into this somewhat strange world can be found at the http://www.sundials.co.uk website. ** e brain contains numerous clocks. e most precise clock for short time intervals, the internal interval timer, is more accurate than o en imagined, especially when trained. For time periods between a few tenths of a second, as necessary for music, and a few minutes, humans can achieve accuracies of a few per cent.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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TA B L E 6 Properties of Galilean time
I
P
M
D
Can be distinguished
distinguishability element of set
Page 646
Can be put in order
sequence
order
Page 1214
De ne duration
measurability
metricity
Page 1205
Can have vanishing duration continuity
denseness, completeness Page 1214
Allow durations to be added additivity
metricity
Page 1205
Don’t harbour surprises translation invariance homogeneity
Page 154
Don’t end
in nity
unboundedness
Page 647
Are equal for all observers absoluteness
uniqueness
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 34 n
When Galileo studied motion in the seventeenth century, there were as yet no stopwatches. He thus had to build one himself, in order to measure times in the range between a fraction and a few seconds. Can you guess how he did it?
We will have quite some fun with Galilean time in the rst two chapters. However, hundreds of years of close scrutiny have shown that every single property of time just listed is approximate, and none is strictly correct. is story is told in the subsequent chapters.
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Challenge 35 n
“ ” What time is it at the North Pole now?
All rotational motions in our society, such as athletic races, horse, bicycle or ice skating races, turn anticlockwise. Likewise, every supermarket leads its guests anticlockwise through the hall. Mathematicians call this the positive rotation sense. Why? Most people are right-handed, and the right hand has more freedom at the outside of a circle. erefore thousands of years ago chariot races in stadia went anticlockwise. As a result, all races still do so to this day. at is why runners move anticlockwise. For the same reason, helical stairs in castles are built in such a way that defending right-handers, usually from above, have that hand on the outside.
On the other hand, the clock imitates the shadow of sundials; obviously, this is true on the northern hemisphere only, and only for sundials on the ground, which were the most common ones. ( e old trick to determine south by pointing the hour hand of an horizontal watch to the Sun and halving the angle between it and the direction of o’clock does not work on the southern hemisphere.) So every clock implicitly continues to state on which hemisphere it was invented. In addition, it also tells us that sundials on walls came in use much later than those on the oor.
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Page 554 Ref. 31
Challenge 36 e
Ref. 32
“Wir können keinen Vorgang mit dem ‘Ablauf der Zeit’ vergleichen – diesen gibt es nicht –, sondern nur mit einem anderen Vorgang (etwa
dem Gang des Chronometers).*
” Ludwig Wittgenstein, Tractatus, .
e expression ‘the ow of time’ is o en used to convey that in nature change follows
a er change, in a steady and continuous manner. But though the hands of a clock ‘ ow’,
time itself does not. Time is a concept introduced specially to describe the ow of events
around us; it does not itself ow, it describes ow. Time does not advance. Time is neither
linear nor cyclic. e idea that time ows is as hindering to understanding nature as is
the idea that mirrors exchange right and le .
e misleading use of the expression ‘ ow of time’, propagated rst by some Greek
thinkers and then again by Newton, continues. Aristotle ( / –
), careful to
think logically, pointed out its misconception, and many did so a er him. Nevertheless,
expressions such as ‘time reversal’, the ‘irreversibility of time’, and the much-abused ‘time’s
arrow’ are still common. Just read a popular science magazine chosen at random. e fact
is: time cannot be reversed, only motion can, or more precisely, only velocities of objects;
time has no arrow, only motion has; it is not the ow of time that humans are unable
to stop, but the motion of all the objects in nature. Incredibly, there are even books writ-
ten by respected physicists that study di erent types of ‘time’s arrows’ and compare them
with each other. Predictably, no tangible or new result is extracted. Time does not ow.
In the same manner, colloquial expressions such as ‘the start (or end) of time’ should be
avoided. A motion expert translates them straight away into ‘the start (or end) of motion’.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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“ e introduction of numbers as coordinates [...] is an act of violence [...]. Hermann Weyl, Philosophie der Mathematik ” und Naturwissenscha .**
Whenever we distinguish two objects from each other, such as two stars, we rst of all distinguish their positions. Distinguishing positions is the main ability of our sense of sight. Position is therefore an important aspect of the physical state of an object. A position is taken by only one object at a time. Positions are limited. e set of all available positions, called (physical) space, acts as both a container and a background.
Closely related to space and position is size, the set of positions an objects occupies. Small objects occupy only subsets of the positions occupied by large ones. We will discuss size shortly.
How do we deduce space from observations? During childhood, humans (and most higher animals) learn to bring together the various perceptions of space, namely the
* We cannot compare a process with ‘the passage of time’ – there is no such thing – but only with another process (such as the working of a chronometer). ** Hermann Weyl (1885–1955) was one of the most important mathematicians of his time, as well as an important theoretical physicist. He was one of the last universalists in both elds, a contributor to quantum theory and relativity, father of the term ‘gauge’ theory, and author of many popular texts.
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Challenge 37 n
Challenge 38 n Challenge 39 n
visual, the tactile, the auditory, the kinesthetic, the vestibular etc., into one coherent set
of experiences and description. e result of this learning process is a certain ‘image’ of
space in the brain. Indeed, the question ‘where?’ can be asked and answered in all lan-
guages of the world. Being more precise, adults derive space from distance measurements.
e concepts of length, area, volume, angle and solid angle are all deduced with their help.
Geometers, surveyors, architects, astronomers, carpet salesmen and producers of metre
sticks base their trade on distance measurements. Space is a concept formed to summar-
ize all the distance relations between objects for a precise description of observations.
Metre sticks work well only if they are straight. But when humans lived in the jungle,
there were no straight objects around them. No straight rulers, no straight tools, noth-
ing. Today, a cityscape is essentially a collection of straight lines. Can you describe how
humans achieved this?
Once humans came out of the jungle with their newly built metre sticks, they collec-
ted a wealth of results. e main ones are listed in Table ; they are easily con rmed
by personal experience. Objects can take positions in an apparently continuous manner:
there indeed are more positions than can be counted.* Size is captured by de ning the
distance between various positions, called length, or by using the eld of view an object
takes when touched, called its surface. Length and surface can be measured with the help
of a metre stick. Selected measurement results are given in Table . e length of objects
is independent of the person measuring it, of the position of the objects and of their ori-
entation. In daily life the sum of angles in any triangle is equal to two right angles. ere
are no limits in space.
Experience shows us that space has three di-
mensions; we can de ne sequences of positions
in precisely three independent ways. Indeed,
the inner ear of (practically) all vertebrates has
three semicircular canals that sense the body’s
position in the three dimensions of space, as
shown in Figure .** Similarly, each human
eye is moved by three pairs of muscles. (Why
three?) Another proof that space has three di-
mensions is provided by shoelaces: if space had more than three dimensions, shoelaces would not be useful, because knots exist only in threedimensional space. But why does space have
F I G U R E 10 Two proofs of the three-dimensionality of space: a knot and the inner ear of a mammal
three dimensions? is is probably the most di cult question of physics; it will be
answered only in the very last part of our walk.
It is o en said that thinking in four dimensions is impossible. at is wrong. Just try.
For example, can you con rm that in four dimensions knots are impossible?
Like time intervals, length intervals can be described most precisely with the help of
real numbers. In order to simplify communication, standard units are used, so that every-
body uses the same numbers for the same length. Units allow us to explore the general
* For a de nition of uncountability, see page 649. ** Note that saying that space has three dimensions implies that space is continuous; the Dutch mathematician and philosopher Luitzen Brouwer (b. 1881 Overschie, d. 1966 Blaricum) showed that dimensionality is only a useful concept for continuous sets.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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TA B L E 7 Properties of Galilean space
P
P
M
Can be distinguished
distinguishability
Can be lined up if on one line sequence
Can form shapes
shape
Lie along three independent possibility of knots directions
Can have vanishing distance continuity
De ne distances Allow adding translations De ne angles Don’t harbour surprises Can beat any limit De ned for all observers
measurability additivity scalar product translation invariance in nity absoluteness
element of set order topology 3-dimensionality
denseness, completeness metricity metricity Euclidean space homogeneity unboundedness uniqueness
D
-
Page 646 Page 1214 Page 1213 Page 1204
Page 1214
Page 1205 Page 1205 Page 69
Page 647 Page 52
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
properties of Galilean space experimentally: space, the container of objects, is continuous, three-dimensional, isotropic, homogeneous, in nite, Euclidean and unique or ‘absolute’. In mathematics, a structure or mathematical concept with all the properties just mentioned is called a three-dimensional Euclidean space. Its elements, (mathematical) points, are described by three real parameters. ey are usually written as
(x, y, z)
(1)
Challenge 40 n
and are called coordinates. ey specify and order the location of a point in space. (For the precise de nition of Euclidean spaces, see page .)
What is described here in just half a page actually took years to be worked out, mainly because the concepts of ‘real number’ and ‘coordinate’ had to be discovered rst.
e rst person to describe points of space in this way was the famous mathematician and philosopher René Descartes*, a er whom the coordinates of expression ( ) are named Cartesian.
Like time, space is a necessary concept to describe the world. Indeed, space is automatically introduced when we describe situations with many objects. For example, when many spheres lie on a billiard table, we cannot avoid using space to describe the relations between them. ere is no way to avoid using spatial concepts when talking about nature.
Even though we need space to talk about nature, it is still interesting to ask why this is possible. For example, since length measurement methods do exist, there must be a natural or ideal way to measure distances, sizes and straightness. Can you nd it?
* René Descartes or Cartesius (1596–1650), French mathematician and philosopher, author of the famous statement ‘je pense, donc je suis’, which he translated into ‘cogito ergo sum’ – I think therefore I am. In his view this is the only statement one can be sure of.
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TA B L E 8 Some measured distance values
O
Galaxy Compton wavelength Planck length, the shortest measurable length Proton diameter Electron Compton wavelength Hydrogen atom size Smallest eardrum oscillation detectable by human ear Wavelength of visible light Size of small bacterium Point: diameter of smallest object visible with naked eye Diameter of human hair (thin to thick) Total length of DNA in each human cell Largest living thing, the fungus Armillaria ostoyae Length of Earth’s Equator Total length of human nerve cells Average distance to Sun Light year Distance to typical star at night Size of galaxy Distance to Andromeda galaxy Most distant visible object
D
− m (calculated only)
−m
fm
.
( ) pm
pm
pm
0.4 to . µm
µm
µm
30 to µm
m
km
. ( )m
ë km
( )m
. Pm
Em
Zm
Zm
Ym
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
As in the case of time, each of the properties of space just lis-
ted has to be checked. And again, careful observations will show
that each property is an approximation. In simpler and more
drastic words, all of them are wrong. is con rms Weyl’s state-
ment at the beginning of this section. In fact, the story about the
violence connected with the introduction of numbers is told by
every forest in the world, and of course also by the one at the foot
of Motion Mountain. To hear it, we need only listen carefully to
what the trees have to tell.
“Μέτρον ἄριστον.*
” Cleobulus
A
?
René Descartes
In everyday life, the concepts of Galilean space and time include two opposing aspects; the contrast has coloured every discussion for several centuries. On the one hand, space and time express something invariant and permanent; they both act like big containers for
* ‘Measure is the best (thing).’ Cleobulus (Κλεοβουλος) of Lindos, (c. 620–550 BCE) was another of the proverbial seven sages.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 41 e Ref. 33
all the objects and events found in nature. Seen this way, space and time have an existence of their own. In this sense one can say that they are fundamental or absolute. On the other hand, space and time are tools of description that allow us to talk about relations between objects. In this view, they do not have any meaning when separated from objects, and only result from the relations between objects; they are derived, relational or relative. Which of these viewpoints do you prefer? e results of physics have alternately favoured one viewpoint or the other. We will repeat this alternation throughout our adventure, until we nd the solution. And obviously, it will turn out to be a third option.
S–
,
A central aspect of objects is their size. As a small child, under
school age, every human learns how to use the properties of size
and space in their actions. As adults seeking precision, the de n-
ition of distance as the di erence between coordinates allows us
to de ne length in a reliable way. It took hundreds of years to dis-
cover that this is not the case. Several investigations in physics
and mathematics led to complications.
e physical issues started with an astonishingly simple ques-
tion asked by Lewis Richardson:* How long is the western coast-
line of Britain?
Following the coastline on a map using an odometer, a device
shown in Figure , Richardson found that the length l of the
coastline depends on the scale s (say :
or :
) of
the map used:
l=l s .
F I G U R E 11 A curvemeter or odometer
(2)
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Challenge 42 e
(Richardson found other numbers for other coasts.) e number l is the length at scale : . e main result is that the larger the map, the longer the coastline. What would hap-
pen if the scale of the map were increased even beyond the size of the original? e length would increase beyond all bounds. Can a coastline really have in nite length? Yes, it can. In fact, mathematicians have described many such curves; they are called fractals. An in nite number of them exist, and Figure shows one example.** Can you construct another?
Length has other strange properties. e Italian mathematician Giuseppe Vitali was the rst to discover that it is possible to cut a line segment of length into pieces that can be reassembled – merely by shi ing them in the direction of the segment – into a
* Lewis Fray Richardson (1881–1953), English physicist and psychologist.
** Most of these curves are self-similar, i.e. they follow scaling laws similar to the above-mentioned. e term
‘fractal’ is due to the Polish mathematician Benoît Mandelbrot and refers to a strange property: in a certain
sense, they have a non-integral number D of dimensions, despite being one-dimensional by construction.
Mandelbrot saw that the non-integer dimension was related to the exponent e of Richardson by D = + e,
thus giving D = . in the example above.
Coastlines and other fractals are beautifully presented in H -O P
,H
J-
&D
S , Fractals for the Classroom, Springer Verlag, 1992, pp. 232–245. It is also available
in several other languages.
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n = 1
•. n = 2
– n = 3
n = ∞
F I G U R E 12 A fractal: a self-similar curve of infinite length (far right), and its construction
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Challenge 43 d
Challenge 44 n Page 178
Challenge 45 n
line segment of length . Are you able to nd such a division using the hint that it is only possible using in nitely many pieces?
To sum up, length is well de ned for lines that are straight or nicely curved, but not for intricate lines, or for lines made of in nitely many pieces. We therefore avoid fractals and other strangely shaped curves in the following, and we take special care when we talk about in nitely small segments. ese are the central assumptions in the rst two parts of this adventure, and we should never forget them. We will come back to these assumptions in the third part.
In fact, all these problems pale when compared with the following problem. Commonly, area and volume are de ned using length. You think that it is easy? You’re wrong, as well as being a victim of prejudices spread by schools around the world. To de ne area and volume with precision, their de nitions must have two properties: the values must be additive, i.e. for nite and in nite sets of objects, the total area and volume have to be the sum of the areas and volumes of each element of the set; and they must be rigid, i.e. if one cuts an area or a volume into pieces and then rearranges the pieces, the value remains the same. Do such concepts exist?
For areas in a plane, one proceeds in the following standard way: one de nes the area A of a rectangle of sides a and b as A = ab; since any polygon can be rearranged into a rectangle with a nite number of straight cuts, one can then de ne an area value for all polygons. Subsequently, one can de ne area for nicely curved shapes as the limit of the sum of in nitely many polygons. is method is called integration; it is introduced in detail in the section on physical action.
However, integration does not allow us to de ne area for arbitrarily bounded regions. (Can you imagine such a region?) For a complete de nition, more sophisticated tools are needed. ey were discovered in by the famous mathematician Stefan Banach.* He proved that one can indeed de ne an area for any set of points whatsoever, even if the border is not nicely curved but extremely complicated, such as the fractal curve previously mentioned. Today this generalized concept of area, technically a ‘ nitely additive isometrically invariant measure,’ is called a Banach measure in his honour. Mathematicians sum up this discussion by saying that since in two dimensions there is a Banach measure, there is a way to de ne the concept of area – an additive and rigid measure – for any set of points whatsoever.**
What is the situation in three dimensions, i.e. for volume? We can start in the same way as for area, by de ning the volume V of a rectangular polyhedron with sides a, b,
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* Stefan Banach (Krakow, 1892–Lvov, 1945), important Polish mathematician. ** Actually, this is true only for sets on the plane. For curved surfaces, such as the surface of a sphere, there are complications that will not be discussed here. In addition, the problems mentioned in the de nition of length of fractals also reappear for area if the surface to be measured is not at but full of hills and valleys. A typical example is the area of the human lung: depending on the level of details examined, one nds area values from a few up to over a hundred square metres.
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Challenge 46 n Ref. 34
Challenge 47 n
c as V = abc. But then we encounter a rst problem: a general polyhedron cannot be
cut into a cube by straight cuts! e limitation was discovered in and by Max
Dehn.* He found that the possibility depends on the values of the edge angles, or dihedral
angles, as the mathematicians call them. If one ascribes to every edge of a general poly-
hedron a number given by its length l times a special function (α) of its dihedral angle
α, then Dehn found that the sum of all the numbers for all the edges of a solid does not
change under dissection, provided that the function ful ls (α + β) = (α) + (β) and
(π) = . An example of such a strange function is the one assigning the value to
any rational multiple of π and the value to a basis set of irrational multiples of π. e
values for all other dihedral angles of the polyhedron can then be constructed by com-
bination of rational multiples of these basis angles. Using this function, you may then
deduce for yourself that a cube cannot be dissected into a regular tetrahedron because
their respective Dehn invariants are di erent.**
Despite the problems with Dehn invariants, one
can de ne a rigid and additive concept of volume for
polyhedra, since for all polyhedra and, in general, for
all ‘nicely curved’ shapes, one can again use integra-
tion for the de nition of their volume.
Now let us consider general shapes and general
cuts in three dimensions, not just the ‘nice’ ones
dihedral
mentioned so far. We then stumble on the famous
angle
Banach–Tarski theorem (or paradox). In , Stefan
Banach and Alfred Tarski*** proved that it is pos-
sible to cut one sphere into ve pieces that can be
recombined to give two spheres, each the size of the original. is counter-intuitive result is the Banach–
F I G U R E 13 A polyhedron with one of its dihedral angles (© Luca Gastaldi)
Tarski theorem. Even worse, another version of the theorem states: take any two sets not
extending to in nity and containing a solid sphere each; then it is always possible to dis-
sect one into the other with a nite number of cuts. In particular it is possible to dissect
a pea into the Earth, or vice versa. Size does not count!**** Volume is thus not a useful
concept at all.
e Banach–Tarski theorem raises two questions: rst, can the result be applied to gold
or bread? at would solve many problems. Second, can it be applied to empty space?
In other words, are matter and empty space continuous? Both topics will be explored
later in our walk; each issue will have its own, special consequences. For the moment, we
eliminate this troubling issue by restricting our interest to smoothly curved shapes (and
cutting knives). With this restriction, volumes of matter and of empty space do behave
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Ref. 35
* Max Dehn (1878–1952), German mathematician, student of David Hilbert.
** is is also told in the beautiful book by M. A
& G.M. Z
, Proofs from the Book, Springer
Verlag, 1999. e title is due to the famous habit of the great mathematician Paul Erdös to imagine that all
beautiful mathematical proofs can be assembled in the ‘book of proofs’.
*** Alfred Tarski (b. 1902 Warsaw, d. 1983 Berkeley), Polish mathematician.
**** e proof of the result does not need much mathematics; it is explained beautifully by Ian Stewart in
Paradox of the spheres, New Scientist, 14 January 1995, pp. 28–31. e Banach–Tarski paradox also exists in
four dimensions, as it does in any higher dimension. More mathematical detail can be found in the beautiful
book by Steve Wagon.
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nicely: they are additive and rigid, and show no paradoxes. Indeed, the cuts required for the Banach–Tarski paradox are not smooth; it is not possible to perform them with an everyday knife, as they require (in nitely many) in nitely sharp bends performed with an in nitely sharp knife. Such a knife does not exist. Nevertheless, we keep in the back of our mind that the size of an object or of a piece of empty space is a tricky quantity – and that we need to be careful whenever we talk about it.
W
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Challenge 48 n
When you see a solid object with a straight edge, it is a %-safe bet that it is man-made.* e contrast between the objects seen in a city – buildings, furniture, cars, electricity poles,
boxes, books – and the objects seen in a forest – trees, plants, stones, clouds – is evident: in the forest nothing is straight or at, in the city most objects are. How is it possible for humans to produce straight objects while there are none to be found in nature?
Any forest teaches us the origin of straightness; it presents tall tree trunks and rays of daylight entering from above through the leaves. For this reason we call a line straight if it touches either a plumb-line or a light ray along its whole length. In fact, the two de nitions are equivalent. Can you con rm this? Can you nd another de nition? Obviously, we call a surface at if for any chosen orientation and position it touches a plumb-line or a light ray along its whole extension.
In summary, the concept of straightness – and thus also of atness – is de ned with the help of bodies or radiation. In fact, all spatial concepts, like all temporal concepts, require motion for their de nition.
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A
E?
Challenge 49 n Challenge 50 e
Page 495
Space and straightness pose subtle challenges. Some strange people maintain that all humans live on the inside of a sphere; they (usually) call this the hollow Earth theory. ey claim that the Moon, the Sun and the stars are all near the centre of the hollow sphere.
ey also explain that light follows curved paths in the sky and that when conventional physicists talk about a distance r from the centre of the Earth, the real hollow Earth distance is rhe = REarth r. Can you show that this model is wrong? Roman Sexl** used to ask this question to his students and fellow physicists. e answer is simple: if you think you have an argument to show that this view is wrong, you are mistaken! ere is no way of showing that such a view is wrong. It is possible to explain the horizon, the appearance of day and night, as well as the satellite photographs of the round Earth, such as Figure . To explain what happened during a ight to the Moon is also fun. A coherent hollow Earth view is fully equivalent to the usual picture of an in nitely extended space. We will come back to this problem in the section on general relativity.
Ref. 36
* e most common counter-examples are numerous crystalline minerals, where the straightness is related to the atomic structure. Another famous exception is the well-known Irish geological formation called the Giant’s Causeway. Other candidates that might come to mind, such as certain bacteria which have (almost) square or (almost) triangular shapes are not counter-examples, as the shapes are only approximate. ** Roman Sexl, (1939–1986), important Austrian physicist, author of several in uential textbooks on gravitation and relativity.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 14 A photograph of the Earth – seen from the direction of the Sun
C Space and time lead to many thought-provoking questions.
Challenge 51 n
**
How does one measure the speed of a gun bullet with a stop watch, in a space of m , without electronics? Hint: the same method can also be used to measure the speed of light.
**
Imagine a black spot on a white surface. What is the colour of the line separating the spot Challenge 52 n from the background? is question is o en called Peirce’s puzzle.
**
Also bread is an (approximate) irregular fractal. Challenge 53 n 2.7. Try to measure it.
e fractal dimension of bread is around
Challenge 54 n
**
Motoring poses many mathematical problems. A central one is the following parking issue: what is the shortest distance d from the car in front necessary to leave a parking spot without using reverse gear? (Assume that you know the geometry of your car, as shown in Figure 16, and its smallest outer turning radius R, which is known for every
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 15 A model illustrating the hollow Earth theory, showing how day and night appear (© Helmut Diehl)
d
b
w
L F I G U R E 16 Leaving a parking space
Challenge 55 n Challenge 56 n
car.) Next question: what is the smallest gap required when you are allowed to manoeuvre back and forward as o en as you like? Now a problem to which no solution seems to be available in the literature: How does the gap depend on the number, n, of times you use reverse gear? ( e author o ers 50 euro for the rst well-explained solution sent to him.)
Challenge 57 n
**
How o en in 24 hours do the hour and minute hands of a clock lie on top of each other? For clocks that also have a second hand, how o en do all three hands lie on top of each other?
** How many times in twelve hours can the two hands of a clock be exchanged with the
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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TA B L E 9 The exponential notation: how to write small and large numbers
N
E
N
E
1 0.1 0.2 0.324 0.01 0.001 0.000 1 0.000 01
−
ë− . ë−
−
−
−
− etc.
10 20 32.4 100 1000 10 000 100 000
ë .ë
etc.
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Challenge 58 n result that the new situation shows a valid time? What happens for clocks that also have a third hand for seconds?
** Challenge 59 n How many minutes does the Earth rotate in one minute?
**
What is the highest speed achieved by throwing (with and without a racket)? What was Challenge 60 n the projectile used?
**
A rope is put around the Earth, on the Equator, as tightly as possible. Challenge 61 n lengthened then by m. Can a mouse slip under it?
e rope is
Challenge 62 n
**
Jack was rowing his boat on a river. When he was under a bridge, he dropped a ball into the river. Jack continued to row in the same direction for 10 minutes a er he dropped the ball. He then turned around and rowed back. When he reached the ball, the ball had
oated m from the bridge. How fast was the river owing?
**
Adam and Bert are brothers. Adam is 18 years old. Bert is twice as old as at the time when Adam was the age that Bert is now. How old is Bert?
Ref. 37 Challenge 63 n
**
Scientists use a special way to write large and small numbers, explained in Table 9. In 1996 the smallest experimentally probed distance was − m, achieved between
quarks at Fermilab. (To savour the distance value, write it down without the exponent.) What does this measurement mean for the continuity of space?
**
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‘Where am I?’ is a common question; ‘When am I?’ is never asked, not even in other Challenge 64 n languages. Why?
** Challenge 65 n Is there a smallest time interval in nature? A smallest distance?
**
Given that you know what straightness is, how would you characterize or de ne the Challenge 66 n curvature of a curved line using numbers? And that of a surface?
** Challenge 67 n What is the speed of your eyelid?
**
e surface area of the human body is about m . Can you say where this large number Challenge 68 n comes from?
Challenge 69 n
**
Fractals in three dimensions bear many surprises. Take a regular tetrahedron; then glue on every one of its triangular faces a smaller regular tetrahedron, so that the surface of the body is again made up of many equal regular triangles. Repeat the process, gluing still smaller tetrahedrons to these new (more numerous) triangular surfaces. What is the shape of the nal fractal, a er an in nite number of steps?
Challenge 70 n
**
Zeno re ected on what happens to a moving object at a given instant of time. To discuss with him, you decide to build the fastest possible shutter for a photographic camera that you can imagine. You have all the money you want. What is the shortest shutter time you would achieve?
**
Can you prove Pythagoras’ theorem by geometrical means alone, without using Challenge 71 n coordinates? ( ere are more than 30 possibilities.)
** Challenge 72 n Why are most planets and moons (almost) spherical?
**
A rubber band connects the tips of the two hands of a clock. What is the path followed Challenge 73 n by the mid-point of the band?
**
ere are two important quantities connected to angles. As shown in Figure 17, what is usually called a (plane) angle is de ned as the ratio between the lengths of the arc and the radius. A right angle is π radian (or π rad) or °.
e solid angle is the ratio between area and the square of the radius. An eighth of a
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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a α
r α = −a
r
F I G U R E 17 The definition of plane and solid angles
rA Ω
Ω = A−r2
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earth horizon sky
sky
horizon earth
F I G U R E 18 How the apparent size of the Moon and the Sun changes
sphere is π or steradian π sr. As a result, a small solid angle shaped like a cone and Challenge 74 n the angle of the cone tip are di erent. Can you nd the relationship?
**
e de nition of angle helps to determine the size of a rework display. Measure the time
T, in seconds, between the moment that you see the rocket explode in the sky and the
moment you hear the explosion, measure the (plane) angle α of the ball with your hand.
e diameter D is
D s°Tα.
(3)
Challenge 75 e Why? For more about reworks, see the http://cc.oulu. /~kempmp website. By the way, the angular distance between the knuckles of an extended st are about °, ° and °, the
Challenge 76 n size of an extended hand °. Can you determine the other angles related to your hand?
**
Measuring angular size with the eye only is tricky. For example, can you say whether the Challenge 77 e Moon is larger or smaller than the nail of your thumb at the end of your extended arm?
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 19 How the apparent size of the Moon changes during its orbit (© Anthony Ayiomamitis)
F I G U R E 20 A vernier/nonius/clavius
Challenge 78 n
Angular size is not an intuitive quantity; it requires measurement instruments. A famous example, shown in Figure 18, illustrates the di culty of estimating angles.
Both the Sun and the Moon seem larger when they are on the horizon. In ancient times, Ptolemy explained this illusion by an unconscious apparent distance change induced by the human brain. In fact, the Moon is even further away from the observer when it is just above the horizon, and thus its image is smaller than it was a few hours earlier, when it was high in the sky. Can you con rm this?
In fact, the Moon’s size changes much more due to another e ect: the orbit of the Moon is elliptical. An example of this is shown in Figure 19.
Challenge 79 n
**
Cylinders can be used to roll a at object over the oor; they keep the object plane always at the same distance from the oor. What cross-sections other than circular allow you to realize the same feat? How many examples can you nd?
Challenge 80 d
**
Galileo also made mistakes. In his famous book, the Dialogues, he says that the curve formed by a thin chain hanging between two nails is a parabola, i.e. the curve de ned by y = x . at is not correct. What is the correct curve? You can observe the shape (approximately) in the shape of suspension bridges.
**
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Challenge 81 n
How does a vernier work? It is called nonius in other languages. e rst name is derived from a French military engineer* who did not invent it, the second is a play of words on the Latinized name of the Portuguese inventor of a more elaborate device** and the Latin word for ‘nine’. In fact, the device as we know it today – shown in Figure 20 – was designed around 1600 by Christophonius Clavius,*** the same astronomer who made the studies that formed the basis of the Gregorian calendar reform of 1582. Are you able to design a vernier/nonius/clavius that, instead of increasing the precision tenfold, does so by an arbitrary factor? Is there a limit to the attainable precision?
Challenge 82 n
**
Draw three circles, of di erent sizes, that touch each other. Now draw a fourth circle in the space between, touching the outer three. What simple relation do the inverse radii of the four circles obey?
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Challenge 83 n
**
Take a tetrahedron OABC whose triangular sides OAB, OBC and OAC are rectangular in O. In other words, OA, OB and OC are all perpendicular to each other. In the tetrahedron, the areas of the triangles OAB, OBC and OAC are respectively 8, 4 and 1. What is the area of triangle ABC?
Challenge 84 n
**
With two rulers, you can add and subtract numbers by lying them side by side. Are you able to design rulers that allow you to multiply and divide in the same manner? More elaborate devices using this principle were called slide rules and were the precursors of electronic calculators; they were in use all over the world until the 1970s.
**
How many days would a year have if the Earth turned the other way with the same rotaChallenge 85 n tion frequency?
** Challenge 86 n Where is the Sun in the spectacular situation shown in Figure 21?
**
Ref. 38 Could a two-dimensional universe exist? Alexander Dewdney described such a universe Challenge 87 d in a book. Can you explain why a two-dimensional universe is impossible?
* Pierre Vernier (1580–1637), French military o cer interested in cartography. ** Pedro Nuñes or Peter Nonnius (1502–1578), Portuguese mathematician and cartographer. *** Christophonius Clavius or Schlüssel (1537–1612), Bavarian astronomer.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 21 Anticrepuscular rays (© Peggy Peterson)
H
–
“La loso a è scritta in questo grandissimo libro che continuamente ci sta aperto innanzi agli occhi (io dico l’universo) ... Egli è scritto in lingua matematica.* ” Galileo Galilei, Il saggiatore VI.
Experiments show that the properties of Galilean time and space are extracted from the environment by most higher animals and by young children. Later, when children learn to speak, they put these experiences into concepts, as was just done above. With the help of these concepts, grown-up children then say that motion is change of position with time.
is description is illustrated by rapidly ipping the lower le corners of this book, starting at page . Each page simulates an instant of time, and the only change that takes place during motion is in the position of the object, represented by the dark spot. e other variations from one picture to the next, which are due to the imperfections of printing techniques, can be taken to simulate the inevitable measurement errors.
It is evident that calling ‘motion’ the change of position with time is neither an explanation nor a de nition, since both the concepts of time and position are deduced from motion itself. It is only a description of motion. Still, the description is useful, because it allows for high precision, as we will nd out by exploring gravitation and electrodynamics. A er all, precision is our guiding principle during this promenade. erefore the detailed description of changes in position has a special name: it is called kinematics.
e set of all positions taken by an object over time forms a path or trajectory. e origin of this concept is evident when one watches reworks** or again the previously mentioned ip movie in the lower le corners a er page . With the description of space and time by real numbers, a trajectory can be described by specifying its three coordinates (x, y, z) – one for each dimension – as continuous functions of time t. (Functions are
* Science is written in this huge book that is continuously open before our eyes (I mean the universe) ... It is
written in mathematical language.
** On the world of reworks, see the frequently asked questions list of the usenet group rec.pyrotechnics, or
search the web. A simple introduction is the article by J.A. C
, Pyrotechnics, Scienti c American
pp. 66–73, July 1990.
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collision F I G U R E 22 Two ways to test that the time of free fall does not depend on horizontal velocity
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de ned in detail on page .) is is usually written as x = x(t) = (x(t), y(t), z(t)).
For example, observation shows that the height z of any thrown or falling stone changes
as
z(t) = z + v (t − t ) − (t − to)
(4)
Ref. 39 Challenge 88 n Challenge 89 n
Ref. 40 Challenge 90 n
where t is the time the fall starts, z is the initial height, v is the initial velocity in the vertical direction and = . m s is a constant that is found to be the same, within about one part in , for all falling bodies on all points of the surface of the Earth. Where do the value . m s and its slight variations come from? A preliminary answer will be given shortly, but the complete elucidation will occupy us during the larger part of this hike.
Equation ( ) allows us to determine the depth of a well, given the time a stone takes to reach its bottom. e equation also gives the speed v with which one hits the ground
a er jumping from a tree, namely v = h . A height of m yields a velocity of km h. e velocity is thus proportional only to the square root of the height. Does this mean
that one’s strong fear of falling results from an overestimation of its actual e ects? Galileo was the rst to state an important result about free fall: the motions in the
horizontal and vertical directions are independent. He showed that the time it takes for a cannon ball that is shot exactly horizontally to fall is independent of the strength of the gunpowder, as shown in Figure . Many great thinkers did not agree with this statement even a er his death: in the Academia del Cimento even organized an experiment to check this assertion, by comparing the ying cannon ball with one that simply fell vertically. Can you imagine how they checked the simultaneity? Figure also shows how you can check this at home. In this experiment, whatever the powder load of the cannon, the two bodies will always collide, thus proving the assertion.
In other words, a canon ball is not accelerated in the horizontal direction. Its horizontal motion is simply unchanging. By extending the description of equation ( ) with the two expressions for the horizontal coordinates x and y, namely
x(t) = x + vx (t − t )
y(t) = y + vy (t − t ) ,
(5)
a complete description for the path followed by thrown stones results. A path of this shape is called a parabola; it is shown in Figures , and .*A parabolic shape is also used for
* Apart from the graphs shown in Figure 23, there is also the con guration space spanned by the coordinates
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configuration space
z
•.
space-time diagrams
z
– hodograph
vz
phase space graph
mv z
x
t
x
vx mv x
z
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t
x
F I G U R E 23 Various types of graphs describing the same path of a thrown stone
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 24 Three superimposed images of a frass pellet shot away by a caterpillar (© Stanley Caveney)
Challenge 91 n light re ectors inside pocket lamps or car headlights. Can you show why?
T e kinematic description of motion is useful for answering a whole range of questions.
Ref. 41 Challenge 92 n
**
Numerous species of moth and butter y caterpillars shoot away their frass – to put it more crudely: their shit – so that its smell does not help predators to locate them. Stanley Caveney and his team took photographs of this process. Figure 24 shows a caterpillar (yellow) of the skipper Calpodes ethlius inside a rolled up green leaf caught in the act. Given that the record distance observed is . m (though by another species, Epargyreus clarus), what is the ejection speed? How do caterpillars achieve it?
of all particles of a system; only for a single particle it is equal to the real space. e phase space diagram is also called state space diagram.
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y
derivative: dy/dt
slope: Δ y/Δ t
Δt
Δy t
F I G U R E 25 Derivatives
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**
What is the horizontal distance one can reach with a stone, given the speed and the angle Challenge 93 n from the horizontal at which it is thrown?
**
Challenge 94 n How can the speed of falling rain be measured using an umbrella? Page 276 be used to measure the speed of light.
is method can also
** Challenge 95 n What is the maximum numbers of balls that could be juggled at the same time?
Ref. 42 Ref. 43 Challenge 96 n
**
Finding an upper limit for the long jump is interesting. e running speed world record in 1997 was m s km h by Ben Johnson, and the women’s record was
m s km h. In fact, long jumpers never run much faster than about . m s. How much extra jump distance could they achieve if they could run full speed? How could they achieve that? In addition, long jumpers take o at angles of about °, as they are not able to achieve a higher angle at the speed they are running. How much would they gain if they could achieve °?
**
Challenge 97 n Is it true that rain drops would kill if it weren’t for the air resistance of the atmosphere? What about ice?
** Challenge 98 n Are bullets red from a gun falling back a er being red into the air dangerous?
Challenge 99 n
e last two issues arise because equation ( ) does not hold in all cases. For example, leaves or potato crisps do not follow it. As Galileo already knew, this is a consequence of air resistance; we will discuss it shortly. In fact, even without air resistance, the path of a stone is not always a parabola; can you nd such a situation?
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In the Galilean description of nature, motion and rest are opposites. In other words, a body is at rest when its position, i.e. its coordinates, do not change with time. In other words, (Galilean) rest is de ned as
x(t) = const .
(6)
Later we will see that this de nition, contrary to rst impressions, is not much use and will have to be modi ed. e de nition of rest implies that non-resting objects can be distinguished by comparing the rapidity of their displacement. One thus can de ne the velocity v of an object as the change of its position x with time t. is is usually written as
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v
=
dx dt
.
(7)
In this expression, valid for each coordinate separately, d dt means ‘change with time’; one can thus say that velocity is the derivative of position with respect to time. e speed v is the name given to the magnitude of the velocity v. Derivatives are written as fractions in order to remind the reader that they are derived from the idea of slope. e expression
dy dt
is meant as an abbreviation of
lim
∆t
∆y ∆t
,
(8)
a shorthand for saying that the derivative at a point is the limit of the slopes in the neighChallenge 100 e bourhood of the point, as shown in Figure . is de nition implies the working rules
d(y + dt
z)
=
dy dt
+
dz dt
,
d(c y) dt
=
c
dy dt
,
d dt
dy dt
=
dy dt
,
d( y z) dt
=
dy dt
z+y
dz dt
,(9)
c being any number. is is all one ever needs to know about derivatives. e quantities dt and dy, sometimes useful by themselves, are called di erentials. ese concepts are due to Gottfried Wilhelm Leibniz.* Derivatives lie at the basis of all calculations based on the continuity of space and time. Leibniz was the person who made it possible to describe and use velocity in physical formulae and, in particular, to use the idea of velocity at a given point in time or space for calculations.
e de nition of velocity assumes that it makes sense to take the limit ∆t . In other words, it is assumed that in nitely small time intervals do exist in nature. e de nition of velocity with derivatives is possible only because both space and time are described by sets which are continuous, or in mathematical language, connected and complete. In the rest of our walk we shall not forget that from the beginning of classical physics, in nities are present in its description of nature. e in nitely small is part of our de nition of
* Gottfried Wilhelm Leibniz (b. 1646 Leipzig, d. 1716 Hannover), Saxon lawyer, physicist, mathematician, philosopher, diplomat and historian. He was one of the great minds of mankind; he invented the di erential calculus (before Newton) and published many successful books in the various elds he explored, among them De arte combinatoria, Hypothesis physica nova, Discours de métaphysique, Nouveaux essais sur l’entendement humain, the éodicée and the Monadologia.
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Challenge 101 e
Page 1001
velocity. Indeed, di erential calculus can be de ned as the study of in nity and its uses.
We thus discover that the appearance of in nity does not automatically render a descrip-
tion impossible or imprecise. In order to remain precise, physicists use only the smallest
two of the various possible types of in nities. eir precise de nition and an overview of
other types are introduced in the intermezzo following this chapter.
e appearance of in nity in the usual description of motion
was rst criticized in his famous ironical arguments by Zeno of
Elea (around
), a disciple of Parmenides. In his so-called
third argument, Zeno explains that since at every instant a given
object occupies a part of space corresponding to its size, the notion
of velocity at a given instant makes no sense; he provokingly con-
cludes that therefore motion does not exist. Nowadays we would
not call this an argument against the existence of motion, but
against its usual description, in particular against the use of in n-
itely divisible space and time. (Do you agree?) Nevertheless, the Gottfried Leibniz description criticized by Zeno actually works quite well in every-
day life. e reason is simple but deep: in daily life, changes are indeed continuous.
Large changes in nature are made up of many small changes. is property of nature is
not obvious. For example, we note that we have tacitly assumed that the path of an object
is not a fractal or some other badly behaved entity. In everyday life this is correct; in other
domains of nature it is not. e doubts of Zeno will be partly rehabilitated later in our
walk, and increasingly so the more we proceed. e rehabilitation is only partial, as the
solution will be di erent from that which he envisaged; on the other hand, the doubts
about the idea of ‘velocity at a point’ will turn out to be well-founded. For the moment
though, we have no choice: we continue with the basic assumption that in nature changes
happen smoothly.
Why is velocity necessary as a concept? Aiming for precision in the description of
motion, we need to nd the complete list of aspects necessary to specify the state of an
object. e concept of velocity is obviously on this list. Continuing along the same lines,
we call acceleration a of a body the change of velocity v with time, or
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a
=
dv dt
=
dx dt
.
(10)
Acceleration is what we feel when the Earth trembles, an aeroplane takes o , or a bicycle goes round a corner. More examples are given in Table . Like velocity, acceleration has both a magnitude and a direction, properties indicated by the use of bold letters for their abbreviations. *
Challenge 102 n
* Such physical quantities are called vectors. In more precise, mathematical language, a vector is an element
of a set, called vector space, in which the following properties hold for all vectors a and b and for all numbers
c and d:
c(a + b) = ca + cb , (c + d)a = ca + da , (cd)a = c(da) and a = a .
(11)
Another example of vector space is the set of all positions of an object. Does the set of all rotations form a vector space? All vector spaces allow the de nition of a unique null vector and of a single negative vector for each vector in it.
Note that vectors do not have speci ed points at which they start: two arrows with same direction and
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
TA B L E 10 Some measured acceleration values O
A
-
What is the lowest you can nd? Acceleration of the galaxy M82 by its ejected jet Acceleration of a young star by an ejected jet Acceleration of the Sun in its orbit around the Milky Way Unexplained deceleration of the Pioneer satellites Acceleration at Equator due to Earth’s rotation Centrifugal acceleration due to the Earth’s rotation Electron acceleration in household electricity wire due to alternating current Gravitational acceleration on the Moon Gravitational acceleration on the Earth’s surface, depending on location Standard gravitational acceleration Highest acceleration for a car or motorbike with engine-driven wheels Gravitational acceleration on Jupiter’s surface Acceleration of cheetah Acceleration that triggers air bags in cars Fastest leg-powered acceleration (by the froghopper, Philaenus spumarius, an insect) Tennis ball against wall Bullet acceleration in ri e Fastest centrifuges Acceleration of protons in large accelerator Acceleration of protons inside nucleus Highest possible acceleration in nature
Challenge 105 n
fm s pm s . nm s . nm s . mm s mm s mm s
. ms
. .ms
.
ms
ms
ms
ms
ms
km s
. Mm s Mm s . Gm s
Tm s ms ms
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Higher derivatives than acceleration can also be de ned in the same manner. ey Challenge 104 n add little to the description of nature, because as we will show shortly neither these nor
even acceleration itself are useful for the description of the state of motion of a system.
length are the same vector, even if they start at di erent points in space. In many vector spaces the concept of length (specifying the ‘magnitude’) can be introduced, usually via
an intermediate step. A vector space is called Euclidean if one can de ne for it a scalar product between two vectors, a number ab satisfying
aa , ab = ba , (a + a′)b = ab + a′b , a(b + b′) = ab + ab′ and (ca)b = a(cb) = c(ab) . (12)
Challenge 103 n
In Cartesian coordinate notation, the standard scalar product is given by ab = axbx + ayby + azbz. Whenever it vanishes the two vectors are orthogonal. e length or norm of a vector can then be de ned as the square root of the scalar product of a vector with itself: a = aa .
e scalar product is also useful for specifying directions. Indeed, the scalar product between two vectors encodes the angle between them. Can you deduce this important relation?
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Challenge 106 e
Ref. 46 Challenge 107 n
“Wenn ich den Gegenstand kenne, so kenne ich auch sämtliche Möglichkeiten seines Vorkommens in Sachverhalten.* ” Ludwig Wittgenstein, Tractatus, .
One aim of the study of motion is to nd a complete and precise description of both states and objects. With the help of the concept of space, the description of objects can be re ned considerably. In particular, one knows from experience that all objects seen in daily life have an important property: they can be divided into parts. O en this observation is expressed by saying that all objects, or bodies, have two properties. First, they are made out of matter,** de ned as that aspect of an object responsible for its impenetrability, i.e. the property preventing two objects from being in the same place. Secondly, bodies have a certain form or shape, de ned as the precise way in which this impenetrability is distributed in space.
In order to describe motion as accurately as possible, it is convenient to start with those bodies that are as simple as possible. In general, the smaller a body, the simpler it is. A body that is so small that its parts no longer need to be taken into account is called a particle. ( e older term corpuscle has fallen out of fashion.) Particles are thus idealized small stones. e extreme case, a particle whose size is negligible compared with the dimensions of its motion, so that its position is described completely by a single triplet of coordinates, is called a point particle or a point mass. In equation ( ), the stone was assumed to be such a point particle.
Do point-like objects, i.e. objects smaller than anything one can measure, exist in daily life? Yes and no. e most notable examples are the stars. At present, angular sizes as small as µrad can be measured, a limit given by the uctuations of the air in the atmosphere. In space, such as for the Hubble telescope orbiting the Earth, the angular limit is due to the diameter of the telescope and is of the order of nrad. Practically all stars seen from Earth are smaller than that, and are thus e ectively ‘point-like’, even when seen with the most powerful telescopes.
As an exception to the general rule, the size of a few large and nearby stars, of red giant type, can be measured with special instruments.*** Betelgeuse, the higher of the two shoulders of Orion shown in Figure , Mira in Cetus, Antares in Scorpio, Aldebaran in Taurus and Sirius in Canis Major are examples of stars whose size has been measured; they are all only a few light years from Earth. Of course, like the Sun, all other stars have a nite size, but one cannot prove this by measuring dimensions in photographs. (True?)
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Ref. 45
* If I know an object I also know all its possible occurrences in states of a airs.
** Matter is a word derived from the Latin ‘materia’, which originally meant ‘wood’ and was derived via
intermediate steps from ‘mater’, meaning ‘mother’.
*** e website http://www.astro.uiuc.edu/~kaler/sow/sowlist.html gives an introduction to the di erent
types of stars. e http://www.astro.wisc.edu/~dolan/constellations/constellations.html website provides de-
tailed and interesting information about constellations.
For an overview of the planets, see the beautiful book by K.R. L & C.A. W
, Vagabonds de
l’espace – Exploration et découverte dans le système solaire, Springer Verlag, 1993. e most beautiful pictures
of the stars can be found in D. M , A View of the Universe, Sky Publishing and Cambridge University
Press, 1993.
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α
Betelgeuse
γ
Bellatrix
ε δ Mintaka ζ Alnilam
Alnitag
κ
Saiph
β
Rigel
F I G U R E 26 Orion (in natural colours) and Betelgeuse
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Challenge 108 e Challenge 109 n Challenge 110 n Challenge 111 n
Ref. 47 Page 704
e di erence between ‘point-like’ and nite size sources can be seen with the naked eye: at night, stars twinkle, but planets do not. (Check it!) is e ect is due to the turbulence of air. Turbulence has an e ect on the almost point-like stars because it de ects light rays by small amounts. On the other hand, air turbulence is too weak to lead to twinkling of sources of larger angular size, such as planets or arti cial satellites,* because the de ection is averaged out in this case.
An object is point-like for the naked eye if its angular size is smaller than about ′= . mrad. Can you estimate the size of a ‘point-like’ dust particle? By the way, an object is invisible to the naked eye if it is point-like and if its luminosity, i.e. the intensity of the light from the object reaching the eye, is below some critical value. Can you estimate whether there are any man-made objects visible from the Moon, or from the space shuttle?
e above de nition of ‘point-like’ in everyday life is obviously misleading. Do proper, real point particles exist? In fact, is it at all possible to show that a particle has vanishing size? is question will be central in the last two parts of our walk. In the same way, we need to ask and check whether points in space do exist. Our walk will lead us to the astonishing result that all the answers to these questions are negative. Can you imagine why? Do not be disappointed if you nd this issue di cult; many brilliant minds have had the same problem.
However, many particles, such as electrons, quarks or photons are point-like for all practical purposes. Once one knows how to describe the motion of point particles, one can also describe the motion of extended bodies, rigid or deformable, by assuming that they are made of parts. is is the same approach as describing the motion of an animal as a whole by combining the motion of its various body parts. e simplest description, the continuum approximation, describes extended bodies as an in nite collection of point particles. It allows us to understand and to predict the motion of milk and honey, the motion of the air in hurricanes and of perfume in rooms. e motion of re and all other gaseous bodies, the bending of bamboo in the wind, the shape changes of chewing gum, and the growth of plants and animals can also be described in this way.
A more precise description than the continuum approximation is given below. Nev-
* A satellite is an object circling a planet, like the Moon; an arti cial satellite is a system put into orbit by humans, like the Sputniks.
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F I G U R E 27 How an object can rotate continuously without tangling up the connection to a second object
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Figure still to be added F I G U R E 28 Legs and ‘wheels’ in living beings
ertheless, all observations so far have con rmed that the motion of large bodies can be described to high precision as the result of the motion of their parts. is approach will guide us through the rst two parts of our mountain ascent. Only in the third part will we discover that, at a fundamental scale, this decomposition is impossible.
L
Appendix D Ref. 48
Challenge 112 n Challenge 113 n Challenge 114 n
Ref. 49
e parts of a body determine its shape. Shape is an important aspect of bodies: among other things, it tells us how to count them. In particular, living beings are always made of a single body. is is not an empty statement: from this fact we can deduce that animals cannot have wheels or propellers, but only legs, ns, or wings. Why?
Living beings have only one surface; simply put, they have only one piece of skin. Mathematically speaking, animals are connected. is is o en assumed to be obvious, and it is o en mentioned that the blood supply, the nerves and the lymphatic connections to a rotating part would get tangled up. However, this argument is not so simple, as Figure shows. It shows that it is indeed possible to rotate a body continuously against a second one, without tangling up the connections. Can you nd an example for this kind of motion in your own body? Are you able to see how many cables may be attached to the rotating body of the gure without hindering the rotation?
Despite the possibility of animals having rotating parts, the method of Figure still cannot be used to make a practical wheel or propeller. Can you see why? Evolution had no choice: it had to avoid animals with parts rotating around axles. at is the reason that propellers and wheels do not exist in nature. Of course, this limitation does not rule out that living bodies move by rotation as a whole: tumbleweed, seeds from various trees, some insects, certain other animals, children and dancers occasionally move by rolling or rotating as a whole.
Single bodies, and thus all living beings, can only move through deformation of their shape: therefore they are limited to walking, running, crawling or apping wings or ns,
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Page 972 Ref. 50
as shown in Figure . In contrast, systems of several bodies, such as bicycles, pedal boats or other machines, can move without any change of shape of their components, thus enabling the use of axles with wheels, propellers or other rotating devices.*
In summary, whenever we observe a construction in which some part is turning continuously (and without the ‘wiring’ of the gure) we know immediately that it is an artefact: it is a machine, not a living being (but built by one). However, like so many statements about living creatures, this one also has exceptions. e distinction between one and two bodies is poorly de ned if the whole system is made of only a few molecules. is happens most clearly inside bacteria. Organisms such as Escherichia coli, the well-known bacterium found in the human gut, or bacteria from the Salmonella family, all swim using
agella. Flagella are thin laments, similar to tiny hairs that stick out of the cell membrane. In the s it was shown that each agellum, made of one or a few long molecules with a diameter of a few tens of nanometres, does in fact turn about its axis. A bacterium is able to turn its agella in both clockwise and anticlockwise directions, can achieve more than turns per second, and can turn all its agella in perfect synchronization. ( ese wheels are so tiny that they do not need a mechanical connection.) erefore wheels actually do exist in living beings, albeit only tiny ones. But let us now continue with our study of simple objects.
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Ref. 51
Walking through a forest we observe two rather di erent types of motion: the breeze moves the leaves, and at the same time their shadows move on the ground. Shadows are a simple type of image. Both objects and images are able to move. Running tigers, falling snow akes, and material ejected by volcanoes are examples of motion, since they all change position over time. For the same reason, the shadow following our body, the beam of light circling the tower of a lighthouse on a misty night, and the rainbow that constantly keeps the same apparent distance from the hiker are examples of motion.
Everybody who has ever seen an animated cartoon knows that images can move in more surprising ways than objects. Images can change their size, shape and even colour, a feat only few objects are able to perform.** Images can appear and disappear without trace, multiply, interpenetrate, go backwards in time and defy gravity or any other force.
Challenge 115 n Challenge 116 n
Ref. 52
* Despite the disadvantage of not being able to use rotating parts and of being restricted to one piece only,
nature’s moving constructions, usually called animals, o en outperform human built machines. As an ex-
ample, compare the size of the smallest ying systems built by evolution with those built by humans. (See,
e.g. http://pixelito.reference.be.) ere are two reasons for this discrepancy. First, nature’s systems have in-
tegrated repair and maintenance systems. Second, nature can build large structures inside containers with
small openings. In fact, nature is very good at what people do when they build sailing ships inside glass
bottles. e human body is full of such examples; can you name a few?
** Excluding very slow changes such as the change of colour of leaves in the Fall, in nature only certain
crystals, the octopus, the chameleon and a few other animals achieve this. Of man-made objects, television,
computer displays, heated objects and certain lasers can do it. Do you know more examples? An excellent
source of information on the topic of colour is the book by K. N
, e Physics and Chemistry of Colour
– the een causes of colour, J. Wiley & Sons, 1983. In the popular science domain, the most beautiful book is
the classic work by the Flemish astronomer M
G.J. M
, Light and Colour in the Outdoors,
Springer, 1993, an updated version based on his wonderful book series, De natuurkunde van ‘t vrije veld,
ieme & Cie, Zutphen. Reading it is a must for all natural scientists. On the web, there is also the – much
simpler – http://webexhibits.org/causesofcolour website.
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F I G U R E 29 In which direction does the bicycle turn?
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 53
Page 787
Challenge 117 n Challenge 118 n
Ref. 54
Images, even ordinary shadows, can move faster than light. Images can oat in space and keep the same distance from approaching objects. Objects can do almost none of this. In general, the ‘laws of cartoon physics’ are rather di erent from those in nature. In fact, the motion of images does not seem to follow any rules, in contrast to the motion of objects. On the other hand, both objects and images di er from their environment in that they have boundaries de ning their size and shape. We feel the need for precise criteria allowing the two cases to be distinguished.
Making a clear distinction between images and objects is performed using the same method that children or animals use when they stand in front of a mirror for the rst time: they try to touch what they see. Indeed, if we are able to touch what we see – or more precisely, if we are able to move it – we call it an object, otherwise an image.* Images cannot be touched, but objects can. Images cannot hit each other, but objects can. And as everybody knows, touching something means feeling that it resists movement. Certain bodies, such as butter ies, pose little resistance and are moved with ease, others, such as ships, resist more, and are moved with more di culty. is resistance to motion – more precisely, to change of motion – is called inertia, and the di culty with which a body can be moved is called its (inertial) mass. Images have neither inertia nor mass.
Summing up, for the description of motion we must distinguish bodies, which can be touched and are impenetrable, from images, which cannot and are not. Everything visible is either an object or an image; there is no third possibility. (Do you agree?) If the object is so far away that it cannot be touched, such as a star or a comet, it can be di cult to decide whether one is dealing with an image or an object; we will encounter this di culty repeatedly. For example, how would you show that comets are objects and not images?
In the same way that objects are made of matter, images are made of radiation. Images are the domain of shadow theatre, cinema, television, computer graphics, belief systems and drug experts. Photographs, motion pictures, ghosts, angels, dreams and many hallucinations are images (sometimes coupled with brain malfunction). To understand images, we need to study radiation (plus the eye and the brain). However, due to the importance of objects – a er all we are objects ourselves – we study the latter rst.
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Ref. 55
Democritus a rms that there is only one type of movement: at resulting from collision.
“ ” Aetius, Opinions.
* One could propose including the requirement that objects may be rotated; however, this requirement gives di culties in the case of atoms, as explained on page 751, and with elementary particles, so that rotation is not made a separate requirement.
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v1
v2
v'1
v'2
F I G U R E 30 Collisions define mass
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F I G U R E 31 The standard kilogram (© BIPM)
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
When a child rides a monocycle, she or he makes use of a general rule in our world: one body acting on another puts it in motion. Indeed, in about six hours, anybody can learn to ride and enjoy a monocycle. As in all of life’s pleasures, such as toys, animals, women, machines, children, men, the sea, wind, cinema, juggling, rambling and loving, something pushes something else. us our rst challenge is to describe this transfer of motion in more precise terms.
Contact is not the only way to put something into motion; a counter-example is an apple falling from a tree or one magnet pulling another. Non-contact in uences are more fascinating: nothing is hidden, but nevertheless something mysterious happens. Contact motion seems easier to grasp, and that is why one usually starts with it. However, despite this choice, non-contact forces are not easily avoided. Taking this choice one has a similar experience to that of cyclists. (See Figure .) If you ride a bicycle at a sustained speed and try to turn le by pushing the right-hand steering bar, you will turn right.* In other words, despite our choice the rest of our walk will rapidly force us to study non-contact interactions as well.
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W
?
∆ός µοι ποῦ στω καὶ κινῶ τὴν γῆν. Da ubi consistam, et terram movebo.*
“ Archimedes ” When we push something we are unfamiliar with, such as when we kick an object on
the street, we automatically pay attention to the same aspect that children explore when
they stand in front of a mirror for the rst time, or when they see a red laser spot for
the rst time. We check whether the unknown entity can be pushed and pay attention to
how the unknown object moves under our in uence. e high precision version of the
experiment is shown in Figure . Repeating the experiment with various pairs of objects,
we nd – as in everyday life – that a xed quantity mi can be ascribed to every object i. e more di cult it is to move an object, the higher the quantity; it is determined by the
relation
m m
=
−
∆v ∆v
(13)
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Ref. 45
where ∆v is the velocity change produced by the collision. e number mi is called the mass of the object i.
In order to have mass values that are common to everybody, the mass value for one par-
ticular, selected object has to be xed in advance. is special object, shown in Figure
is called the standard kilogram and is kept with great care under vacuum in a glass con-
tainer in Sèvres near Paris. It is touched only once every few years because otherwise dust,
humidity, or scratches would change its mass. rough the standard kilogram the value
of the mass of every other object in the world is determined.
e mass thus measures the di culty of getting something moving. High masses are
harder to move than low masses. Obviously, only objects have mass; images don’t. (By the
way, the word ‘mass’ is derived, via Latin, from the Greek µαζα – bread – or the Hebrew
‘mazza’ – unleavened bread – quite a change in meaning.)
Experiments with everyday life objects also show that throughout any collision, the
sum of all masses is conserved:
mi = const .
(14)
i
e principle of conservation of mass was rst stated by Antoine-Laurent Lavoisier.** Conservation of mass implies that the mass of a composite system is the sum of the mass of the components. In short, Galilean mass is a measure for the quantity of matter.
Challenge 119 n Ref. 56
* is surprising e ect obviously works only above a certain minimal speed. Can you determine what this speed is? Be careful! Too strong a push will make you fall. * ‘Give me a place to stand, and I’ll move the Earth.’ Archimedes (c. 283–212), Greek scientist and engineer.
is phrase was attributed to him by Pappus. Already Archimedes knew that the distinction used by lawyers between movable and immovable property made no sense. ** Antoine-Laurent Lavoisier (1743–1794), French chemist and a genius. Lavoisier was the rst to understand that combustion is a reaction with oxygen; he discovered the components of water and introduced mass measurements into chemistry. When he was (unjustly) sentenced to the guillotine during the French revolution, he decided to use the experience for a scienti c experiment; he decided to blink his eyes as frequently as possible a er his head was cut o , in order to show others how long it takes to lose consciousness. Lavoisier managed to blink eleven times.
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F I G U R E 32 Is this dangerous?
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Antoine Lavoisier
e de nition of mass can also be given in another way. We can
ascribe a number mi to every object i such that for collisions free of outside interference the following sum is unchanged throughout the
collision:
mi vi = const .
(15)
i
e product of the velocity vi and the mass mi is called the momentum of the body. e sum, or total momentum of the system, is the same before and a er the collision; it is a conserved quantity. Momentum conservation de nes mass. e two conservation principles Christiaan Huygens ( ) and ( ) were rst stated in this way by the important Dutch physicist Christiaan Huygens.* Some typical momentum values are given in Table .
Momentum conservation implies that when a moving sphere hits a resting one of the same mass, a simple rule determines the angle between the directions the two spheres take
* Christiaan Huygens (b. 1629 ’s Gravenhage, d. 1695 Hofwyck) was one of the main physicists and mathematicians of his time. Huygens clari ed the concepts of mechanics; he also was one of the rst to show that light is a wave. He wrote in uential books on probability theory, clock mechanisms, optics and astronomy. Among other achievements, Huygens showed that the Orion Nebula consists of stars, discovered Titan, the moon of Saturn, and showed that the rings of Saturn consist of rock. ( is is in contrast to Saturn itself, whose density is lower than that of water.)
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TA B L E 11 Some measured momentum values
O
Green photon momentum Average momentum of oxygen molecule in air X-ray photon momentum γ photon momentum Highest particle momentum in accelerators Planck momentum Fast billiard ball Flying ri e bullet Box punch Comfortably walking human Car on highway Impact of meteorite with km diameter Momentum of a galaxy in galaxy collision
M
ë − Ns − Ns − Ns − Ns f Ns . Ns Ns Ns 15 to Ns Ns kNs
TNs up to Ns
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Challenge 120 n Challenge 121 n
a er the collision. Can you nd this rule? It is particularly useful when playing billiards. We will nd out later that it is not valid in special relativity.
Another consequence is shown in Figure : a man lying on a bed of nails with two large blocks of concrete on his stomach. Another man is hitting the concrete with a heavy sledgehammer. As the impact is mostly absorbed by the concrete, there is no pain and no danger – unless the concrete is missed. Why?
e above de nition of mass has been generalized by the physicist and philosopher Ernst Mach* in such a way that it is valid even if the two objects interact without contact, as long as they do so along the line connecting their positions. e mass ratio between two bodies is de ned as a negative inverse acceleration ratio, thus as
m m
=
−
a a
,
(16)
where a is the acceleration of each body during the interaction. is de nition has been studied in much detail in the physics community, mainly in the nineteenth century. A few points sum up the results:
— e de nition of mass implies the conservation of momentum mv. Momentum conservation is not a separate principle. Conservation of momentum cannot be checked experimentally, because mass is de ned in such a way that the principle holds.
* Ernst Mach (1838 Chrlice–1916 Vaterstetten), Austrian physicist and philosopher. e mach unit for aeroplane speed as a multiple of the speed of sound in air (about . km s) is named a er him. He developed the so-called Mach–Zehnder interferometer; he also studied the basis of mechanics. His thoughts about mass and inertia in uenced the development of general relativity, and led to Mach’s principle, which will appear later on. He was also proud to be the last scientist denying – humorously, and against all evidence – the existence of atoms.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
— e de nition of mass implies the equality of the products m a and −m a . Such products are called forces. e equality of acting and reacting forces is not a separate principle; mass is de ned in such a way that the principle holds.
— e de nition of mass is independent of whether contact is involved or not, and whether the origin of the accelerations is due to electricity, gravitation, or other interactions.* Since the interaction does not enter the de nition of mass, mass values de ned with the help of the electric, nuclear or gravitational interaction all agree, as long as momentum is conserved. All known interactions conserve momentum. For some unfortunate historical reasons, the mass value measured with the electric or nuclear interactions is called the ‘inertial’ mass and the mass measured using gravity is called the ‘gravitational’ mass. As it turns out, this arti cial distinction has no real meaning; this becomes especially clear when one takes an observation point that is far away from all the bodies concerned.
— e de nition of mass is valid only for observers at rest or in inertial motion. More about this issue later.
Challenge 123 n Challenge 124 n
By measuring the masses of bodies around us, as given in Table , we can explore the science and art of experiments. We also discover the main properties of mass. It is additive in everyday life, as the mass of two bodies combined is equal to the sum of the two separate masses. Furthermore, mass is continuous; it can seemingly take any positive value. Finally, mass is conserved; the mass of a system, de ned as the sum of the mass of all constituents, does not change over time if the system is kept isolated from the rest of the world. Mass is not only conserved in collisions but also during melting, evaporation, digestion and all other processes.
Later we will nd that in the case of mass all these properties, summarized in Table , are only approximate. Precise experiments show that none of them are correct.** For the moment we continue with the present, Galilean concept of mass, as we have not yet a better one at our disposal.
In a famous experiment in the sixteenth century, for several weeks Santorio Santorio (Sanctorius) ( – ), friend of Galileo, lived with all his food and drink supply, and also his toilet, on a large balance. He wanted to test mass conservation. How did the measured weight change with time?
e de nition of mass through momentum conservation implies that when an object falls, the Earth is accelerated upwards by a tiny amount. If one could measure this tiny amount, one could determine the mass of the Earth. Unfortunately, this measurement is impossible. Can you nd a better way to determine the mass of the Earth?
Summarizing Table , the mass of a body is thus most precisely described by a positive real number, o en abbreviated m or M. is is a direct consequence of the impenetrability of matter. Indeed, a negative (inertial) mass would mean that such a body would move in the opposite direction of any applied force or acceleration. Such a body could not be kept
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Page 90 Challenge 122 n
* As mentioned above, only central forces obey the relation (16) used to de ne mass. Central forces act between the centre of mass of bodies. We give a precise de nition later. However, since all fundamental forces are central, this is not a restriction. ere seems to be one notable exception: magnetism. Is the de nition of mass valid in this case? ** In particular, in order to de ne mass we must be able to distinguish bodies. is seems a trivial requirement, but we discover that this is not always possible in nature.
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TA B L E 12 Some measured mass values
O
Mass increase due to absorption of one green photon Lightest known object: electron Atom of argon Lightest object ever weighed (a gold particle) Human at early age (fertilized egg) Water adsorbed on to a kilogram metal weight Planck mass Fingerprint Typical ant Water droplet Honey bee Heaviest living things, such as the fungus Armillaria ostoyae or a large Sequoia Sequoiadendron giganteum Largest ocean-going ship Largest object moved by man (Troll gas rig) Large antarctic iceberg Water on Earth Solar mass Our galaxy Total mass visible in the universe
M
. ë − kg
.
( ) ë − kg
.
( )u= .
. ag
−g
−g
. ë −g
−g
−g
mg
.g
kg
ë kg . ë kg kg kg . ë kg kg kg
( ) yg
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
TA B L E 13 Properties of Galilean mass
M
P
M
D
Can be distinguished Can be ordered Can be compared Can change gradually Can be added Beat any limit Do not change Do not disappear
distinguishability sequence measurability continuity quantity of matter in nity conservation impenetrability
element of set order metricity completeness additivity unboundedness, openness invariance positivity
Page 646 Page 1195 Page 1205 Page 1214 Page 69 Page 647
m = const m
Challenge 125 e
in a box; it would break through any wall trying to stop it. Strangely enough, negative mass bodies would still fall downwards in the eld of a large positive mass (though more slowly than an equivalent positive mass). Are you able to con rm this? However, a small positive mass object would oat away from a large negative-mass body, as you can easily deduce by comparing the various accelerations involved. A positive and a negative mass of the
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Challenge 126 e Page 315, page 759
same value would stay at constant distance and spontaneously accelerate away along the line connecting the two masses. Note that both energy and momentum are conserved in all these situations.* Negative-mass bodies have never been observed. Antimatter, which will be discussed later, also has positive mass.
I
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Challenge 129 n Challenge 130 n
“Every body continues in the state of rest or of uniform motion in a straight line except in so far as it doesn’t. ” Arthur Eddington**
e product p = mv of mass and velocity is called the momentum of a particle; it describes the tendency of an object to keep moving during collisions. e larger it is, the harder it is to stop the object. Like velocity, momentum has a direction and a magnitude: it is a vector. In French, momentum is called ‘quantity of motion’, a more appropriate term. In the old days, the term ‘motion’ was used instead of ‘momentum’, for example by Newton. Relation ( ), the conservation of momentum, therefore expresses the conservation of motion during interactions.
Momentum and energy are extensive quantities. at means that it can be said of both that they ow from one body to the other, and that they can be accumulated in bodies, in the same way that water ows and can be accumulated in containers. Imagining momentum as something that can be exchanged between bodies in collisions is always useful when thinking about the description of moving objects.
Momentum is conserved. at explains the limitations you might experience when being on a perfectly frictionless surface, such as ice or a polished, oil covered marble: you cannot propel yourself forward by patting your own back. (Have you ever tried to put a cat on such a marble surface? It is not even able to stand on its four legs. Neither are humans. Can you imagine why?) Momentum conservation also answers the puzzles of Figure .
e conservation of momentum and mass also means that teleportation (‘beam me up’) is impossible in nature. Can you explain this to a non-physicist?
Momentum conservation implies that momentum can be imagined to be like an invisible uid. In an interaction, the invisible uid is transferred from one object to another. However, the sum is always constant.
Momentum conservation implies that motion never stops; it is only exchanged. On the other hand, motion o en ‘disappears’ in our environment, as in the case of a stone dropped to the ground, or of a ball le rolling on grass. Moreover, in daily life we o en observe the creation of motion, such as every time we open a hand. How do these examples
t with the conservation of momentum?
Page 83 Challenge 127 e
Challenge 128 n
* For more curiosities, see R.H. P , Negative mass can be positively amusing, American Journal of Physics 61, pp. 216–217, 1993. Negative mass particles in a box would heat up a box made of positive mass while traversing its walls, and accelerating, i.e. losing energy, at the same time. ey would allow one to build a perpetuum mobile of the second kind, i.e. a device circumventing the second principle of thermodynamics. Moreover, such a system would have no thermodynamic equilibrium, because its energy could decrease forever. e more one thinks about negative mass, the more one nds strange properties contradicting observations. By the way, what is the range of possible mass values for tachyons? ** Arthur Eddington (1882–1944), British astrophysicist.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
It turns out that the answer lies in the
microscopic aspects of these systems. A
muscle only transforms one type of mo-
tion, namely that of the electrons in cer-
tain chemical compounds* into another,
the motion of the ngers. e working of
muscles is similar to that of a car engine
wine
transforming the motion of electrons in
the fuel into motion of the wheels. Both
systems need fuel and get warm in the process.
cork
We must also study the microscopic be-
haviour when a ball rolls on grass until
it stops. e disappearance of motion is
wine
called friction. Studying the situation care-
fully, one nds that the grass and the ball
heat up a little during this process. Dur-
stone
ing friction, visible motion is transformed
into heat. Later, when we discover the F I G U RE 33 What happens?
structure of matter, it will become clear
that heat is the disorganized motion of the microscopic constituents of every material.
When these constituents all move in the same direction, the object as a whole moves;
when they oscillate randomly, the object is at rest, but is warm. Heat is a form of motion.
Friction thus only seems to be disappearance of motion; in fact it is a transformation of
ordered into unordered motion.
Despite momentum conservation, macroscopic perpetual motion does not exist, since
friction cannot be completely eliminated.** Motion is eternal only at the microscopic
scale. In other words, the disappearance and also the spontaneous appearance of motion
in everyday life is an illusion due to the limitations of our senses. For example, the mo-
tion proper of every living being exists before its birth, and stays a er its death. e same
happens with its energy. is result is probably the closest one can get to the idea of ever-
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Ref. 57
Ref. 58 Challenge 131 n
* Usually adenosine triphosphate (ATP), the fuel of most processes in animals.
** Some funny examples of past attempts to built a perpetual motion machine are described in S
M
, Perpetuum mobile, VDI Verlag, 1976. Interestingly, the idea of eternal motion came to Europe
from India, via the Islamic world, around the year 1200, and became popular as it opposed the then standard
view that all motion on Earth disappears over time. See also the http://www.geocities.com/mercutio78_99/
pmm.html and the http://www.lhup.edu/~dsimanek/museum/unwork.htm websites. e conceptual mis-
take made by eccentrics and used by crooks is always the same: the hope of overcoming friction. (In fact,
this applied only to the perpetual motion machines of the second kind; those of the rst kind – which are
even more in contrast with observation – even try to generate energy from nothing.)
If the machine is well constructed, i.e. with little friction, it can take the little energy it needs for the
sustenance of its motion from very subtle environmental e ects. For example, in the Victoria and Albert
Museum in London one can admire a beautiful clock powered by the variations of air pressure over time.
Low friction means that motion takes a long time to stop. One immediately thinks of the motion of the
planets. In fact, there is friction between the Earth and the Sun. (Can you guess one of the mechanisms?)
But the value is so small that the Earth has already circled around the Sun for thousands of millions of years,
and will do so for quite some time more.
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lasting life from evidence collected by observation. It is perhaps less than a coincidence that energy used to be called vis viva, or ‘living force’, by Leibniz and many others.
Since motion is conserved, it has no origin. erefore, at this stage of our walk we cannot answer the fundamental questions: Why does motion exist? What is its origin?
e end of our adventure is nowhere near.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
M
–
When collisions are studied in detail, a second conserved quantity turns up. Experiments show that in the case of perfect, or elastic collisions – collisions without friction – the following quantity, called the kinetic energy T of the system, is also conserved:
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T = mi vi = mivi = const .
(17)
i
i
Challenge 132 n Challenge 133 e
Kinetic energy thus depends on the mass and on the square of the speed v of a body. Kinetic energy is the ability that a body has to induce change in bodies it hits. e full name ‘kinetic energy’ was introduced by Gustave-Gaspard Coriolis.* Coriolis also introduced the factor / , in order that the relation dT dv = p would be obeyed. (Why?) Energy is a word taken from ancient Greek; originally it was used to describe character, and meant ‘intellectual or moral vigour’. It was taken into physics by omas Young ( – ) in
because its literal meaning is ‘force within’. ( e letters E, W, A and several others are also used to denote energy.) Another, equivalent de nition of energy will become clear later: energy is what can be transformed into heat.
(Physical) energy is the measure of the ability to generate motion. A body has a lot of energy if it has the ability to move many other bodies. Energy is a number; it has no direction. e total momentum of two equal masses moving with opposite velocities is zero; their total energy increases with velocity. Energy thus also measures motion, but in a di erent way than momentum. Energy measures motion in a more global way. An equivalent de nition is the following. Energy is the ability to perform work. Here, the physical concept of work is just the precise version of what is meant by work in everyday life.**
Do not be surprised if you do not grasp the di erence between momentum and energy straight away: physicists took about two centuries to gure it out. For some time they even insisted on using the same word for both, and o en they didn’t know which situation required which concept. So you are allowed to take a few minutes to get used to it.
Both energy and momentum measure how systems change. Momentum tells how systems change over distance, energy measures how systems change over time. Momentum is needed to compare motion here and there. Energy is needed to compare motion now and later. Some measured energy values are given in Table .
One way to express the di erence between energy and momentum is to think about the following challenges. Is it more di cult to stop a running man with mass m and speed v, or one with mass m and speed v, or one with mass m and speed v? You may want to ask a rugby-playing friend for con rmation.
* Gustave-Gaspard Coriolis (b. 1792 Paris, d. 1843 Paris), French engineer and mathematician. ** (Physical) work is the product of force and distance in direction of the force.
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TA B L E 14 Some measured energy values
O
Average kinetic energy of oxygen molecule in air Green photon energy X-ray photon energy γ photon energy Highest particle energy in accelerators Comfortably walking human Flying arrow Right hook in boxing Energy in torch battery Flying ri e bullet Apple digestion Car on highway Highest laser pulse energy Lightning ash Planck energy Small nuclear bomb ( ktonne) Earthquake of magnitude 7 Largest nuclear bomb ( Mtonne) Impact of meteorite with km diameter Yearly machine energy use Rotation energy of Earth Supernova explosion Gamma ray burst Energy content E = mc of Sun’s mass Energy content of Galaxy’s central black hole
E
ë −J .ë − J −J −J −J
J J J kJ kJ . MJ MJ . MJ up to GJ . GJ TJ PJ PJ EJ
EJ ëJ
J up to J .ë J
ëJ
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 134 n Challenge 135 n
Ref. 59
Another distinction is illustrated by athletics: the real long jump world record, almost m, is still kept by an athlete who in the early twentieth century ran with two weights in his hands, and then threw the weights behind him at the moment he took o . Can you explain the feat?
When a car travelling at m s runs head-on into a parked car of the same kind and make, which car receives the greatest damage? What changes if the parked car has its brakes on?
To get a better feeling for energy, here is an additional approach. Robert Mayer e world consumption of energy by human machines (coming from solar, geothermal, biomass, wind, nuclear, hydro, gas, oil, coal, or animal sources) in the year was about EJ,* for a world population of about million people. To see
Page 1154 * For the explanation of the abbreviation E, see Appendix B.
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Challenge 136 n
what this energy consumption means, we translate it into a personal power consumption; we get about . kW. e watt W is the unit of power, and is simply de ned as W = J s, re ecting the de nition of (physical) power as energy used per unit time. As a working person can produce mechanical work of about W, the average human energy consumption corresponds to about humans working hours a day. (See Table for some power values found in nature.) In particular, if we look at the energy consumption in First World countries, the average inhabitant there has machines working for them equivalent to several hundred ‘servants’. Can you point out some of these machines?
Kinetic energy is thus not conserved in everyday life. For example, in non-elastic collisions, such as that of a piece of chewing gum hitting a wall, kinetic energy is lost. Friction destroys kinetic energy. At the same time, friction produces heat. It was one of the important conceptual discoveries of physics that total energy is conserved if one includes the discovery that heat is a form of energy. Friction is thus in fact a process transforming kinetic energy, i.e. the energy connected with the motion of a body, into heat. On a microscopic scale, energy is conserved.* Indeed, without energy conservation, the concept of time would not be de nable. We will show this connection shortly.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
I
?–T
Page 393 Challenge 137 e
Why don’t we feel all the motions of the Earth? e two parts of the answer were already given in . First of all, as Galileo explained, we do not feel the accelerations of the Earth because the e ects they produce are too small to be detected by our senses. Indeed, many of the mentioned accelerations do induce measurable e ects in high-precision experiments, e.g. in atomic clocks.
But the second point made by Galileo is equally important. We do not feel translational, unaccelerated motions because this is impossible in principle. We cannot feel that we are moving! Galileo discussed the issue by comparing the observations of two observers: one on the ground and another on the most modern means of transportation of the time, a ship. Galileo asked whether a man on the ground and a man in a ship moving at constant speed experience (or ‘feel’) anything di erent. Einstein used observers in trains. Later it became fashionable to use travellers in rockets. (What will come next?) Galileo explained that only relative velocities between bodies produce e ects, not the absolute values of the velocities. For the senses, there is no di erence between constant, undisturbed motion, however rapid it may be, and rest. is is now called Galileo’s principle of relativity. In everyday life we feel motion only if the means of transportation trembles (thus if it accelerates), or if we move against the air. erefore Galileo concludes that two observers in straight and undisturbed motion against each other cannot say who is ‘really’ moving. Whatever their relative speed, neither of them ‘feels’ in motion.**
* In fact, the conservation of energy was stated in its full generality in public only in 1842, by Julius Robert Mayer. He was a medical doctor by training, and the journal Annalen der Physik refused to publish his paper, as it supposedly contained ‘fundamental errors’. What the editors called errors were in fact mostly – but not only – contradictions of their prejudices. Later on, Helmholtz, Kelvin, Joule and many others acknowledged Mayer’s genius. However, the rst to have stated energy conservation in its modern form was the French physicist Sadi Carnot (1796–1832) in 1820. To him the issue was so clear that he did not publish the result. In fact he went on and discovered the second ‘law’ of thermodynamics. Today, energy conservation, also called the rst ‘law’ of thermodynamics, is one of the pillars of physics, as it is valid in all its domains. ** In 1632, in his Dialogo, Galileo writes: ‘Shut yourself up with some friend in the main cabin below decks
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TA B L E 15 Some measured power values
O
Power of agellar motor in bacterium Incandescent light bulb light output Incandescent light bulb electricity consumption A human, during one work shi of eight hours One horse, for one shi of eight hours Eddy Merckx, the great bicycle athlete, during one hour O cial horse power power unit Large motorbike Electrical power station output World’s electrical power production in 2000 Power used by the geodynamo Input on Earth surface: Sun’s irradiation of Earth Ref. 60 Input on Earth surface: thermal energy from inside of the Earth Input on Earth surface: power from tides (i.e. from Earth’s rotation) Input on Earth surface: power generated by man from fossil fuels Lost from Earth surface: power stored by plants’ photosynthesis World’s record laser power Output of Earth surface: sunlight re ected into space Output of Earth surface: power radiated into space at K Sun’s output Maximum power in nature, c G
P
. pW 1 to W 25 to W
W W W W kW 0.1 to GW GW 200 to GW . EW TW TW 8 to TW TW PW . EW . EW . YW .ë W
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
on some large ship, and have with you there some ies, butter ies, and other small ying animals. Have a large bowl of water with some sh in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals y with equal speed to all sides of the cabin. e sh swim indi erently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal: jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though there is no doubt that when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not uctuating this way and that, you will discover not the least change in all the e ects named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, you will pass on the oor the same spaces as before, nor will you make larger jumps toward the stern than toward the prow even though the ship is moving quite rapidly, despite the fact that during the time you are in the air the oor under you will be going in a direction opposite to your jump. In throwing something to your companion, you will need no more force to get it to him whether he is in the direction of the bow or the stern, with yourself situated opposite. e droplets will fall as before into the vessel beneath without dropping toward the stern, although while the drops are in the air the ship runs many spans. e sh in their water will swim toward the front of their bowl with no more e ort than toward the back, and will go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butter ies and ies will continue their ights indi erently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during long
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Rest is relative. Or more clearly: rest is an observer-dependent concept. is result of Galilean physics is so important that Poincaré introduced the expression ‘theory of relativity’ and Einstein repeated the principle explicitly when he published his famous theory of special relativity. However, these names are awkward. Galilean physics is also a theory of relativity! e relativity of rest is common to all of physics; it is an essential aspect of motion.
Undisturbed or uniform motion has no observable e ect; only change of motion does. As a result, every physicist can deduce something simple about the following statement by Wittgenstein:
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Daß die Sonne morgen aufgehen wird, ist eine Hypothese; und das heißt: wir wissen nicht, ob sie aufgehen wird.*
e statement is wrong. Can you explain why Wittgenstein erred here, despite his strong Challenge 138 n desire not to?
R
Rotation keeps us alive. Without the change of day and night, we would be either fried or frozen to death, depending on our location on our planet. A short summary of rotation is thus appropriate. We saw before that a body is described by its reluctance to move; similarly, a body also has a reluctance to turn. is quantity is called its moment of inertia and is o en abbreviated Θ. e speed or rate of rotation is described by angular velocity, usually abbreviated ω. A few values found in nature are given in Table .
e observables that describe rotation are similar to those describing linear motion, as shown in Table . Like mass, the moment of inertia is de ned in such a way that the sum of angular momenta L – the product of moment of inertia and angular velocity – is conserved in systems that do not interact with the outside world:
Θi ωi = Li = const .
(18)
i
i
In the same way that linear momentum conservation de nes mass, angular momentum conservation de nes the moment of inertia.
e moment of inertia can be related to the mass and shape of a body. If the body is imagined to consist of small parts or mass elements, the resulting expression is
Θ = mnrn ,
(19)
n
intervals by keeping themselves in the air. And if smoke is made by burning some incense, it will be seen going up in the form of a little cloud, remaining still and moving no more toward one side than the other.
e cause of all these correspondences of e ects is the fact that the ship’s motion is common to all the things contained in it, and to the air also. at is why I said you should be below decks; for if this took place above in the open air, which would not follow the course of the ship, more or less noticeable di erences would be seen in some of the e ects noted.’ * ‘It is a hypothesis that the Sun will rise tomorrow; and this means that we do not know whether it will rise.’
is well-known statement is found in Ludwig Wittgenstein, Tractatus, 6.36311.
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TA B L E 16 Some measured rotation frequencies
O
A
ω= π T
Galactic rotation
πë . ë − s= π ë a
Average Sun rotation around its axis
πë . ë − s= π d
Typical lighthouse
πë . s
Pirouetting ballet dancer
πë s
Ship’s diesel engine
πë s
Helicopter motor
πë . s
Washing machine
up to π ë s
Bacterial agella
πë s
Racing car engine
up to π ë s
Fastest turbine built
πë s
Fastest pulsars (rotating stars)
up to at least π ë s
Ultracentrifuge
πë ë s
Dental drill
up to π ë ë s
Proton rotation
πë s
Highest possible, Planck angular velocity πë s
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
TA B L E 17 Correspondence between linear and rotational motion
Q
L
R
State
Motion Reluctance to move Motion change
time position momentum energy velocity acceleration mass force
t x p = mv mv v a m ma
time angle angular momentum energy angular velocity angular acceleration moment of inertia torque
t φ L = Θω Θω ω α Θ Θα
Challenge 139 e Challenge 140 n
where rn is the distance from the mass element mn to the axis of rotation. Can you conrm the expression? erefore, the moment of inertia of a body depends on the chosen
axis of rotation. Can you con rm that this is so for a brick? Obviously, the value of the moment of inertia also depends on the location of the axis
used for its de nition. For each axis direction, one distinguishes an intrinsic moment of inertia, when the axis passes through the centre of mass of the body, from an extrinsic moment of inertia, when it does not.* In the same way, one distinguishes intrinsic and
* Extrinsic and intrinsic moment of inertia are related by Θext = Θint + md ,
where d is the distance between the centre of mass and the axis of extrinsic rotation.
(20) is relation is called
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L
r A p
middle finger: "r x p"
thumb: "r"
index: "p"
fingers in rotation sense; thumb shows angular momentum
F I G U R E 34 Angular momentum and the two versions of the right-hand rule
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 142 n Challenge 143 n
extrinsic angular momenta. (By the way, the centre of mass of a body is that imaginary point which moves straight during vertical fall, even if the body is rotating. Can you nd a way to determine its location for a speci c body?)
Every object that has an orientation also has an intrinsic angular momentum. (What about a sphere?) erefore, point particles do not have intrinsic angular momenta – at least in rst approximation. ( is conclusion will change in quantum theory.) e extrinsic angular momentum L of a point particle is given by
L=r
p=
A(T )m T
so that
L=rp=
A(T )m T
(21)
where p is the momentum of the particle, A(T) is the surface swept by the position vector r of the particle during time T.* e angular momentum thus points along the rotation axis, following the right-hand rule, as shown in Figure .
We then de ne a corresponding rotational energy as
Erot =
Θω
=L . Θ
(23)
e expression is similar to the expression for the kinetic energy of a particle. Can you
Challenge 141 n Challenge 144 e
Steiner’s parallel axis theorem. Are you able to deduce it? * For the curious, the result of the cross product or vector product a b between two vectors a and b is de ned as that vector that is orthogonal to both, whose orientation is given by the right-hand rule, and whose length is given by ab sin ∢(a, b), i.e. by the surface area of the parallelogram spanned by the two vectors. From the de nition you can show that the vector product has the properties
a b = −b a , a (b + c) = a b + a c , λa b = λ(a b) = a λb , a a = , a(b c) = b(c a) = c(a b) , a (b c) = b(ac) − c(ab) , (a b)(c d) = a(b (c d)) = (ac)(bd) − (bc)(ad) , (a b) (c d) = c((a b)d) − d((a b)c) , a (b c) + b (c a) + c (a b) = . (22)
Page 1203 Challenge 145 e
e vector product exists (almost) only in three-dimensional vector spaces. (See Appendix D.) e cross product vanishes if and only if the vectors are parallel. e parallelepiped spanned by three vectors a, b and c has the volume V = c(a b). e pyramid or tetrahedron formed by the three vectors has one sixth of that volume.
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frictionless axis
F I G U R E 35 How a snake turns itself around its axis
F I G U R E 36 Can the ape reach the banana?
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 146 n
Ref. 61 Challenge 147 d
Ref. 3 Challenge 148 n
guess how much larger the rotational energy of the Earth is compared with the yearly electricity usage of humanity? In fact, if you can nd a way to harness this energy, you will become famous.
As in the case of linear motion, rotational energy and angular momentum are not always conserved in the macroscopic world: rotational energy can change due to friction, and angular momentum can change due to external forces (torques). However, for closed systems both quantities always conserved on the microscopic scale.
On a frictionless surface, as approximated by smooth ice or by a marble oor covered by a layer of oil, it is impossible to move forward. In order to move, we need to push against something. Is this also the case for rotation?
Surprisingly, it is possible to turn even without pushing against something. You can check this on a well-oiled rotating o ce chair: simply rotate an arm above the head. A er each turn of the hand, the orientation of the chair has changed by a small amount. Indeed, conservation of angular momentum and of rotational energy do not prevent bodies from changing their orientation. Cats learn this in their youth. A er they have learned the trick, if they are dropped legs up, they can turn themselves in such a way that they always land feet rst. Snakes also know how to rotate themselves, as Figure shows. During the Olympic Games one can watch board divers and gymnasts perform similar tricks. Rotation is thus di erent from translation in this aspect. (Why?)
Angular momentum is conserved. is statement is valid for any axis, provided that friction plays no role. external forces (torques) play no role. To make the point, Jean-Marc Lévy-Leblond poses the problem of Figure . Can the ape reach the banana without leaving the plate, assuming that the plate on which the ape rests can turn around the axis without friction?
R
Rotation is an interesting phenomenon in many ways. A rolling wheel does not turn
around its axis, but around its point of contact. Let us show this.
A wheel of radius R is rolling if the speed of the axis vaxis is related to the angular
velocity ω by
ω=
vaxis R
.
(24)
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•.
ω r ω d = vp
P
ω R
r
ω R = vaxis
d
R
C
F I G U R E 37 The velocities and unit vectors for a rolling wheel
–
F I G U R E 38 A simulated photograph of a rolling wheel with spokes
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 149 e
For any point P on the wheel, with distance r from the axis, the velocity vP is the sum of the motion of the axis and the motion around the axis. Figure shows that vP is orthogonal to d, the distance between the point P and the contact point of the wheel. e
gure also shows that the length ratio between vP and d is the same as between vaxis and R. As a result, we can write
vP = ω d ,
(25)
Challenge 150 n Ref. 62 Ref. 63
Challenge 151 d
which shows that a rolling wheel does indeed rotate about its point of contact with the ground.
Surprisingly, when a wheel rolls, some points on it move towards the wheel’s axis, some stay at a xed distance and others move away from it. Can you determine where these various points are located? Together, they lead to an interesting pattern when a rolling wheel with spokes, such as a bicycle wheel, is photographed.
With these results you can tackle the following beautiful challenge. When a turning bicycle wheel is put on a slippery surface, it will slip for a while and then end up rolling. How does the nal speed depend on the initial speed and on the friction?
H
?
Ref. 17
“Golf is a good walk spoiled.
” Mark Twain
Why do we move our arms when walking or running? To conserve energy. In fact, when
a body movement is performed with as little energy as possible, it is natural and graceful.
( is can indeed be taken as the actual de nition of grace. e connection is common
knowledge in the world of dance; it is also a central aspect of the methods used by actors
to learn how to move their bodies as beautifully as possible.)
To convince yourself about the energy savings, try walking or running with your arms
xed or moving in the opposite direction to usual: the e ort required is considerably
higher. In fact, when a leg is moved, it produces a torque around the body axis which
has to be counterbalanced. e method using the least energy is the swinging of arms.
Since the arms are lighter than the legs, they must move further from the axis of the body,
to compensate for the momentum; evolution has therefore moved the attachment of the
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 39 The measured motion of a walking human (© Ray McCoy)
rotating
Moon
sky
Earth
or
and
Sun
stars
N
F I G U R E 40 The parallaxis – not drawn to scale
Ref. 64 Challenge 152 e
arms, the shoulders, farther apart than those of the legs, the hips. Animals on two legs but no arms, such as penguins or pigeons, have more di culty walking; they have to move their whole torso with every step.
Which muscles do most of the work when walking, the motion that experts call gait? In , Serge Gracovetsky found that in human gait most power comes from the muscles
along the spine, not from the legs. (Indeed, people without legs are also able to walk.) When you take a step, the lumbar muscles straighten the spine; this automatically makes it turn a bit to one side, so that the knee of the leg on that side automatically comes forward. When the foot is moved, the lumbar muscles can relax, and then straighten again for the next step. In fact, one can experience the increase in tension in the back muscles when walking without moving the arms, thus con rming where the human engine is located.
Human legs di er from those of apes in a fundamental aspect: humans are able to run.
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In fact the whole human body has been optimized for running, an ability that no other primate has. e human body has shed most of its hair to achieve better cooling, has evolved the ability to run while keeping the head stable, has evolved the right length of arms for proper balance when running, and even has a special ligament in the back that works as a shock absorber while running. In other words, running is the most human of all forms of motion.
I
E
?
Ref. 65 Page 276
“Eppur si muove!
” Anonymous*
e search for answers to this question gives a beautiful cross section of the history of
classical physics. Around the year
, the Greek thinker Aristarchos of Samos main-
tained that the Earth rotates. He had measured the parallax of the Moon (today known to
be up to . °) and of the Sun (today known to be . ′).** e parallax is an interesting
e ect; it is the angle describing the di erence between the directions of a body in the sky
when seen by an observer on the surface of the Earth and when seen by a hypothetical
observer at the Earth’s centre. (See Figure .) Aristarchos noticed that the Moon and the
Sun wobble across the sky, and this wobble has a period of hours. He concluded that
the Earth rotates.
Measurements of the aberration of light
also show the rotation of the Earth; it can be detected with a telescope while looking at
sphere 5 km
the stars. e aberration is a change of the ex-
pected light direction, which we will discuss Earth
shortly. At the Equator, Earth rotation adds an angular deviation of . ′, changing sign 5 km
Equator
5 km
every hours, to the aberration due to the
motion of the Earth around the Sun, about
. ′. In modern times, astronomers have
found a number of additional proofs, but
5 km
none is accessible to the man on the street. F I G U R E 41 Earth’s deviation from spherical
Furthermore, the measurements showing shape due to its rotation
that the Earth is not a sphere, but is attened
at the poles, con rmed the rotation of the Earth. Again, however, this eighteenth century
measurement by Maupertuis*** is not accessible to everyday observation.
en, in the years to in Bologna, Giovanni Battista Guglielmini ( – )
nally succeeded in measuring what Galileo and Newton had predicted to be the simplest
proof for the Earth’s rotation. On the Earth, objects do not fall vertically, but are slightly
* ‘And yet she moves’ is the sentence falsely attributed to Galileo about the Earth. It is true, however, that at his trial he was forced to publicly retract the idea of a moving Earth to save his life (see the footnote on page 220). ** For the de nition of angles see page 61 and for the de nition of angle units see Appendix B. *** Pierre Louis Moreau de Maupertuis (1698–1759), French physicist and mathematician. He was one of the key gures in the quest for the principle of least action, which he named in this way. He was also founding president of the Berlin Academy of Sciences.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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v = ω (R+h) h v=ω R
N E
Equator
Nh ϕ
S
F I G U R E 42 The deviations of free fall towards the east and towards the Equator due to the rotation of the Earth
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
deviated to the east. is deviation appears because an object keeps the larger horizontal velocity it had at the height from which it started falling, as shown in Figure . Guglielmini’s result was the rst non-astronomical proof of the Earth’s rotation. e experiments were repeated in by Johann Friedrich Benzenberg ( – ). Using metal balls which he dropped from the Michaelis tower in Hamburg – a height of m – Benzenberg found that the deviation to the east was . mm. Can you con rm that the value measured by Benzenberg almost agrees with the assumption that the Earth turns Challenge 153 ny once every hours? ( ere is also a much smaller deviation towards the Equator, not measured by Guglielmini, Benzenberg or anybody a er them up to this day; however, it completes the list of e ects on free fall by the rotation of the Earth.) Both deviations are easily understood if we remember that falling objects describe an ellipse around the centre of the rotating Earth. e elliptical shape shows that the path of a thrown stone does not lie on a plane for an observer standing on Earth; for such an observer, the exact path thus cannot be drawn on a piece of paper.
In , the engineer and mathematician Gustave-Gaspard Coriolis ( – ), the Frenchman who also introduced the modern concepts of ‘work’ and of ‘kinetic energy’, found a closely related e ect that nobody had as yet noticed in everyday life. An object travelling in a rotating background does not move in a straight line. If the rotation is anticlockwise, as is the case for the Earth on the northern hemisphere, the velocity of an object is slightly turned to the right, while its magnitude stays constant. is so-called Coriolis acceleration (or Coriolis force) is due to the change of distance to the rotation Challenge 154 ny axis. Can you deduce the analytical expression for it, namely aC = ω v?
e Coriolis acceleration determines the handedness of many large-scale phenomena with a spiral shape, such as the directions of cyclones and anticyclones in meteorology, the general wind patterns on Earth and the de ection of ocean currents and tides. Most beautifully, the Coriolis acceleration explains why icebergs do not follow the direction Ref. 66 of the wind as they dri away from the polar caps. e Coriolis acceleration also plays a role in the ight of cannon balls (that was the original interest of Coriolis), in satellite Ref. 67 launches, in the motion of sunspots and even in the motion of electrons in molecules. All these phenomena are of opposite sign on the northern and southern hemispheres and thus prove the rotation of the Earth. (In the First World War, many naval guns missed
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•.
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ψ N
ψ1 ψ2
Earth's
centre ϕ
Equator
F I G U R E 43 The turning motion of a pendulum showing the rotation of the Earth
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Ref. 68 Ref. 68 Challenge 156 d
their targets in the southern hemisphere because the engineers had compensated them for the Coriolis e ect in the northern hemisphere.)
Only in , a er several earlier attempts by other researchers, Asher Shapiro was the rst to verify that the Coriolis e ect has a tiny in uence on the direction of the vortex formed by the water owing out of a bathtub. Instead of a normal bathtub, he had to use a carefully designed experimental set-up because, contrary to an o en-heard assertion, no such e ect can be seen in a real bathtub. He succeeded only by carefully eliminating all disturbances from the system; for example, he waited hours a er the lling of the reservoir (and never actually stepped in or out of it!) in order to avoid any le -over motion of water that would disturb the e ect, and built a carefully designed, completely rotationallysymmetric opening mechanism. Others have repeated the experiment in the southern hemisphere, con rming the result. In other words, the handedness of usual bathtub vortices is not caused by the rotation of the Earth, but results from the way the water starts to ow out. But let us go on with the story about the Earth’s rotation.
Finally, in , the French physician-turned-physicist Jean Bernard Léon Foucault (b. Paris, d. Paris) performed an experiment that removed all doubts and rendered him world-famous practically overnight. He suspended a m long pendulum* in the Panthéon in Paris and showed the astonished public that the direction of its swing changed over time, rotating slowly. To anybody with a few minutes of patience to watch the change of direction, the experiment proved that the Earth rotates. If the Earth did not rotate, the swing of the pendulum would always continue in the same direction. On a rotating Earth, the direction changes to the le , as shown in Figure , unless the pendulum is located at the Equator.** e time over which the orientation of the swing performs a full turn – the precession time – can be calculated. Study a pendulum swinging in the North–South direction and you will nd that the precession time TFoucault is given by
Challenge 155 d Ref. 69
* Why was such a long pendulum necessary? Understanding the reasons allows one to repeat the experiment at home, using a pendulum as short as cm, with the help of a few tricks. ** e discovery also shows how precision and genius go together. In fact, the rst person to observe the e ect was Vincenzo Viviani, a student of Galilei, as early as 1661! Indeed, Foucault had read about Viviani’s work in the publications of the Academia dei Lincei. But it took Foucault’s genius to connect the e ect to the rotation of the Earth; nobody had done so before him.
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m
m
F I G U R E 45 Showing the rotation of the Earth through the rotation of an axis
N r E
W
S
F I G U R E 46 Demonstrating the rotation of the Earth with water
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
TFoucault
=
h sin φ
(26)
Challenge 157 n
Challenge 159 n Ref. 70
Challenge 160 n
where φ is the latitude of the location of the pendulum, e.g. ° at the Equator and ° at
the North Pole. is formula is one of the most beautiful results of Galilean kinematics.*
Foucault was also the inventor and namer of the gyroscope. He
built the device, shown in Figure , in , one year a er his pen-
dulum. With it, he again demonstrated the rotation of the Earth.
Once a gyroscope rotates, the axis stays xed in space – but only
when seen from distant stars or galaxies. ( is is not the same as
talking about absolute space. Why?) For an observer on Earth, the
axis direction changes regularly with a period of hours. Gyro-
scopes are now routinely used in ships and in aeroplanes to give
the direction of north, because they are more precise and more re-
liable than magnetic compasses. In the most modern versions, one
uses laser light running in circles instead of rotating masses.**
F I G U R E 44 The
In , Roland von Eőtvős measured a simple e ect: due to the gyroscope
rotation of the Earth, the weight of an object depends on the direction in which it moves.
As a result, a balance in rotation around the vertical axis does not stay perfectly horizontal:
the balance starts to oscillate slightly. Can you explain the origin of the e ect?
In , John Hagen published the results of an even simpler experiment, proposed
by Louis Poinsot in . Two masses are put on a horizontal bar that can turn around a
vertical axis, a so-called isotomeograph. If the two masses are slowly moved towards the
support, as shown in Figure , and if the friction is kept low enough, the bar rotates.
Obviously, this would not happen if the Earth were not rotating. Can you explain the
observation? is little-known e ect is also useful for winning bets between physicists.
* e calculation of the period of Foucault’s pendulum assumed that the precession rate is constant during a rotation. is is only an approximation (though usually a good one). Challenge 158 n ** Can you guess how rotation is detected in this case?
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Ref. 82 Challenge 161 d
Page 728 Challenge 162 n
In , Arthur Compton showed that a closed tube lled with water and some small oating particles (or bubbles) can be used to show the rotation of the Earth. e device is called a Compton tube or Compton wheel. Compton showed that when a horizontal tube lled with water is rotated by °, something happens that allows one to prove that the Earth rotates. e experiment, shown in Figure , even allows measurement of the latitude of the point where the experiment is made. Can you guess what happens?
In , Albert Michelson* and his collaborators in Illinois constructed a vacuum interferometer with the incredible perimeter of . km. Interferometers produce bright and dark fringes of light; the position of the fringes depends on the speed at which the interferometers rotates. e fringe shi is due to an e ect rst measured in by the French physicist Georges Sagnac: the rotation of a complete ring interferometer with angular frequency (vector) Ω produces a fringe shi of angular phase ∆φ given by
Dvipsbugw
∆φ =
π ΩA cλ
(27)
where A is the area (vector) enclosed by the two interfering light rays, λ their wavelength and c the speed of light. e e ect is now called the Sagnac e ect, even though it had Ref. 71 already been predicted years earlier by Oliver Lodge.** Michelson and his team found a fringe shi with a period of hours and of exactly the magnitude predicted by the rotation of the Earth. Modern high precision versions use ring lasers with areas of only a few square metres, but which are able to measure variations of the rotation rates of the Earth of less than one part per million. Indeed, over the course of a year the length of a day varies irregularly by a few milliseconds, mostly due to in uences from the Sun or the Ref. 72 Moon, due to weather changes and due to hot magma ows deep inside the Earth.*** But also earthquakes, the el Ninño e ect in the climate and the lling of large water dams have e ects on the rotation of the Earth. All these e ects can be studied with such precision interferometers; these can also be used for research into the motion of the soil due to lunar tides or earthquakes, and for checks on the theory of relativity.
In summary, observations show that the Earth surface rotates at m s at the Equator, a larger value than that of the speed of sound in air – about m s in usual conditions – and that we are in fact whirling through the universe.
H
E
?
Ref. 73
Is the rotation of the Earth constant over geological time scales? at is a hard question. If you nd a method leading to an answer, publish it! ( e same is true for the question whether the length of the year is constant.) Only a few methods are known, as we will
nd out shortly.
* Albert Abraham Michelson (b. 1852 Strelno, d. 1931 Pasadena) Prussian–Polish–US-American physicist, obsessed by the precise measurement of the speed of light, received the Nobel Prize for Physics in 1907. ** Oliver Lodge (1851–1940) was a British physicist who studied electromagnetic waves and tried to communicate with the dead. A strange but in uential gure, his ideas are o en cited when fun needs to be made of physicists; for example, he was one of those (rare) physicists who believed that at the end of the nineteenth century physics was complete. *** e growth of leaves on trees and the consequent change in the Earth’s moment of inertia, already studied in 1916 by Harold Je reys, is too small to be seen, so far.
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year 15000: North pole is Vega in Lyra
nutation period is 18.6 years
precession N
equatorial bulge
Equator
year 2000: North pole is Polaris in the Ursa minor
Moon’s path
Moon
equatorial bulge
S Earth’s path F I G U R E 47 The precession and the nutation of the Earth’s axis
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Ref. 74 Ref. 75
e rotation of the Earth is not even constant during a human lifespan. It varies by a few parts in . In particular, on a ‘secular’ time scale, the length of the day increases by about to ms per century, mainly because of the friction by the Moon and the melting of the polar ice caps. is was deduced by studying historical astronomical observations of the ancient Babylonian and Arab astronomers. Additional ‘decadic’ changes have an amplitude of or ms and are due to the motion of the liquid part of the Earth’s core.
e seasonal and biannual changes of the length of the day – with an amplitude of . ms over six months, another . ms over the year, and . ms over to months – are mainly due to the e ects of the atmosphere. In the s the availability of precision measurements showed that there is even a and day period with an amplitude of . ms, due to the Moon. In the s, when wind oscillations with a length scale of about
days were discovered, they were also found to alter the length of the day, with an amplitude of about . ms. However, these last variations are quite irregular.
But why does the Earth rotate at all? e rotation derives from the rotating gas cloud at the origin of the solar system. is connection explains that the Sun and all planets, except one, turn around themselves in the same direction, and that they also all turn around the Sun in that same direction. But the complete story is outside the scope of this text.
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Ref. 76
Page 105, page 600 Ref. 77
e rotation around its axis is not the only motion of the Earth; it performs other
motions as well. is was already known long ago. In
, the Greek astronomer
Hipparchos discovered what is today called the (equinoctial) precession. He compared a
measurement he made himself with another made years before. Hipparchos found
that the Earth’s axis points to di erent stars at di erent times. He concluded that the sky
was moving. Today we prefer to say that the axis of the Earth is moving. During a period
of
years the axis draws a cone with an opening angle of . °. is motion, shown
in Figure , is generated by the tidal forces of the Moon and the Sun on the equatorial
bulge of the Earth that results form its attening. e Sun and the Moon try to align the
axis of the Earth at right angles to the Earth’s path; this torque leads to the precession of
the Earth’s axis. ( e same e ect appears for any spinning top or in the experiment with
the suspended wheel shown on page .)
In addition, the axis of the Earth is not even xed relative to the Earth’s surface. In ,
by measuring the exact angle above the horizon of the celestial North Pole, Friedrich Küst-
ner ( – ) found that the axis of the Earth moves with respect to the Earth’s crust, as
Bessel had suggested years earlier. As a consequence of Küstner’s discovery, the Inter-
national Latitude Service was created. e polar motion Küstner discovered turned out to
consist of three components: a small linear dri – not yet understood – a yearly elliptical
motion due to seasonal changes of the air and water masses, and a circular motion* with
a period of about . years due to uctuations in the pressure at the bottom of the oceans.
In practice, the North Pole moves with an amplitude of m around an average central
position.
In , the German meteorologist and geophysicist Alfred Wegener ( – ) dis-
covered an even larger e ect. A er studying the shapes of the continental shelves and the
geological layers on both sides of the Atlantic, he conjectured that the continents move,
and that they are all fragments of a single continent that broke up million years ago.**
Even though at rst derided across the world, Wegener’s discoveries were correct. Mod-
ern satellite measurements, shown in Figure , con rm this model. For example, the
American continent moves away from the European continent by about mm every
year. ere are also speculations that this velocity may have been much higher at certain
periods in the past. e way to check this is to look at the magnetization of sedimental
rocks. At present, this is still a hot topic of research. Following the modern version of the
model, called plate tectonics, the continents (with a density of . ë kg m ) oat on the
uid mantle of the Earth (with a density of . ë kg m ) like pieces of cork on water,
and the convection inside the mantle provides the driving mechanism for the motion.
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D
E
?
e centre of the Earth is not at rest in the universe. In the third century Aristarchos of Samos maintained that the Earth turns around the Sun. However, a fundamental di -
* e circular motion, a wobble, was predicted by the great Swiss mathematician Leonhard Euler (1707– 1783). Using this prediction and Küstner’s data, in 1891 Seth Chandler claimed to be the discoverer of the circular component. ** In this old continent, called Gondwanaland, there was a huge river that owed westwards from the Chad to Guayaquil in Ecuador. A er the continent split up, this river still owed to the west. When the Andes appeared, the water was blocked, and many millions of years later, it owed back. Today, the river still ows eastwards and is called the Amazonas.
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F I G U R E 48 The continental plates are the objects of tectonic motion
Challenge 163 n Page 276
culty of the heliocentric system is that the stars look the same all year long. How can this be, if the Earth travels around the Sun? e distance between the Earth and the Sun has been known since the seventeenth century, but it was only in that Friedrich Wilhelm Bessel* became the rst to observe the parallax of a star. is was a result of extremely careful measurements and complex calculations: he discovered the Bessel functions in order to realize it. He was able to nd a star, Cygni, whose apparent position changed with the month of the year. Seen over the whole year, the star describes a small ellipse in the sky, with an opening of . ′′ (this is the modern value). A er carefully eliminating all other possible explanations, he deduced that the change of position was due to the motion of the Earth around the Sun, and from the size of the ellipse he determined the distance to the star to be Pm, or . light years.
Bessel had thus managed for the rst time to measure the distance of a star. By doing so he also proved that the Earth is not xed with respect to the stars in the sky and that the Earth indeed revolves around the Sun. e motion itself was not a surprise. It con rmed the result of the mentioned aberration of light, discovered in by James Bradley** and to be discussed shortly; the Earth moves around the Sun.
With the improvement of telescopes, other motions of the Earth were discovered. In , James Bradley announced that there is a small regular change of the precession,
which he called nutation, with a period of . years and an angular amplitude of . ′′.
* Friedrich Wilhelm Bessel (1784–1846), Westphalian astronomer who le a successful business career to dedicate his life to the stars, and became the foremost astronomer of his time. ** James Bradley, (1693–1762), English astronomer. He was one of the rst astronomers to understand the value of precise measurement, and thoroughly modernized Greenwich. He discovered the aberration of light, a discovery that showed that the Earth moves and also allowed him to measure the speed of light; he also discovered the nutation of the Earth.
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precession rotation axis
Earth Sun
rotation tilt change axis
Earth
Sun
– ellipticity change
Sun perihelion shift
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Earth
orbital inclination change Sun
P Sun
P
F I G U R E 49 Changes in the Earth’s motion around the Sun
Challenge 164 ny Ref. 78
Nutation occurs because the plane of the Moon’s orbit around the Earth is not exactly the
same as the plane of the Earth’s orbit around the Sun. Are you able to con rm that this
situation produces nutation?
Astronomers also discovered that the . ° tilt – or obliquity – of the Earth’s axis, the
angle between its intrinsic and its orbital angular momentum, actually changes from . °
to . ° with a period of
years. is motion is due to the attraction of the Sun and
the deviations of the Earth from a spherical shape. During the Second World War, in
, the Serbian astronomer Milutin Milankovitch ( – ) retreated into solitude
and studied the consequences. In his studies he realized that this
year period of
the tilt, together with an average period of
years due to precession,* gives rise to
the more than ice ages in the last million years. is happens through stronger or
weaker irradiation of the poles by the Sun. e changing amounts of melted ice then lead
to changes in average temperature. e last ice age had is peak about
years ago and
ended around
years ago; the next is still far away. A spectacular con rmation of the
ice age cycles, in addition to the many geological proofs, came through measurements of
oxygen isotope ratios in sea sediments, which allow the average temperature over the past
million years to be tracked.
e Earth’s orbit also changes its eccentricity with time, from completely circular to
slightly oval and back. However, this happens in very complex ways, not with periodic
* In fact, the 25 800 year precession leads to three insolation periods, of 23 700, 22 400 and 19 000 years, due to the interaction between precession and perihelion shi .
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our galaxy
120 000 al = 1.2 Zm
orbit of our local star system
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500 al = 5 Em
Sun's path 50 000 al = 500 Em
F I G U R E 50 The motion of the Sun around the galaxy
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Ref. 79
regularity, and is due to the in uence of the large planets of the solar system on the Earth’s
orbit. e typical time scale is
to
years.
In addition, the Earth’s orbit changes in inclination with respect to the orbits of the
other planets; this seems to happen regularly every
years. In this period the in-
clination changes from + . ° to − . ° and back.
Even the direction in which the ellipse points changes with time. is so-called peri-
helion shi is due in large part to the in uence of the other planets; a small remaining
part will be important in the chapter on general relativity. It was the rst piece of data
con rming the theory.
Obviously, the length of the year also changes with time. e measured variations
are of the order of a few parts in or about ms per year. However, knowledge of
these changes and of their origins is much less detailed than for the changes in the Earth’s
rotation.
e next step is to ask whether the Sun itself moves. Indeed it does. Locally, it moves
with a speed of . km s towards the constellation of Hercules. is was shown by Wil-
liam Herschel in . But globally, the motion is even more interesting. e diameter of
the galaxy is at least
light years, and we are located
light years from the
centre. ( is has been known since ; the centre of the galaxy is located in the direc-
tion of Sagittarius.) At our position, the galaxy is light years thick; presently, we are
light years ‘above’ the centre plane. e Sun, and with it the solar system, takes about
million years to turn once around the galactic centre, its orbital velocity being around
km s. It seems that the Sun will continue moving away from the galaxy plane until it
is about light years above the plane, and then move back, as shown in Figure . e
oscillation period is estimated to be around million years, and has been suggested as
the mechanism for the mass extinctions of animal life on Earth, possibly because some
gas cloud may be encountered on the way. e issue is still a hot topic of research.
We turn around the galaxy centre because the formation of galaxies, like that of solar
systems, always happens in a whirl. By the way, can you con rm from your own observa-
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Challenge 165 n Ref. 80
tion that our galaxy itself rotates? Finally, we can ask whether the galaxy itself moves. Its motion can indeed be observed
because it is possible to give a value for the motion of the Sun through the universe, dening it as the motion against the background radiation. is value has been measured
to be km s. ( e velocity of the Earth through the background radiation of course depends on the season.) is value is a combination of the motion of the Sun around the galaxy centre and of the motion of the galaxy itself. is latter motion is due to the gravitational attraction of the other, nearby galaxies in our local group of galaxies.*
In summary, the Earth really moves, and it does so in rather complex ways. As Henri Poincaré would say, if we are in a given spot today, say the Panthéon in Paris, and come back to the same spot tomorrow at the the same time, we are in fact million kilometres away. is state of a airs would make time travel extremely di cult even if it were possible (which it is not); whenever you went back to the past, you would have to get to the old spot exactly!
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
I
?
When we turn rapidly, our arms li . Why does this happen? How can our body detect whether we are rotating or not? ere are two possible answers. e rst approach, promoted by Newton, is to say that there is an absolute space; whenever we rotate against this space, the system reacts. e other answer is to note that whenever the arms li , the stars also rotate, and in exactly the same manner. In other words, our body detects rotation because we move against the average mass distribution in space.
e most cited discussion of this question is due to Newton. Instead of arms, he explored the water in a rotating bucket. As usual for philosophical issues, Newton’s answer was guided by the mysticism triggered by his father’s early death. Newton saw absolute space as a religious concept and was not even able to conceive an alternative. Newton thus sees rotation as an absolute concept. Most modern scientist have fewer problems and more common sense than Newton; as a result, today’s consensus is that rotation effects are due to the mass distribution in the universe: rotation is relative. However, we have to be honest; the question cannot be settled by Galilean physics. We will need general relativity.
C
“It is a mathematical fact that the casting of this pebble from my hand alters the centre of gravity of the universe.
Here are a few facts to ponder about motion.
” omas Carlyle,** Sartor Resartus III.
** A car at a certain speed uses 7 litres of gasoline per km. What is the combined air and
* is is roughly the end of the ladder. Note that the expansion of the universe, to be studied later, produces no motion. Challenge 166 n ** omas Carlyle (1797–1881). Do you agree with the quotation?
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F I G U R E 51 Is it safe to let the cork go?
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 167 n rolling resistance? (Assume that the engine has an e ciency of 25%.)
Challenge 168 e
**
When travelling in the train, you can test Galileo’s statement about everyday relativity o fmotion. Close your eyes and ask somebody to turn you around many times: are you able to say in which direction the train is running?
Challenge 169 e
**
A train starts to travel at a constant speed of m s between two cities A and B, km apart. e train will take one hour for the journey. At the same time as the train, a fast dove starts to y from A to B, at m s. Being faster than the train, the dove arrives at B rst. e dove then ies back towards A; when it meets the train, it backturns again to city B. It goes on ying back and forward until the train reaches B. What distance did the dove cover?
**
A good bathroom scales, used to determine the weight of objects, does not show a conChallenge 170 n stant weight when you step on it and stay motionless. Why not?
Challenge 171 n
**
A cork is attached to a thin string a metre long. e string is passed over a long rod held horizontally, and a wine glass is attached at the other end. If you let go the cork in Figure 51, nothing breaks. Why not? And what happens?
Challenge 172 n
**
In 1901, Duncan MacDougalls, a medical doctor, measured the weight of dying people, in the hope to see whether death leads to a mass change. He found a sudden change of about
g at the moment of death, with large variations from person to person. He attributed it to the soul. Is this explanation satisfactory? (If you know a better one, publish it!)
Challenge 173 n
**
e Earth’s crust is less dense ( . kg l) than the Earth’s mantle ( . kg l) and oats on it. As a result, the lighter crust below a mountain ridge must be much deeper than below a plain. If a mountain rises km above the plain, how much deeper must the crust be below it? e simple block model shown in Figure 52 works fairly well; rst, it explains why, near
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plain
mountain h
continental crust
d mantle magma
F I G U R E 52 A simple model for continents and mountains
ocean
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
mountains, measurements of the deviation of free fall from the vertical line lead to so much lower values than those expected without a deep crust. Later, sound measurements have con rmed directly that the continental crust is indeed thicker beneath mountains.
**
Balance a pencil vertically (tip upwards!) on a piece of paper near the edge of a table. How Challenge 174 e can you pull out the paper without letting the pencil fall?
Challenge 175 e
**
Take a pile of coins. One can push out the coins, starting with the one at the bottom, by shooting another coin over the table surface. e method also helps to visualize twodimensional momentum conservation.
Ref. 81
**
In early 2004, two men and a woman earned £ 1.2 million in a single evening in a London casino. ey did so by applying the formulae of Galilean mechanics. ey used the method pioneered by various physicists in the 1950s who built various small computers that could predict the outcome of a roulette ball from the initial velocity imparted by the croupier. In the case in Britain, the group added a laser scanner to a smart phone that measured the path of a roulette ball and predicted the numbers where it would arrive. In this way, they increased the odds from 1 in 37 to about 1 in 6. A er six months of investigations, Scotland Yard ruled that they could keep the money they won.
**
Is a return ight by plane – from a point A to B and back to A – faster if the wind blows Challenge 176 e or not?
**
e toy of Figure 53 shows interesting behaviour: when a number of spheres are li ed and dropped to hit the resting ones, the same number of spheres detach on the other side, whereas the previously dropped spheres remain motionless. At rst sight, all this seems to follow from energy and momentum conservation. However, energy and momentum conservation provide only two equations, which are insu cient to explain or determine
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before the hit
observed after the hit
F I G U R E 53 A well-known toy
before the hit
V=0
v
observed after the hit
V‘
v’ 0
2 L , 2 M
L, M
F I G U R E 54 An elastic collision that seems not to obey energy conservation
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 55 The centre of mass defines stability
wall
ladder
F I G U R E 56 How does the ladder fall?
the behaviour of ve spheres. Why then do the spheres behave in this way? And why do Challenge 177 d they all swing in phase when a longer time has passed?
Ref. 83 Challenge 178 d
**
A surprising e ect is used in home tools such as hammer drills. We remember that when a small ball elastically hits a large one at rest, both balls move a er the hit, and the small one obviously moves faster than the large one. Despite this result, when a short cylinder hits a long one of the same diameter and material, but with a length that is some integer multiple of that of the short one, something strange happens. A er the hit, the small cylinder remains almost at rest, whereas the large one moves, as shown in Figure 54. Even though the collision is elastic, conservation of energy seems not to hold in this case. (In fact this is the reason that demonstrations of elastic collisions in schools are always performed with spheres.) What happens to the energy?
** Does a wall get a stronger jolt when it is hit by a ball rebounding from it or when it is hit
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Challenge 179 n by a ball that remains stuck to it?
Challenge 180 n
**
Housewives know how to extract a cork of a wine bottle using a cloth. Can you imagine how? ey also know how to extract the cork with the cloth if the cork has fallen inside the bottle. How?
Challenge 181 ny
**
e sliding ladder problem, shown schematically in Figure 56, asks for the detailed motion of the ladder over time. e problem is more di cult than it looks, even if friction is not taken into account. Can you say whether the lower end always touches the oor?
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Ref. 84 Challenge 182 n
**
A common y on the stern of a
ton ship of m length tilts it by less than the
diameter of an atom. Today, distances that small are easily measured. Can you think of at
least two methods, one of which should not cost more than 2000 euro?
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
**
e level of acceleration a human can survive depends on the duration over which one is subjected to it. For a tenth of a second, = m s , as generated by an ejector seat in an aeroplane, is acceptable. (It seems that the record acceleration a human has survived is about = m s .) But as a rule of thumb it is said that accelerations of
= m s or more are fatal.
Ref. 85
**
e highest microscopic accelerations are observed in particle collisions, where one gets values up to m s . e highest macroscopic accelerations are probably found in the collapsing interiors of supernovae, exploding stars which can be so bright as to be visible in the sky even during the daytime. A candidate on Earth is the interior of collapsing bubbles in liquids, a process called cavitation. Cavitation o en produces light, an e ect discovered by Frenzel and Schulte in 1934 and called sonoluminescence. (See Figure 57.) It appears most prominently when air bubbles in water are expanded and contracted by underwater loudspeakers at around kHz and allows precise measurements of bubble motion. At a certain threshold intensity, the bubble radius changes at m s in as little as a few µm, giving an acceleration of several m s .
** Challenge 183 n If a gun located at the Equator shoots a bullet vertically, where does the bullet fall?
** Challenge 184 n Why are most rocket launch sites as near as possible to the Equator?
**
Would travelling through interplanetary space be healthy? People o en fantasize about long trips through the cosmos. Experiments have shown that on trips of long duration, cosmic radiation, bone weakening and muscle degeneration are the biggest dangers.
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no figure yet
F I G U R E 57 Observation of sonoluminescence and a diagram of the experimental set-up
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 86
Many medical experts question the viability of space travel lasting longer than a couple of years. Other dangers are rapid sunburn, at least near the Sun, and exposure to the vacuum. So far only one man has experienced vacuum without protection. He lost consciousness a er 14 seconds, but survived unharmed.
** Challenge 185 n How does the kinetic energy of a ri e bullet compare to that of a running man?
** Challenge 186 n In which direction does a ame lean if it burns inside a jar on a rotating turntable?
Challenge 187 n
**
A ping-pong ball is attached by a string to a stone, and the whole is put under water in a jar. e set-up is shown in Figure 58. Now the jar is accelerated horizontally. In which direction does the ball move? What do you deduce for a jar at rest?
** Challenge 188 n What happens to the size of an egg when one places it in a jar of vinegar for a few days?
Page 677 Challenge 189 n
**
Does centrifugal acceleration exist? Most university students go through the shock of meeting a teacher who says that it doesn’t because it is a ‘ ctitious’ quantity, in the face of what one experiences every day in a car when driving around a bend. Simply ask the teacher who denies it to de ne ‘existence’. ( e de nition physicists usually use is given in the Intermezzo following this chapter.) en check whether the de nition applies to the term and make up your own mind.
Challenge 190 e
**
Rotation holds a surprise for anybody who studies it carefully. Angular momentum is a quantity with a magnitude and a direction. However, it is not a vector, as any mirror shows. e angular momentum of a body circling in a plane parallel to a mirror behaves in a di erent way from a usual arrow: its mirror image is not re ected if it points towards the mirror! You can easily check this for yourself. For this reason, angular momentum
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ping-pong ball string stone
F I G U R E 58 How does the ball move when the jar is accelerated?
–
F I G U R E 59 The famous Celtic stone and a version made with a spoon
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
is called a pseudovector. e fact has no important consequences in classical physics; but we have to keep it in mind for later occasions.
**
What is the best way to transport a number of full co ee or tea cups while at the same Challenge 191 n time avoiding spilling any precious liquid?
**
e Moon recedes from the Earth by . cm a year, due to friction. Can you nd the Challenge 192 n mechanism responsible?
**
What is the amplitude of a pendulum oscillating in such a way that the absolute value of Challenge 193 ny its acceleration at the lowest point and at the return point are equal?
**
Can you con rm that the value of the acceleration of a drop of water falling through Challenge 194 ny vapour is ?
**
What are earthquakes? Earthquakes are large examples of the same process that make a door squeak. e continental plates correspond to the metal surfaces in the joints of the door.
Earthquakes can be described as energy sources. e Richter scale is a direct measure of this energy. e Richter magnitude Ms of an earthquake, a pure number, is de ned from its energy E in joule via
Ms = log(E
J) − .
.
.
(28)
e strange numbers in the expression have been chosen to put the earthquake values as
near as possible to the older, qualitative Mercalli scale (now called EMS98) that classi es
the intensity of earthquakes. However, this is not fully possible; the most sensitive instruments today detect earthquakes with magnitudes of − . e highest value every meas-
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ured was a Richter magnitude of 10, in Chile in 1960. Magnitudes above 12 are probably Challenge 195 n impossible. (Can you show why?)
Ref. 83 Challenge 196 d
**
Figure 59 shows the so-called Celtic wiggle stone, a stone that starts rotating on a plane surface when it is put into oscillation. e size can vary between a few centimetres and a few metres. By simply bending a spoon one can realize a primitive form of this strange device, if the bend is not completely symmetrical. e rotation is always in the same direction. If the stone is put into rotation in the wrong direction, a er a while it stops and starts rotating in the other sense! Can you explain the e ect?
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**
What is the motion of the point on the surface of the Earth that has Sun in its zenith (i.e., Challenge 197 ny vertically above it), when seen on a map of the Earth during one day, and day a er day?
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 198 n
**
e moment of inertia of a body depends on the shape of the body; usually, angular momentum and the angular velocity do not point in the same direction. Can you con rm this with an example?
**
Can it happen that a satellite dish for geostationary TV satellites focuses the sunshine Challenge 199 n onto the receiver?
**
Why is it di cult to re a rocket from an aeroplane in the direction opposite to the motion Challenge 200 n of the plane?
Challenge 201 ny
**
You have two hollow spheres: they have the same weight, the same size and are painted in the same colour. One is made of copper, the other of aluminium. Obviously, they fall with the same speed and acceleration. What happens if they both roll down a tilted plane?
Challenge 202 n
**
An ape hangs on a rope. e rope hangs over a wheel and is attached to a mass of equal weight hanging down on the other side, as shown in Figure 60. e rope and the wheel are massless and frictionless. What happens when the ape climbs the rope?
** Challenge 203 ny What is the shape of a rope when rope jumping?
** Challenge 204 n How can you determine the speed of a ri e bullet with only a scale and a metre stick?
** Why does a gun make a hole in a door but cannot push it open, in exact contrast to what
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 60 How does the ape move?
Challenge 205 e a nger can do?
** Challenge 206 n Can a water skier move with a higher speed than the boat pulling him?
**
Take two cans of the same size and weight, one full of ravioli and one full of peas. Which Challenge 207 e one rolls faster on an inclined plane?
** Challenge 208 n What is the moment of inertia of a homogeneous sphere?
Challenge 209 n
**
e moment of inertia is determined by the values of its three principal axes. ese are all equal for a sphere and for a cube. Does it mean that it is impossible to distinguish a sphere from a cube by their inertial behaviour?
**
You might know the ‘Dynabee’, a hand-held gyroscopic device that can be accelerated to Challenge 210 d high speed by proper movements of the hand. How does it work?
**
It is possible to make a spinning top with a metal paperclip. It is even possible to make Challenge 211 n one of those tops with turn on their head when spinning. Can you nd out how?
** Is it true that the Moon in the rst quarter in the northern hemisphere looks like the
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 61 A long exposure of the stars at night - over the Mauna Kea telescope in Hawaii (Gemini)
Challenge 212 n Moon in the last quarter in the southern hemisphere?
**
An impressive con rmation that the Earth is round can be seen at sunset, if one turns, against usual habits, one’s back on the Sun. On the eastern sky one can see the impressive rise of the Earth’s shadow. (In fact, more precise investigations show that it is not the shadow of the Earth alone, but the shadow of its ionosphere.) One can admire a vast shadow rising over the whole horizon, clearly having the shape of a segment of a huge circle.
** Challenge 213 n How would Figure 61 look if taken at the Equator?
Challenge 214 e
**
Since the Earth is round, there are many ways to drive from one point on the Earth to another along a circle segment. is has interesting consequences for volley balls and for girl-watching. Take a volleyball and look at its air inlet. If you want to move the inlet to a di erent position with a simple rotation, you can choose the rotation axis in may di erent ways. Can you con rm this? In other words, when we look in a given direction and then want to look in another, the eye can accomplish this change in di erent ways. e option chosen by the human eye had already been studied by medical scientists in the eighteenth
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•.
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Challenge 215 n
century. It is called Listing’s ‘law’.* It states that all axes that nature chooses lie in one plane. Can you imagine its position in space? Men have a real interest that this mechanism is strictly followed; if not, looking at girls on the beach could cause the muscles moving the eyes to get knotted up.
L
?–A
Ref. 87 Challenge 216 n
Ref. 88
e acceleration and deceleration of standard wheel-driven cars is never much greater
than about = . m s , the acceleration due to gravity on our planet. Higher acceler-
ations are achieved by motorbikes and racing cars through the use of suspensions that
divert weight to the axes and by the use of spoilers, so that the car is pushed downwards
with more than its own weight. Modern spoilers are so e cient in pushing a car towards
the track that racing cars could race on the roof of a tunnel without falling down.
rough the use of special tyres these downwards forces are transformed into grip;
modern racing tyres allow forward, backward and sideways accelerations (necessary for
speed increase, for braking and for turning corners) of about . to . times the load. En-
gineers once believed that a factor was the theoretical limit and this limit is still some-
times found in textbooks; but advances in tyre technology, mostly by making clever use
of interlocking between the tyre and the road surface as in a gear mechanism, have al-
lowed engineers to achieve these higher values. e highest accelerations, around ,
are achieved when part of the tyre melts and glues to the surface. Special tyres designed
to make this happen are used for dragsters, but high performance radio-controlled model
cars also achieve such values.
How do all these e orts compare to using legs? High jump athletes can achieve peak
accelerations of about to , cheetahs over , bushbabies up to , locusts about
, and eas have been measured to accelerate about . e maximum accelera-
tion known for animals is that of click beetles, a small insect able to accelerate at over
ms=
, about the same as an airgun pellet when red. Legs are thus de nit-
ively more e cient accelerating devices than wheels – a cheetah can easily beat any car
or motorbike – and evolution developed legs, instead of wheels, to improve the chances
of an animal in danger getting to safety. In short, legs outperform wheels.
ere are other reasons for using legs instead of wheels. (Can you name some?) For
example, legs, unlike wheels, allow walking on water. Most famous for this ability is the
basilisk,** a lizard living in Central America. is reptile is about cm long and has a
mass of about g. It looks like a miniature Tyrannosaurus rex and is able to run over
water surfaces on its hind legs. e motion has been studied in detail with high-speed
cameras and by measurements using aluminium models of the animal’s feet. e experi-
ments show that the feet slapping on the water provides only % of the force necessary
to run above water; the other % is provided by a pocket of compressed air that the
basilisks create between their feet and the water once the feet are inside the water. In fact,
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* If you are interested in learning in more detail how nature and the eye cope with the complexities of three dimensions, see the http://schorlab.berkeley.edu/vilis/whatisLL.htm and http://www.med.uwo.ca/physiology/ courses/LLConsequencesWeb/ListingsLaw/perceptual2.htm websites. ** In the Middle Ages, the term ‘basilisk’ referred to a mythical monster supposed to appear shortly before the end of the world. Today, it is a small reptile in the Americas.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 62 A basilisk lizard (Basiliscus basiliscus) running on water, showing how the propulsing leg pushes into the water (© TERRA)
Challenge 217 n Challenge 218 n
Challenge 219 e
basilisks mainly walk on air.* It was calculated that humans are also able to walk on water,
provided their feet hit the water with a speed of km h using the simultaneous physical
power of sprinters. Quite a feat for all those who ever did so.
ere is a second method of walking and running on water;
this second method even allows its users to remain immobile
on top of the water surface. is is what water striders, insects
of the family Gerridae with a overall length of up to mm, are
able to do (together with several species of spiders). Like all in-
sects, the water strider has six legs (spiders have eight). e wa-
ter strider uses the back and front legs to hover over the surface,
helped by thousands of tiny hairs attached to its body. e hairs,
together with the surface tension of water, prevent the strider from getting wet. If you put shampoo into the water, the wa-
F I G U R E 63 A water strider (© Charles Lewallen)
ter strider sinks and can no longer move. e water strider uses its large middle legs as
oars to advance over the surface, reaching speeds of up to m s doing so. In short, water
striders actually row over water.
Legs pose many interesting problems. Engineers know that a staircase is comfortable
to walk only if for each step the depth l plus twice the height h is a constant: l + h =
. . m. is is the so-called staircase formula. Why does it hold?
All animals have an even number of legs. Do you know an exception? Why not? In
fact, one can argue that no animal has less than four legs. Why is this the case?
On the other hand, all animals with two legs have the legs side by side, whereas systems
with wheels have them one behind the other. Why is this not the other way round?
But let us continue with the study of motion transmitted over distance, without the
use of any contact at all.
Ref. 89 * Both e ects used by basilisks are also found in fast canoeing.
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D
Page 677
“Caddi come corpo morto cade. Dante, Inferno, c. V, v.
”.*
e rst and main contact-free method to generate motion we discover in our environ-
ment is height. Waterfalls, snow, rain and falling apples all rely on it. It was one of the
fundamental discoveries of physics that height has this property because there is an in-
teraction between every body and the Earth. Gravitation produces an acceleration along
the line connecting the centres of gravity of the body and the Earth. Note that in order to
make this statement, it is necessary to realize that the Earth is a body in the same way as
a stone or the Moon, that this body is nite and that therefore it has a centre and a mass.
Today, these statements are common knowledge, but they are by no means evident from
everyday personal experience.**
How does gravitation change when two bodies are far apart? e experts on distant ob-
jects are the astronomers. Over the years they have performed numerous measurements
of the movements of the Moon and the planets. e most industrious of all was Tycho
Brahe,*** who organized an industrial-scale search for astronomical facts sponsored by
his king. His measurements were the basis for the research of his young assistant, the Swa-
bian astronomer Johannes Kepler**** who found the rst precise description of planetary
motion. In , all observations of planets and stones were condensed into an astonish-
ingly simple result by the English physicist Robert Hooke:***** every body of mass M
attracts any other body towards its centre with an acceleration whose magnitude a is
given by
a
=
G
M r
(29)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 220 n Ref. 90
Challenge 221 n Challenge 222 n
* ‘I fell like dead bodies fall.’ Dante Alighieri (1265, Firenze–1321, Ravenna), the powerful Italian poet. ** In several myths about the creation or the organization of the world, such as the biblical one or the Indian one, the Earth is not an object, but an imprecisely de ned entity, such as an island oating or surrounded by water with unclear boundaries and unclear method of suspension. Are you able to convince a friend that the Earth is round and not at? Can you nd another argument apart from the roundness of the Earth’s shadow when it is visible on the Moon?
A famous crook, Robert Peary, claimed to have reached the North Pole in 1909. (In fact, Roald Amundsen reached the both the South and the North Pole rst.) Peary claimed to have taken a picture there, but that picture, which went round the world, turned out to be the proof that he had not been there. Can you imagine how?
By the way, if the Earth is round, the top of two buildings is further apart than their base. Can this e ect be measured? *** Tycho Brahe (1546–1601), famous Danish astronomer, builder of Uraniaborg, the astronomical castle. He consumed almost 10 % of the Danish gross national product for his research, which produced the rst star catalogue and the rst precise position measurements of planets. **** Johannes Kepler (1571 Weil der Stadt–1630 Regensburg); a er helping his mother successfully defend herself in a trial where she was accused of witchcra , he studied Protestant theology and became a teacher of mathematics, astronomy and rhetoric. His rst book on astronomy made him famous, and he became assistant to Tycho Brahe and then, at his teacher’s death, the Imperial Mathematician. He was the rst to use mathematics in the description of astronomical observations, and introduced the concept and eld of ‘celestial physics’. ***** Robert Hooke, (1635–1703), important English physicist and secretary of the Royal Society. He also wrote the Micrographia, a beautifully illustrated exploration of the world of the very small.
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Ref. 91
where r is the centre-to-centre distance of the two bodies. is is called the universal
‘law’ of gravitation, or universal gravity, because it is valid in general. e proportionality
constant G is called the gravitational constant; it is one of the fundamental constants of
nature, like the speed of light or the quantum of action. More about it will be said shortly.
e e ect of gravity thus decreases with increasing distance; gravity depends on the in-
verse distance squared of the bodies under consideration. If bodies are small compared
with the distance r, or if they are spherical, expression ( ) is correct as it stands; for
non-spherical shapes the acceleration has to be calculated separately for each part of the
bodies and then added together.
is inverse square dependence is o en called Newton’s ‘law’ of gravitation, because
the English physicist Isaac Newton proved more elegantly than Hooke that it agreed with
all astronomical and terrestrial observations. Above all, however, he organized a better
public relations campaign, in which he falsely claimed to be the originator of the idea.
Newton published a simple proof showing that this description of astronomical mo-
tion also gives the correct description for stones thrown through the air, down here on
‘father Earth’. To achieve this, he compared the acceleration am of the Moon with that of stones . For the ratio between these two accelerations, the inverse square relation pre-
dicts a value am = R dm, where R is the radius of the Earth and dm the distance of the Moon. e Moon’s distance can be measured by triangulation, comparing the pos-
ition of the Moon against the starry background from two di erent points on Earth.*
e result is dm R =
, depending on the orbital position of the Moon, so that an
average ratio am = . ë is predicted from universal gravity. But both accelerations
can also be measured directly. At the surface of the Earth, stones are subject to an accele-
ration due to gravitation with magnitude = . m s , as determined by measuring the
time that stones need to fall a given distance. For the Moon, the de nition of acceleration,
a = dv dt, in the case of circular motion – roughly correct here – gives am = dm( π T) , where T = . Ms is the time the Moon takes for one orbit around the Earth.** e meas-
urement of the radius of the Earth*** yields R = . Mm, so that the average Earth–Moon
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Challenge 223 n
Ref. 92 Challenge 224 ny
Ref. 93 Page 62 Challenge 225 ny
* e rst precise – but not the rst – measurement was achieved in 1752 by the French astronomers Lalande and La Caille, who simultaneously measured the position of the Moon seen from Berlin and from Le Cap. ** is is deduced easily by noting that for an object in circular motion, the magnitude v of the velocity v = dx dt is given as v = πr T. e drawing of the vector v over time, the so-called hodograph, shows that it behaves exactly like the position of the object. erefore the magnitude a of the acceleration a = dv dt is given by the corresponding expression, namely a = πv T. *** is is the hardest quantity to measure oneself. e most surprising way to determine the Earth’s size is the following: watch a sunset in the garden of a house, with a stopwatch in hand. When the last ray of the Sun disappears, start the stopwatch and run upstairs. ere, the Sun is still visible; stop the stopwatch when the Sun disappears again and note the time t. Measure the height distance h of the two eye positions where the Sun was observed. e Earth’s radius R is then given by R = k h t , with k = ë s .
ere is also a simple way to measure the distance to the Moon, once the size of the Earth is known. Take a photograph of the Moon when it is high in the sky, and call θ its zenith angle, i.e. its angle from the vertical. Make another photograph of the Moon a few hours later, when it is just above the horizon. On this picture, unlike a common optical illusion, the Moon is smaller, because it is further away. With a sketch the reason for this becomes immediately clear. If q is the ratio of the two angular diameters, the Earth–Moon distance dm is given by the relation dm = R + [ Rq cos θ ( − q )] . Enjoy nding its derivation from the sketch.
Another possibility is to determine the size of the Moon by comparing it with the size of the shadow of the Earth during an eclipse. e distance to the Moon is then computed from its angular size, about . °.
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Earth
figure to be inserted
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F I G U R E 64 A physicist’s and an artist’s view of the fall of the Moon: a diagram by Christiaan Huygens (not to scale) and a marble statue by Auguste Rodin
Challenge 226 n Ref. 94
distance is dm = . Gm. One thus has am = . ë , in agreement with the above prediction. With this famous ‘Moon calculation’ we have thus shown that the inverse square property of gravitation indeed describes both the motion of the Moon and that of stones. You might want to deduce the value of GM.
From the observation that on the Earth all motion eventually comes to rest, whereas in the sky all motion is eternal, Aristotle and many others had concluded that motion in the sublunar world has di erent properties from motion in the translunar world. Several thinkers had criticized this distinction, notably the French philosopher and rector of the University of Paris, Jean Buridan.* e Moon calculation was the most important result showing this distinction to be wrong. is is the reason for calling the expression ( ) the universal ‘law’ of gravitation.
is result allows us to answer another old question. Why does the Moon not fall from the sky? Well, the preceding discussion showed that fall is motion due to gravitation.
erefore the Moon actually is falling, with the peculiarity that instead of falling towards the Earth, it is continuously falling around it. Figure illustrates the idea. e Moon is continuously missing the Earth.**
Universal gravity also explains why the Earth and most planets are (almost) spherical. Since gravity increases with decreasing distance, a liquid body in space will always try to form a spherical shape. Seen on a large scale, the Earth is indeed liquid. We also know that the Earth is cooling down – that is how the crust and the continents formed. e sphericity of smaller solid objects encountered in space, such as the Moon, thus means that they used to be liquid in older times.
Ref. 95 Challenge 227 d
* Jean Buridan (c. 1295 to c. 1366) was also one of the rst modern thinkers to speculate on a rotation of the Earth about an axis. ** Another way to put it is to use the answer of the Dutch physicist Christiaan Huygens (1629–1695): the Moon does not fall from the sky because of the centrifugal acceleration. As explained on page 109, this explanation is nowadays out of favour at most universities.
ere is a beautiful problem connected to the le part of the gure: Which points on the surface of the Earth can be hit by shooting from a mountain? And which points can be hit by shooting horizontally?
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Challenge 228 ny
Gravitation implies that the path of a stone is not a parabola, as stated earlier, but actually an ellipse around the centre of the Earth. is happens for exactly the same reason that the planets move in ellipses around the Sun. Are you able to con rm this statement?
Universal gravitation allows us to solve a mystery. e puzzling acceleration value = . m s we encountered in equation ( ) is thus due to the relation
= GMEarth REarth .
(30)
Challenge 229 n Challenge 230 n Challenge 231 n
e equation can be deduced from equation ( ) by taking the Earth to be spherical. e
everyday acceleration of gravity thus results from the size of the Earth, its mass, and the universal constant of gravitation G. Obviously, the value for is almost constant on the surface of the Earth because the Earth is almost a sphere. Expression ( ) also explains why gets smaller as one rises above the Earth, and the deviations of the shape of the Earth from sphericity explain why is di erent at the poles and higher on a plateau.
(What would it be on the Moon? On Mars? On Jupiter?) By the way, it is possible to devise a simple machine, other than a yo-yo, that slows
down the e ective acceleration of gravity by a known amount, so that one can measure
its value more easily. Can you imagine it? Note that . is roughly π . is is not a coincidence: the metre has been chosen in
such a way to make this correct. e period T of a swinging pendulum, i.e. a back and forward swing, is given by*
T= π
l ,
(31)
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Challenge 233 e
where l is the length of the pendulum, and is the gravitational acceleration. ( e pen-
dulum is assumed to be made of a mass attached to a string of negligible mass.) e
oscillation time of a pendulum depends only on the length of the string and the planet it
is located on. If the metre had been de ned such that T = s, the value of the normal
acceleration would have been exactly π m s . is was the rst proposal for the de n-
ition of the metre; it was made in by Huygens and repeated in by Talleyrand,
but was rejected by the conference that de ned the metre because variations in the value
of with geographical position and temperature-induced variations of the length of a
pendulum induce errors that are too large to yield a de nition of useful precision.
Finally, the proposal was made to de ne the metre as
of the circum-
ference of the Earth through the poles, a so-called meridian. is proposal was almost
identical to – but much more precise than – the pendulum proposal. e meridian de n-
ition of the metre was then adopted by the French national assembly on March ,
Challenge 232 n
* Formula (31) is noteworthy mainly for all that is missing. e period of a pendulum does not depend on the mass of the swinging body. In addition, the period of a pendulum does not depend on the amplitude. ( is is true as long as the oscillation angle is smaller than about °.) Galileo discovered this as a student, when observing a chandelier hanging on a long rope in the dome of Pisa. Using his heartbeat as a clock he found that even though the amplitude of the swing got smaller and smaller, the time for the swing stayed the same.
A leg also moves like a pendulum, when one walks normally. Why then do taller people tend to walk faster?
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Challenge 234 n Challenge 235 n
with the statement that ‘a meridian passes under the feet of every human being, and all meridians are equal’. (Nevertheless, the distance from Equator to the poles is not exactly
Mm; that is a strange story. One of the two geographers who determined the size of the rst metre stick was dishonest. e data he gave for his measurements – the general method of which is shown in Figure – was fabricated. us the rst o cial metre stick in Paris was shorter than it should be.)
But we can still ask: Why does the Earth have the mass and size it has? And why does G have the value it has? e
rst question asks for a history of the solar system; it is still unanswered and is topic of research. e second question is addressed in Appendix B.
If all objects attract each other, it should also be the case for objects in everyday life. Gravity must also work sideways. is is indeed the case, even though the e ects are so small that they were measured only long a er universal gravity had predicted them. Measuring this e ect allows the gravitational constant G to be determined.
Note that measuring the gravitational constant G is also the only way to determine the mass of the Earth. e
rst to do so, in , was the English physicist Henry Cavendish; he used the machine, ideas and method of John Michell who died when attempting the experiment. Michell and Cavendish* called the aim and result of their experiments ‘weighing the Earth’. Are you able to imagine how they did it? e value found in modern experiments is
F I G U R E 65 The measurements
G = . ë − Nm kg = . ë − m kg s . (32) that lead to the definition of the
metre (© Ken Alder)
Cavendish’s experiment was thus the rst to con rm that gravity also works sideways.
For example, two average people m apart feel an acceleration towards each other that is less than that exerted by a common y when landing on the skin. erefore we usually do not notice the attraction to other people. When we notice it, it is much stronger than that. is simple calculation thus proves that gravitation cannot be the true cause of people falling in love, and that sexual attraction is not of gravitational origin, but of a di erent source. e basis for this other interaction, love, will be studied later in our walk: it is called electromagnetism.
But gravity has more interesting properties to o er. e e ects of gravitation can also be described by another observable, namely the (gravitational) potential φ. We then have
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* Henry Cavendish (1731–1810) was one of the great geniuses of physics; rich and solitary, he found many rules of nature, but never published them. Had he done so, his name would be much more well known.
John Michell (1724–1793) was church minister, geologist and amateur astronomer.
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the simple relation that the acceleration is given by the gradient of the potential
a = −∇φ or a = −grad φ .
(33)
e gradient is just a learned term for ‘slope along the steepest direction’. It is de ned
for any point on a slope, is large for a steep one and small for a shallow one and it
points in the direction of steepest ascent, as shown in Figure . e gradient is abbreviated ∇, pronounced ‘nabla’ and is mathematically de ned as the vector ∇φ = (∂φ ∂x, ∂φ ∂y, ∂φ ∂z) = grad φ. e minus sign in ( ) is introduced by convention, in order to have higher potential values at larger heights.* For a point-like or a spherical
body of mass M, the potential φ is
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φ = −G
M r
.
(34)
A potential considerably simpli es the description of motion, since a potential is additive:
given the potential of a point particle, one can calculate the potential and then the motion
around any other irregularly shaped object.**
e potential φ is an interesting quantity; with a single number at every position in
space we can describe the vector aspects of gravitational acceleration. It automatically
gives that gravity in New Zealand acts in the opposite direction to gravity in Paris. In
addition, the potential suggests the introduction of the so-called potential energy U by
setting
U = mφ
(36)
and thus allowing us to determine the change of kinetic energy T of a body falling from a point to a point via
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
T − T = U − U or m v − m v = mφ − mφ .
(37)
Challenge 236 ny
* In two or more dimensions slopes are written ∂φ ∂z – where ∂ is still pronounced ‘d’ – because in those
cases the expression dφ dz has a slightly di erent meaning. e details lie outside the scope of this walk.
** Alternatively, for a general, extended body, the potential is found by requiring that the divergence of its
gradient is given by the mass (or charge) density times some proportionality constant. More precisely, one
has
∆φ = πGρ
(35)
where ρ = ρ(x, t) is the mass volume density of the body and the operator ∆, pronounced ‘delta’, is de ned as ∆ f = ∇∇ f = ∂ f ∂x + ∂ f ∂y + ∂ f ∂z . Equation (35) is called the Poisson equation for the potential φ. It is named a er Siméon-Denis Poisson (1781–1840), eminent French mathematician and physicist. e positions at which ρ is not zero are called the sources of the potential. e so-called source term ∆φ of a function is a measure for how much the function φ(x) at a point x di ers from the average value in a region around that point. (Can you show this, by showing that ∆φ φ¯ − φ(x)?) In other words, the Poisson equation (35) implies that the actual value of the potential at a point is the same as the average value around that point minus the mass density multiplied by πG. In particular, in the case of empty space the potential at a point is equal to the average of the potential around that point.
O en the concept of gravitational eld is introduced, de ned as g = −∇φ. We avoid this in our walk, because we will discover that, following the theory of relativity, gravity is not due to a eld at all; in fact even
the concept of gravitational potential turns out to be only an approximation.
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Challenge 237 n Ref. 99
Ref. 100 Ref. 101 Challenge 238 ny
Page 94
Appendix B
In other words, the total energy, de ned as the sum of kinetic and potential energy, is
conserved in motion due to gravity. is is a characteristic property of gravitation. Not
all accelerations can be derived from a potential; systems with this property are called
conservative. e accelerations due to friction are not conservative, but those due to elec-
tromagnetism are.
Interestingly, the number of dimensions of space
d is coded into the potential of a spherical mass: its
f (x,y)
dependence on the radius r is in fact rd− . e ex-
ponent d − has been checked experimentally to
high precision; no deviation of d from has ever
been found.
e concept of potential helps in understanding y
the shape of the Earth. Since most of the Earth is
x
still liquid when seen on a large scale, its surface is
grad f
always horizontal with respect to the direction de- F I G U RE 66 The potential and the termined by the combination of the accelerations gradient of gravity and rotation. In short, the Earth is not a
sphere. It is not an ellipsoid either. e mathematical shape de ned by the equilibrium
requirement is called a geoid. e geoid shape di ers from a suitably chosen ellipsoid
by at most m. Can you describe the geoid mathematically? e geoid is an excellent
approximation to the actual shape of the Earth; sea level di ers from it by less than
metres. e di erences can be measured with satellite radar and are of great interest to
geologists and geographers. For example, it turns out that the South Pole is nearer to the
equatorial plane than the North Pole by about m. is is probably due to the large land
masses in the northern hemisphere.
Above we saw how the inertia of matter,
through the so-called ‘centrifugal force’,
increases the radius of the Earth at the
Equator. In other words, the Earth is
attened at the poles. e Equator has
a radius a of . Mm, whereas the dis-
tance b from the poles to the centre of
the Earth is . Mm. e precise atten-
ing (a − b) a has the value
.=
. . As a result, the top of Mount Chim-
borazo in Ecuador, even though its height
is only m above sea level, is about
km farther away from the centre of the
Earth than the top of Mount Sagarmatha*
in Nepal, whose height above sea level is
m. e top of Mount Chimborazo is
in fact the point on the surface most dis- F I G U R E 67 The shape of the Earth, with
tant from the centre of the Earth. As a consequence, if the Earth stopped
exaggerated height scale (© GeoForschungsZentrum Potsdam)
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* Mount Sagarmatha is sometimes also called Mount Everest.
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Page 88
rotating (but kept its shape), the water of the oceans would ow north; all of Europe would be under water, except for the few mountains of the Alps that are higher than about km.
e northern parts of Europe would be covered by between km and km of water. Mount Sagarmatha would be over km above sea level. If one takes into account the resulting change of shape of the Earth, the numbers come out smaller. In addition, the change in shape would produce extremely strong earthquakes and storms. As long as there are none of these e ects, we can be sure that the Sun will indeed rise tomorrow, despite what some philosophers might pretend.
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Challenge 239 n Page 176
Let us give a short summary. If a body can move only along a (possibly curved) line, the concepts of kinetic and potential energy are su cient to determine the way it moves. In short, motion in one dimension follows directly from energy conservation.
If more than two spatial dimensions are involved, energy conservation is insu cient to determine how a body moves. If a body can move in two dimensions, and if the forces involved are internal (which is always the case in theory, but not in practice), the conservation of angular momentum can be used. e full motion in two dimensions thus follows from energy and angular momentum conservation. For example, all properties of free fall follow from energy and angular momentum conservation. (Are you able to show this?)
In the case of motion in three dimensions, a more general rule for determining motion is necessary. It turns out that all motion follows from a simple principle: the time average of the di erence between kinetic and potential energy must be as small as possible. is is called the least action principle. We will explain the details of this calculation method later.
For simple gravitational motions, motion is two-dimensional. Most threedimensional problems are outside the scope of this text; in fact, some of these problems are still subjects of research. In this adventure, we will explore three-dimensional motion only for selected cases that provide important insights.
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G
e expression for the acceleration due to gravity a = GM r also describes the motion of all the planets around the Sun. Anyone can check that the planets always stay within the zodiac, a narrow stripe across the sky. e centre line of the zodiac gives the path of the Sun and is called the ecliptic, since the Moon must be located on it to produce an eclipse. But the detailed motion of the planets is not easy to describe.* A few generations before Hooke, the Swabian astronomer Johannes Kepler had deduced several ‘laws’ in his painstaking research about the movements of the planets in the zodiac. e three main ones are as follows:
. Planets move on ellipses with the Sun located at one focus ( ). . Planets sweep out equal areas in equal times ( ).
* e apparent height of the ecliptic changes with the time of the year and is the reason for the changing seasons. erefore seasons are a gravitational e ect as well.
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Challenge 240 ny Ref. 24
Challenge 241 n
. All planets have the same ratio T d between the orbit duration T and the semimajor axis d ( ).
e main results are given in Figure . e sheer
work required to deduce the three ‘laws’ was enorm-
ous. Kepler had no calculating machine available, not
even a slide rule. e calculation technology he used was the recently discovered logarithms. Anyone who has used tables of logarithms to perform calculations
d
Sun
d
can get a feeling for the amount of work behind these three discoveries.
planet
e second ‘law’ about equal swept areas implies
that planets move faster when they are near the Sun. It is a way to state the conservation of angular momentum. But now comes the central point. e huge volume of work by Brahe and Kepler can be summar-
F I G U R E 68 The motion of a planet around the Sun, showing its semimajor axis d, which is also the spatial average of its distance from the Sun
ized in the expression a = GM r . Can you con rm
that all three laws follow from Hooke’s expression of universal gravity? Publishing this
result was the main achievement of Newton. Try to repeat his achievement; it will show
you not only the di culties, but also the possibilities of physics, and the joy that puzzles
give.
Newton solved the puzzle with geometric drawing. Newton was not able to write down,
let alone handle, di erential equations at the time he published his results on gravitation.
In fact, it is well known that Newton’s notation and calculation methods were poor. (Much
poorer than yours!) e English mathematician Godfrey Hardy* used to say that the
insistence on using Newton’s integral and di erential notation, rather than the earlier
and better method, still common today, due to his rival Leibniz – threw back English
mathematics by years.
Kepler, Hooke and Newton became famous because they brought order to the descrip-
tion of planetary motion. is achievement, though of small practical signi cance, was
widely publicized because of the age-old prejudices linked with astrology.
However, there is more to gravitation. Universal gravity explains the motion and shape
of the Milky Way and of the other galaxies, the motion of many weather phenomena and
explains why the Earth has an atmosphere but the Moon does not. (Can you explain this?)
In fact, universal gravity explains much more about the Moon.
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TM
How long is a day on the Moon? e answer is roughly Earth-days. at is the time that it takes for the Moon to see the Sun again in the same position.
One o en hears that the Moon always shows the same side to the Earth. But this is wrong. As one can check with the naked eye, a given feature in the centre of the face of the Moon at full Moon is not at the centre one week later. e various motions leading
* Godfrey Harold Hardy (1877–1947) was an important English number theorist, and the author of the well-known A Mathematician’s Apology. He also ‘discovered’ the famous Indian mathematician Srinivasa Ramanujan, bringing him to Britain.
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F I G U R E 69 The change of the moon during the month, showing its libration (© Martin Elsässer)
to this change are called librations; they are shown in the movie in Figure .* e motions appear mainly because the Moon does not describe a circular, but an elliptical orbit around the Earth and because the axis of the Moon is slightly inclined, compared with that of its rotation around the Earth. As a result, only around % of the Moon’s surface is permanently hidden from Earth.
e rst photographs of the hidden area were taken in the s by a Soviet arti cial satellite; modern satellites provided exact maps, as shown in Figure . e hidden surface is much more irregular than the visible one, as the hidden side is the one that intercepts most asteroids attracted by the Earth. us the gravitation of the Moon helps to de ect asteroids from the Earth. e number of animal life extinctions is thus reduced to a small, but not negligible number. In other words, the gravitational attraction of the Moon has saved the human race from extinction many times over.**
e trips to the Moon in the s also showed that the Moon originated from the Earth itself: long ago, an object hit the Earth almost tangentially and threw a sizeable
* e movie is in DivX 5 AVI format and requires a so ware plug-in in Acrobat Reader that can play it. ** e web pages http://cfa-www.harvard.edu/iau/lists/Closest.html and InnerPlot.html give an impression of the number of objects that almost hit the Earth every year. Without the Moon, we would have many additional catastrophes.
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F I G U R E 70 Maps (not photographs) of the near side (left) and far side (right) of the moon, showing how often the latter saved the Earth from meteorite impacts (courtesy USGS)
Ref. 102 Ref. 103 Challenge 242 n
Ref. 104
Ref. 105
Ref. 106 Ref. 107 Page 100
fraction of material up into the sky. is is the only mechanism able to explain the large size of the Moon, its low iron content, as well as its general material composition.
e Moon is receding from the Earth at . cm a year. is result con rms the old deduction that the tides slow down the Earth’s rotation. Can you imagine how this measurement was performed? Since the Moon slows down the Earth, the Earth also changes shape due to this e ect. (Remember that the shape of the Earth depends on its speed of rotation.) ese changes in shape in uence the tectonic activity of the Earth, and maybe also the dri of the continents.
e Moon has many e ects on animal life. A famous example is the midge Clunio, which lives on coasts with pronounced tides. Clunio spends between six and twelve weeks as a larva, then hatches and lives for only one or two hours as an adult ying insect, during which time it reproduces. e midges will only reproduce if they hatch during the low tide phase of a spring tide. Spring tides are the especially strong tides during the full and new moons, when the solar and lunar e ects combine, and occur only every . days. In
, Dietrich Neumann showed that the larvae have two built-in clocks, a circadian and a circalunar one, which together control the hatching to precisely those few hours when the insect can reproduce. He also showed that the circalunar clock is synchronized by the brightness of the Moon at night. In other words, the larvae monitor the Moon at night and then decide when to hatch: they are the smallest known astronomers.
If insects can have circalunar cycles, it should come as no surprise that women also have such a cycle. However, in this case the origin of the cycle length is still unknown.
e Moon also helps to stabilize the tilt of the Earth’s axis, keeping it more or less xed relative to the plane of motion around the Sun. Without the Moon, the axis would change its direction irregularly, we would not have a regular day and night rhythm, we would have extremely large climate changes, and the evolution of life would have been impossible. Without the Moon, the Earth would also rotate much faster and we would have much less clement weather. e Moon’s main remaining e ect on the Earth, the precession of its axis, is responsible for the ice ages.
Furthermore, the Moon shields the Earth from cosmic radiation by greatly increasing
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Ref. 108
the Earth’s magnetic eld. In other words, the Moon is of central importance for the evolution of life. Understanding how o en Earth-sized planets have Moon-sized satellites is thus important for the estimation of the probability that life exists on other planets. So far, it seems that large satellites are rare; there are only four known moons that are larger than that of the Earth, but they circle much larger planets, namely Jupiter and Saturn. Indeed, the formation of satellites is still an area of research. But let us return to the e ects of gravitation in the sky.
O
Challenge 243 e Challenge 244 ny
e path of a body orbiting another under the in uence of gravity is an ellipse with the
central body at one focus. A circular orbit is also possible, a circle being a special case
of an ellipse. Single encounters of two objects can also be parabolas or hyperbolas, as
shown in Figure . Circles, ellipses, parabolas and hyperbolas are collectively known as
conic sections. Indeed each of these curves can be produced by cutting a cone with a knife.
Are you able to con rm this?
If orbits are mostly ellipses, it follows that comets
return. e English astronomer Edmund Halley ( – ) was the rst to draw this conclusion and
hyperbola
to predict the return of a comet. It arrived at the pre-
dicted date in , and is now named a er him. e
period of Halley’s comet is between and years; the rst recorded sighting was centuries ago, and
parabola
it has been seen at every one of its passages since,
the last time in .
mass
Depending on the initial energy and the initial angular momentum of the body with respect to the cent-
circle
ellipse
ral planet, there are two additional possibilities: para-
bolic paths and hyperbolic paths. Can you determ-
ine the conditions of the energy and the angular mo-
mentum needed for these paths to appear?
In practice, parabolic paths do not exist in nature.
( ough some comets seem to approach this case
when moving around the Sun; almost all comets fol- F I G U RE 71 The possible orbits due to
low elliptical paths). Hyperbolic paths do exist; arti - universal gravity
cial satellites follow them when they are shot towards
a planet, usually with the aim of changing the direction of the satellite’s journey across
the solar system.
Why does the inverse square law lead to conic sections? First, for two bodies, the total
angular momentum L is a constant:
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L = mr φ˙
(38)
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and therefore the motion lies in a plane. Also the energy E is a constant
E=
m(
dr dt
)
+
m(r
dφ dt
)
−
G
mM r
.
(39)
Challenge 245 ny Together, the two equations imply that
r=
L Gm
M
+
.
+
G
EL mM
cos φ
Now, any curve de ned by the general expression
(40) Dvipsbugw
r=
C + e cos φ
or
r=
C − e cos φ
(41)
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Challenge 246 e
Page 80 Challenge 247 ny Challenge 248 ny
is an ellipse for < e < , a parabola for e = and a hyperbola for e , one focus being at the origin. e quantity e, called the eccentricity, describes how squeezed the curve is. In other words, a body in orbit around a central mass follows a conic section.
In all orbits, also the heavy mass moves. In fact, both bodies orbit around the common centre of mass. Both bodies follow the same type of curve (ellipsis, parabola or hyperbola), but the dimensions of the two curves di er.
If more than two objects move under mutual gravitation, many additional possibilities for motions appear. e classi cation and the motions are quite complex. In fact, this socalled many-body problem is still a topic of research, and the results are mathematically fascinating. Let us look at a few examples.
When several planets circle a star, they also attract each other. Planets thus do not move in perfect ellipses. e largest deviation is a perihelion shi , as shown in Figure . It is observed for Mercury and a few other planets, including the Earth. Other deviations from elliptical paths appear during a single orbit. In , the observed deviations of the motion of the planet Uranus from the path predicted by universal gravity were used to predict the existence of another planet, Neptune, which was discovered shortly a erwards.
We have seen that mass is always positive and that gravitation is thus always attractive; there is no antigravity. Can gravity be used for levitation nevertheless, maybe using more than two bodies? Yes; there are two examples.* e rst are the geostationary satellites, which are used for easy transmission of television and other signals from and towards Earth.
e Lagrangian libration points are the second example. Named a er their discoverer, these are points in space near a two-body system, such as Moon–Earth or Earth–Sun, in which small objects have a stable equilibrium position. A general overview is given in Figure . Can you nd their precise position, remembering to take rotation into account?
ere are three additional Lagrangian points on the Earth–Moon line. How many of them are stable?
ere are thousands of asteroids, called Trojan asteroids, at and around the Lagrangian points of the Sun–Jupiter system. In , a Trojan asteroid for the Mars–Sun system was
* Levitation is discussed in detail on page 602.
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Ref. 109 Ref. 110
discovered. Finally, in , an ‘almost Trojan’ asteroid was found that follows the Earth
on its way around the Sun (it is only transitionary and follows a somewhat more complex
orbit). is ‘second companion’ of the Earth has a diameter of km. Similarly, on the
main Lagrangian points of the Earth–Moon system a high concentration of dust has been
observed.
To sum up, the single equation a = −GMr r cor-
rectly describes a large number of phenomena in the
sky. e rst person to make clear that this expression
describes everything happening in the sky was Pierre Simon Laplace* in his famous treatise Traité de méca-
planet (or Sun)
nique céleste. When Napoleon told him that he found no mention about the creator in the book, Laplace gave
L5
π/3
a famous, one sentence summary of his book: Je n’ai pas
eu besoin de cette hypothèse. ’I had no need for this hypothesis.’ In particular, Laplace studied the stability of the solar system, the eccentricity of the lunar orbit, and the eccentricities of the planetary orbits, always getting full agreement between calculation and measurement.
ese results are quite a feat for the simple expres-
π/3
π/3
π/3
L4 moon (or planet)
F I G U R E 72 The two stable Lagrangian points
sion of universal gravitation; they also explain why it is called ‘universal’. But how pre-
cise is the formula? Since astronomy allows the most precise measurements of gravita-
tional motion, it also provides the most stringent tests. In , Urbain Le Verrier con-
cluded a er intensive study that there was only one known example of a discrepancy
between observation and universal gravity, namely one observation for the planet Mer-
cury. (Nowadays a few more are known.) e point of least distance to the Sun of the
orbit of planet Mercury, its perihelion, changes at a rate that is slightly less than that pre-
dicted: he found a tiny di erence, around ′′ per century. ( is was corrected to ′′ per
century in by Simon Newcomb.) Le Verrier thought that the di erence was due to
a planet between Mercury and the Sun, Vulcan, which he chased for many years without
success. e study of motion had to wait for Albert Einstein to nd the correct explana-
tion of the di erence.
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Challenge 249 n Ref. 39 Ref. 111
Why do physics texts always talk about tides? Because, as general relativity will show, tides prove that space is curved! It is thus useful to study them in a bit more detail. Gravitation explains the sea tides as results of the attraction of the ocean water by the Moon and the Sun. Tides are interesting; even though the amplitude of the tides is only about . m on the open sea, it can be up to m at special places near the coast. Can you imagine why? e soil is also li ed and lowered by the Sun and the Moon, by about . m, as satellite measurements show. Even the atmosphere is subject to tides, and the corresponding pressure variations can be ltered out from the weather pressure measurements.
* Pierre Simon Laplace (b. 1749 Beaumont-en-Auge, d. 1827 Paris), important French mathematician. His treatise appeared in ve volumes between 1798 and 1825. He was the rst to propose that the solar system was formed from a rotating gas cloud, and one of the rst people to imagine and explore black holes.
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Sun t = t1 : deformed
t = 0 : spherical
F I G U R E 73 Tidal deformations due to gravity
– before
after
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F I G U R E 74 The origin of tides
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Tides appear for any extended body moving in the gravitational eld of another. To understand the origin of tides, picture a body in orbit, like the Earth, and imagine its components, such as the segments of Figure , as being held together by springs. Universal gravity implies that orbits are slower the more distant they are from a central body. As a result, the segment on the outside of the orbit would like to be slower than the central one; but it is pulled by the rest of the body through the springs. In contrast, the inside segment would like to orbit more rapidly but is retained by the others. Being slowed down, the inside segments want to fall towards the Sun. In sum, both segments feel a pull away from the centre of the body, limited by the springs that stop the deformation. erefore, extended bodies are deformed in the direction of the eld inhomogeneity.
For example, as a result of tidal forces, the Moon always has (roughly) the same face to the Earth. In addition, its radius in direction of the Earth is larger by about m than the radius perpendicular to it. If the inner springs are too weak, the body is torn into pieces; in this way a ring of fragments can form, such as the asteroid ring between Mars and Jupiter or the rings around Saturn.
Let us return to the Earth. If a body is surrounded by water, it will form bulges in the direction of the applied gravitational eld. In order to measure and compare the strength of the tides from the Sun and the Moon, we reduce tidal e ects to their bare minimum. As shown in Figure , we can study the deformation of a body due to gravity by studying the deformation of four pieces. We can study it in free fall, because orbital motion and free fall are equivalent. Now, gravity makes some of the pieces approach and others diverge, depending on their relative positions. e gure makes clear that the strength of the deformation – water has no built-in springs – depends on the change of gravitational acceleration with distance; in other words, the relative acceleration that leads to the tides is proportional to the derivative of the gravitational acceleration.
Using the numbers from Appendix B, the gravitational accelerations from the Sun and
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
the Moon measured on Earth are
aSun
=
G MSun dSun
=
.
mm s
aMoon
=
G MMoon dMoon
=
.
mm s
(42)
Challenge 250 e
and thus the attraction from the Moon is about times weaker than that from the Sun.
When two nearby bodies fall near a large mass, the relative acceleration is proportional
to their distance, and follows da = da dr dr. e proportionality factor da dr = ∇a,
called the tidal acceleration (gradient), is the true measure of tidal e ects. Near a large
spherical mass M, it is given by
da dr
=
−
GM r
(43)
which yields the values
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d aSun dr
=
−
G MSun dSun
=−
.
ë
−
s
d aMoon dr
=
−
G MMoon dMoon
=−
.
ë
−
s.
(44)
Ref. 72 Page 451 Challenge 251 n
Page 377
In other words, despite the much weaker pull of the Moon, its tides are predicted to be over twice as strong as the tides from the Sun; this is indeed observed. When Sun, Moon and Earth are aligned, the two tides add up; these so-called spring tides are especially strong and happen every . days, at full and new moon.
Tides also produce friction. e friction leads to a slowing of the Earth’s rotation. Nowadays, the slowdown can be measured by precise clocks (even though short time variations due to other e ects, such as the weather, are o en larger). e results t well with fossil results showing that million years ago, in the Devonian period, a year had
days, and a day about hours. It is also estimated that million years ago, each of the days of a year were . hours long. e friction at the basis of this slowdown also results in an increase in the distance of the Moon from the Earth by about . cm per year. Are you able to explain why?
As mentioned above, the tidal motion of the soil is also responsible for the triggering of earthquakes. us the MoonMoon, dangers of can have also dangerous e ects on Earth. e most fascinating example of tidal e ects is seen on Jupiter’s satellite Io. Its tides are so strong that they induce intense volcanic activity, as shown in Figure , with eruption plumes as high as km. If tides are even stronger, they can destroy the body altogether, as happened to the body between Mars and Jupiter that formed the planetoids, or (possibly) to the moons that led to Saturn’s rings.
In summary, tides are due to relative accelerations of nearby mass points. is has an important consequence. In the chapter on general relativity we will nd that time multiplied by the speed of light plays the same role as length. Time then becomes an additional dimension, as shown in Figure . Using this similarity, two free particles moving in the
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F I G U R E 75 A spectacular result of tides: volcanism on Io (NASA)
x t1
t2 t
F I G U R E 76 Particles falling side by side approach over time
α b M
F I G U R E 77 Masses bend light
Page 418
same direction correspond to parallel lines in space-time. Two particles falling side-byside also correspond to parallel lines. Tides show that such particles approach each other. In other words, tides imply that parallel lines approach each other. But parallel lines can approach each other only if space-time is curved. In short, tides imply curved space-time and space. is simple reasoning could have been performed in the eighteenth century; however, it took another years and Albert Einstein’s genius to uncover it.
C
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Die Maxime, jederzeit selbst zu denken, ist die Aufklärung.
“ ” Immanuel Kant*
* e maxim to think at all times for oneself is the enlightenment.
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Page 275 Ref. 112
Challenge 252 ny
Towards the end of the seventeenth century people discovered that light has a nite velo-
city – a story which we will tell in detail later. An entity that moves with in nite velocity
cannot be a ected by gravity, as there is no time to produce an e ect. An entity with a
nite speed, however, should feel gravity and thus fall.
Does its speed increase when light reaches the surface of the Earth? For almost three
centuries people had no means of detecting any such e ect; so the question was not in-
vestigated. en, in , the Prussian astronomer Johann Soldner ( – ) was the
rst to put the question in a di erent way. Being an astronomer, he was used to measur-
ing stars and their observation angles. He realized that light passing near a massive body
would be de ected due to gravity.
Soldner studied a body on a hyperbolic path, moving with velocity c past a spherical
mass M at distance b (measured from the centre), as shown in Figure . Soldner deduced
the de ection angle
αuniv.
grav.
=
GM bc
.
(45)
Page 409
One sees that the angle is largest when the motion is just grazing the mass M. For light de ected by the mass of the Sun, the angle turns out to be at most a tiny . ′′= . µrad. In Soldner’s time, this angle was too small to be measured. us the issue was forgotten. Had it been pursued, general relativity would have begun as an experimental science, and not as the theoretical e ort of Albert Einstein! Why? e value just calculated is di erent from the measured value. e rst measurement took place in ;* it found the correct dependence on the distance, but found a de ection of up to . ′′, exactly double that of expression ( ). e reason is not easy to nd; in fact, it is due to the curvature of space, as we will see. In summary, light can fall, but the issue hides some surprises.
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W
?–A
Mass describes how an object interacts with others. In our walk, we have encountered two of its aspects. Inertial mass is the property that keeps objects moving and that o ers resistance to a change in their motion. Gravitational mass is the property responsible for the acceleration of bodies nearby (the active aspect) or of being accelerated by objects nearby (the passive aspect). For example, the active aspect of the mass of the Earth determines the surface acceleration of bodies; the passive aspect of the bodies allows us to weigh them in order to measure their mass using distances only, e.g. on a scale or a balance. e gravitational mass is the basis of weight, the di culty of li ing things.**
Is the gravitational mass of a body equal to its inertial mass? A rough answer is given by the experience that an object that is di cult to move is also di cult to li . e simplest experiment is to take two bodies of di erent masses and let them fall. If the acceleration is the same for all bodies, inertial mass is equal to (passive) gravitational mass, because in the relation ma = ∇(GMm r) the le -hand m is actually the inertial mass, and the right-hand m is actually the gravitational mass.
Challenge 253 ny * By the way, how would you measure the de ection of light near the bright Sun? Challenge 254 ny ** What are the values shown by a balance for a person of kg juggling three balls of . kg each?
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Ref. 113 Page 77 Challenge 255 ny
Page 395
But in the seventeenth century Galileo had made widely known an even older argument showing without a single experiment that the acceleration is indeed the same for all bodies. If larger masses fell more rapidly than smaller ones, then the following paradox would appear. Any body can be seen as being composed of a large fragment attached to a small fragment. If small bodies really fell less rapidly, the small fragment would slow the large fragment down, so that the complete body would have to fall less rapidly than the larger fragment (or break into pieces). At the same time, the body being larger than its fragment, it should fall more rapidly than that fragment. is is obviously impossible: all masses must fall with the same acceleration.
Many accurate experiments have been performed since Galileo’s original discussion. In all of them the independence of the acceleration of free fall from mass and material composition has been con rmed with the precision they allowed. In other words, as far as we can tell, gravitational mass and inertial mass are identical. What is the origin of this mysterious equality?
is so-called ‘mystery’ is a typical example of disinformation, now common across the whole world of physics education. Let us go back to the de nition of mass as a negative inverse acceleration ratio. We mentioned that the physical origins of the accelerations do not play a role in the de nition because the origin does not appear in the expression. In other words, the value of the mass is by de nition independent of the interaction. at means in particular that inertial mass, based on electromagnetic interaction, and gravitational mass are identical by de nition.
We also note that we have never de ned a separate concept of ‘passive gravitational mass’. e mass being accelerated by gravitation is the inertial mass. Worse, there is no way to de ne a ‘passive gravitational mass’. Try it! All methods, such as weighing an object, cannot be distinguished from those that determine inertial mass from its reaction to acceleration. Indeed, all methods of measuring mass use non-gravitational mechanisms. Scales are a good example.
If the ‘passive gravitational mass’ were di erent from the inertial mass, we would have strange consequences. For those bodies for which it were di erent we would get into trouble with energy conservation. Also assuming that ‘active gravitational mass’ di ers from inertial mass gets us into trouble.
Another way of looking at the issue is as follows. How could ‘gravitational mass’ differ from inertial mass? Would the di erence depend on relative velocity, time, position, composition or on mass itself? Each of these possibilities contradicts either energy or momentum conservation.
No wonder that all measurements con rm the equality of all mass types. e issue is usually resurrected in general relativity, with no new results. ‘Both’ masses remain equal; mass is a unique property of bodies. Another issue remains, though. What is the origin of mass? Why does it exist? is simple but deep question cannot be answered by classical physics. We will need some patience to nd out.
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F I G U R E 78 Brooms fall more rapidly than stones (© Luca Gastaldi)
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Fallen ist weder gefährlich noch eine Schande; Liegen bleiben ist beides.*
“ Konrad Adenauer ” * *
Gravity on the Moon is only one sixth of that on the Earth. Why does this imply that it is di cult to walk quickly and to run on the Moon (as can be seen in the TV images recorded there)?
**
e inverse square expression of universal gravity has a limitation: it does not allow one to make sensible statements about the matter in the universe. Universal gravity predicts that a homogeneous mass distribution is unstable; indeed, an inhomogeneous distribution is observed. However, universal gravity does not predict the average mass density, the darkness at night, the observed speeds of the distant galaxies, etc. In fact, not a single property of the universe is predicted. To do this, we need general relativity.
Challenge 256 e
**
Imagine that you have twelve coins of identical appearance, of which one is a forgery. e forged one has a di erent mass from the eleven genuine ones. How can you decide which is the forged one and whether it is lighter or heavier, using a simple balance only three times?
**
For a physicist, antigravity is repulsive gravity; it does not exist in nature. Nevertheless, the term ‘antigravity’ is used incorrectly by many people, as a short search on the internet shows. Some people call any e ect that overcomes gravity, ‘antigravity’. However, this de nition implies that tables and chairs are antigravity devices. Following the de nition,
* ‘Falling is neither dangerous nor a shame; to keep lying is both.’ Konrad Adenauer (b. 1876 Köln, d. 1967 Rhöndorf), German chancellor.
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M
1000 km
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F I G U R E 79 The starting situation for a bungee jumper
M F I G U R E 80 An honest balance?
most of the wood, steel and concrete producers are in the antigravity business. e internet de nition makes absolutely no sense.
Challenge 257 n
**
Do all objects on Earth fall with the same acceleration of . m s , assuming that air resistance can be neglected? No; every housekeeper knows that. You can check this by yourself. A broom angled at around ° hits the oor before a stone, as the sounds of impact con rm. Are you able to explain why?
**
Also Bungee jumpers are accelerated more strongly than . For a rubber of mass m and a jumper of mass M, the maximum acceleration a is
a=
+
m M
(
+
m M
)
.
(46)
Challenge 258 n Can you deduce the relation from Figure 79?
** Challenge 259 n Guess: What is the weight of a ball of cork with a radius of m?
** Challenge 260 n Guess: A heap of 1000 mm diameter iron balls is collected. What is its mass?
** How can you use your observations made during your travels to show that the Earth is
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 261 n not at?
**
Is the acceleration due to gravity constant? Not really. Every day, it is estimated that kg of material fall onto the Earth in the form of meteorites.
Challenge 262 n
**
Both the Earth and the Moon attract bodies. e centre of mass of the Earth–Moon system is km away from the centre of the Earth, quite near its surface. Why do bodies on Earth still fall towards the centre of the Earth?
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Challenge 263 ny
**
Does every spherical body fall with the same acceleration? No. If the weight of the object is comparable to that of the Earth, the distance decreases in a di erent way. Can you con-
rm this statement? What then is wrong about Galileo’s argument about the constancy of acceleration of free fall?
**
It is easy to li a mass of a kilogram from the oor on a table. Twenty kilograms is harder. Challenge 264 n A thousand is impossible. However, ë kg is easy. Why?
**
e ratio of the strengths of the tides of Moon and Sun is roughly Challenge 265 ny is also the ratio between the mass densities of the two bodies?
. Is it true that this
Challenge 266 ny Challenge 267 n
**
e friction between the Earth and the Moon slows down the rotation of both. e Moon stopped rotating millions of years ago, and the Earth is on its way to doing so as well. When the Earth stops rotating, the Moon will stop moving away from Earth. How far will the Moon be from the Earth at that time? A erwards however, even further in the future, the Moon will move back towards the Earth, due to the friction between the Earth–Moon system and the Sun. Even though this e ect would only take place if the Sun burned for ever, which is known to be false, can you explain it?
Challenge 268 ny
**
When you run towards the east, you lose weight. ere are two di erent reasons for this: the ‘centrifugal’ acceleration increases so that the force with which you are pulled down diminishes, and the Coriolis force appears, with a similar result. Can you estimate the size of the two e ects?
Challenge 269 n
**
What is the time ratio between a stone falling through a distance l and a pendulum swinging though half a circle of radius l? ( is problem is due to Galileo.) How many digits of the number π can one expect to determine in this way?
**
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Earth
Moon
Earth Moon
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Sun F I G U R E 81 Which of the two Moon paths is correct?
Challenge 270 n Ref. 97
Why can a spacecra accelerate through the slingshot e ect when going round a planet, despite momentum conservation? It is speculated that the same e ect is also the reason for the few exceptionally fast stars that are observed in the galaxy. For example, the star HE0457-5439 moves with km s, which is much higher than the 100 to km s of most stars in the Milky way. It seems that the role of the accelerating centre was taken by a black hole.
**
Ref. 98 e orbit of a planet around the Sun has many interesting properties. What is the hodoChallenge 271 n graph of the orbit? What is the hodograph for parabolic and hyperbolic orbits?
**
A simple, but di cult question: if all bodies attract each other, why don’t or didn’t all stars Challenge 272 n fall towards each other?
Ref. 114 Challenge 273 n
**
e acceleration due to gravity at a depth of km is . m s , over 2 % more than at the surface of the Earth. How is this possible? Also, on the Tibetan plateau, is higher than the sea level value of . m s , even though the plateau is more distant from the centre of the Earth than sea level is. How is this possible?
Challenge 274 n
**
When the Moon circles the Sun, does its path have sections concave towards the Sun, as shown at the right of Figure 81, or not, as shown on the le ? (Independent of this issue, both paths in the diagram disguise that fact that the Moon path does not lie in the same plane as the path of the Earth around the Sun.)
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You can prove that objects attract each other (and that they are not only attracted by the Earth) with a simple experiment that anybody can perform at home, as described on the http://www.fourmilab.ch/gravitation/foobar/ website.
Challenge 275 n
**
It is instructive to calculate the escape velocity of the Earth, i.e. that velocity with which a body must be thrown so that it never falls back. It turns out to be km s. What is the escape velocity for the solar system? By the way, the escape velocity of our galaxy is
km s. What would happen if a planet or a system were so heavy that its escape velocity would be larger than the speed of light?
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For bodies of irregular shape, the centre of gravity of a body is not the same as the centre Challenge 276 n of mass. Are you able to con rm this? (Hint: Find and use the simplest example possible.)
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
**
Can gravity produce repulsion? What happens to a small test body on the inside of a large Challenge 277 ny C-shaped mass? Is it pushed towards the centre of mass?
**
Ref. 115 e shape of the Earth is not a sphere. As a consequence, a plumb line usually does not Challenge 278 ny point to the centre of the Earth. What is the largest deviation in degrees?
** Challenge 279 n What is the largest asteroid one can escape from by jumping?
**
If you look at the sky every day at 6 a.m., the Sun’s position varies during the year. e result of photographing the Sun on the same lm is shown in Figure 82. e curve, called the analemma, is due to the inclination of the Earth’s axis, as well as the elliptical shape of the path around the Sun. e shape of the analemma is also built into high quality sundials. e top and the (hidden) bottom points correspond to the solstices.
Page 102
**
e constellation in which the Sun stands at noon (at the centre of the time zone) is supposedly called the ‘zodiacal sign’ of that day. Astrologers say there are twelve of them, namely Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpius, Sagittarius, Capricornus, Aquarius and Pisces and that each takes (quite precisely) a twel h of a year or a twel h of the ecliptic. Any check with a calendar shows that at present, the midday Sun is never in the zodiacal sign during the days usually connected to it. e relation has shi ed by about a month since it was de ned, due to the precession of the Earth’s axis. A check with a map of the star sky shows that the twelve constellations do not have the same length and that on the ecliptic there are fourteen of them, not twelve. ere is Ophiuchus, the snake constellation, between Scorpius and Sagittarius, and Cetus, the whale, between Aquarius and Pisces. In fact, not a single astronomical statement about zodiacal signs is
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F I G U R E 82 The analemma over Delphi, between January and December 2002 (© Anthony Ayiomamitis)
dm r m R
dM
F I G U R E 83 The vanishing of gravitational force inside a spherical shell of matter
Ref. 116 correct. To put it clearly, astrology, in contrast to its name, is not about stars. (In some languages, the term for ‘crook’ is derived from the word ‘astrologer’.)
Ref. 117 Challenge 280 n
**
e gravitational acceleration for a particle inside a spherical shell is zero. e vanishing of gravity in this case is independent of the particle shape and its position, and independent of the thickness of the shell.Can you nd the argument using Figure 83? is works only because of the r dependence of gravity. Can you show that the result does not hold for non-spherical shells? Note that the vanishing of gravity inside a spherical shell
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usually does not hold if other matter is found outside the shell. How could one eliminate Challenge 281 n the e ects of outside matter?
Ref. 118 Ref. 119
**
For a long time, it was thought that there is no additional planet in our solar system outside Neptune and Pluto, because their orbits show no disturbances from another body. Today, the view has changed. It is known that there are only eight planets: Pluto is not a planet, but the rst of a set of smaller objects beyond them, in the so-called Kuiper belt and Oort cloud. (Astronomers have also agreed to continue to call Pluto a ‘planet’ despite this evidence, to avoid debates.) Kuiper belt objects are regularly discovered. In 2003, an object, called Sedna, was found that is almost as large as Pluto but three times farther from the Sun.
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Ref. 120
**
In astronomy new examples of motion are regularly discovered even in the present century. Sometimes there are also false alarms. One example was the alleged fall of mini comets on the Earth. ey were supposedly made of a few dozen kilograms of ice and hitting the Earth every few seconds. It is now known not to happen. On the other hand, it is known that many tons of asteroids fall on the Earth every day, in the form of tiny particles. Incidentally, discovering objects hitting the Earth is not at all easy. Astronomers like to point out that an asteroid as large as the one that led to the extinction of the dinosaurs could hit the Earth without any astronomer noticing in advance, if the direction is slightly unusual, such as from the south, where few telescopes are located.
Challenge 282 n
**
Universal gravity allows only elliptical, parabolic or hyperbolic orbits. It is impossible for a small object approaching a large one to be captured. At least, that is what we have learned so far. Nevertheless, all astronomy books tell stories of capture in our solar system; for example, several outer satellites of Saturn have been captured. How is this possible?
Challenge 283 n
**
How would a tunnel have to be shaped in order that a stone would fall through it without touching the walls? (Assume constant density.) If the Earth did not rotate, the tunnel would be a straight line through its centre, and the stone would fall down and up again, in a oscillating motion. For a rotating Earth, the problem is much more di cult. What is the shape when the tunnel starts at the Equator?
Challenge 284 e
**
e International Space Station circles the Earth every 90 minutes at an altitude of about km. You can see where it is from the website http://www.heavens-above.com. By the
way, whenever it is just above the horizon, the station is the third brightest object in the night sky, superseded only by the Moon and Venus. Have a look at it.
**
Is it true that the centre of mass of the solar system, its barycentre, is always inside the Challenge 285 n Sun? Even though a star or the Sun move very little when planets move around them,
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this motion can be detected with precision measurements making use of the Doppler e ect for light or radio waves. Jupiter, for example, produces a speed change of m s in the Sun, the Earth m s. e rst planets outside the solar system, around the pulsar PSR1257+12 and the star Pegasi 51, was discovered in this way, in 1992 and 1995. In the meantime, over 150 planets have been discovered with this method. So far, the smallest planet discovered has 7 times the mass of the Earth.
**
Not all points on the Earth receive the same number of daylight hours during a year. e Challenge 286 d e ects are di cult to spot, though. Can you nd one?
**
Can the phase of the Moon have a measurable e ect on the human body? What about Challenge 287 n the tidal e ects of the Moon?
Challenge 288 n
**
ere is an important di erence between the heliocentric system and the old idea that all planets turn around the Earth. e heliocentric system states that certain planets, such as Mars and Venus, can be between the Earth and the Sun at certain times, and behind the Sun at other times. In contrast, the geocentric system states that they are always in between. Why did such an important di erence not immediately invalidate the geocentric system?
Ref. 121
**
e strangest reformulation of the description of motion given by ma = ∇U is the almost
absurd looking equation
∇v = dv ds
(47)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
where s is the motion path length. It is called the ray form of Newton’s equation of motion. Challenge 289 n Can you nd an example of its application?
**
Seen from Neptune, the size of the Sun is the same as that of Jupiter seen from the Earth Challenge 290 n at the time of its closest approach. True?
**
What is gravity? is is not a simple question. In 1690, Nicolas Fatio de Duillier and in Ref. 122 1747, Georges-Louis Lesage proposed an explanation for the r dependence. Lesage ar-
gued that the world is full of small particles – he called them ‘corpuscules ultra-mondains’ – ying around randomly and hitting all objects. Single objects do not feel the hits, since they are hit continuously and randomly from all directions. But when two objects are near to each other, they produce shadows for part of the ux to the other body, resulting Challenge 291 ny in an attraction. Can you show that such an attraction has a r dependence?
However, Lesage’s proposal has a number of problems. e argument only works if the collisions are inelastic. (Why?) However, that would mean that all bodies would heat up Ref. 3 with time, as Jean-Marc Lévy-Leblond explains.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
ere are two additional problems with the idea of Lesage. First, a moving body in free space would be hit by more or faster particles in the front than in the back; as a result, the body should be decelerated. Second, gravity would depend on size, but in a strange way. In particular, three bodies lying on a line should not produce shadows, as no such shadows are observed; but the naive model predicts such shadows.
Despite all the criticisms, this famous idea has regularly resurfaced in physics ever since, even though such particles have never been found. Only in the third part of our mountain ascent will we settle the issue.
**
Challenge 292 ny For which bodies does gravity decrease as you approach them?
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**
Could one put a satellite into orbit using a cannon? Does the answer depend on the dirChallenge 293 ny ection in which one shoots?
**
Two computer users share experiences. ‘I threw my Pentium III and Pentium IV out of the window.’ ‘And?’ ‘ e Pentium III was faster.’
**
Challenge 294 n How o en does the Earth rise and fall when seen from the Moon? Does the Earth show phases?
** Challenge 295 ny What is the weight of the Moon? How does it compare with the weight of the Alps?
Challenge 296 n
**
Owing to the slightly attened shape of the Earth, the source of the Mississippi is about km nearer to the centre of the Earth than its mouth; the water e ectively runs uphill.
How can this be?
**
If a star is made of high density material, the speed of a planet orbiting near to it could Challenge 297 n be greater than the speed of light. How does nature avoid this strange possibility?
Ref. 123 Page 259
**
What will happen to the solar system in the future? is question is surprisingly hard to answer. e main expert of this topic, French planetary scientist Jacques Laskar, simulated a few hundred million years of evolution using computer-aided calculus. He found that the planetary orbits are stable, but that there is clear evidence of chaos in the evolution of the solar system, at a small level. e various planets in uence each other in subtle and still poorly understood ways. E ects in the past are also being studied, such as the energy change of Jupiter due to its ejection of smaller asteroids from the solar system, or energy gains of Neptune. ere is still a lot of research to be done in this eld.
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TA B L E 18 An unexplained property of nature: planet distances and the values resulting from the Titius–Bode rule
P
n
AU
Mercury −
0.4
Venus
0
0.7
Earth
1
1.0
Mars
2
1.6
Planetoids 3
2.8
Jupiter
4
5.2
Saturn
5
10.0
Uranus
6
19.6
Neptune
7
38.8
Pluto
8
77.2
0.4 0.7 1.0 1.5 2.2 to 3.2 5.2 9.5 19.2 30.1 39.5
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
**
One of the open problems of the solar system is the description of planet distances dis-
covered in 1766 by Johann Daniel Titius (1729–1796) and publicized by Johann Elert
Bode (1747–1826). Titius discovered that planetary distances d from the Sun can be ap-
proximated by
d = a + n b with a = . AU , b = . AU
(48)
Ref. 124 Ref. 125
where distances are measured in astronomical units and n is the number of the planet. e resulting approximation is compared with observations in Table 18. Interestingly, the last three planets, as well as the planetoids, were discovered a er
Bode’s and Titius’ deaths; the rule had successfully predicted Uranus’ distance, as well as that of the planetoids. Despite these successes – and the failure for the last two planets – nobody has yet found a model for the formation of the planets that explains Titius’ rule.
e large satellites of Jupiter and of Uranus have regular spacing, but not according to the Titius–Bode rule.
Explaining or disproving the rule is one of the challenges that remains in classical mechanics. Some researchers maintain that the rule is a consequence of scale invariance, others maintain that it is a accident or even a red herring. e last interpretation is also suggested by the non-Titius–Bode behaviour of practically all extrasolar planets. e issue is not closed.
**
Around 3000 years ago, the Babylonians had measured the orbital times of the seven celestial bodies. Ordered from longest to shortest, they wrote them down in Table 19.
e Babylonians also introduced the week and the division of the day into 24 hours. e Babylonians dedicated every one of the 168 hours of the week to a celestial body, following the order of the Table. ey also dedicated the whole day to that celestial body
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TA B L E 19 The orbital periods known to the Babylonians
B
P
Saturn
a
Jupiter
a
Mars
d
Sun
d
Venus
d
Mercury
d
Moon
d
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 84 A solar eclipse (11 August 1999, photographed from the Russian Mir station)
Challenge 298 e Ref. 126
that corresponds to the rst hour of that day. e rst day of the week was dedicated to Saturn; the present ordering of the other days of the week then follows from Table 19. is story was told by Cassius Dio (c. 160 to c. 230). Towards the end of Antiquity, the ordering was taken up by the Roman empire. In Germanic languages, including English, the Latin names of the celestial bodies were replaced by the corresponding Germanic gods.
e order Saturday, Sunday, Monday, Tuesday, Wednesday, ursday and Friday is thus a consequence of both the astronomical measurements and the astrological superstitions of the ancients.
**
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 299 n Challenge 300 n
In 1722, the great mathematician Leonhard Euler made a mistake in his calculation that led him to conclude that if a tunnel were built from one pole of the Earth to the other, a stone falling into it would arrive at the Earth’s centre and then immediately turn and go back up. Voltaire made fun of this conclusion for many years. Can you correct Euler and show that the real motion is an oscillation from one pole to the other, and can you calculate the time a pole-to-pole fall would take (assuming homogeneous density)?
What would be the oscillation time for an arbitrary straight surface-to-surface tunnel of length l, thus not going from pole to pole?
Challenge 301 n
**
Figure 84 shows a photograph of the 1999 solar eclipse taken by the Russian space station Mir. It clearly shows that a global view of a phenomenon can be quite di erent from a local one. What is the speed of the shadow?
Ref. 127
**
In 2005, satellite measurements have shown that the water in the Amazon river presses down the land up to mm more in the season when it is full of water than in the season when it is almost empty.
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W
?
All types of motion that can be described when the mass of a body is its only permanent property form what is called mechanics. e same name is also given to the experts studying the eld. We can think of mechanics as the athletic part of physics;* both in athletics and in mechanics only lengths, times and masses are measured.
More speci cally, our topic of investigation so far is called classical mechanics, to distinguish it from quantum mechanics. e main di erence is that in classical physics arbitrary small values are assumed to exist, whereas this is not the case in quantum physics.
e use of real numbers for observable quantities is thus central to classical physics. Classical mechanics is o en also called Galilean physics or Newtonian physics. e basis of classical mechanics, the description of motion using only space and time, is called kinematics. An example is the description of free fall by z(t) = z + v (t − t ) − (t − t ) . e other, main part of classical mechanics is the description of motion as a consequence of interactions between bodies; it is called dynamics. An example of dynamics is the formula of universal gravity.
e distinction between kinematics and dynamics can also be made in relativity, thermodynamics and electrodynamics. Even though we have not explored these elds of enquiry yet, we know that there is more to the world than gravity. A simple observation makes the point: friction. Friction cannot be due to gravity, because friction is not observed in the skies, where motion follows gravity rules only.** Moreover, on Earth, fric-
Page 402
* is is in contrast to the actual origin of the term ‘mechanics’, which means ‘machine science’. It derives from the Greek µηκανή, which means ‘machine’ and even lies at the origin of the English word ‘machine’ itself. Sometimes the term ‘mechanics’ is used for the study of motion of solid bodies only, excluding, e.g., hydrodynamics. is use fell out of favour in physics in the last century. ** is is not completely correct: in the 1980s, the rst case of gravitational friction was discovered: the emission of gravity waves. We discuss it in detail later.
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TA B L E 20 Some measured force values
O
F
Value measured in a magnetic resonance force microscope zN
Maximum force exerted by teeth
. kN
Typical force exerted by sledgehammer
kN
Force exerted by quadricpes
up to kN
Force sustained by cm of a good adhesive
up to kN
Force needed to tear a good rope used in rock climbing
kN
Maximum force measurable in nature
.ë N
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Challenge 302 e tion is independent of gravity, as you might want to check. ere must be another interaction responsible for friction. We shall study it shortly. But one issue merits a discussion right away.
S
?
“ e direct use of force is such a poor solution to any problem, it is generally employed only by small children and large nations. ” David Friedman
Everybody has to take a stand on this question, even students of physics. Indeed, many types of forces are used and observed in daily life. One speaks of muscular, gravitational, psychic, sexual, satanic, supernatural, social, political, economic and many others. Physicists see things in a simpler way. ey call the di erent types of forces observed between objects interactions. e study of the details of all these interactions will show that, in everyday life, they are of electrical origin.
For physicists, all change is due to motion. e term force then also takes on a more restrictive de nition. (Physical) force is de ned as the change of momentum, i.e. as
F=
dp dt
.
(49)
Force is the change or ow of motion. If a force acts on a body, momentum ows into
it. Indeed, momentum can be imagined to be some invisible and intangible liquid. Force
measures how much of this liquid ows from one body to another per unit time.
Using the Galilean de nition of linear momentum p = mv, we can rewrite the de ni-
tion of force (for constant mass) as
F = ma ,
(50)
where F = F(t, x) is the force acting on an object of mass m and where a = a(t, x) = dv dt = d x dt is the acceleration of the same object, that is to say its change of velocity.*
* is equation was rst written down by the Swiss mathematician and physicist Leonhard Euler (1707– 1783) in 1747, over 70 years a er Newton’s rst law and 20 years a er the death of Newton, to whom it is
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e expression states in precise terms that force is what changes the velocity of masses. e
quantity is called ‘force’ because it corresponds in many, but not all aspects to muscular
force. For example, the more force is used, the further a stone can be thrown.
However, whenever the concept of force is used, it should be remembered that physical
force is di erent from everyday force or everyday e ort. E ort is probably best approxim-
ated by the concept of (physical) power, usually abbreviated P, and de ned (for constant
force) as
P=
dW dt
=Fëv
(51)
Challenge 303 n Challenge 304 n Challenge 305 d
Ref. 128
in which (physical) work W is de ned as W = F ë s. Physical work is a form of energy, as you might want to check. Note that a man who walks carrying a heavy rucksack is hardly doing any work; why then does he get tired? Work, as a form of energy, has to be taken into account when the conservation of energy is checked.
With the de nition of work just given you can solve the following puzzles. What happens to the electricity consumption of an escalator if you walk on it instead of standing still? What is the e ect of the de nition of power for the salary of scientists?
When students in exams say that the force acting on a thrown stone is least at the highest point of the trajectory, it is customary to say that they are using an incorrect view, namely the so-called Aristotelian view, in which force is proportional to velocity. Sometimes it is even said that they are using a di erent concept of state of motion. Critics then add, with a tone of superiority, how wrong all this is. is is a typical example of intellectual disinformation. Every student knows from riding a bicycle, from throwing a stone or from pulling an object that increased e ort results in increased speed. e student is right; those theoreticians who deduce that the student has a mistaken concept of force are wrong. In fact, instead of the physical concept of force, the student is just using the everyday version, namely e ort. Indeed, the e ort exerted by gravity on a ying stone is least at the highest point of the trajectory. Understanding the di erence between physical force and everyday e ort is the main hurdle in learning mechanics.*
O en the ow of momentum, equation ( ), is not recognized as the de nition of force. is is mainly due to an everyday observation: there seem to be forces without any associated acceleration or change in momentum, such as in a string under tension or in water at high pressure. When one pushes against a tree, there is no motion, yet a force is applied. If force is momentum ow, where does the momentum go? It ows into the slight deformations of the arms and the tree. In fact, when one starts pushing and thus deforming, the associated momentum change of the molecules, the atoms, or the electrons of the two bodies can be observed. A er the deformation is established, and looking at even higher magni cation, one can indeed nd that a continuous and equal
ow of momentum is going on in both directions. e nature of this ow will be clari ed
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Ref. 24
usually and falsely ascribed; it was Euler, not Newton, who rst understood that this de nition of force is useful in every case of motion, whatever the appearance, be it for point particles or extended objects, and be it rigid, deformable or uid bodies. Surprisingly and in contrast to frequently-made statements, equation (50) is even correct in relativity, as shown on page 323. * is stepping stone is so high that many professional physicists do not really take it themselves; this is con rmed by the innumerable comments in papers that state that physical force is de ned using mass, and, at the same time, that mass is de ned using force (the latter part of the sentence being a fundamental mistake).
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 307 e
Challenge 308 n Ref. 129
in the part on quantum theory. As force is net momentum ow, it is only needed as a separate concept in everyday life,
where it is useful in situations where net momentum ows are less than the total ows. At the microscopic level, momentum alone su ces for the description of motion. For example, the concept of weight describes the ow of momentum due to gravity. us we will hardly ever use the term ‘weight’ in the microscopic part of our adventure.
rough its de nition, the concept of force is distinguished clearly from ‘mass’, ‘momentum’, ‘energy’ and ‘power’. But where do forces originate? In other words, which effects in nature have the capacity to accelerate bodies by pumping momentum into objects? Table gives an overview.
Every example of motion, from the one that lets us choose the direction of our gaze to the one that carries a butter y through the landscape, can be put into one of the two le -most columns of Table . Physically, the two columns are separated by the following criterion: in the rst class, the acceleration of a body can be in a di erent direction from its velocity. e second class of examples produces only accelerations that are exactly opposed to the velocity of the moving body, as seen from the frame of reference of the braking medium. Such a resisting force is called friction, drag or a damping. All examples in the second class are types of friction. Just check.
Friction can be so strong that all motion of a body against its environment is made impossible. is type of friction, called static friction or sticking friction, is common and important: without it, turning the wheels of bicycles, trains or cars would have no e ect. Not a single screw would stay tightened. We could neither run nor walk in a forest, as the soil would be more slippery than polished ice. In fact not only our own motion, but all voluntary motion of living beings is based on friction. e same is the case for selfmoving machines. Without static friction, the propellers in ships, aeroplanes and helicopters would not have any e ect and the wings of aeroplanes would produce no li to keep them in the air. (Why?) In short, static friction is required whenever we want to move relative to our environment.
Once an object moves through its environment, it is hindered by another type of friction; it is called dynamic friction and acts between bodies in relative motion. Without it, falling bodies would always rebound to the same height, without ever coming to a stop; neither parachutes nor brakes would work; worse, we would have no memory, as we will see later.*
As the motion examples in the second column of Table include friction, in those examples macroscopic energy is not conserved: the systems are dissipative. In the rst column, macroscopic energy is constant: the systems are conservative.
e rst two columns can also be distinguished using a more abstract, mathematical criterion: on the le are accelerations that can be derived from a potential, on the right, decelerations that can not. As in the case of gravitation, the description of any kind of motion is much simpli ed by the use of a potential: at every position in space, one needs only the single value of the potential to calculate the trajectory of an object, instead of the three values of the acceleration or the force. Moreover, the magnitude of the velocity of
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* Recent research suggest that maybe in certain crystalline systems, such as tungsten bodies on silicon, under ideal conditions gliding friction can be extremely small and possibly even vanish in certain directions of Ref. 130 motion. is so-called superlubrication is presently a topic of research.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•.
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TA B L E 21 Selected processes and devices changing the motion of bodies
S
S
M
-
-
piezoelectricity quartz under applied voltage
gravitation falling
collisions satellite in planet encounter growth of mountains
magnetic e ects compass needle near magnet magnetostriction current in wire near magnet
electric e ects rubbed comb near hair bombs television tube
light levitating objects by light solar sail for satellites
elasticity bow and arrow bent trees standing up again
osmosis water rising in trees electro-osmosis
heat & pressure freezing champagne bottle tea kettle barometer earthquakes attraction of passing trains
nuclei radioactivity
biology bamboo growth
thermoluminescence
walking piezo tripod
emission of gravity waves pulley
car crash meteorite crash
rocket motor swimming of larvae
electromagnetic braking transformer losses electric heating
electromagnetic gun linear motor galvanometer
friction between solids re
electron microscope
electrostatic motor muscles, sperm agella Brownian motor
light bath stopping atoms (true) light mill light pressure inside stars solar cell
trouser suspenders pillow, air bag
ultrasound motor bimorphs
salt conservation of food
osmotic pendulum tunable X-ray screening
surfboard water resistance quicksand parachute sliding resistance shock absorbers
hydraulic engines steam engine air gun, sail seismometer water turbine
plunging into the Sun
supernova explosion
nd example! Challenge 306 ny molecular motors
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an object at any point can be calculated directly from energy conservation.
e processes from the second column cannot
be described by a potential. ese are the cases where we necessarily have to use force if we want
ideal shape, cw = 0.0168
to describe the motion of the system. For example,
the force F due to wind resistance of a body is
roughly given by
F = cwρAv
typical passenger aeroplane, cw = 0.03
(52)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 309 ny
Ref. 132 Challenge 311 ny
where A is the area of its cross-section and v its
velocity relative to the air, ρ is the density of air;
the drag coe cient cw is a pure number that depends on the shape of the moving object. (A few ex-
typical sports car, cw = 0.44
amples are given in Figure .) You may check that
aerodynamic resistance cannot be derived from a
potential.*
dolphin
e drag coe cient cw is found experimentally to be always larger than . , which corres-
ponds to the optimally streamlined tear shape. An
aerodynamic car has a value of . to . ; but
F I G U R E 85 Shapes and air/water resistance
many sports cars share with vans values of .
and higher.**
Wind resistance is also of importance to humans, in particular in athletics. It is estim-
ated that m sprinters spend between % and % of their power overcoming drag.
is leads to varying sprint times tw when wind of speed w is involved, related by the expression
t tw
=
.
−.
− wtw ,
(53)
Challenge 312 ny Challenge 313 n
where the more conservative estimate of % is used. An opposing wind speed of − m s gives an increase in time of . s, enough to change a potential world record into an ‘only’ excellent result. (Are you able to deduce the cw value for running humans from the formula?)
Likewise, parachuting exists due to wind resistance. Can you determine how the speed of a falling body changes with time, assuming constant shape and drag coe cient?
Challenge 310 n Ref. 131
* Such a statement about friction is correct only in three dimensions, as is the case in nature; in the case of a single dimension, a potential can always be found. ** Calculating drag coe cients in computers, given the shape of the body and the properties of the uid, is one of the most di cult tasks of science; the problem is still not fully solved.
e topic of aerodynamic shapes is even more interesting for uid bodies. ey are kept together by surface tension. For example, surface tension keeps the hairs of a wet brush together. Surface tension also determines the shape of rain drops. Experiments show that it is spherical for drops smaller than mm diameter, and that larger rain drops are lens shaped, with the at part towards the bottom. e usual tear shape is not encountered in nature; something vaguely similar to it appears during drop detachment, but never during drop fall.
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Ref. 133
In contrast, static friction has di erent properties. It is proportional to the force press-
ing the two bodies together. Why? Studying the situation in more detail, sticking friction
is found to be proportional to the actual contact area. It turns out that putting two solids
into contact is rather like turning Switzerland upside down and putting it onto Austria;
the area of contact is much smaller than that estimated macroscopically. e important
point is that the area of actual contact is proportional to the normal force. e study of
what happens in that contact area is still a topic of research; researchers are investigating
the issues using instruments such as atomic force microscopes, lateral force microscopes
and triboscopes. ese e orts resulted in computer hard discs which last longer, as the
friction between disc and the reading head is a central quantity in determining the life-
time.
All forms of friction are accompanied by an increase in the temperature of the mov-
ing body. e reason became clear a er the discovery of atoms. Friction is not observed
in few – e.g. , , or – particle systems. Friction only appears in systems with many
particles, usually millions or more. Such systems are called dissipative. Both the tem-
perature changes and friction itself are due to the motion of large numbers of micro-
scopic particles against each other. is motion is not included in the Galilean descrip-
tion. When it is included, friction and energy loss disappear, and potentials can then be
used throughout. Positive accelerations – of microscopic magnitude – then also appear,
and motion is found to be conserved. As a result, all motion is conservative on a micro-
scopic scale. erefore, on a microscopic scale it is possible to describe all motion without
the concept of force.* e moral of the story is that one should use force only in one situ-
ation: in the case of friction, and only when one does not want to go into the microscopic
details.**
“Et qu’avons-nous besoin de ce moteur, quand l’étude ré échie de la nature nous prouve que le mouvement perpétuel est la première de ses
lois ?***
Donatien de Sade Justine, ou les malheurs de la
”vertu.
C
–
Ref. 134 Ref. 56
*
Quid sit futurum cras, fuge quaerere ...**** Horace, Odi, lib. I, ode , v. .
“ ” e rst scientist who eliminated force from the description of nature was Heinrich Rudolf Hertz (b. 1857
Hamburg, d. 1894 Bonn), the famous discoverer of electromagnetic waves, in his textbook on mechanics,
Die Prinzipien der Mechanik, Barth, 1894, republished by Wissenscha liche Buchgesellscha , Darmstadt,
1963. His idea was strongly criticized at that time; only a generation later, when quantum mechanics quietly
got rid of the concept for good, did the idea become commonly accepted. (Many have speculated about
the role Hertz would have played in the development of quantum mechanics and general relativity, had he
not died so young.) In his book, Hertz also formulated the principle of the straightest path: particles follow
geodesics. is same description is one of the pillars of general relativity, as we will see later on.
** In the case of human relations the evaluation should be somewhat more discerning, as the research by
James Gilligan shows.
*** ‘And whatfor do we need this motor, when the reasoned study of nature proves to us that perpetual
motion is the rst of its laws?’
**** ‘What future will be tomorrow, never ask ...’ Horace is Quintus Horatius Flaccus (65–8 ), the great
Roman poet.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
We o en describe the motion of a body by specifying the time dependence of its position, for example as
x(t) = x + v (t − t ) + a (t − t ) + j (t − t ) + ... .
(54)
Page 39
Page 71 Challenge 314 ny Challenge 315 n Challenge 316 n Challenge 317 n Challenge 318 n
e quantities with an index , such as the starting position x , the starting velocity v , etc., are called initial conditions. Initial conditions are necessary for any description of motion. Di erent physical systems have di erent initial conditions. Initial conditions thus specify the individuality of a given system. Initial conditions also allow us to distinguish the present situation of a system from that at any previous time: initial conditions specify the changing aspects of a system. In other words, they summarize the past of a system.
Initial conditions are thus precisely the properties we have been seeking for a description of the state of a system. To nd a complete description of states we thus need only a complete description of initial conditions. It turns out that for gravitation, as for all other microscopic interactions, there is no need for initial acceleration a , initial jerk j , or higher-order initial quantities. In nature, acceleration and jerk depend only on the properties of objects and their environment; they do not depend on the past. For example, the expression a = GM r , giving the acceleration of a small body near a large one, does not depend on the past, but only on the environment. e same happens for the other fundamental interactions, as we will nd out shortly.
e complete state of a moving mass point is thus described by specifying its position and its momentum at all instants of time. us we have achieved a complete description of the intrinsic properties of point objects, namely by their mass, and of their states of motion, namely by their momentum, energy, position and time. For extended rigid objects we also need orientation, angular velocity and angular momentum. Can you specify the necessary quantities in the case of extended elastic bodies or uids?
e set of all possible states of a system is given a special name: it is called the phase space. We will use the concept repeatedly. Like any space, it has a number of dimensions. Can you specify it for a system consisting of N point particles?
However, there are situations in nature where the motion of an object depends on characteristics other than its mass; motion can depend on its colour (can you nd an example?), on its temperature, and on a few other properties that we will soon discover. Can you give an example of an intrinsic property that we have so far missed? And for each intrinsic property there are state variables to discover. ese new properties are the basis of the eld of physical enquiry beyond mechanics. We must therefore conclude that as yet we do not have a complete description of motion.
It is interesting to recall an older challenge and ask again: does the universe have initial conditions? Does it have a phase space? As a hint, recall that when a stone is thrown, the initial conditions summarize the e ects of the thrower, his history, the way he got there etc.; in other words, initial conditions summarize the e ects that the environment had during the history of a system.
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An optimist is somebody who thinks that the
“ ” future is uncertain.
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•.
–
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
D
?I
?
Challenge 319 e
“Die Ereignisse der Zukun können wir nicht aus den gegenwärtigen erschließen. Der Glaube an den Kausalnexus ist ein Aberglaube.*
Ludwig Wittgenstein, Tractatus, .
“Freedom is the recognition of necessity. Friedrich Engels ( –
” ”)
If, a er climbing a tree, we jump down, we cannot halt the jump in the middle of the
trajectory; once the jump has begun, it is unavoidable and determined, like all passive
motion. However, when we begin to move an arm, we can stop or change its motion
from a hit to a caress. Voluntary motion does not seem unavoidable or predetermined.
Which of these two cases is the general one?
Let us start with the example that we can describe most precisely so far: the fall of a
body. Once the potential φ acting on a particle is given and taken into account, using
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a(x) = −∇φ = −GMr r ,
(55)
and the state at a given time is given by initial conditions such as
x(t ) = x and v(t ) = v ,
(56)
we then can determine the motion in advance. e complete trajectory x(t) can be calculated with these two pieces of information. Owing to this possibility, an equation such as ( ) is called an evolution equation for the motion of the object. (Note that the term ‘evolution’ has di erent meanings in physics and in biology.) An evolution equation always expresses the observation that not all types of change are observed in nature, but only certain speci c cases. Not all imaginable sequences of events are observed, but only a limited number of them. In particular, equation ( ) expresses that from one instant to the next, objects change their motion based on the potential acting on them. us, given an evolution equation and initial state, the whole motion of a system is uniquely xed; this property of motion is o en called determinism. Since this term is o en used with di erent meanings, let us distinguish it carefully from several similar concepts, to avoid misunderstandings.
Motion can be deterministic and at the same time still be unpredictable. e latter property can have four origins: an impracticably large number of particles involved, the complexity of the evolution equations, insu cient information about initial conditions, and strange shapes of space-time. e weather is an example where the rst three conditions are ful lled at the same time.** Nevertheless, its motion is still deterministic. Near black holes all four cases apply together. We will discuss black holes in the section on general relativity. Nevertheless, near black holes, motion is still deterministic.
Motion can be both deterministic and time random, i.e. with di erent outcomes in
* We cannot infer the events of the future from those of the present. Superstition is nothing but belief in the causal nexus. ** For a beautiful view of clouds, see the http://www.goes.noass.gov website.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 320 n Challenge 321 n
similar experiments. A roulette ball’s motion is deterministic, but it is also random.* As we will see later, quantum-mechanical situations fall into this category, as do all examples of irreversible motion, such as a drop of ink spreading out in clear water. In all such cases the randomness and the irreproducibility are only apparent; they disappear when the description of states and initial conditions in the microscopic domain are included. In short, determinism does not contradict (macroscopic) irreversibility. However, on the microscopic scale, deterministic motion is always reversible.
A nal concept to be distinguished from determinism is acausality. Causality is the requirement that a cause must precede the e ect. is is trivial in Galilean physics, but becomes of importance in special relativity, where causality implies that the speed of light is a limit for the spreading of e ects. Indeed, it seems impossible to have deterministic motion (of matter and energy) which is acausal, i.e. faster than light. Can you con rm this? is topic will be looked at more deeply in the section on special relativity.
Saying that motion is ‘deterministic’ means that it is xed in the future and also in the past. It is sometimes stated that predictions of future observations are the crucial test for a successful description of nature. Owing to our o en impressive ability to in uence the future, this is not necessarily a good test. Any theory must, rst of all, describe past observations correctly. It is our lack of freedom to change the past that results in our lack of choice in the description of nature that is so central to physics. In this sense, the term ‘initial condition’ is an unfortunate choice, because it automatically leads us to search for the initial condition of the universe and to look there for answers to questions that can be answered without that knowledge. e central ingredient of a deterministic description is that all motion can be reduced to an evolution equation plus one speci c state. is state can be either initial, intermediate, or nal. Deterministic motion is uniquely speci ed into the past and into the future.
To get a clear concept of determinism, it is useful to remind ourselves why the concept of ‘time’ is introduced in our description of the world. We introduce time because we observe rst that we are able to de ne sequences in observations, and second, that unrestricted change is impossible. is is in contrast to lms, where one person can walk through a door and exit into another continent or another century. In nature we do not observe metamorphoses, such as people changing into toasters or dogs into toothbrushes. We are able to introduce ‘time’ only because the sequential changes we observe are extremely restricted. If nature were not reproducible, time could not be used. In short, determinism expresses the observation that sequential changes are restricted to a single possibility.
Since determinism is connected to the use of the concept of time, new questions arise whenever the concept of time changes, as happens in special relativity, in general relativity and in theoretical high energy physics. ere is a lot of fun ahead.
In summary, every description of nature that uses the concept of time, such as that of everyday life, that of classical physics and that of quantum mechanics, is intrinsically and inescapably deterministic, since it connects observations of the past and the future, eliminating alternatives. In short, the use of time implies determinism, and vice versa. When
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Ref. 135
* Mathematicians have developed a large number of tests to determine whether a collection of numbers may be called random; roulette results pass all these tests – in honest casinos only, however. Such tests typically check the equal distribution of numbers, of pairs of numbers, of triples of numbers, etc. Other tests are the χ test, the Monte Carlo test(s), and the gorilla test.
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Page 831
Challenge 322 n Ref. 136
drawing metaphysical conclusions, as is so popular nowadays when discussing quantum theory, one should never forget this connection. Whoever uses clocks but denies determinism is nurturing a split personality!*
e idea that motion is determined o en produces fear, because we are taught to associate determinism with lack of freedom. On the other hand, we do experience freedom in our actions and call it free will. We know that it is necessary for our creativity and for our happiness. erefore it seems that determinism is opposed to happiness.
But what precisely is free will? Much ink has been consumed trying to nd a precise de nition. One can try to de ne free will as the arbitrariness of the choice of initial conditions. However, initial conditions must themselves result from the evolution equations, so that there is in fact no freedom in their choice. One can try to de ne free will from the idea of unpredictability, or from similar properties, such as uncomputability. But these de nitions face the same simple problem: whatever the de nition, there is no way to prove experimentally that an action was performed freely. e possible de nitions are useless. In short, free will cannot be observed. (Psychologists also have a lot of their own data to support this, but that is another topic.)
No process that is gradual – in contrast to sudden – can be due to free will; gradual processes are described by time and are deterministic. In this sense, the question about free will becomes one about the existence of sudden changes in nature. is will be a recurring topic in the rest of this walk. Does nature have the ability to surprise? In everyday life, nature does not. Sudden changes are not observed. Of course, we still have to investigate this question in other domains, in the very small and in the very large. Indeed, we will change our opinion several times. e lack of surprises in everyday life is built deep into our body: the concept of curiosity is based on the idea that everything discovered is useful a erwards. If nature continually surprised us, curiosity would make no sense.
Another observation contradicts the existence of surprises: in the beginning of our walk we de ned time using the continuity of motion; later on we expressed this by saying that time is a consequence of the conservation of energy. Conservation is the opposite of surprise. By the way, a challenge remains: can you show that time would not be de nable even if surprises existed only rarely?
In summary, so far we have no evidence that surprises exist in nature. Time exists because nature is deterministic. Free will cannot be de ned with the precision required by physics. Given that there are no sudden changes, there is only one consistent de nition of free will: it is a feeling, in particular of independence of others, of independence from fear and of accepting the consequences of one’s actions. Free will is a feeling of satisfaction.
is solves the apparent paradox; free will, being a feeling, exists as a human experience, even though all objects move without any possibility of choice. ere is no contradiction.**
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Challenge 323 e
* at can be a lot of fun though. ** at free will is a feeling can also be con rmed by careful introspection. e idea of free will always appears a er an action has been started. It is a beautiful experiment to sit down in a quiet environment, with the intention to make, within an unspeci ed number of minutes, a small gesture, such as closing a hand. If you carefully observe, in all detail, what happens inside yourself around the very moment of decision, you
nd either a mechanism that led to the decision, or a di use, unclear mist. You never nd free will. Such an experiment is a beautiful way to experience deeply the wonders of the self. Experiences of this kind might also be one of the origins of human spirituality, as they show the connection everybody has with the rest of
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Ref. 137 Challenge 324 e
Even if human action is determined, it is still authentic. So why is determinism so frightening? at is a question everybody has to ask themself. What di erence does determinism imply for your life, for the actions, the choices, the responsibilities and the pleasures you encounter?* If you conclude that being determined is di erent from being free, you should change your life! Fear of determinism usually stems from refusal to take the world the way it is. Paradoxically, it is precisely he who insists on the existence of free will who is running away from responsibility.
You do have the ability to surprise yourself.
“ ” Richard Bandler and John Grinder
A
Challenge 326 n Challenge 327 n
Darum kann es in der Logik auch nie Überraschungen geben.**
“ Ludwig Wittgenstein, Tractatus, . ” Classical mechanics describes nature in a rather simple way. Objects are permanent and
massive entities localized in space-time. States are changing properties of objects, described by position in space and instant in time, by energy and momentum, and by their rotational equivalents. Time is the relation between events measured by a clock. Clocks are devices in undisturbed motion whose position can be observed. Space and position is the relation between objects measured by a metre stick. Metre sticks are devices whose shape is subdivided by some marks, xed in an invariant and observable manner. Motion is change of position with time (times mass); it is determined, does not show surprises, is conserved (even in death), and is due to gravitation and other interactions.
Even though this description works rather well, it contains a circular de nition. Can you spot it? Each of the two central concepts of motion is de ned with the help of the other. Physicists worked for about years on classical mechanics without noticing or wanting to notice the situation. Even thinkers with an interest in discrediting science did not point it out. Can an exact science be based on a circular de nition? Obviously, physics has done quite well so far. Some even say the situation is unavoidable in principle. Despite these opinions, undoing this logical loop is one of the aims of the rest of our walk. To achieve it, we need to increase substantially the level of precision in our description of motion.
Whenever precision is increased, imagination is restricted. We will discover that many types of motion that seem possible are not. Motion is limited. Nature limits speed, size, acceleration, mass, force, power and many other quantities. Continue reading only if you are prepared to exchange fantasy for precision. It will be no loss, as you will gain something else: the workings of nature will fascinate you.
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Challenge 325 n
nature. * If nature’s ‘laws’ are deterministic, are they in contrast with moral or ethical ‘laws’? Can people still be held responsible for their actions? ** Hence there can never be surprises in logic.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
B
Aiunt enim multum legendum esse, non multa.
“ ” Plinius, Epistulae.*
Page 678
1 For a history of science in antiquity, see L
R , La rivoluzione dimenticata, Fel-
trinelli , also available in several other languages. Cited on page .
2 An overview of motion illusions can be found on the excellent website http://wwww. mittelbach.de/ot. Cited on page .
3 A beautiful book explaining physics and its many applications in nature and technology
vividly and thoroughly is P G. H
,J S
&L
A. H
,
Conceptual Physical Science, Bejamin/Cummings, .
A book famous for its passion for curiosity is R
P. F
,R.B. L
& M. S , e Feynman Lectures on Physics, Addison Wesley, .
A lot can be learned about motion from quiz books. One of the best is the well-structured
collection of beautiful problems that require no mathematics, written by J -M
L -L
, La physique en questions – mécanique, Vuibert, .
Another excellent quiz collection is Y
P
, Oh, la physique, Dunod, ,
a translation from the Russian original.
A good problem book is W.G. R , Physics by Example: 200 Problems and Solutions,
Cambridge University Press, .
A good history of physical ideas is given in the excellent text by D
P , e How
and the Why, Princeton University Press, .
An excellent introduction into physics, is R
P , Pohl’s Einführung in die Physik,
Klaus Lüders & Robert O. Pohl editors, Springer, , in two volumes with CDs. It is a new
edition of a book that is over years old; but the didactic quality, in particular of the exper-
imental side of physics, is unsurpassed. Cited on pages , , , and .
4 A well-known principle in the social sciences states that, given a question, for every possible
answer, however weird it may seem, there is somebody – and o en a whole group – who
holds it as his opinion. One just has to go through literature (or the internet) to con rm
this.
About group behaviour in general, see R. A
, e Evolution of Cooperation,
Harper Collins, . e propagation and acceptance of ideas, such as those of physics,
are also an example of human cooperation, with all its potential dangers and weaknesses.
Cited on page .
5 All the known texts by Parmenides and Heraclitos can be found in J -P D
,
Les écoles présocratiques, Folio-Gallimard, . Views about the non-existence of motion
have also been put forward by much more modern and much more contemptible authors,
such as in by Berkeley. Cited on page .
6 An example of people worried by Zeno is given by W
ML
, Resolving
Zeno’s paradoxes, Scienti c American pp. – , November . e actual argument was
not about a hand slapping a face, but about an arrow hitting the target. See also Ref. .
Cited on page .
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* ‘Read much, but not anything.’ Ep. 7, 9, 15. Gaius Plinius Secundus (b. 23/4 Novum Comum, d. 79 Vesuvius eruption), Roman writer, especially famous for his large, mainly scienti c work Historia naturalis, which has been translated and read for almost 2000 years.
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7 e full text of La Beauté and the other poems from Les eurs du mal, one of the nest books of poetry ever written, can be found at the http://hypermedia.univ-paris .fr/bibliotheque/ Baudelaire/Spleen.html website. Cited on page .
8 e most famous text is J
W
, e Flying Circus of Physics, Wiley, . For
more interesting physical e ects in everyday life, see E
F
, Hundertfünfzig
Physikrätsel, Ernst Klett Verlag, . e book also covers several clock puzzles, in puzzle
numbers to . Cited on page .
9 A concise and informative introduction into the history of classical physics is given in the
rst chapter of the book by F.K. R
, E.H. K
& J.N. C
, Intro-
duction to Modern Physics, McGraw–Hill, . Cited on page .
10 A good overview over the arguments used to prove the existence of god from motion is given
by M
B
, Motion and Motion’s God, Princeton University Press, . e
intensity of the battles waged around these failed attempts is one of the tragicomic chapters
of history. Cited on page .
11 T
A
, Summa eologiae or Summa eologica, – , online in Latin
at http://www.newadvent.org/summa, in English on several other servers. Cited on page
.
12 For an exploration of ‘inner’ motions, see the beautiful text by R
S
, In-
ternal Family Systems erapy, e Guilford Press, . Cited on page .
13 See e.g. the fascinating text by D
G. C
, e Campaigns of Napoleon - e
Mind and Method of History’s Greatest Soldier, Macmillan, . Cited on page .
14 R
M
, American Roulette, St Martin’s Press,
Cited on page .
, a thriller and a true story.
15 A good and funny book on behaviour change is the well-known text by R. B
, Using
Your Brain for a Change, Real People Press, . See also R
B
&J
G
, Frogs into princes – Neuro Linguistic Programming, Eden Grove Editions, .
Cited on pages and .
16 A beautiful book about the mechanisms of human growth from the original cell to full size
is L
W
, e Triumph of the Embryo, Oxford University Press, . Cited on
page .
17 On the topic of grace and poise, see e.g. the numerous books on the Alexander technique,
such as M. G , Body Learning – An Introduction to the Alexander Technique, Aurum
Press, , and R
B
, Introduction to the Alexander Technique, Little
Brown and Company, . Among others, the idea of the Alexander technique is to re-
turn to the situation that the muscle groups for sustention and those for motion are used
only for their respective function, and not vice versa. Any unnecessary muscle tension, such
as neck sti ness, is a waste of energy due to the use of sustention muscles for movement and
of motion muscles for sustention. e technique teaches the way to return to the natural use
of muscles.
Motion of animals was discussed extensively already in the seventeenth century by
G. B
, De motu animalium, . An example of a more modern approach is J.J.
C
& I. S
, Hexapodal gaits and coupled nonlinear oscillator models, Biolo-
gical Cybernetics 68, pp. – , . See also I. S
& M. G
, Fearful
Symmetry, Blackwell, . Cited on pages and .
18 e results on the development of children mentioned here and in the following have been drawn mainly from the studies initiated by Jean Piaget; for more details on child development, see the Intermezzo following this chapter, on page . At http://www.piaget.org you
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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can nd the website maintained by the Jean Piaget Society. Cited on pages , , and .
19 e reptilian brain (eat? ee? ignore?), also called the R-complex, includes the brain stem, the cerebellum, the basal ganglia and the thalamus; the old mammalian (emotions) brain, also called the limbic system, contains the amygdala, the hypothalamus and the hippocampus; the human (and primate) (rational) brain, called the neocortex, consists of the famous grey matter. For images of the brain, see the atlas by J N , e Human Brain: An Introduction to its Functional Anatomy, Mosby, fourth edition, . Cited on page .
20 e lower le corner movie can be reproduced on a computer a er typing the following lines in the Mathematica so ware package: Cited on page .
« Graphics‘Animation‘ Nxpixels=72; Nypixels=54; Nframes=Nxpixels 4/3; Nxwind=Round[Nxpixels/4]; Nywind=Round[Nypixels/3]; front=Table[Round[Random[]],{y,1,Nypixels},{x,1,Nxpixels}]; back =Table[Round[Random[]],{y,1,Nypixels},{x,1,Nxpixels}]; frame=Table[front,{nf,1,Nframes}]; Do[ If[ x>n-Nxwind && x<n && y>Nywind && y<2Nywind,
frame[[n,y,x]]=back[[y,x-n]] ], {x,1,Nxpixels}, {y,1,Nypixels}, {n,1,Nframes}];
film=Table[ListDensityPlot[frame[[nf]], Mesh-> False, Frame-> False, AspectRatio-> N[Nypixels/Nxpixels], DisplayFunction-> Identity], {nf,1,Nframes}]
ShowAnimation[film]
But our motion detection system is much more powerful than the example shown in the lower le corners. e following, di erent movie makes the point.
« Graphics‘Animation‘ Nxpixels=72; Nypixels=54; Nframes=Nxpixels 4/3; Nxwind=Round[Nxpixels/4]; Nywind=Round[Nypixels/3]; front=Table[Round[Random[]],{y,1,Nypixels},{x,1,Nxpixels}]; back =Table[Round[Random[]],{y,1,Nypixels},{x,1,Nxpixels}]; frame=Table[front,{nf,1,Nframes}]; Do[ If[ x>n-Nxwind && x<n && y>Nywind && y<2Nywind,
frame[[n,y,x]]=back[[y,x]] ], {x,1,Nxpixels}, {y,1,Nypixels}, {n,1,Nframes}];
film=Table[ListDensityPlot[frame[[nf]], Mesh-> False, Frame-> False, AspectRatio-> N[Nypixels/Nxpixels], DisplayFunction-> Identity], {nf,1,Nframes}]
ShowAnimation[film]
Similar experiments, e.g. using randomly changing random patterns, show that the eye per-
ceives motion even in cases where all Fourier components of the image are practically zero;
such image motion is called dri -balanced or non-Fourier motion. Several examples are
presented in J. Z
, Modelling human motion perception I: Classical stimuli, Natur-
wissenscha en 81, pp. – , , and J. Z
, Modelling human motion perception
II: Beyond Fourier motion stimuli, Naturwissenscha en 81, pp. – , .
21 An introduction into perception research is E. B
G
Books/Cole, th edition, . Cited on pages and .
, Perception,
22 All fragments from Heraclitus are from J Early Greek Philosophy, Houghton Mu n
M
R
, An Introduction to
, chapter . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 328 n
23 An introduction to Newton the alchemist are the two books by B
JT
D,
e Foundations of Newton’s Alchemy, Cambridge University Press, , and e Janus Face
of Genius, Cambridge University Press, . Newton is found to be a sort of highly intellec-
tual magician, desperately looking for examples of processes where gods interact with the
material world. An intense but tragic tale. A good overview is provided by R.G. K
,
Essay Review: Newton’s Alchemy, Contemporary Physics 36, pp. – , .
Newton’s infantile theology, typical for god seekers who grew up without a father, can be
found in the many books summarizing the letter exchanges between Clarke, his secretary,
and Leibniz, Newton’s rival for fame. Cited on page .
24 An introduction to the story of classical mechanics, which also destroys a few of the myths
surrounding it – such as the idea that Newton could solve di erential equations or that he
introduced the expression F = ma – is given by C
A. T
, Essays in the
History of Mechanics, Springer, . Cited on pages , , and .
25 C. L , Z. D
, C.H. B
& L. V
H , Observation of coher-
ent optical information storage in an atomic medium using halted light pulses, Nature 409,
pp. – , . ere is also a comment on the paper by E.A. C
, Stopping light
in its track, 409, pp. – , . However, despite the claim, the light pulses of course
have not been halted. Can you give at least two reasons without even reading the paper, and
maybe a third a er reading it?
e work was an improvement on the previous experiment where a group velocity of
light of m s had been achieved, in an ultracold gas of sodium atoms, at nanokelvin tem-
peratures. is was reported by L. V
H , S.E. H
, Z. D
&
C.H. B
, Light speed reduction to meters per second in an ultracold atomic gas,
Nature 397, pp. – , . Cited on pages and .
26 R
F
, Biologie in Zahlen – Eine Datensammlung in Tabellen mit über 10.000
Einzelwerten, Spektrum Akademischer Verlag, . Cited on page .
27 Two jets with that speed have been observed by I.F. M superluminal source in the Galaxy, Nature 371, pp. – , on p. . Cited on page .
& L.F. R
,A
, as well as the comments
28 A beautiful introduction to the e ects of the slowest motions is D
B
,J
K
&I
S
, Landscha sformen der Erde – Bildatlas der Geomorphologie,
Primus Verlag, . Cited on page .
29 An introduction to the sense of time as a result of clocks in the brain is found in R.B. I
& R. S
, e neural representation of time, Current Opinion in Neurobiology 14,
pp. – , . e chemical clocks in our body are described in J D. P
,e
Living Clock, Oxford University Press, , or in A. A
& F. H
, Cycles of
Nature: An Introduction to Biological Rhythms, National Science Teachers Association, .
See also the http://www.msi.umn.edu/~halberg/introd/ website. Cited on page .
30 is has been shown among others by the work of Anna Wierzbicka mentioned in more de-
tail in the Intermezzo following this chapter, on page . e passionate best seller by the
Chomskian author S
P
, e Language Instinct – How the Mind Creates Lan-
guage, Harper Perennial, , also discusses issues related to this matter, refuting amongst
others on page the o en repeated false statement that the Hopi language is an exception.
Cited on page .
31 Aristotle rejects the idea of the ow of time in chapter IV of his Physics. See the full text on the http://classics.mit.edu/Aristotle/physics. .iv.html website. Cited on page .
32 Perhaps the most informative of the books about the ‘arrow of time’ is H D
Z,
e Physical Basis of the Direction of Time, Springer Verlag, th edition, . It is still the
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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best book on the topic. Most other texts – have a look on the internet – lack clarity of ideas.
A typical conference proceeding is J.J. H
, J. P –M
&W -
H. Z
, Physical Origins of Time Asymmetry, Cambridge University Press, .
Cited on page .
33 On the issue of absolute and relative motion there are many books about few issues. Ex-
amples are J B
, Absolute or Relative Motion? Vol. 1: A Study from the Machian
Point of View of the Discovery and the Structure of Spacetime eories, Cambridge Univer-
sity Press, , J B
, Absolute or Relative Motion? Vol. 2: e Deep Structure
of General Relativity, Oxford University Press, , or J E
, World Enough and
Spacetime: Absolute vs Relational eories of Spacetime, MIT Press, . Cited on page .
34 R. D
& M. F
, Banach–Tarski decompositions using sets with the
property of Baire, Journal of the American Mathematical Society 7, pp. – , . See
also A
L.T. P
, Amenability, American Mathematical Society, , and
R
M. F
, e Banach–Tarski theorem, e Mathematical Intelligencer 10,
pp. – , . Finally, there are the books by B
R. G
& J M.H.
O
, counter-examples in Analysis, Holden–Day, , and their eorems and
counter-examples in Mathematics, Springer, . Cited on page .
35 e beautiful but not easy text is S
W
, e Banach Tarski Paradox, Cambridge
University Press, . Cited on pages and .
36 About the shapes of salt water bacteria, see the corresponding section in the interesting book
by B
D , Power Unseen – How Microbes Rule the World, W.H. Freeman, .
e book has about sections, in which as many microorganisms are vividly presented.
Cited on page .
37 e smallest distances are probed in particle accelerators; the distance can be determined from the energy of the particle beam. In , the value of − m (for the upper limit of the size of quarks) was taken from the experiments described in F. A & al., Measurement of dijet angular distributions by the collider detector at Fermilab, Physical Review Letters 77, pp. – , . Cited on page .
38 A
K. D
, e Planiverse – Computer Contact with a Two-dimensional
World, Poseidon Books/Simon & Schuster, . Several other ction authors had explored
the option of a two-dimensional universe before, always answering, incorrectly, in the af-
rmative. Cited on page .
39 ere is a whole story behind the variations of . It can be found in C T
, Grav-
ity, Allen & Unwin, , or in W
T , Gravimetry, de Gruyter, , or in
M
B
&K
P , e Gravity Field and the Dynamics of the Earth, Springer,
. e variation of the height of the soil by around . m due to the Moon is one of the
interesting e ects found by these investigations. Cited on pages and .
40 A ano,
F
, La sica sotto il naso – 44 pezzi facili, Biblioteca Universale Rizzoli, Mil-
. Cited on page .
41 e study of shooting faeces (i.e., shit) and its mechanisms is a part of modern biology. e
reason that caterpillars do this was determined by M. W , Good housekeeping: why do
shelter-dwelling caterpillars ing their frass?, Ecology Letters 6, pp. – , , who also
gives the present record of . m for the mg pellets of Epargyreus clarus. e picture of
the ying frass is from S. C
, H. M L & D. S
, Faecal ring in a skipper
caterpillar is pressure-driven, e Journal of Experimental Biology 201, pp. – , .
Cited on page .
42 is was discussed in the Frankfurter Allgemeine Zeitung, nd of August, , at the time of the world athletics championship. e values are for the fastest part of the race of a m
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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sprinter; the exact values cited were called the running speed world records in , and were given as . m s = . km h by Ben Johnson for men, and . m s = . km h for women. Cited on page .
43 Long jump data and literature can be found in three articles all entitled Is a good long jumper a good high jumper?, in the American Journal of Physics 69, pp. – , . In particular, world class long jumpers run at . . m s, with vertical take-o speeds of . . m s, giving take-o angles of about (only) °. A new technique for achieving higher take-o angles would allow the world long jump record to increase dramatically. Cited on page .
44 e arguments of Zeno can be found in A
, Physics, VI, . It can be found trans-
lated in almost any language. e http://classics.mit.edu/Aristotle/physics. .vi.html website
provides an online version in English. Cited on pages and .
45 Etymology can be a fascinating topic, e.g. when research discovers the origin of the German word ‘Weib’ (‘woman’, related to English ‘wife’). It was discovered, via a few texts in Tocharian – an extinct Indo-European language from a region inside modern China – to mean originally ‘shame’. It was used for the female genital region in an expression meaning ‘place of shame’. With time, this expression became to mean ‘woman’ in general, while being shortened to the second term only. is connection was discovered by the German linguist Klaus T. Schmidt; it explains in particular why the word is not feminine but neutral, i.e. why it uses the article ‘das’ instead of ‘die’. Julia Simon, private communication. Etymology can also be simple and plain fun, for example when one discovers in the Oxford English Dictionary that ‘testimony’ and ‘testicle’ have the same origin; indeed in Latin the same word ‘testis’ was used for both concepts. Cited on pages and .
46 An overview of the latest developments is given by J.T. A
, D.J. H
, K.J.
J
& D. M
, Stellar optical interferometry in the s, Physics
Today pp. – , May . More than stellar diameters were known already in .
Several dedicated powerful instruments are being planned. Cited on page .
47 A good biology textbook on growth is A
F. H
ations of Animal Deveopment, Oxford University Press,
&N
H. H , Found-
. Cited on page .
48 is is discussed for example in C.L. S
, e amateur scientist – how to supply electric
power to something which is turning, Scienti c American pp. – , December . It
also discusses how to make a still picture of something rotating simply by using a few prisms,
the so-called Dove prisms. Other examples of attaching something to a rotating body are
given by E. R
, Some mechanisms related to Dirac’s strings, American Journal of
Physics 47, pp. – , . Cited on page .
49 J
A. Y
, Tumbleweed, Scienti c American 264, pp. – , March . e
tumbleweed is in fact quite rare, except in in Hollywood westerns, where all directors feel
obliged to give it a special appearance. Cited on page .
50 e rst experiments to prove the rotation of the agella were by M. S
& M.I.
S
, Flagellar rotation and the mechanism of bacterial motility, Nature 249, pp. – ,
. For some pretty pictures of the molecules involved, see K. N
, A biological mo-
lecular machine: bacterial agellar motor and lament, Wear 168, pp. – , . e
present record speed of rotation, rotations per second, is reported by Y. M
-
, S. S
, K. M
, Y. M
, I. K
, Y. I & S.
K , Very fast agellar rotation, Nature 371, p. , .
More on bacteria can be learned from D
D
, Life at a Small Scale, Sci-
enti c American Library, . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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51 On shadows, see the agreeable popular text by R
C
, Alla scoperta dell’ombra
– Da Platone a Galileo la storia di un enigma che ha a ascinato le grandi menti dell’umanità,
Oscar Mondadori, , and his websites located at http://www.shadowmill.com and http://
roberto.casati.free.fr/casati/roberto.htm. Cited on page .
52 ere is also the beautiful book by P . Cited on page .
F
, Colour in Nature, Blandford,
53 e laws of cartoon physics can easily be found using any search engine on the internet. Cited on page .
54 For the curious, an overview of the illusions used in the cinema and in television, which
lead to some of the strange behaviour of images mentioned above, is given in B
W
, e Technique of Special E ects in Television, Focal Press, , and his other books,
or in the Cinefex magazine. Cited on page .
55 A
, Opinions, I, XXIII, . See J -P D
Essais, Gallimard, p. , . Cited on page .
, Les écoles présocratiques, Folio
56 G
F
, Chi l’ha detto?, Hoepli, . Cited on pages and .
57 For the role and chemistry of adenosine triphosphate (ATP) in cells and in living beings, see any chemistry book, or search the internet. e uncovering of the mechanisms around ATP has led to Nobel Prizes in chemistry in and in . Cited on page .
58 A picture of this unique clock can be found in the article by A. G
, Perpetual motion
– a delicious delirium, Physics World pp. – , December . Cited on page .
59 A Shell study estimated the world’s total energy consumption in
to be EJ. e
US Department of Energy estimated it to be around EJ. We took the lower value here.
A discussion and a breakdown into electricity usage ( EJ) and other energy forms, with
variations per country, can be found in S. B
, e energy challenge, Physics Today
55, pp. – , April , and in E.J. M
& M.A. K
, Meeting energy
challenges: technology and policy, Physics Today 55, pp. – , April . Cited on page
.
60 For an overview, see the paper by J.F. M
& H.G. H , An unpublished lecture
by Heinrich Hertz: ‘On the energy balance of the Earth’, American Journal of Physics 65,
pp. – , . Cited on page .
61 For a beautiful photograph of this feline feat, see the cover of the journal and the article of
J. D
, A tale of a falling cat, Nature 308, p. , . Cited on page .
62 N
L. S
17, pp. – ,
, A new observation about rolling motion, European Journal of Physics . Cited on page .
63 C. S , When physical intuition fails, American Journal of Physics 70, pp. – , . Cited on page .
64 S
G
, e Spinal Engine, Springer Verlag, . It is now also knon that
human gait is chaotic. is is explained by M. P , e dynamics of human gait, European
Journal of Physics 26, pp. – , . Cited on page .
65 T
H
, Aristarchus of Samos – the Ancient Copernicus, Dover, , reprinted
from the original edition. Aristarchos’ treaty is given in Greek and English. Aristarchos
was the rst proposer of the heliocentric system. Aristarchos had measured the length of the
day (in fact, by determining the number of days per year) to the astonishing precision of
less than one second. is excellent book also gives an overview of Greek astronomy before
Aristarchos, explained in detail for each Greek thinker. Aristarchos’ text is also reprinted
in A
, On the sizes and the distances of the Sun and the Moon, c.
in
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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M Dover,
J. C
, eories of the World From Antiquity to the Copernican Revolution,
, especially on pp. – . Cited on page .
66 e in uence of the Coriolis e ect on icebergs was studied most thoroughly by the Swedish physicist turned oceanographer Walfrid Ekman ( – ); the topic was suggested by the great explorer Fridtjof Nansen, who also made the rst observations. In his honour, one speaks of the Ekman layer, Ekman transport and Ekman spirals. Any text on oceanography or physical geography will give more details about them. Cited on page .
67 An overview of the e ects of the Coriolis acceleration a = − ω v in the rotating frame is
given by E
A. D
, Classical Mechanics, Volume , John Wiley & Sons, .
Even the so-called Gulf Stream, the current of warm water owing from the Caribbean to
the North Sea, is in uenced by it. Cited on page .
68 e original publication is by A.H. S
, Bath-tub vortex, Nature 196, pp. – ,
. He also produced two movies of the experiment. e experiment has been repeated
many times in the northern and in the southern hemisphere, where the water drains clock-
wise; the rst southern hemisphere test was L.M. T
& al., e bath-tub vortex
in the southern hemisphere, Nature 201, pp. – , . A complete literature list is
found in the letters to the editor of the American Journal of Physics 62, p. , . Cited
on page .
69 e tricks are explained by H. R
C
, Short Foucault pendulum: a way to
eliminate the precession due to ellipticity, American Journal of Physics 49, pp. – ,
, and particularly in H. R
C
, Foucault pendulum wall clock, American
Journal of Physics 63, pp. – , . e Foucault pendulum was also the topic of the thesis
of H K
O , Nieuwe bewijzen der aswenteling der aarde, Universiteit
Groningen, . Cited on page .
70 e reference is J.G. H
, La rotation de la terre : ses preuves mécaniques anciennes et
nouvelles, Sp. Astr. Vaticana Second. App. Rome, . His other experiment is published as
J.G. H
, How Atwood’s machine shows the rotation of the Earth even quantitatively,
International Congress of Mathematics, Aug. . Cited on page .
71 R. A
, H.R. B
& G.E. S
, e Sagnac-e ect: a century of Earth-
rotated interferometers, American Journal of Physics 62, pp. – , .
See also the clear and extensive paper by G.E. S
, Ring laser tests of funda-
mental physics and geophysics, Reports on Progress in Physics 60, pp. – , . Cited
on page .
72 About the length of the day, see the http://maia.usno.navy.mil website, or the books by
K. L
, e Earth’s Variable Rotation: Geophysical Causes and Consequences, Cam-
bridge University Press, , and by W.H. M & G.J.F. M D
, e Rotation
of the Earth, Cambridge University Press, . Cited on pages and .
73 One example of data is by C.P. S
, E.P. K , A. Z
, M.A. C &
T.M. D
, Late proterozoic and paleozoic tides, retreat of the moon, and rotation of
the Earth, Science 273, pp. – , July . ey deduce from tidal sediment analysis
that days were only to hours long in the Proterozoic, i.e. million years ago; they
assume that the year was million seconds long from then to today. Another determination
was by G.E. W
, Precambrian tidal and glacial clastic deposits: implications for
precambrian Earth–Moon dynamics and palaeoclimate, Sedimentary Geology 120, pp. –
, . Using a geological formation called tidal rhythmites, he deduced that about
million years ago there were months per year and a day had hours. Cited on page .
74 e story of this combination of history and astronomy is told in R Historical Eclispes and Earth’s Rotation, Cambridge University Press,
S
,
. Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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75 On the rotation and history of the solar system, see S. B , eories of the origin of the solar system – , Reviews of Modern Physics 62, pp. – , . Cited on page .
76 e website http://maia.usno.navy.mil shows the motion of the Earth’s axis over the last ten years. e International Latitude Service founded by Küstner is now part of the International Earth Rotation Service; more information can be found on the http://www.iers.org website. e latest idea is that two-thirds of the circular component of the polar motion, which in the USA is called ‘Chandler wobble’ a er the person who attributed to himself the discovery by Küstner, is due to uctuations of the ocean pressure at the bottom of the oceans and onethird is due to pressure changes in the atmosphere of the Earth. is is explained by R.S. G , e excitation of the Chandler wobble, Geophysical Physics Letters 27, pp. – , . Cited on page .
77 For more information about Alfred Wegener, see the (simple) text by K R
,
Alfred Wegener – Erforscher der wandernden Kontinente, Verlag Freies Geistesleben, ;
about plate tectonics, see the http://www.scotese.com website. About earthquakes, see the
http://www.geo.ed.ac.uk/quakexe/quakes and the http://www.iris.edu/seismon website. See
the http://vulcan.wr.usgs.gov and the http://www.dartmouth.edu/~volcano/ websites for in-
formation about volcanoes. Cited on page .
78 J.D. H , J. I
& N.J. S
, Variations in the Earth’s orbit: pacemaker of
the ice ages, Science 194, pp. – , . ey found the result by literally digging in the
mud that covers the ocean oor in certain places. Note that the web is full of information
on the ice ages. Just look up ‘Milankovitch’ in a search engine. Cited on page .
79 R
H
&J
Astronomical Journal 110, pp.
L
, e sun’s distance above the galactic plane,
– , November . Cited on page .
80 C.L. B
, M.S. T
& M. W , e cosmic rosetta stone, Physics Today 50,
pp. – , November . Cited on page .
81 A good roulette prediction story from the s is told by T
A. B , e Eu-
daemonic Pie also published under the title e Newtonian Casino, Backinprint, . An
overview up to is given in the paper E
O. T
, e invention of the rst
wearable computer, Proceedings of the Second International Symposium on Wearable Com-
puters (ISWC ), - October , Pittsburgh, Pennsylvania, USA (IEEE Computer Soci-
ety), pp. – , , downloadable at http://csdl.computer.org/comp/proceedings/iswc/ /
/ / toc.htm. Cited on pages and .
82 e original papers are A.H. C
, A laboratory method of demonstrating the Earth’s
rotation, Science 37, pp. – , , A.H. C
, Watching the Earth revolve, Sci-
enti c American Supplement no. , pp. – , , and A.H. C
, A determina-
tion of latitude, azimuth and the length of the day independent of astronomical observations,
Physical Review (second series) 5, pp. – , . Cited on page .
83 is and many other physics surprises are described in the beautiful lecture script by J
Z
, Physik im Alltag, the notes of his lectures held in / at the Universität Re-
gensburg. Cited on pages and .
84 e equilibrium of ships, so important in car ferries, is an interesting part of shipbuilding;
an introduction was already given by L
E , Scientia navalis, . Cited on
page .
85 K.R. W
, B.P. B
& S.J. P
, Pulsed Mie scattering measure-
ments of the collapse of a sonoluminescing bubble, Physical Review Letters 78, pp. –
, . Cited on page .
86 On http://www.s .net/people/geo rey.landis/vacuum.html you can read a description of
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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what happened. See also the http://www.s .net/people/geo rey.landis/ebullism.html and
http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/
.html websites. ey all give
details on the e ects of vacuum on humans. Cited on page .
87 R. M N. A
, Leg design and jumping technique for humans, other vertebrates
and insects, Philosophical Transactions of the Royal Society in London B 347, pp. – ,
. Cited on page .
88 J. W. G
& T. A. M M
, A hydrodynamic model of locomotion in the
basilisk lizard, Nature 380, pp. – , For pictures, see also New Scientist, p. , March
, or Scienti c American, pp. – , September , or the website by the author at
http://rjf .biol.berkeley.edu/Full_Lab/FL_Personnel/J_Glasheen/J_Glasheen.html.
Several shore birds also have the ability to run over water, using the same mechanism.
Cited on page .
89 A. F
–N
& F.J.
N
, About the propulsion system of a kayak
and of Basiliscus basiliscus, European Journal of Physics 19, pp. – , . Cited on page
.
90 e material on the shadow discussion is from the book by R
M. P , Cook and
Peary, Stackpole Books, . See also W H
, e Noose of Laurels, Doubleday
. e sad story of Robert Peary is also told in the centenary number of National Geo-
graphic, September . Since the National Geographic Society had nanced Peary in his
attempt and had supported him until the US Congress had declared him the rst man at the
Pole, the (partial) retraction is noteworthy. ( e magazine then changed its mind again later
on, to sell more copies.) By the way, the photographs of Cook, who claimed to have been
at the North Pole even before Peary, have the same problem with the shadow length. Both
men have a history of cheating about their ‘exploits’. As a result, the rst man at the North
Pole was probably Roald Amundsen, who arrived there a few years later, and who was also
the rst man at the South Pole. Cited on page .
91 e story is told in M. N ican Journal of Physics 62,
, Hooke, orbital motion, and Newton’s Principia, Amer, pp. – . Cited on page .
92 More details are given by D. R
, in Doubling your sunsets or how anyone can meas-
ure the Earth’s size with wristwatch and meter stick, American Journal of Physics 47, ,
pp. – . Another simple measurement of the Earth radius, using only a sextant, is given
by R. O’K
& B. G
–A
, in e World Trade Centre and the distance
to the world’s centre, American Journal of Physics 60, pp. – , . Cited on page .
93 More details on astronomical distance measurements can be found in the beautiful little
book by A. H
, Measuring the Universe, University of Chicago Press, , and in
NH
&H
C
, e Guide to the Galaxy, Cambridge University
Press, . Cited on page .
94 A lot of details can be found in M. J
, Concepts of Mass in Classical and Modern Phys-
ics, reprinted by Dover, , and in Concepts of Force, a Study in the Foundations of Mechan-
ics, Harvard University Press, . ese eclectic and thoroughly researched texts provide
numerous details and explain various philosophical viewpoints, but lack clear statements
and conclusions on the accurate description of nature; thus are not of help on fundamental
issues.
Jean Buridan (c. to c. ) criticizes the distinction of sublunar and translunar mo-
tion in his book De Caelo, one of his numerous works. Cited on page .
95 D. T
& D.E. V
, An analysis of Newton’s projectile diagram, European
Journal of Physics 20, pp. – , . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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96 e absurd story of the metre is told in the historical novel by K A , e Measure of All ings : e Seven-Year Odyssey and Hidden Error that Transformed the World, e Free Press, . Cited on page .
97 H. E -
L,
, R. N
, U. H , N. C
& D. R
, HE
: an unbound hyper-velocity B-type star, e Astrophysical Journal 634, pp. L –
. Cited on page .
98 is is explained for example by D.K. F
& I.V. A
, e planets, a er all, may
run only in perfect circles – but in the velocity space!, European Journal of Physics 14, pp. –
, . Cited on pages and .
99 About the measurement of spatial dimensions via gravity – and the failure to nd any hint
for a number di erent from three – see the review by E.G. A
, B.R. H
& A.E. N
, Tests of the gravitational inverse-square law, Annual Review of Nuclear
and Particle Science 53, pp. – , , also http://www.arxiv.org/abs/hep-ph/
, or
the review by J.A. H
& M. S
, Particle physics probes of extra spacetime
dimensions, Annual Review of Nuclear and Particle Science 52, pp. – , , http://
www.arxiv.org/abs/hep-ph/
. Cited on page .
100 ere are many books explaining the origin of the precise shape of the Earth, such as the
pocket book S. A
, Weil die Erde rotiert, Verlag Harri Deutsch, . Cited on page
.
101 e shape of the Earth is described most precisely with the World Geodetic System. For an extensive presentation, see the http://www.eurocontrol.be/projects/eatchip/wgs /start. html website. See also the website of the International Earth Rotation Service at http://hpiers. obspm.fr. Cited on page .
102 W.K. H
, R.J. P
& G.J. T
Planetary Institute, . Cited on page .
, editors, Origin of the Moon, Lunar and
103 If you want to read about the motion of the Moon in all its fascinating details, have a look
at M
C. G
, Moon–Earth–Sun: the oldest three body problem, Reviews
of Modern Physics 70, pp. – , . Cited on page .
104 D
N
, Physiologische Uhren von Insekten – Zur Ökophysiologie lun-
arperiodisch kontrollierter Fortp anzungszeiten, Naturwissenscha en 82, pp. – , .
Cited on page .
105 e origin of the duration of the menstrual cycle is not yet settled; however, there are ex-
planations on how it becomes synchronized with other cycles. For a general explanation see
A
P
,M
R
&J
K
, Synchronization: A
Universal Concept in Nonlinear Science, Cambridge University Press, . Cited on page
.
106 J. L
, F. J
& P. R
, Stability of the Earth’s obliquity by the moon,
Nature 361, pp. – , . However, the question is not completely settled, and other
opinions exist. Cited on page .
107 N F. C
, What if the Moon Did not Exist? – Voyages to Earths that Might Have
Been, Harper Collins, . Cited on page .
108 See for example the discussion by J.J. L 389, pp. – , . Cited on page .
, It is not easy to make the moon, Nature
109 P A. W
,K
A. I
&S
M
, An asteroidal compan-
ion to the Earth, Nature 387, pp. – , June , together with the comment on
pp. – . Details on the orbit and on the fact that Lagrangian points do not always form
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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equilateral triangles can be found in F. N
,A.A. C
& C.D. M
, Co-
orbital dynamics at large eccentricity and inclination, Physical Review Letters 83, pp. –
, . Cited on page .
110 S
N
on page .
, Astronomical Papers of the American Ephemeris 1, p. , . Cited
111 A beautiful introduction is the classic G. F & W. R
, Mechanik, Relativität, Grav-
itation – ein Lehrbuch, Springer Verlag, Dritte Au age, . Cited on page .
112 J. S .
, Berliner Astronomisches Jahrbuch auf das Jahr 1804, , p. . Cited on page
113 e equality was rst tested with precision by R
E
, Annalen der Physik
& Chemie 59, p. , , and by R.
E
, V. P
& E. F
, Beiträge
zum Gesetz der Proportionalität von Trägheit und Gravität, Annalen der Physik 4, Leipzig
68, pp. – , . He found agreement to parts in . More experiments were performed
by P.G. R , R. K
& R.H. D , e equivalence of inertial and passive grav-
itational mass, Annals of Physics (NY) 26, pp. – , , one of the most interesting and
entertaining research articles in experimental physics, and by V.B. B
& V.I.
P
, Soviet Physics – JETP 34, pp. – , . Modern results, with errors less than
one part in , are by Y. S & al., New tests of the universality of free fall, Physical Review
D50, pp. – , . Several experiments have been proposed to test the equality in
space to less than one part in . Cited on page .
114 See L. H
, Gravitational eld strength inside the Earth, American Journal of Physics
59, pp. – , . Cited on page .
115 P. M
& M.C. J
of Physics 68, pp. – ,
, Plumb line and the shape of the Earth, American Journal . Cited on page .
116 From N
G
T
. Cited on page .
, e Universe Down to Earth, Columbia University Press,
117 is is a small example from the beautiful text by M P. S
, And Yet It Moves:
Strange Systems and Subtle Questions in Physics, Cambridge University Press, . It is a
treasure chest for anybody interested in the details of physics. Cited on page .
118 G.D. Q
, Planet X: a myth exposed, Nature 363, pp. – , . Cited on page .
119 See http://en.wikipedia.org/wiki/ _Sedna. Cited on page .
120 See R
M
, Not a snowball’s chance ..., New Scientist July , pp. – .
e original claim is by L
A. F
, J.B. S
& J.D. C
, On the in-
ux of small comets into the Earth’s upper atmosphere, parts I and II, Geophysical Research
Letters 13, pp. – , pp. – , . e latest observations have disproved the claim.
Cited on page .
121 e ray form is beautifully explained by J. E American Journal of Physics 61, pp. – ,
, e ray form of Newton’s law of motion, . Cited on page .
122 G.-L. L
, Lucrèce Newtonien, Nouveaux mémoires de l’Académie Royale des Sciences
et Belles Lettres pp. – , , or http://www .bbaw.de/bibliothek/digital/struktur/
-nouv/ /jpg- /
.htm. Cited on page .
123 J. L
, A numerical experiment on the chaotic behaviour of the solar system, Nature
338, pp. – , , and J. L
, e chaotic motion of the solar system - A numer-
ical estimate of the size of the chaotic zones, Icarus 88, pp. – , . e work by Laskar
was later expanded by Jack Wisdom, using specially built computers, following only the plan-
ets, without taking into account the smaller objects. For more details, see G.J. S
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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& J. W
, Chaotic Evolution of the Solar System, Science 257, pp. – , . Today,
such calculations can be performed on your home PC with computer code freely available
on the internet. Cited on page .
124 B. D
& F. G
, Titius-Bode laws in the solar system. : Scale invariance ex-
plains everything, Astronomy and Astrophysics 282, pp. – , , and Titius-Bode laws
in the solar system. : Build your own law from disk models, Astronomy and Astrophysics
282, pp. –- , . Cited on page .
125 M. L
, Bode’s Law, Nature 242, pp. – , , and M. H
, A comment on
“ e resonant structure of the solar system” by A.M. Molchanov, Icarus 11, pp. – , .
Cited on page .
126 C
D , Historia Romana, c. , book , . For an English translation, see the
site http://penelope.uchicago.edu/ ayer/E/Roman/Texts/Cassius_Dio/ *.html. Cited on
page .
127 M. B , D. A
, E. K
, L.P. F
, B. F
, R. M
& J.
B
, Seasonal uctuations in the mass of the Amazon River system and Earth’s elastic
response, Geophysical Research Letters 32, p. L , . Cited on page .
128 D. H
, M. W & G. S
, Force concept inventory, Physics Teacher
30, pp. – , . e authors developed tests to check the understanding of the concept
of physical force in students; the work has attracted a lot of attention in the eld of physics
teaching. Cited on page .
129 For a general overview on friction, from physics to economics, architecture and organiza-
tional theory, see N. Å
, editor, e Necessity of Friction – Nineteen Essays on a
Vital Force, Springer Verlag, . Cited on page .
130 See M. H
, K. S
,R. K
& Y. M
, Observation of superlubricity
by scanning tunneling microscopy, Physical Review Letters 78, pp. – , . See also
the discussion of their results by S
F
, Superlubricity: when friction stops,
Physics World pp. – , May . Cited on page .
131 C. D
A
, Meteorology Today: An Introduction to the Weather, Climate, and
the Environment, West Publishing Company, . Cited on page .
132 is topic is discussed with lucidity by J.R. M
, What really are the best m per-
formances?, Athletics: Canada’s National Track and Field Running Magazine, July . It can
also be found as http://www.arxiv.org/abs/physics/
, together with other papers on
similar topics by the same author. Cited on page .
133 F.P. B
& D. T
Part I, revised edition,
, e Friction and Lubrication of Solids, Oxford University Press, , and part II, . Cited on page .
134 A powerful book on human violence is J
G
, Violence – Our Deadly Epidemic
and its Causes, Grosset/Putnam, . Cited on page .
135 e main tests of randomness of number series – among them the gorilla test – can be found
in the authoritative paper by G. M
& W.W. T , Some di cult-to-pass tests
of randomness, Journal of Statistical So ware 7, p. , . It can also be downloaded from
the http://www.jstatso .org/v /i /tu ests.pdf website. Cited on page .
136 For one aspect of the issue, see for example the captivating book by B H
,
Zweierlei Glück, Carl Auer Systeme Verlag, . e author explains how to live serenely
and with the highest possible responsibility for one’s actions, by reducing entanglements
with the destiny of others. He describes a simple powerful technique to realise this goal.
A completely di erent viewpoint is given by Nobel Peace Price winner A
S S K , Freedom from Fear, Penguin, . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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137 H
W
, Neurophilosophie der Willensfreiheit, Mentis Verlag, Paderborn .
Also available in English translation. Cited on page .
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–
A
B
F I G U R E 86 What shape of rail allows the black stone to glide most rapidly from point A to the lower point B?
F I G U R E 87 Can motion be described in a manner common to all observers?
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.
–
“Πλεῖν ἀνάγκε, ζῆν οὐκ ἀνάγκη.*
” Pompeius
A over the Earth – even in Australia – people observe that stones fall ‘down’. is ncient observation led to the discovery of the universal law of gravity. To nd it, ll that was necessary was to look for a description of gravity that was valid globally.
e only additional observation that needs to be recognized in order to deduce the result
a = GM r is the variation of gravity with height.
In short, thinking globally helps us to make our description of motion more precise.
How can we describe motion as globally as possible? It turns out that there are six ap-
proaches to this question, each of which will be helpful on our way to the top of Motion
Mountain. We will start with an overview, and then explore the details of each approach.
Challenge 329 d
— e rst global approach to motion arises from a limitation of what we have learned so far. When we predict the motion of a particle from its current acceleration, we are using the most local description of motion possible. For example, whenever we use an evolution equation we use the acceleration of a particle at a certain place and time to determine its position and motion just a er that moment and in the immediate neighbourhood of that place. Evolution equations thus have a mental ’horizon’ of radius zero. e opposite approach is illustrated in the famous problem of Figure 86. e challenge is to nd the path that allows the fastest possible gliding motion from a high point to a distant low point. To solve this we need to consider the motion as a whole, for all times and positions. e global approach required by questions such as this one will lead us to a description of motion which is simple, precise and fascinating: the so-called principle of cosmic laziness, also known as the principle of least action.
— e second global approach to motion emerges when we compare the various descriptions of the same system produced by di erent observers. For example, the observations by somebody falling from a cli , a passenger in a roller coaster, and an observer
* Navigare necesse, vivere non necesse. ‘To navigate is necessary, to live is not.’ Gnaeus Pompeius Magnus Ref. 138 (106–48 ), as cited by Plutarchus (c. 45 to c. 125).
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bicycle
rope
wheel
F I G U R E 88 What happens when one rope is cut?
•.
a
b
b
F
C
P
a
b
b
F I G U R E 89 How to draw a straight line with a compass: fix point F, put a pencil into joint P and move C with a compass along a circle
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F I G U R E 90 A south-pointing carriage
Challenge 330 ny Ref. 139
Challenge 331 d
Ref. 140
on the ground will usually di er. e relationships between these observations lead us to a global description, valid for everybody. is approach leads us to the theory of relativity. — e third global approach to motion is to exploring the motion of extended and rigid bodies, rather than mass points. e counter-intuitive result of the experiment in Figure 88 shows why this is worthwhile.
In order to design machines, it is essential to understand how a group of rigid bodies interact with one another. As an example, the mechanism in Figure 89 connects the motion of points C and P. It implicitly de nes a circle such that one always has the relation rC = rP between the distances of C and P from its centre. Can you nd that circle?
Another famous challenge is to devise a wooden carriage, with gearwheels that connect the wheels to an arrow in such a way that whatever path the carriage takes, the arrow always points south (see Figure 90). e solution to this is useful in helping us to understand general relativity, as we will see.
Another interesting example of rigid motion is the way that human movements, such as the general motions of an arm, are composed from a small number of basic motions. All these examples are from the fascinating eld of engineering; unfortunately, we will have little time to explore this topic in our hike. — e fourth global approach to motion is the description of non-rigid extended bodies. For example, uid mechanics studies the ow of uids (like honey, water or air) around solid bodies (like spoons, ships, sails or wings). Fluid mechanics thus seeks to explain how insects, birds and aeroplanes y,* why sailboats can sail against the wind, what
* e mechanisms of insect ight are still a subject of active research. Traditionally, uid dynamics has
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? or ?
F I G U R E 91 How and where does a falling brick chimney break?
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F I G U R E 92 Why do hot-air balloons stay inflated? How can you measure the weight of a bicycle rider using only a ruler?
F I G U R E 93 What determines the number of petals in a daisy?
Ref. 141 Challenge 332 n
Challenge 333 n
happens when a hard-boiled egg is made to spin on a thin layer of water, or how a bottle full of wine can be emptied in the fastest way possible.
As well as uids, we can study the behaviour of deformable solids. is area of research is called continuum mechanics. It deals with deformations and oscillations of extended structures. It seeks to explain, for example, why bells are made in particular shapes; how large bodies – such as falling chimneys – break when under stress; and how cats can turn themselves the right way up as they fall. During the course of our journey we will repeatedly encounter issues from this eld, which impinges even upon general relativity and the world of elementary particles. — e h global approach to motion is the study of the motion of huge numbers of particles. is is called statistical mechanics. e concepts needed to describe gases, such as temperature and pressure (see Figure 92), will be our rst steps towards the understanding of black holes. — e sixth global approach to motion involves all of the above-mentioned viewpoints at the same time. Such an approach is needed to understand everyday experience, and life itself. Why does a ower form a speci c number of petals? How does an embryo di erentiate in the womb? What makes our hearts beat? How do mountains ridges and cloud patterns emerge? How do stars and galaxies evolve? How are sea waves formed by the wind?
concentrated on large systems, like boats, ships and aeroplanes. Indeed, the smallest human-made object that can y in a controlled way – say, a radio-controlled plane or helicopter – is much larger and heavier than many ying objects that evolution has engineered. It turns out that controlling the ight of small things requires more knowledge and more tricks than controlling the ight of large things. ere is more about this topic on page 969.
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All these are examples of self-organization; life scientists simply speak of growth. Whatever we call these processes, they are characterized by the spontaneous appearance of patterns, shapes and cycles. Such processes are a common research theme across many disciplines, including biology, chemistry, medicine, geology and engineering.
We will now give a short introduction to these six global approaches to motion. We will begin with the rst approach, namely, the global description of moving point-like objects. e beautiful method described below was the result of several centuries of collective e ort, and is the highlight of mechanics. It also provides the basis for all the further descriptions of motion that we will meet later on.
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M
Motion can be described by numbers. For a single particle, the relations between the spatial and temporal coordinates describe the motion. e realization that expressions like (x(t), y(t), z(t)) could be used to describe the path of a moving particle was a milestone in the development of modern physics.
We can go further. Motion is a type of change. And this change can itself be usefully described by numbers. In fact, change can be measured by a single number. is realization was the next important milestone. Physicists took almost two centuries of attempts to uncover the way to describe change. As a result, the quantity that measures change has a strange name: it is called (physical) action.* To remember the connection of ’action’ with change, just think about a Hollywood movie: a lot of action means a large amount of change.
Imagine taking two snapshots of a system at di erent times. How could you de ne the amount of change that occurred in between? When do things change a lot, and when do they change only a little? First of all, a system with a lot of motion shows a lot of change. So it makes sense that the action of a system composed of independent subsystems should be the sum of the actions of these subsystems.
Secondly, change o en – but not always – builds up over time; in other cases, recent change can compensate for previous change. Change can thus increase or decrease with time.
irdly, for a system in which motion is stored, transformed or shi ed from one subsystem to another, the change is smaller than for a system where this is not the case.
Ref. 142
* Note that this ‘action’ is not the same as the ‘action’ appearing in statements such as ‘every action has an equal and opposite reaction’. is last usage, coined by Newton, has not stuck; therefore the term has been recycled. A er Newton, the term ‘action’ was rst used with an intermediate meaning, before it was nally given the modern meaning used here. is last meaning is the only meaning used in this text.
Another term that has been recycled is the ‘principle of least action’. In old books it used to have a di erent meaning from the one in this chapter. Nowadays, it refers to what used to be called Hamilton’s principle in the Anglo-Saxon world, even though it is (mostly) due to others, especially Leibniz. e old names and meanings are falling into disuse and are not continued here.
Behind these shi s in terminology is the story of an intense two-centuries-long attempt to describe motion with so-called extremal or variational principles: the objective was to complete and improve the work initiated by Leibniz. ese principles are only of historical interest today, because all are special cases of the principle of least action described here.
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TA B L E 22 Some action values for changes either observed or imagined
C
A
Smallest measurable change Exposure of photographic lm Wing beat of a fruit y Flower opening in the morning Getting a red face Held versus dropped glass Tree bent by the wind from one side to the other Making a white rabbit vanish by ‘real’ magic Hiding a white rabbit Maximum brain change in a minute Levitating yourself within a minute by m Car crash Birth Change due to a human life Driving car stops within the blink of an eye Large earthquake Driving car disappears within the blink of an eye Sunrise Gamma ray burster before and a er explosion Universe a er one second has elapsed
. ë − Js . ë − Js to − Js c. pJs c. nJs c. mJs . Js
Js PJs c. . Js c. Js c. kJs c. kJs c. kJs c. EJs kJs c. PJs ZJs c. . ZJs c. Js unde ned and unde nable
L L(t) = T − U
average L
integral ∫ L(t)dt
ti
∆t tm
elapsed time
t tf
F I G U R E 94 Defining a total effect as an accumulation (addition, or integral) of small effects over time
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Challenge 334 e Page 120
Challenge 335 e
e mentioned properties imply that the natural measure of change is the average di erence between kinetic and potential energy multiplied by the elapsed time. is quantity has all the right properties: it is (usually) the sum of the corresponding quantities for all subsystems if these are independent; it generally increases with time (unless the evolution compensates for something that happened earlier); and it decreases if the system transforms motion into potential energy.
us the (physical) action S, measuring the change in a system, is de ned as
Joseph Lagrange
∫ ∫ tf
tf
S = L ë (tf − ti) = T − U ë (tf − ti) = (T − U) dt = L dt ,
ti
ti
(57)
where T is the kinetic energy, U the potential energy we already know, L is the di er-
ence between these, and the overbar indicates a time average. e quantity L is called the
Lagrangian (function) of the system,* describes what is being added over time, whenever
things change. e sign ∫ is a stretched ‘S’, for ‘sum’, and is pronounced ‘integral of ’. In
intuitive terms it designates the operation (called integration) of adding up the values of
a varying quantity in in nitesimal time steps dt. e initial and the nal times are writ-
ten below and above the integration sign, respectively. Figure illustrates the idea: the
integral is simply the size of the dark area below the curve L(t).
Mathematically, the integral of the curve L(t) is de ned as
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∫ tf
f
L(t) dt = lim L(tm)∆t = L ë (tf − ti) .
(58)
ti
∆t m=i
In other words, the integral is the limit, as the time slices get smaller, of the sum of the areas of the individual rectangular strips that approximate the function.** Since the sign also means a sum, and since an in nitesimal ∆t is written dt, we can understand the notation used for integration. Integration is a sum over slices. e notation was developed by Gottfried Leibniz to make exactly this point. Physically speaking, the integral of the Lagrangian measures the e ect that L builds up over time. Indeed, action is called ‘e ect’ in some languages, such as German.
In short, then, action is the integral of the Lagrangian over time. e unit of action, and thus of physical change, is the unit of energy (the Joule), times the unit of time (the second). us change is measured in Js. A large value means a big change. Table shows some approximate values of actions.
* It is named a er Giuseppe Lodovico Lagrangia (b. 1736 Torino, d. 1813 Paris), better known as Joseph Louis Lagrange. He was the most important mathematician of his time; he started his career in Turin, then worked for 20 years in Berlin, and nally for 26 years in Paris. Among other things he worked on number theory and analytical mechanics, where he developed most of the mathematical tools used nowadays for calculations in classical mechanics and classical gravitation. He applied them successfully to many motions in the solar system. ** For more details on integration see Appendix D.
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F I G U R E 95 The minimum of a curve has vanishing slope
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To understand the de nition of action in more detail, we will start with the simplest case: a system for which the potential energy is zero, such as a particle moving freely. Obviously, a large kinetic energy means a lot of change. If we observe the particle at two instants, the more distant they are the larger the change. Furthermore, the observed change is larger if the particle moves more rapidly, as its kinetic energy is larger. is is not surprising.
Next, we explore a single particle moving in a potential. For example, a falling stone loses potential energy in exchange for a gain in kinetic energy. e more energy is exchanged, the more change there is. Hence the minus sign in the de nition of L. If we explore a particle that is rst thrown up in the air and then falls, the curve for L(t) rst is below the times axis, then above. We note that the de nition of integration makes us count the grey surface below the time axis negatively. Change can thus be negative, and be compensated by subsequent change, as expected.
To measure change for a system made of several independent components, we simply add all the kinetic energies and subtract all the potential energies. is technique allows us to de ne actions for gases, liquids and solid matter. Even if the components interact, we still get a sensible result. In short, action is an additive quantity.
Physical action thus measures, in a single number, the change observed in a system between two instants of time. e observation may be anything at all: an explosion, a caress or a colour change. We will discover later that this idea is also applicable in relativity and quantum theory. Any change going on in any system of nature can be measured with a single number.
T
Challenge 336 e
We now have a precise measure of change, which, as it turns out, allows a simple and powerful description of motion. In nature, the change happening between two instants is always the smallest possible. In nature, action is minimal.* Of all possible motions, nature always chooses for which the change is minimal. Let us study a few examples.
In the simple case of a free particle, when no potentials are involved, the principle of minimal action implies that the particle moves in a straight line with constant velocity. All other paths would lead to larger actions. Can you verify this?
* In fact, in some pathological situations the action is maximal, so that the snobbish form of the principle is that the action is ‘stationary,’ or an ‘extremum,’ meaning minimal or maximal. e condition of vanishing
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Challenge 337 e
When gravity is present, a thrown stone ies along a parabola (or more precisely, along an ellipse) because any other path, say one in which the stone makes a loop in the air, would imply a larger action. Again you might want to verify this for yourself.
All observations support this simple and basic statement: things always move in a way that produces the smallest possible value for the action. is statement applies to the full path and to any of its segments. Betrand Russell called it the ‘law of cosmic laziness’.
It is customary to express the idea of minimal change in a di erent way. e action varies when the path is varied. e actual path is the one with the smallest action. You will recall from school that at a minimum the derivative of a quantity vanishes: a minimum has a horizontal slope. In the present case, we do not vary a quantity, but a complete path; hence we do not speak of a derivative or slope, but of a variation. It is customary to write the variation of action as δS. e principle of least action thus states:
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e actual trajectory between speci ed end points satis es δS = .
(59)
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Mathematicians call this a variational principle. Note that the end points have to be speci ed: we have to compare motions with the same initial and nal situations.
Before discussing the principle further, we can check that it is equivalent to the evolution equation.* To do this, we can use a standard procedure, part of the so-called calculus
Page 149 Challenge 338 ny
Ref. 143
variation, given below, encompasses both cases.
* For those interested, here are a few comments on the equivalence of Lagrangians and evolution equations.
First of all, Lagrangians do not exist for non-conservative, or dissipative systems. We saw that there is no
potential for any motion involving friction (and more than one dimension); therefore there is no action in
these cases. One approach to overcome this limitation is to use a generalized formulation of the principle
of least action. Whenever there is no potential, we can express the work variation δW between di erent
trajectories xi as
δW = mi x¨i δxi .
(60)
i
Motion is then described in the following way:
∫ e actual trajectory sati es tf (δT + δW)dt = provided δx(ti) = δx(tf ) = . (61) ti
e quantity being varied has no name; it represents a generalized notion of change. You might want to check that it leads to the correct evolution equations. us, although proper Lagrangian descriptions exist only for conservative systems, for dissipative systems the principle can be generalized and remains useful.
Many physicists will prefer another approach. What a mathematician calls a generalization is a special case for a physicist: the principle (61) hides the fact that all friction results from the usual principle of minimal action, if we include the complete microscopic details. ere is no friction in the microscopic domain. Friction is an approximate, macroscopic concept.
Nevertheless, more mathematical viewpoints are useful. For example, they lead to interesting limitations for the use of Lagrangians. ese limitations, which apply only if the world is viewed as purely classical – which it isn’t – were discovered about a hundred years ago. In those times computers where not available, and the exploration of new calculation techniques was important. Here is a summary.
e coordinates used in connection with Lagrangians are not necessarily the Cartesian ones. Generalized coordinates are especially useful when there are constraints on the motion. is is the case for a pendulum, where the weight always has to be at the same distance from the suspension, or for an ice skater, where the skate has to move in the direction in which it is pointing. Generalized coordinates may even be mixtures of positions and momenta. ey can be divided into a few general types.
Generalized coordinates are called holonomic–scleronomic if they are related to Cartesian coordinates in a xed way, independently of time: physical systems described by such coordinates include the pendulum
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of variations. e condition δS = implies that the action, i.e. the area under the curve
in Figure , is a minimum. A little bit of thinking shows that if the Lagrangian is of the Challenge 339 ny form L(xn, vn) = T(vn) − U(xn), then
d dt
∂T ∂vn
=
∂U ∂xn
(62)
where n counts all coordinates of all particles.* For a single particle, these Lagrange’s equa-
Challenge 340 e tions of motion reduce to
ma = ∇U .
(64)
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Challenge 341 n Challenge 342 ny
is is the evolution equation: it says that the force on a particle is the gradient of the potential energy U. e principle of least action thus implies the equation of motion. (Can you show the converse?)
In other words, all systems evolve in such a way that the change is as small as possible. Nature is economical. Nature is thus the opposite of a Hollywood thriller, in which the action is maximized; nature is more like a wise old man who keeps his actions to a minimum.
e principle of minimal action also states that the actual trajectory is the one for which the average of the Lagrangian over the whole trajectory is minimal (see Figure ). Nature is a Dr. Dolittle. Can you verify this? is viewpoint allows one to deduce Lagrange’s equations ( ) directly.
e principle of least action distinguishes the actual trajectory from all other imaginable ones. is observation lead Leibniz to his famous interpretation that the actual world
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Page 242 Page 559
Ref. 144
and a particle in a potential. Coordinates are called holonomic–rheonomic if the dependence involves time.
An example of a rheonomic systems would be a pendulum whose length depends on time. e two terms
rheonomic and scleronomic are due to Ludwig Boltzmann. ese two cases, which concern systems that are
only described by their geometry, are grouped together as holonomic systems. e term is due to Heinrich
Hertz.
e more general situation is called anholonomic, or nonholonomic. Lagrangians work well only for holo-
nomic systems. Unfortunately, the meaning of the term ‘nonholonomic’ has changed. Nowadays, the term
is also used for certain rheonomic systems. e modern use calls nonholonomic any system which involves
velocities. erefore, an ice skater or a rolling disc is o en called a nonholonomic system. Care is thus ne-
cessary to decide what is meant by nonholonomic in any particular context.
Even though the use of Lagrangians, and of action, has its limitations, these need not bother us at micro-
scopic level, since microscopic systems are always conservative, holonomic and scleronomic. At the funda-
mental level, evolution equations and Lagrangians are indeed equivalent.
* e most general form for a Lagrangian L(qn, q˙n, t), using generalized holonomic coordinates qn, leads to Lagrange equations of the form
d dt
∂L ∂q˙n
=
∂L ∂qn
.
(63)
In order to deduce these equations, we also need the relation δq˙ = d dt(δq). is relation is valid only for holonomic coordinates introduced in the previous footnote and explains their importance.
It should also be noted that the Lagrangian for a moving system is not unique; however, the study of how the various Lagrangians for a given moving system are related is not part of this walk.
By the way, the letter q for position and p for momentum were introduced in physics by the mathematician Carl Jacobi (b. 1804 Potsdam, d. 1851 Berlin).
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Challenge 343 n Challenge 344 ny
Page 195 Ref. 142
Challenge 345 n
is the ‘best of all possible worlds.’* We may dismiss this as metaphysical speculation, but we should still be able to feel the fascination of the issue. Leibniz was so excited about the principle of least action because it was the rst time that actual observations were distinguished from all other imaginable possibilities. For the rst time, the search for reasons why things are the way they are became a part of physical investigation. Could the world be di erent from what it is? In the principle of least action, we have a hint of a negative answer. (What do you think?) e nal answer will emerge only in the last part of our adventure.
As a way to describe motion, the Lagrangian has several advantages over the evolution equation. First of all, the Lagrangian is usually more compact than writing the corresponding evolution equations. For example, only one Lagrangian is needed for one system, however many particles it includes. One makes fewer mistakes, especially sign mistakes, as one rapidly learns when performing calculations. Just try to write down the evolution equations for a chain of masses connected by springs; then compare the e ort with a derivation using a Lagrangian. ( e system behaves like a chain of atoms.) We will encounter another example shortly: David Hilbert took only a few weeks to deduce the equations of motion of general relativity using a Lagrangian, whereas Albert Einstein had worked for ten years searching for them directly.
In addition, the description with a Lagrangian is valid with any set of coordinates describing the objects of investigation. e coordinates do not have to be Cartesian; they can be chosen as one prefers: cylindrical, spherical, hyperbolic, etc. ese so-called generalized coordinates allow one to rapidly calculate the behaviour of many mechanical systems that are in practice too complicated to be described with Cartesian coordinates. For example, for programming the motion of robot arms, the angles of the joints provide a clearer description than Cartesian coordinates of the ends of the arms. Angles are nonCartesian coordinates. ey simplify calculations considerably: the task of nding the most economical way to move the hand of a robot from one point to another can be solved much more easily with angular variables.
More importantly, the Lagrangian allows one to quickly deduce the essential properties of a system, namely, its symmetries and its conserved quantities. We will develop this important idea shortly, and use it regularly throughout our walk.
Finally, the Lagrangian formulation can be generalized to encompass all types of interactions. Since the concepts of kinetic and potential energy are general, the principle of least action can be used in electricity, magnetism and optics as well as mechanics. e principle of least action is central to general relativity and to quantum theory, and allows one to easily relate both elds to classical mechanics.
As the principle of least action became well known, people applied it to an ever-increasing number of problems. Today, Lagrangians are used in everything from the study of elementary particle collisions to the programming of robot motion in arti cial intelligence. However, we should not forget that despite its remarkable simplicity and usefulness, the Lagrangian formulation is equivalent to the evolution equations. It is neither more general nor more speci c. In particular, it is not an explanation for any type of motion, but only a view of it. In fact, the search of a new physical ‘law’ of motion is just the search
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* is idea was ridiculed by the French philosopher Voltaire (1694–1778) in his lucid writings, notably in the brilliant book Candide, written in 1759, and still widely available.
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Ref. 145 Challenge 346 ny
Ref. 146
for a new Lagrangian. is makes sense, as the description of nature always requires the description of change. Change in nature is always described by actions and Lagrangians.
e principle of least action states that the action is minimal when the end point of the motion, and in particular the time between them, are xed. It is less well known that the reciprocal principle also holds: if the action is kept xed, the elapsed time is maximal. Can you show this?
Even though the principle of least action is not an explanation of motion, it somehow calls for one. We need some patience, though. Why nature follows the principle of least action, and how it does so, will become clear when we explore quantum theory.
Never confuse movement with action.
“ ” Ernest Hemingway
W
?
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Page 785 Challenge 347 n
e optimist thinks this is the best of all possible worlds, and the pessimist knows it.
“ Robert Oppenheimer ” Looking around ourselves on Earth and in the sky, we nd that matter is not evenly dis-
tributed. Matter tends to be near other matter: it is lumped together in aggregates. Some major examples of aggregates are given in Figure and Table . In the mass–size diagram of Figure , both scales are logarithmic. One notes three straight lines: a line m l extending from the Planck mass* upwards, via black holes, to the universe itself; a line m l extending from the Planck mass downwards, to the lightest possible aggregate; and the usual matter line with m l , extending from atoms upwards, via the Earth and the Sun. e rst of the lines, the black hole limit, is explained by general relativity; the last two, the aggregate limit and the common matter line, by quantum theory.**
e aggregates outside the common matter line also show that the stronger the interaction that keeps the components together, the smaller the aggregate. But why is matter mainly found in lumps?
First of all, aggregates form because of the existence of attractive interactions between objects. Secondly, they form because of friction: when two components approach, an aggregate can only be formed if the released energy can be changed into heat. irdly, aggregates have a nite size because of repulsive e ects that prevent the components from collapsing completely. Together, these three factors ensure that bound motion is much more common than unbound, ‘free’ motion.
Only three types of attraction lead to aggregates: gravity, the attraction of electric charges, and the strong nuclear interaction. Similarly, only three types of repulsion are observed: rotation, pressure, and the Pauli exclusion principle (which we will encounter later on). Of the nine possible combinations of attraction and repulsion, not all appear in nature. Can you nd out which ones are missing from Figure and Table , and why?
Together, attraction, friction and repulsion imply that change and action are minimized when objects come and stay together. e principle of least action thus implies the
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* e Planck mass is given by mPl = ħc G = . ( ) µg. ** Figure 96 suggests that domains beyond physics exist; we will discover later on that this is not the case, as mass and size are not de nable in those domains.
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•.
mass [kg]
universe
1040 1020 100 10-20 10-40 10-60
beyond science: beyond Planck length limit common matter line
beyond science: undefined
black holes
galaxy star cluster
beyboenydonscdiethneceb:lack hole limit
neutron star
Sun Earth
mountain
human
Planck mass
cell
heavy
DNA
nucleus
uranium
muon
hydrogen
proton
electron
neutrino
microscopic
Elementary particles
Aggregates aggregate limit
lightest imaginable aggregate
10-40
10-20
100
F I G U R E 96 Aggregates in nature
1020 size [m]
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Challenge 348 ny
stability of aggregates. By the way, formation history also explains why so many aggregates rotate. Can you tell why?
But why does friction exist at all? And why do attractive and repulsive interactions exist? And why is it – as it would appear from the above – that in some distant past matter was not found in lumps? In order to answer these questions, we must rst study another global property of motion: symmetry.
TA B L E 23 Some major aggregates observed in nature
A
S
O
(
)
.C .
gravitationally bound aggregates
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A
matter across universe
quasar supercluster of galaxies galaxy cluster galaxy group or cluster our local galaxy group general galaxy
our galaxy
interstellar clouds solar system a our solar system
S
(
)
c. Ym
to m c. Ym c. Zm c. Zm
Zm . to Zm
. ( . ) Zm
up to Em unknown
Pm
Oort cloud Kuiper belt star b
to Pm Tm km to Gm
our star planet a (Jupiter, Earth)
. Gm Mm, . Mm
planetoids (Varuna, etc)
to km
moons neutron stars
to km km
electromagnetically bound aggregates c
asteroids, mountains d
m to km
comets
cm to km
planetoids, solids, liquids, nm to
km
gases, cheese
animals, plants, ke r
µm to km
brain
.m
cells:
smallest (nanobacteria) c. µm
amoeba
µm
largest (whale nerve,
c. m
single-celled plants)
molecules:
O .C .
superclusters of galaxies, hydrogen
andhelium atoms
ë baryons and leptons
galaxy groups and clusters
ë
to galaxies
to over galaxies
c. galaxies
.ë
to ë stars, dust and gas
clouds, probably solar systems
stars, dust and gas clouds, solar
systems
hydrogen, ice and dust
star, planets
Sun, planets (Pluto’s orbit’s diameter:
. Tm), moons, planetoids, comets,
asteroids, dust, gas
comets, dust
planetoids, comets, dust
ionized gas: protons, neutrons,
electrons, neutrinos, photons
+c. solids, liquids, gases; in particular, heavy atoms
c.
solids
(est. )
c.
solids
c.
mainly neutrons
( estimated) solids, usually monolithic ice and dust n.a. molecules, atoms
organs, cells neurons and other cell types organelles, membranes, molecules molecules molecules molecules
c.
atoms
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•.
A
S
O .C
(
)
.
H DNA (human) atoms, ions
c. pm m (total per cell) pm to pm
atoms atoms electrons and nuclei
aggregates bound by the weak interaction c none
aggregates bound by the strong interaction c
nucleus
−m
nucleon (proton, neutron) c. − m
mesons
c. − m
n.a.
neutron stars: see above
nucleons quarks quarks
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Ref. 148
Page 466 Ref. 149
a. Only in was the rst evidence found for objects circling stars other than our Sun; of over extrasolar planets found so far, most are found around F, G and K stars, including neutron stars. For example, three objects circle the pulsar PSR + , and a matter ring circles the star β Pictoris. e objects seem to be dark stars, brown dwarfs or large gas planets like Jupiter. Due to the limitations of observation systems, none of the systems found so far form solar systems of the type we live in. In fact, only a few Earth-like planets have been found so far. b. e Sun is among the brightest % of stars. Of all stars, %, are red M dwarfs, % are orange K dwarfs, and % are white D dwarfs: these are all faint. Almost all stars visible in the night sky belong to the bright
%. Some of these are from the rare blue O class or blue B class (such as Spica, Regulus and Riga); . % consist of the bright, white A class (such as Sirius, Vega and Altair); % are of the yellow–white F class (such as Canopus, Procyon and Polaris); . % are of the yellow G class (like Alpha Centauri, Capella or the Sun). Exceptions include the few visible K giants, such as Arcturus and Aldebaran, and the rare M supergiants, such as Betelgeuse and Antares. More on stars later on. c. For more details on microscopic aggregates, see the table of composites in Appendix C.
d. It is estimated that there are about asteroids (or planetoids) larger than km and about that are
heavier than kg. By the way, no asteroids between Mercury and the Sun – the hypothetical Vulcanoids –
have been found so far.
C
L
Lagrangians and variational principles form a fascinating topic, which has charmed physicists for the last four centuries.
**
When Lagrange published his book Mécanique analytique, in 1788, it formed one of the high points in the history of mechanics. He was proud of having written a systematic exposition of mechanics without a single gure. Obviously the book was di cult to read and was not a sales success. erefore his methods took another generation to come into general use.
** Given that action is the basic quantity describing motion, we can de ne energy as action
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per unit time, and momentum as action per unit distance. e energy of a system thus describes how much it changes over time, and the momentum how much it changes over Challenge 349 n distance. What are angular momentum and rotational energy?
Challenge 350 n
**
‘In nature, e ects of telekinesis or prayer are impossible, as in most cases the change inside the brain is much smaller than the change claimed in the outside world.’ Is this argument correct?
**
In Galilean physics, the Lagrangian is the di erence between kinetic and potential energy. Later on, this de nition will be generalized in a way that sharpens our understanding of this distinction: the Lagrangian becomes the di erence between a term for free particles and a term due to their interactions. In other words, particle motion is a continuous compromise between what the particle would do if it were free and what other particles want it to do. In this respect, particles behave a lot like humans beings.
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** Challenge 351 ny Explain: why is T + U constant, whereas T − U is minimal?
Challenge 352 ny
**
In nature, the sum T + U of kinetic and potential energy is constant during motion (for closed systems), whereas the average of the di erence T −U is minimal. Is it possible to deduce, by combining these two facts, that systems tend to a state with minimum potential energy?
Challenge 353 ny
**
ere is a principle of least e ort describing the growth of trees. When a tree – a monopodal phanerophyte – grows and produces leaves, between 40% and 60% of the mass it consists of, namely the water and the minerals, has to be li ed upwards from the ground.*
erefore, a tree gets as many branches as high up in the air as possible using the smallest amount of energy. is is the reason why not all leaves are at the very top of a tree. Can you deduce more details about trees from this principle?
Ref. 150
**
Another minimization principle can be used to understand the construction of animal bodies, especially their size and the proportions of their inner structures. For example, the heart pulse and breathing frequency both vary with animal mass m as m− , and the dissipated power varies as m . It turns out that such exponents result from three properties of living beings. First, they transport energy and material through the organism via a branched network of vessels: a few large ones, and increasingly many smaller ones. Secondly, the vessels all have the same minimum size. And thirdly, the networks are optimized in order to minimize the energy needed for transport. Together, these relations explain many additional scaling rules; they might also explain why animal lifespan
* e rest of the mass comes form the CO in the air.
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•.
α
air
water
β
F I G U R E 97 Refraction of light is due to travel-time optimization
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Ref. 151
scales as m− , or why most mammals have roughly the same number of heart beats in a lifetime.
A competing explanation, using a di erent minimization principle, states that quarter powers arise in any network built in order that the ow arrives to the destination by the most direct path.
Challenge 354 n Challenge 355 n
Challenge 356 n
**
e minimization principle for the motion of light is even more beautiful: light always takes the path that requires the shortest travel time. It was known long ago that this idea describes exactly how light changes direction when it moves from air to water. In water, light moves more slowly; the speed ratio between air and water is called the refractive index of water. e refractive index, usually abbreviated n, is material-dependent. e value for water is about 1.3. is speed ratio, together with the minimum-time principle, leads to the ‘law’ of refraction, a simple relation between the sines of the two angles. Can you deduce it? (In fact, the exact de nition of the refractive index is with respect to vacuum, not to air. But the di erence is negligible: can you imagine why?)
For diamond, the refractive index is 2.4. e high value is one reason for the sparkle of diamonds cut with the 57-face brilliant cut. Can you think of some other reasons?
Challenge 357 n
**
Can you con rm that each of these minimization principles is a special case of the principle of least action? In fact, this is the case for all known minimization principles in nature. Each of them, like the principle of least action, is a principle of least change.
Challenge 358 n
**
In Galilean physics, the value of the action depends on the speed of the observer, but not on his position or orientation. But the action, when properly de ned, should not depend on the observer. All observers should agree on the value of the observed change. Only special relativity will ful l the requirement that action be independent of the observer’s speed. How will the relativistic action be de ned?
**
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F I G U R E 98 Forget-me-not, also called Myosotis (Boraginaceae) (© Markku Savela)
Measuring all the change that is going on in the universe presupposes that the universe Challenge 359 n is a physical system. Is this the case?
M
Challenge 360 n
e second way to describe motion globally is to describe it in such a way that all observers agree. An object under observation is called symmetric if it looks the same when seen from di erent points of view. For example, a forget-me-not ower, shown in Figure , is symmetrical because it looks the same a er turning around it by degrees; many fruit tree owers have the same symmetry. One also says that under change of viewpoint the
ower has an invariant property, namely its shape. If many such viewpoints are possible, one talks about a high symmetry, otherwise a low symmetry. For example, a four-leaf clover has a higher symmetry than a usual, three-leaf one. Di erent points of view imply di erent observers; in physics, the viewpoints are o en called frames of reference and are described mathematically by coordinate systems.
High symmetry means many agreeing observers. At rst sight, not many objects or observations in nature seem to be symmetrical. But this is a mistake. On the contrary, we can deduce that nature as a whole is symmetric from the simple fact that we have the ability to talk about it! Moreover, the symmetry of nature is considerably higher than that of a forget-me-not. We will discover that this high symmetry is at the basis of the famous expression E = mc .
W
?
Ref. 152
e hidden harmony is stronger than the apparent.
“ Heraclitos of Ephesos, about ” Why can we understand somebody when he is talking about the world, even though we
are not in his shoes? We can for two reasons: because most things look similar from di er-
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•.
Challenge 361 n
ent viewpoints, and because most of us have already had similar experiences beforehand. ‘Similar’ means that what we and what others observe somehow correspond. In other
words, many aspects of observations do not depend on viewpoint. For example, the number of petals of a ower has the same value for all observers. We can therefore say that this quantity has the highest possible symmetry. We will see below that mass is another such example. Observables with the highest possible symmetry are called scalars in physics. Other aspects change from observer to observer. For example, the apparent size varies with the distance of observation. However, the actual size is observer-independent. In general terms, any type of viewpoint-independence is a form of symmetry, and the observation that two people looking at the same thing from di erent viewpoints can understand each other proves that nature is symmetric. We start to explore the details of this symmetry in this section and we will continue during most of the rest of our hike.
In the world around us, we note another general property: not only does the same phenomenon look similar to di erent observers, but di erent phenomena look similar to the same observer. For example, we know that if re burns the nger in the kitchen, it will do so outside the house as well, and also in other places and at other times. Nature shows reproducibility. Nature shows no surprises. In fact, our memory and our thinking are only possible because of this basic property of nature. (Can you con rm this?) As we will see, reproducibility leads to additional strong restrictions on the description of nature.
Without viewpoint-independence and reproducibility, talking to others or to oneself would be impossible. Even more importantly, we will discover that viewpointindependence and reproducibility do more than determine the possibility of talking to each other: they also x the content of what we can say to each other. In other words, we will see that our description of nature follows logically, almost without choice, from the simple fact that we can talk about nature to our friends.
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V
Tolerance ... is the suspicion that the other might be right.
“ Kurt Tucholski ( – ), German writer
Tolerance – a strength one mainly wishes to
” political opponents. “ Wolfram Weidner (b. ) German journalist ” When a young human starts to meet other people in childhood, it quickly nds out that
certain experiences are shared, while others, such as dreams, are not. Learning to make this distinction is one of the adventures of human life. In these pages, we concentrate on a section of the rst type of experiences: physical observations. However, even among these, distinctions are to be made. In daily life we are used to assuming that weights, volumes, lengths and time intervals are independent of the viewpoint of the observer. We can talk about these observed quantities to anybody, and there are no disagreements over their values, provided they have been measured correctly. However, other quantities do depend on the observer. Imagine talking to a friend a er he jumped from one of the trees along our path, while he is still falling downwards. He will say that the forest oor is approaching with high speed, whereas the observer below will maintain that the oor
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Challenge 362 n
is stationary. Obviously, the di erence between the statements is due to their di erent viewpoints. e velocity of an object (in this example that of the forest oor or of the friend himself) is thus a less symmetric property than weight or size. Not all observers agree on its value.
In the case of viewpoint-dependent observations, understanding is still possible with the help of a little e ort: each observer can imagine observing from the point of view of the other, and check whether the imagined result agrees with the statement of the other.* If the statement thus imagined and the actual statement of the other observer agree, the observations are consistent, and the di erence in statements is due only to the di erent viewpoints; otherwise, the di erence is fundamental, and they cannot agree or talk. Using this approach, you can even argue whether human feelings, judgements, or tastes arise from fundamental di erences or not.
e distinction between viewpoint-independent (invariant) and viewpointdependent quantities is an essential one. Invariant quantities, such as mass or shape, describe intrinsic properties, and quantities depending on the observer make up the state of the system. erefore, we must answer the following questions in order to nd a complete description of the state of a physical system:
— Which viewpoints are possible? — How are descriptions transformed from one viewpoint to another? — Which observables do these symmetries admit? — What do these results tell us about motion?
Page 86 Page 200
In the discussion so far, we have studied viewpoints di ering in location, in orientation, in time and, most importantly, in motion. With respect to each other, observers can be at rest, move with constant speed, or accelerate. ese ‘concrete’ changes of viewpoint are those we will study rst. In this case the requirement of consistency of observations made by di erent observers is called the principle of relativity. e symmetries associated with this type of invariance are also called external symmetries. ey are listed in Table .
A second class of fundamental changes of viewpoint concerns ‘abstract’ changes. Viewpoints can di er by the mathematical description used: such changes are called changes of gauge. ey will be introduced rst in the section on electrodynamics. Again, it is required that all statements be consistent across di erent mathematical descriptions. is requirement of consistency is called the principle of gauge invariance. e associated symmetries are called internal symmetries.
e third class of changes, whose importance may not be evident from everyday life, is that of the behaviour of a system under exchange of its parts. e associated invariance is called permutation symmetry. It is a discrete symmetry, and we will encounter it in the second part of our adventure.
e three consistency requirements described above are called ‘principles’ because these basic statements are so strong that they almost completely determine the ‘laws’ of physics, as we will see shortly. Later on we will discover that looking for a complete description of the state of objects will also yield a complete description of their intrinsic
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* Humans develop the ability to imagine that others can be in situations di erent from their own at the Ref. 153 age of about four years. erefore, before the age of four, humans are unable to conceive special relativity;
a erwards, they can.
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•.
properties. But enough of introduction: let us come to the heart of the topic.
S
Since we are looking for a complete description of motion, we need to understand and describe the full set of symmetries of nature. A system is said to be symmetric or to possess a symmetry if it appears identical when observed from di erent viewpoints. We also say that the system possesses an invariance under change from one viewpoint to the other. Viewpoint changes are called symmetry operations or transformations. A symmetry is thus a transformation, or more generally, a set of transformations. However, it is more than that: the successive application of two symmetry operations is another symmetry operation. To be more precise, a symmetry is a set G = a, b, c, ... of elements, the transformations, together with a binary operation called concatenation or multiplication and pronounced ‘a er’ or ‘times’, in which the following properties hold for all elements a, b and c:
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associativity, i.e. (a b) c = a (b c)
a neutral element e exists such that e a = a e = a
an inverse element a− exists such that a− a = a a− = e .
(65)
Challenge 363 n Ref. 154
Challenge 364 n
Any set that ful ls these three de ning properties, or axioms, is called a (mathematical) group. Historically, the notion of group was the rst example of a mathematical structure which was de ned in a completely abstract manner.* Can you give an example of a group taken from daily life? Groups appear frequently in physics and mathematics, because symmetries are almost everywhere, as we will see.** Can you list the symmetry operations of the pattern of Figure ?
R
Challenge 365 e
Looking at a symmetric and composed system such as the one shown in Figure , we notice that each of its parts, for example each red patch, belongs to a set of similar objects, usually called a multiplet. Taken as a whole, the multiplet has (at least) the symmetry properties of the whole system. For some of the coloured patches in Figure we need four objects to make up a full multiplet, whereas for others we need two, or only one, as in the case of the central star. In fact, in any symmetric system each part can be classi ed according to what type of multiplet it belongs to. roughout our mountain ascent we will perform the same classi cation with every part of nature, with ever-increasing precision.
* e term is due to Evariste Galois (1811–1832), the structure to Augustin-Louis Cauchy (1789–1857) and the axiomatic de nition to Arthur Cayley (1821–1895). ** In principle, mathematical groups need not be symmetry groups; but it can be proven that all groups can be seen as transformation groups on some suitably de ned mathematical space, so that in mathematics we can use the terms ‘symmetry group’ and ‘group’ interchangeably.
A group is called Abelian if its concatenation operation is commutative, i.e. if a b = b a for all pairs of elements a and b. In this case the concatenation is sometimes called addition. Do rotations form an abelian group?
A subset G ⊂ G of a group G can itself be a group; one then calls it a subgroup and o en says sloppily that G is larger than G or that G is a higher symmetry group than G .
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Copyright © 1990 Christoph Schiller F I G U R E 99 A Hispano–Arabic ornament from the Governor’s Palace in Sevilla
A multiplet is a set of parts that transform into each other under all symmetry transformations. Mathematicians o en call abstract multiplets representations. By specifying to which multiplet a component belongs, we describe in which way the component is part of the whole system. Let us see how this classi cation is achieved.
In mathematical language, symmetry transformations are o en described by matrices. For example, in the plane, a re ection along the rst diagonal is represented by the matrix
D(re ) =
,
(66)
since every point (x, y) becomes transformed to (y, x) when multiplied by the matrix Challenge 366 e D(re ). erefore, for a mathematician a representation of a symmetry group G is an
assignment of a matrix D(a) to each group element a such that the representation of
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the concatenation of two elements a and b is the product of the representations D of the
elements:
D(a b) = D(a)D(b) .
(67)
Challenge 367 e
For example, the matrix of equation ( ), together with the corresponding matrices for all the other symmetry operations, have this property.*
For every symmetry group, the construction and classi cation of all possible representations is an important task. It corresponds to the classi cation of all possible multiplets a symmetric system can be made of. In this way, understanding the classi cation of all multiplets and parts which can appear in Figure will teach us how to classify all possible parts of which an object or an example of motion can be composed!
A representation D is called unitary if all matrices D(a) are unitary.** Almost all representations appearing in physics, with only a handful of exceptions, are unitary: this term is the most restrictive, since it speci es that the corresponding transformations are one-to-one and invertible, which means that one observer never sees more or less than another. Obviously, if an observer can talk to a second one, the second one can also talk to the rst.
e nal important property of a multiplet, or representation, concerns its structure. If a multiplet can be seen as composed of sub-multiplets, it is called reducible, else irreducible; the same is said about representations. e irreducible representations obviously cannot be decomposed any further. For example, the symmetry group of Figure , commonly called D , has eight elements. It has the general, faithful, unitary and irreducible matrix representation
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* ere are some obvious, but important, side conditions for a representation: the matrices D(a) must be
invertible, or non-singular, and the identity operation of G must be mapped to the unit matrix. In even more
compact language one says that a representation is a homomorphism from G into the group of non-singular or invertible matrices. A matrix D is invertible if its determinant det D is not zero.
In general, if a mapping f from a group G to another G′ satis es
f (a G b) = f (a) G′ f (b) ,
(68)
the mapping f is called an homomorphism. A homomorphism f that is one-to-one (injective) and onto
(surjective) is called a isomorphism. If a representation is also injective, it is called faithful, true or proper.
In the same way as groups, more complex mathematical structures such as rings, elds and associative
algebras may also be represented by suitable classes of matrices. A representation of the eld of complex
numbers is given in Appendix D. ** e transpose AT of a matrix A is de ned element-by-element by (AT )ik = Aki. e complex conjugate A of a matrix A is de ned by (A )ik = (Aik) . e adjoint A† of a matrix A is de ned by A† = (AT ) . A matrix is called symmetric if AT = A, orthogonal if AT = A− , Hermitean or self-adjoint (the two are synonymous in all physical applications) if A† = A (Hermitean matrices have real eigenvalues), and unitary if A† = A− .
Unitary matrices have eigenvalues of norm one. Multiplication by a unitary matrix is a one-to-one mapping;
since the time evolution of physical systems is a mapping from one time to another, evolution is always
described by a unitary matrix. A real matrix obeys A = A, an antisymmetric or skew-symmetric matrix is de ned by AT = −A, an anti-Hermitean matrix by A† = −A and an anti-unitary matrix by A† = −A− .
All the mappings described by these special types of matrices are one-to-one. A matrix is singular, i.e. not one-to-one, if det A = .
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TA B L E 24 Correspondences between the symmetries of an ornament, a flower and nature as a whole
S
H
–A -
F
M
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Structure and components
set of ribbons and patches
System symmetry
pattern symmetry
Mathematical D description of the symmetry group
Invariants
number of multiplet elements
Representations multiplet types of
of the
elements
components
Most symmetric singlet representation
Simplest faithful quartet representation
Least symmetric quartet representation
set of petals, stem ower symmetry
motion path and observables
symmetry of Lagrangian
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C
petal number
multiplet types of components
in Galilean relativity: position, orientation, instant and velocity changes
number of coordinates, magnitude of scalars, vectors and tensors
tensors, including scalars and vectors
part with circular symmetry
quintet
scalar vector
quintet
no limit (tensor of in nite rank)
Challenge 368 ny
cos nπ − sin nπ sin nπ cos nπ
n = .. , −
,
−,
,−− .
(69) e representation is an octet. e complete list of possible irreducible representations
of the group D is given by singlets, doublets and quartets. Can you nd them all? ese representations allow the classi cation of all the white and black ribbons that appear in
the gure, as well as all the coloured patches. e most symmetric elements are singlets, the least symmetric ones are members of the quartets. e complete system is always a singlet as well.
With these concepts we are ready to talk about motion with improved precision.
S
,
G
Every day we experience that we are able to talk to each other about motion. It must therefore be possible to nd an invariant quantity describing it. We already know it: it is the action. Lighting a match is a change. It is the same whether it is lit here or there, in one direction or another, today or tomorrow. Indeed, the (Galilean) action is a number whose value is the same for each observer at rest, independent of his orientation or the
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 370 n Challenge 371 e
time at which he makes his observation. In the case of the Arabic pattern of Figure , the symmetry allows us to deduce the
list of multiplets, or representations, that can be its building blocks. is approach must be possible for motion as well. We deduced the classi cation of the ribbons in the Arabic pattern into singlets, doublets, etc. from the various possible observation viewpoints. For a moving system, the building blocks, corresponding to the ribbons, are the observables. Since we observe that nature is symmetric under many di erent changes of viewpoint, we can classify all observables. To do so, we need to take the list of all viewpoint transformations and deduce the list of all their representations.
Our everyday life shows that the world stays unchanged a er changes in position, orientation and instant of observation. One also speaks of space translation invariance, rotation invariance and time translation invariance. ese transformations are di erent from those of the Arabic pattern in two respects: they are continuous and they are unbounded. As a result, their representations will generally be continuously variable and without bounds: they will be quantities or magnitudes. In other words, observables will be constructed with numbers. In this way we have deduced why numbers are necessary for any description of motion.*
Since observers can di er in orientation, most representations will be objects possessing a direction. To cut a long story short, the symmetry under change of observation position, orientation or instant leads to the result that all observables are either ‘scalars’, ‘vectors’ or higher-order ‘tensors.’**
A scalar is an observable quantity which stays the same for all observers: it corresponds to a singlet. Examples are the mass or the charge of an object, the distance between two points, the distance of the horizon, and many others. eir possible values are (usually) continuous, unbounded and without direction. Other examples of scalars are the potential at a point and the temperature at a point. Velocity is obviously not a scalar; nor is the coordinate of a point. Can you nd more examples and counter-examples?
Energy is a puzzling observable. It is a scalar if only changes of place, orientation and instant of observation are considered. But energy is not a scalar if changes of observer speed are included. Nobody ever searched for a generalization of energy that is a scalar also for moving observers. Only Albert Einstein discovered it, completely by accident. More about this issue shortly.
Any quantity which has a magnitude and a direction and which ‘stays the same’ with respect to the environment when changing viewpoint is a vector. For example, the arrow between two xed points on the oor is a vector. Its length is the same for all observers; its direction changes from observer to observer, but not with respect to its environment. On the other hand, the arrow between a tree and the place where a rainbow touches the Earth is not a vector, since that place does not stay xed with respect to the environment, when the observer changes.
Mathematicians say that vectors are directed entities staying invariant under coordinate transformations. Velocities of objects, accelerations and eld strength are examples of vectors. (Can you con rm this?) e magnitude of a vector is a scalar: it is the same for
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* Only scalars, in contrast to vectors and higher-order tensors, may also be quantities which only take a Challenge 369 e discrete set of values, such as + or − only. In short, only scalars may be discrete observables.
** Later on, spinors will be added to, and complete, this list.
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any observer. By the way, a famous and ba ing result of nineteenth-century experiments is that the velocity of light is not a vector for Galilean transformations. is mystery will be solved shortly.
Tensors are generalized vectors. As an example, take the moment of inertia of an object. Page 89 It speci es the dependence of the angular momentum on the angular velocity. For any
object, doubling the magnitude of angular velocity doubles the magnitude of angular momentum; however, the two vectors are not parallel to each other if the object is not a Page 111 sphere. In general, if any two vector quantities are proportional, in the sense that doubling the magnitude of one vector doubles the magnitude of the other, but without the two vectors being parallel to each other, then the proportionality ‘factor’ is a (second order) tensor. Like all proportionality factors, tensors have a magnitude. In addition, tensors have a direction and a shape: they describe the connection between the vectors they relate. Just as vectors are the simplest quantities with a magnitude and a direction, so tensors are the simplest quantities with a magnitude and with a direction depending on a second, chosen direction. Vectors can be visualized as oriented arrows; tensors can be visualized Challenge 373 n as oriented ellipsoids.* Can you name another example of tensor?
Let us get back to the description of motion. Table shows that in physical systems we always have to distinguish between the symmetry of the whole Lagrangian – corresponding to the symmetry of the complete pattern – and the representation of the observables – corresponding to the ribbon multiplets. Since the action must be a scalar, and since all observables must be tensors, Lagrangians contain sums and products of tensors only in combinations forming scalars. Lagrangians thus contain only scalar products or generalizations thereof. In short, Lagrangians always look like
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L = α aibi + β cjkd jk + γ elmn f lmn + ...
(70)
where the indices attached to the variables a, b, c etc. always come in matching pairs to be
summed over. ( erefore summation signs are usually simply le out.) e Greek letters
represent constants. For example, the action of a free point particle in Galilean physics
was given as
∫ ∫ S = L dt = m v dt
(71)
which is indeed of the form just mentioned. We will encounter many other cases during our study of motion.**
Challenge 372 e Ref. 155
* A rank-n tensor is the proportionality factor between a rank-1 tensor, i.e. between a vector, and an rank(n − ) tensor. Vectors and scalars are rank 1 and rank 0 tensors. Scalars can be pictured as spheres, vectors as arrows, and rank-2 tensors as ellipsoids. Tensors of higher rank correspond to more and more complex shapes.
A vector has the same length and direction for every observer; a tensor (of rank 2) has the same determinant, the same trace, and the same sum of diagonal subdeterminants for all observers.
A vector is described mathematically by a list of components; a tensor (of rank 2) is described by a matrix of components. e rank or order of a tensor thus gives the number of indices the observable has. Can you show this? ** By the way, is the usual list of possible observation viewpoints – namely di erent positions, di erent observation instants, di erent orientations, and di erent velocities – also complete for the action (71)? Surprisingly, the answer is no. One of the rst who noted this fact was Niederer, in 1972. Studying the quantum
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•.
Page 86
Galileo already understood that motion is also invariant under change of viewpoints with di erent velocity. However, the action just given does not re ect this. It took some years to nd out the correct generalization: it is given by the theory of special relativity. But before we study it, we need to nish the present topic.
R
,
N
’
Challenge 375 ny Ref. 156
I will leave my mass, charge and momentum to science.
“ Gra to ” e reproducibility of observations, i.e. the symmetry under change of instant of time
or ‘time translation invariance’, is a case of viewpoint-independence. ( at is not obvious; can you nd its irreducible representations?) e connection has several important consequences. We have seen that symmetry implies invariance. It turns out that for continuous symmetries, such as time translation symmetry, this statement can be made more precise: for any continuous symmetry of the Lagrangian there is an associated conserved constant of motion and vice versa. e exact formulation of this connection is the theorem of Emmy Noether.* She found the result in when helping Albert Einstein and David Hilbert, who were both struggling and competing at constructing general relativity. However, the result applies to any type of Lagrangian.
Noether investigated continuous symmetries depending on a continuous parameter b. A viewpoint transformation is a symmetry if the action S does not depend on the value of b. For example, changing position as
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x x+b
(74)
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
theory of point particles, he found that even the action of a Galilean free point particle is invariant under some additional transformations. If the two observers use the coordinates (t, x) and (τ, ξ), the action (71) Challenge 374 ny is invariant under the transformations
ξ
=
Rx + x γt +
+ δ
vt
and
τ
=
αt γt
+ +
β δ
with
RTR =
and αδ − βγ = .
(72)
where R describes the rotation from the orientation of one observer to the other, v the velocity between the two observers, and x the vector between the two origins at time zero. is group contains two important special cases of transformations:
e connected, static Galilei group ξ = Rx + x + vt and τ = t
e
transformation
group
SL(2,R)
ξ
=
x γt +
δ
and
τ
=
αt γt
+ +
β δ
(73)
e latter, three-parameter group includes spatial inversion, dilations, time translation and a set of timedependent transformations such as ξ = x t, τ = t called expansions. Dilations and expansions are rarely mentioned, as they are symmetries of point particles only, and do not apply to everyday objects and systems.
ey will return to be of importance later on, however. * Emmy Noether (b. 1882 Erlangen, d. 1935 Bryn Mayr), German mathematician. e theorem is only a sideline in her career which she dedicated mostly to number theory. e theorem also applies to gauge symmetries, where it states that to every gauge symmetry corresponds an identity of the equation of motion, and vice versa.
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leaves the action
∫ S = T(v) − U(x) dt
(75)
invariant, since S(b) = S . is situation implies that
∂T ∂v
=
p = const
;
(76)
in short, symmetry under change of position implies conservation of momentum. e converse is also true. Challenge 376 ny In the case of symmetry under shi of observation instant, we nd
T + U = const ;
(77)
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Challenge 377 n
Challenge 378 e Page 39 Page 154
in other words, time translation invariance implies constant energy. Again, the converse is also correct. One also says that energy and momentum are the generators of time and space translations.
e conserved quantity for a continuous symmetry is sometimes called the Noether charge, because the term charge is used in theoretical physics to designate conserved extensive observables. So, energy and momentum are Noether charges. ‘Electric charge’, ‘gravitational charge’ (i.e. mass) and ‘topological charge’ are other common examples. What is the conserved charge for rotation invariance?
We note that the expression ‘energy is conserved’ has several meanings. First of all, it means that the energy of a single free particle is constant in time. Secondly, it means that the total energy of any number of independent particles is constant. Finally, it means that the energy of a system of particles, i.e. including their interactions, is constant in time. Collisions are examples of the latter case. Noether’s theorem makes all of these points at the same time, as you can verify using the corresponding Lagrangians.
But Noether’s theorem also makes, or rather repeats, an even stronger statement: if energy were not conserved, time could not be de ned. e whole description of nature requires the existence of conserved quantities, as we noticed when we introduced the concepts of object, state and environment. For example, we de ned objects as permanent entities, that is, as entities characterized by conserved quantities. We also saw that the introduction of time is possible only because in nature there are ‘no surprises’. Noether’s theorem describes exactly what such a ‘surprise’ would have to be: the non-conservation of energy. However, energy jumps have never been observed – not even at the quantum level.
Since symmetries are so important for the description of nature, Table gives an overview of all the symmetries of nature we will encounter. eir main properties are also listed. Except for those marked as ‘approximate’ or ‘speculative’, an experimental proof of incorrectness of any of them would be a big surprise indeed.
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•.
TA B L E 25 The symmetries of relativity and quantum theory with their properties; also the complete list of logical inductions used in the two fields
S
T
S
G
P - C - V-
M
[
-
-
-
/
-
-
-
-
-
-
-
]
Geometric or space-time, external, symmetries
Time and space R R space, not
scalars,
translation
[ par.] time compact vectors,
momentum yes/yes and energy
Rotation
SO( ) space S [ par.]
tensors angular yes/yes momentum
Galilei boost
R [ par.] space, time
not
scalars,
compact vectors,
tensors
velocity of yes/for
centre of low
mass
speeds
Lorentz
homogen- spaceeous Lie time SO( , ) [ par.]
not
tensors,
compact spinors
energy- yes/yes momentum T µν
Poincaré ISL( ,C)
inhomogeneous Lie [ par.]
spacetime
not
tensors,
compact spinors
energy- yes/yes momentum T µν
Dilation invariance
R+ [ par.] space- ray time
n-dimen. none continuum
yes/no
Special conformal invariance
R [ par.] space- R time
n-dimen. none continuum
yes/no
Conformal invariance
[ par.]
spacetime
involved massless tensors, spinors
none
yes/no
Dynamic, interaction-dependent symmetries: gravity
r gravity
SO( ) con g. as SO( ) vector pair perihelion yes/yes
[ par.] space
direction
Di eomorphism [ par.] space- involved space-
invariance
time
times
local
yes/no
energy–
momentum
Dynamic, classical and quantum-mechanical motion symmetries
allow everyday communication relativity of motion
constant light speed
massless particles massless particles
light cone invariance
closed orbits perihelion shi
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
S
T
S
G
P - C - V-
M
[
-
-
-
/
-
-
-
-
-
-
-
]
Motion(‘time’) discrete inversion T
Hilbert discrete even, odd T-parity or phase space
yes/no
reversibility
Parity(‘spatial’) discrete inversion P
Hilbert discrete even, odd P-parity or phase space
yes/no
mirror world exists
Charge conjugation C
global, Hilbert discrete
antilinear, or phase
anti-
space
Hermitean
even, odd C-parity
yes/no
antiparticles exist
CPT
discrete Hilbert discrete even
CPT-parity yes/yes makes eld
or phase
theory
space
possible
Dynamic, interaction-dependent, gauge symmetries
Electromagnetic [ par.] space of un- im- un-
electric
classical gauge
elds portant important charge
invariance
yes/yes massless light
Electromagnetic abelian Lie Hilbert circle S elds
q.m. gauge inv. U( )
space
[ par.]
electric charge
yes/yes massless photon
Electromagnetic abelian Lie space of circle S abstract
duality
U( )
elds
[ par.]
abstract
yes/no
none
Weak gauge
non-
Hilbert
abelian Lie space
SU( )
[ par.]
as SU( ) particles
weak charge
no/ approx.
Colour gauge
non-
Hilbert
abelian Lie space
SU( )
[ par.]
as SU( ) coloured quarks
colour
yes/yes massless gluons
Chiral symmetry
discrete fermions discrete le , right helicity
approxi- ‘massless’ mately fermionsa
Permutation symmetries
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•.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
S
T
S
G
P - C - V-
M
[
-
-
-
/
-
-
-
-
-
-
-
]
Particle exchange
discrete
Fock space etc.
discrete fermions none and bosons
n.a./yes Gibbs’ paradox
Selected speculative symmetries of nature
GUT
E , SO( ) Hilbert from Lie particles group
from Lie group
yes/no
coupling constant convergence
N-supersymmetryb
R-parity
global discrete
Hilbert Hilbert
discrete
particles, sparticles
+ ,-
Tmn and N no/no spinors c Qimn
‘massless’a particles
R-parity
yes/yes sfermions, gauginos
Braid symmetry discrete own discrete unclear unclear space
yes/maybe unclear
Space-time duality
discrete all
discrete vacuum unclear
yes/maybe xes particle masses
Event symmetry discrete space- discrete nature time
none
yes/no unclear
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For details about the connection between symmetry and induction, see page . e explanation of the terms in the table will be completed in the rest of the walk. e real numbers are denoted as R. a. Only approximate; ‘massless’ means that m  mPl, i.e. that m  µg. b. N = supersymmetry, but not N = supergravity, is probably a good approximation for nature at everyday energies. c. i = .. N.
In summary, since we can talk about nature we can deduce several of its symmetries, in particular its symmetry under time and space translations. From nature’s symmetries, using Noether’s theorem, we can deduce the conserved charges, such as energy or linear and angular momentum. In other words, the de nition of mass, space and time, together with their symmetry properties, is equivalent to the conservation of energy and momentum. Conservation and symmetry are two ways to express the same property of nature. To put it simply, our ability to talk about nature means that energy and momentum are conserved.
In general, the most elegant way to uncover the ‘laws’ of nature is to search for nature’s
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–
symmetries. In many historical cases, once this connection had been understood, physics made rapid progress. For example, Albert Einstein discovered the theory of relativity in this way, and Paul Dirac started o quantum electrodynamics. We will use the same method throughout our walk; in its third part we will uncover some symmetries which are even more mind-boggling than those of relativity. Now, though, we will move on to the next approach to a global description of motion.
C As diet for your brain, a few questions to ponder:
** What is the path followed by four turtles starting on the four angles of a square, if each Challenge 379 ny of them continuously walks at the same speed towards the next one?
** Challenge 380 n What is the symmetry of a simple oscillation? And of a wave?
** Challenge 381 n For what systems is motion reversal a symmetry transformation?
** Challenge 382 ny What is the symmetry of a continuous rotation?
** A sphere has a tensor for the moment of inertia that is diagonal with three equal numbers.
e same is true for a cube. Can you distinguish spheres and cubes by their rotation Challenge 383 ny behaviour?
** Challenge 384 ny Is there a motion in nature whose symmetry is perfect?
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S
–
We de ned action, and thus change, as the integral of the Lagrangian, and the Lagrangian as the di erence between kinetic and potential energy. One of the simplest systems in nature is a mass m attached to a spring. Its Lagrangian is given by
L = mv − kx ,
(78)
Challenge 385 e
where k is a quantity characterizing the spring, the so-called spring constant. e Lagrangian is due to Robert Hooke, in the seventeenth century. Can you con rm it?
e motion that results from this Lagrangian is periodic, as shown in Figure . e Lagrangian describes the oscillation of the spring length. e motion is exactly the same as that of a long pendulum. It is called harmonic motion, because an object vibrating
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•. position
oscillation amplitude
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
time F I G U R E 100 The simplest oscillation
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TA B L E 26 Some mechanical frequency values found in nature
O
Sound frequencies in gas emitted by black holes Precision in measured vibration frequencies of the Sun Vibration frequencies of the Sun Vibration frequencies that disturb gravitational radiation detection Lowest vibration frequency of the Earth Ref. 157 Resonance frequency of stomach and internal organs (giving the ‘sound in the belly’ experience) Wing beat of tiny y Sound audible to young humans Sonar used by bats Sonar used by dolphins Sound frequency used in ultrasound imaging Phonon (sound) frequencies measured in single crystals
F
c. fHz down to nHz down to c. nHz down to µHz
µHz 1 to Hz
c. Hz Hz to kHz
up to over kHz up to kHz up to MHz up to THz and more
Page 205 Challenge 386 ny
rapidly in this way produces a completely pure – or harmonic – musical sound. ( e musical instrument producing the purest harmonic waves is the transverse ute. is instrument thus gives the best idea of how harmonic motion ‘sounds’.) e graph of a harmonic or linear oscillation, shown in Figure , is called a sine curve; it can be seen as the basic building block of all oscillations. All other, non-harmonic oscillations in nature can be composed from sine curves, as we shall see shortly.
Every oscillating motion continuously transforms kinetic energy into potential energy and vice versa. is is the case for the tides, the pendulum, or any radio receiver. But many oscillations also diminish in time: they are damped. Systems with large damping, such as the shock absorbers in cars, are used to avoid oscillations. Systems with small damping are useful for making precise and long-running clocks. e simplest measure of damping is the number of oscillations a system takes to reduce its amplitude to e . times the original value. is characteristic number is the so-called Q-factor, named a er the abbreviation of ‘quality factor’. A poor Q-factor is or less, an extremely good one is
or more. (Can you write down a simple Lagrangian for a damped oscillation with a given Q-factor?) In nature, damped oscillations do not usually keep constant frequency;
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–
Figure yet to be included
F I G U R E 101 Decomposing a general wave or signal into harmonic waves
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however, for the simple pendulum this remains the case to a high degree of accuracy. e reason is that for a pendulum, the frequency does not depend signi cantly on the
amplitude (as long as the amplitude is smaller than about °). is is one reason why pendulums are used as oscillators in mechanical clocks.
Obviously, for a good clock, the driving oscillation must not only show small damping, but must also be independent of temperature and be insensitive to other external in uences. An important development of the twentieth century was the introduction of quartz crystals as oscillators. Technical quartzes are crystals of the size of a few grains of sand; they can be made to oscillate by applying an electric signal. ey have little temperature dependence and a large Q-factor, and therefore low energy consumption, so that precise clocks can now run on small batteries.
All systems that oscillate also emit waves. In fact, oscillations only appear in extended systems, and oscillations are only the simplest of motions of extended systems. e general motion of an extended system is the wave.
W
Challenge 387 e
Waves are travelling imbalances, or, equivalently, travelling oscillations. Waves move, even though the substrate does not move. Every wave can be seen as a superposition of harmonic waves. Can you describe the di erence in wave shape between a pure harmonic tone, a musical sound, a noise and an explosion? Every sound e ect can be thought of as being composed of harmonic waves. Harmonic waves, also called sine waves or linear waves, are the building blocks of which all internal motions of an extended body are constructed.
Every harmonic wave is characterized by an oscillation frequency and a propagation velocity. Low-amplitude water waves show this most clearly.
Waves appear inside all extended bodies, be they solids, liquids, gases or plasmas. Inside uid bodies, waves are longitudinal, meaning that the wave motion is in the same direction as the wave oscillation. Sound in air is an example of a longitudinal wave. Inside solid bodies, waves can also be transverse; in that case the wave oscillation is perpendicular to the travelling direction.
Waves appear also on interfaces between bodies: water–air interfaces are a well-known case. Even a saltwater–freshwater interface, so-called dead water, shows waves: they can appear even if the upper surface of the water is immobile. Any ight in an aero-
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•.
TA B L E 27 Some wave velocities
W
Tsunami Sound in most gases Sound in air at K Sound in air at K Sound in helium at K Sound in most liquids Sound in water at K Sound in water at K Sound in gold Sound in steel Sound in granite Sound in glass Sound in beryllium Sound in boron Sound in diamond Sound in fullerene (C ) Plasma wave velocity in InGaAs Light in vacuum
V
around m s . km s ms ms . km s . km s . km s . km s . km s . km s . km s . km s . km s
up to km s up to km s up to km s
km s . ë ms
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 102 The formation of gravity waves on water
plane provides an opportunity to study the regular cloud arrangements on the interface between warm and cold air layers in the atmosphere. Seismic waves travelling along the boundary between the sea oor and the sea water are also well-known. General surface waves are usually neither longitudinal nor transverse, but of a mixed type.
On water surfaces, one classi es waves according to the force that restores the plane surface. e rst type, surface tension waves, plays a role on scales up to a few centimetres. At longer scales, gravity takes over as the main restoring force and one speaks of gravity waves. is is the type we focus on here. Gravity waves in water, in contrast to surface tension waves, are not sinusoidal. is is because of the special way the water moves in such a wave. As shown in Figure , the surface water moves in circles; this leads to the typical, asymmetrical wave shape with short sharp crests and long shallow troughs. (As long as there is no wind and the oor below the water is horizontal, the waves are also symmetric under front-to-back re ection.)
For water gravity waves, as for many other waves, the speed depends on the wavelength.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
–
Indeed, the speed c of water waves depends on the wavelength λ and on the depth of the water d in the following way:
c=
λ tanh π
πd λ
,
(79)
Challenge 388 e
Challenge 389 n Ref. 158
where is the acceleration due to gravity (and an amplitude much smaller than the
wavelength is assumed). e formula shows two limiting regimes. First, short or deep
waves appear when the water depth is larger than half the wavelength; for deep waves,
the phase velocity is c
λ π , thus wavelength dependent, and the group velocity
is about half the phase velocity. Shorter deep waves are thus slower. Secondly, shallow or
long waves appear when the depth is less than % of the wavelength; in this case, c
d,
there is no dispersion, and the group velocity is about the same as the phase velocity.
e most impressive shallow waves are tsunamis, the large waves triggered by submarine
earthquakes. ( e Japanese name is composed of tsu, meaning harbour, and nami, mean-
ing wave.) Since tsunamis are shallow waves, they show little dispersion and thus travel
over long distances; they can go round the Earth several times. Typical oscillation times
are between and minutes, giving wavelengths between and km and speeds in
the open sea of to m s, similar to that of a jet plane. eir amplitude on the open
sea is o en of the order of cm; however, the amplitude scales with depth d as d and
heights up to m have been measured at the shore. is was the order of magnitude of
the large and disastrous tsunami observed in the Indian Ocean on December .
Waves can also exist in empty space. Both light and gravity waves are examples. e
exploration of electromagnetism and relativity will tell us more about their properties.
Any study of motion must include the study of wave motion. We know from experi-
ence that waves can hit or even damage targets; thus every wave carries energy and mo-
mentum, even though (on average) no matter moves along the wave propagation direc-
tion. e energy E of a wave is the sum of its kinetic and potential energy. e kinetic
energy (density) depends on the temporal change of the displacement u at a given spot:
rapidly changing waves carry a larger kinetic energy. e potential energy (density) de-
pends on the gradient of the displacement, i.e. on its spatial change: steep waves carry
a larger potential energy than shallow ones. (Can you explain why the potential energy
does not depend on the displacement itself?) For harmonic waves propagating along the
direction z, each type of energy is proportional to the square of its respective displace-
ment change:
E
(
∂u ∂t
)
+v
(
∂u ∂z
)
.
(80)
Challenge 390 ny
How is the energy density related to the frequency?
e momentum of a wave is directed along the direction of wave propagation. e
momentum value depends on both the temporal and the spatial change of displacement
u. For harmonic waves, the momentum (density) P is proportional to the product of these
two quantities:
Pz
∂u ∂t
∂u ∂z
.
(81)
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Interference Polarisation Diffraction
•. Refraction
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Damping
Dispersion
F I G U R E 103 The six main properties of the motion of waves
Challenge 391 n Challenge 392 ny
When two linear wave trains collide or interfere, the total momentum is conserved throughout the collision. An important consequence of momentum conservation is that waves that are re ected by an obstacle do so with an outcoming angle equal to minus the infalling angle. What happens to the phase?
Waves, like moving bodies, carry energy and momentum. In simple terms, if you shout against a wall, the wall is hit. is hit, for example, can start avalanches on snowy mountain slopes. In the same way, waves, like bodies, can carry also angular momentum. (What type of wave is necessary for this to be possible?) However, we can distinguish six main properties that set the motion of waves apart from the motion of bodies.
— Waves can add up or cancel each other out; thus they can interpenetrate each other. ese e ects, called superposition and interference, are strongly tied to the linearity of
most waves.
— Transverse waves in three dimensions can oscillate in di erent directions: they show polarization.
— Waves, such as sound, can go around corners. is is called di raction. — Waves change direction when they change medium. is is called refraction. — Waves can have a frequency-dependent propagation speed. is is called dispersion. — O en, the wave amplitude decreases over time: waves show damping.
Material bodies in everyday life do not behave in these ways when they move. ese six
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–
Challenge 393 n
wave e ects appear because wave motion is the motion of extended entities. e famous debate whether electrons or light are waves or particles thus requires us to check whether these e ects speci c to waves can be observed or not. is is one topic of quantum theory. Before we study it, can you give an example of an observation that implies that a motion surely cannot be a wave?
As a result of having a frequency f and a propagation velocity v, all sine waves are characterized by the distance λ between two neighbouring wave crests: this distance is called the wavelength. All waves obey the basic relation
λf =v .
(82)
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In many cases the wave velocity v depends on the wavelength of the wave. For example, this is the case for water waves. is change of speed with wavelength is called dispersion. In contrast, the speed of sound in air does not depend on the wavelength (to a high degree of accuracy). Sound in air shows almost no dispersion. Indeed, if there were dispersion for sound, we could not understand each other’s speech at larger distances.
In everyday life we do not experience light as a wave, because the wavelength is only around one two-thousandth of a millimetre. But light shows all six e ects typical of wave motion. A rainbow, for example, can only be understood fully when the last ve wave e ects are taken into account. Di raction and interference can even be observed with your ngers only. Can you tell how?
Like every anharmonic oscillation, every anharmonic wave can be decomposed into sine waves. Figure gives examples. If the various sine waves contained in a disturbance propagate di erently, the original wave will change in shape while it travels. at is the reason why an echo does not sound exactly like the original sound; for the same reason, a nearby thunder and a far-away one sound di erent.
All systems which oscillate also emit waves. Any radio or TV receiver contains oscillators. As a result, any such receiver is also a (weak) transmitter; indeed, in some countries the authorities search for people who listen to radio without permission listening to the radio waves emitted by these devices. Also, inside the human ear, numerous tiny structures, the hair cells, oscillate. As a result, the ear must also emit sound. is prediction, made in by Tommy Gold, was con rmed only in by David Kemp. ese so-called otoacoustic emissions can be detected with sensitive microphones; they are presently being studied in order to unravel the still unknown workings of the ear and in order to diagnose various ear illnesses without the need for surgery.
Since any travelling disturbance can be decomposed into sine waves, the term ‘wave’ is used by physicists for all travelling disturbances, whether they look like sine waves or not. In fact, the disturbances do not even have to be travelling. Take a standing wave: is it a wave or an oscillation? Standing waves do not travel; they are oscillations. But a standing wave can be seen as the superposition of two waves travelling in opposite directions. Since all oscillations are standing waves (can you con rm this?), we can say that all oscillations are special forms of waves.
e most important travelling disturbances are those that are localized. Figure shows an example of a localized wave group or pulse, together with its decomposition into harmonic waves. Wave groups are extensively used to talk and as signals for communication.
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e properties of our environment o en disclose their full importance only when we ask simple questions. Why can we use the radio? Why can we talk on mobile phones? Why can we listen to each other? It turns out that a central part of the answer to these questions is that the space we live has an odd numbers of dimensions.
In spaces of even dimension, it is impossible to talk, because messages do not stop. is is an important result which is easily checked by throwing a stone into a lake: even a er the stone has disappeared, waves are still emitted from the point at which it entered the water. Yet, when we stop talking, no waves are emitted any more.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
We can also say that Huygens’ principle holds if the wave equation is solved by a circular wave leaving no amplitude behind it. Mathematicians translate this by requiring that the evolving delta function δ(c t − r ) satis es the wave equation, i.e. that ∂t δ = c ∆δ.
e delta function is that strange ‘function’ which is zero everywhere except at the origin, where it is in nite. A few more properties describe the precise way in which this happens.* It turns out that the delta function is a solution of the wave equation only if the space dimension is odd at least three.
In summary, the reason a room gets dark when we switch o the light, is that we live in a space with a number of dimensions which is odd and larger than one.
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A signal is the transport of information. Every signal is motion of energy. Signals can be either objects or waves. A thrown stone can be a signal, as can a whistle. Waves are a more practical form of communication because they do not require transport of matter: it is easier to use electricity in a telephone wire to transport a statement than to send a messenger. Indeed, most modern technological advances can be traced to the separation between signal and matter transport. Instead of transporting an orchestra to transmit music, we can send radio signals. Instead of sending paper letters we write email messages. Instead of going to the library we browse the internet.
e greatest advances in communication have resulted from the use of signals to transport large amounts of energy. at is what electric cables do: they transport energy without transporting any (noticeable) matter. We do not need to attach our kitchen machines to the power station: we can get the energy via a copper wire.
For all these reasons, the term ‘signal’ is o en meant to imply waves only. Voice, sound, electric signals, radio and light signals are the most common examples of wave signals.
Signals are characterized by their speed and their information content. Both quantities turn out to be limited. e limit on speed is the central topic of the theory of special relativity.
A simple limit on information content can be expressed when noting that the information ow is given by the detailed shape of the signal. e shape is characterized by a frequency (or wavelength) and a position in time (or space). For every signal – and every
* e main property is ∫ δxdx = . In mathematically precise terms, the delta ‘function’ is a distribution.
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Figure to be included
F I G U R E 104 The electrical signals measured in a nerve
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wave – there is a relation between the time-of-arrival error ∆t and the angular frequency error ∆ω:
∆t ∆ω .
(83)
Challenge 396 e
is time–frequency indeterminacy relation expresses that, in a signal, it is impossible to specify both the time of arrival and the frequency with full precision. e two errors are (within a numerical factor) the inverse of each other. (One also says that the timebandwidth product is always larger than π.) e limitation appears because on one hand one needs a wave as similar as possible to a sine wave in order to precisely determine the frequency, but on the other hand one needs a signal as narrow as possible to precisely determine its time of arrival. e contrast in the two requirements leads to the limit. e indeterminacy relation is thus a feature of every wave phenomenon. You might want to test this relation with any wave in your environment.
Similarly, there is a relation between the position error ∆x and the wave vector error ∆k = π ∆λ of a signal:
∆x ∆k .
(84)
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Like the previous case, also this indeterminacy relation expresses that it is impossible to specify both the position of a signal and its wavelength with full precision. Also this position–wave-vector indeterminacy relation is a feature of any wave phenomenon.
Every indeterminacy relation is the consequence of a smallest entity. In the case of waves, the smallest entity of the phenomenon is the period (or cycle, as it used to be called). Whenever there is a smallest unit in a natural phenomenon, an indeterminacy relation results. We will encounter other indeterminacy relations both in relativity and in quantum theory. As we will nd out, they are due to smallest entities as well.
Whenever signals are sent, their content can be lost. Each of the six characteristics of waves listed on page can lead to content degradation. Can you provide an example for each case? e energy, the momentum and all other conserved properties of signals are never lost, of course. e disappearance of signals is akin to the disappearance of motion. When motion disappears by friction, it only seems to disappear, and is in fact transformed into heat. Similarly, when a signal disappears, it only seems to disappear, and is in fact transformed into noise. (Physical) noise is a collection of numerous disordered signals, in the same way that heat is a collection of numerous disordered movements.
All signal propagation is described by a wave equation. A famous example is the equa-
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F I G U R E 105 A solitary water wave followed by a motor boat, reconstructing the discovery by Scott Russel (© Dugald Duncan)
Ref. 160
tion found by Hodgkin and Huxley. It is a realistic approximation for the behaviour of electrical potential in nerves. Using facts about the behaviour of potassium and sodium ions, they found an elaborate equation that describes the voltage V in nerves, and thus the way the signals are propagated. e equation accurately describes the characteristic voltage spikes measured in nerves, shown in Figure . e gure clearly shows that these waves di er from sine waves: they are not harmonic. Anharmonicity is one result of nonlinearity. But nonlinearity can lead to even stronger e ects.
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Ref. 161
In August , the Scottish engineer John Scott Russell ( – ) recorded a strange observation in a water canal in the countryside near Edinburgh. When a boat pulled through the channel was suddenly stopped, a strange water wave departed from it. It consisted of a single crest, about m long and . m high, moving at about m s. He followed that crest with his horse for several kilometres: the wave died out only very slowly. He did not observe any dispersion, as is usual in water waves: the width of the crest remained constant. Russell then started producing such waves in his laboratory, and extensively studied their properties. He showed that the speed depended on the amplitude, in contrast to linear waves. He also found that the depth d of the water canal was an important parameter. In fact, the speed v, the amplitude A and the width L of these single-crested waves are related by
v=
d
+A d
and L =
d A
.
(85)
As shown by these expressions, and noted by Russell, high waves are narrow and fast, whereas shallow waves are slow and wide. e shape of the waves is xed during their
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Figure to be inserted F I G U R E 106 Solitons are stable against encounters
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motion. Today, these and all other stable waves with a single crest are called solitary waves. ey appear only where the dispersion and the nonlinearity of the system exactly com-
pensate for each other. Russell also noted that the solitary waves in water channels can cross each other unchanged, even when travelling in opposite directions; solitary waves with this property are called solitons. Solitons are stable against encounters, whereas solitary waves in general are not.
Only sixty years later, in , Korteweg and de Vries found out that solitary waves in water channels have a shape described by
u(x, t) = A sech
x −vt L
where
sech x = ex + e−x ,
(86)
and that the relation found by Russell was due to the wave equation
d
∂u ∂t
+(
+
d
u)
∂u ∂x
+
d
∂u ∂x
=
.
(87)
Ref. 162
is equation for the elongation u is called the Korteweg–de Vries equation in their honour.* e surprising stability of the solitary solutions is due to the opposite e ect of the two terms that distinguish the equation from linear wave equations: for the solitary solutions, the nonlinear term precisely compensates for the dispersion induced by the thirdderivative term.
For many decades such solitary waves were seen as mathematical and physical curiosities. But almost a hundred years later it became clear that the Korteweg–de Vries equation is a universal model for weakly nonlinear waves in the weak dispersion regime, and thus of basic importance. is conclusion was triggered by Kruskal and Zabusky, who in proved mathematically that the solutions ( ) are unchanged in collisions. is discovery prompted them to introduce the term soliton. ese solutions do indeed interpenetrate one another without changing velocity or shape: a collision only produces a small positional shi for each pulse.
* e equation can be simpli ed by transforming the variable u; most concisely, it can be rewritten as ut + uxxx = uux . As long as the solutions are sech functions, this and other transformed versions of the equation
are known by the same name.
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Ref. 163 Ref. 161
Solitary waves play a role in many examples of uid ows. ey are found in ocean currents; and even the red spot on Jupiter, which was a steady feature of Jupiter photographs for many centuries, is an example.
Solitary waves also appear when extremely high-intensity sound is generated in solids. In these cases, they can lead to sound pulses of only a few nanometres in length. Solitary light pulses are also used inside certain optical communication bres, where they provide (almost) lossless signal transmission.
Towards the end of the twentieth century a second wave of interest in the mathematics of solitons arose, when quantum theorists became interested in them. e reason is simple but deep: a soliton is a ‘middle thing’ between a particle and a wave; it has features of both concepts. For this reason, solitons are now an essential part of any description of elementary particles, as we will nd out later on.
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Society is a wave. e wave moves onward, but the water of which it is composed does not.
“ Ralph Waldo Emerson, Self-Reliance. ” Oscillations, waves and signals are a limitless source of fascination.
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Challenge 398 e
When the frequency of a tone is doubled, one says that the tone is higher by an octave. Two tones that di er by an octave, when played together, sound pleasant to the ear. Two other agreeable frequency ratios – or ‘intervals’, as musicians say – are quarts and quints. What are the corresponding frequency ratios? (Note: the answer was one of the oldest discoveries in physics; it is attributed to Pythagoras, around 500 .)
Challenge 399 n
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An orchestra is playing music in a large hall. At a distance of m, somebody is listening to the music. At a distance of km, another person is listening to the music via the radio. Who hears the music rst?
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What is the period of a simple pendulum, i.e. a mass m attached to a massless string of Challenge 400 ny length l? What is the period if the string is much longer than the radius of the Earth?
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What path is followed by a body moving in a plane, but attached by a spring to a xed Challenge 401 n point on the plane?
Challenge 402 e
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Light is a wave, as we will discover later on. As a result, light reaching the Earth from space is refracted when it enters the atmosphere. Can you con rm that as a result, stars appear somewhat higher in the night sky than they really are?
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air
air
water
coin
F I G U R E 107 Shadows and refraction
Ref. 164
What are the highest sea waves? is question has been researched systematically only recently, using satellites. e surprising result is that sea waves with a height of m and more are common: there are a few such waves on the oceans at any given time. is result
con rms the rare stories of experienced ship captains and explains many otherwise ship
sinkings. Surfers may thus get many chances to ride m waves. ( e record is just below this
size.) But maybe the most impressive waves to surf are those of the Pororoca, a series of m waves that move from the sea into the Amazonas River every spring, against the ow
of the river. ese waves can be surfed for tens of kilometres.
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All waves are damped, eventually. is e ect is o en frequency-dependent. Can you Challenge 403 n provide a con rmation of this dependence in the case of sound in air?
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Challenge 404 e
When you make a hole with a needle in black paper, the hole can be used as a magnifying lens. (Try it.) Di raction is responsible for the lens e ect. By the way, the di raction of light by holes was noted by Francesco Grimaldi in the seventeenth century; he deduced that light is a wave. His observations were later discussed by Newton, who wrongly dismissed them.
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Put a empty cup near a lamp, in such a way that the bottom of the cup remains in the shadow. When you ll the cup with water, some of the bottom will be lit, because of the refraction of the light from the lamp. e same e ect allows us to build lenses. e same e ect is at the basis of instruments such as the telescope.
** Challenge 405 n Are water waves transverse or longitudinal?
**
e speed of water waves limits the speeds of ships. A surface ship cannot travel (much)
faster than about vcrit = . l , where = . m s , l is its length, and 0.16 is a number determined experimentally, called the critical Froude number. is relation is valid for all vessels, from large tankers (l = m gives vcrit = m s) down to ducks (l = . m gives vcrit = . m s). e critical speed is that of a wave with the same wavelength as the ship. In fact, moving at higher speeds than the critical value is possible, but requires
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 406 n
much more energy. (A higher speed is also possible if the ship surfs on a wave.) erefore all water animals and ships are faster when they swim below the surface – where the limit due to surface waves does not exist – than when they swim on the surface. For example, ducks can swim three times as fast under water than on the surface.
How far away is the olympic swimming record from the critical value?
**
e group velocity of water waves (in deep water) is less than the velocity of the individual waves. As a result, when a group of wave crests travels, within the group the crests move from the back to the front, appearing at the back, travelling forward and then dying out at the front.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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One can hear the distant sea or a distant highway more clearly in the evening than in the morning. is is an e ect of refraction. Sound speed decreases with temperature. In the evening, the ground cools more quickly than the air above. As a result, sound leaving the ground and travelling upwards is refracted downwards, leading to the long hearing distance. In the morning, usually the air is cold above and warm below. Sound is refracted upwards, and distant sound does not reach a listener on the ground. Refraction thus implies that mornings are quiet, and that one can hear more distant sounds in the evenings. Elephants use the sound situation during evenings to communicate over distances of more than km. ( ey also use sound waves in the ground to communicate, but that is another story.)
Challenge 407 e Ref. 165
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Refraction also implies that there is a sound channel in the ocean, and in the atmosphere. Sound speed decreases with temperature, and increases with pressure. At an ocean depth of km, or at an atmospheric height of 13 to km (that is at the top of the tallest cumulonimbus clouds or equivalently, at the middle of the ozone layer) sound has minimal speed. As a result, sound that starts from that level and tries to leave is channelled back to it. Whales use the sound channel to communicate with each other with beautiful songs; one can nd recordings of these songs on the internet. e military successfully uses microphones placed at the sound channel in the ocean to locate submarines, and microphones on balloons in the atmospheric channel to listen for nuclear explosions. (In fact, sound experiments conducted by the military are the main reason why whales are deafened and lose their orientation, stranding on the shores. Similar experiments in the air with high-altitude balloons are o en mistaken for ying saucers, as in the famous Roswell incident.)
Ref. 166
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Much smaller also animals communicate by sound waves. In 2003, it was found that herring communicate using noises they produce when farting. When they pass wind, the gas creates a ticking sound whose frequency spectrum reaches up to kHz. One can even listen to recordings of this sound on the internet. e details of the communication, such as the di erences between males and females, are still being investigated. It is possible that the sounds may also be used by predators to detect herring, and they might even by
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used by future shing vessels.
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On windy seas, the white wave crests have several important e ects. e noise stems from tiny exploding and imploding water bubbles. e noise of waves on the open sea is thus the superposition of many small explosions. At the same time, white crests are the events where the seas absorb carbon dioxide from the atmosphere, and thus reduce global warming.
** Challenge 408 n Why are there many small holes in the ceilings of many o ce buildings?
**
Which quantity determines the wavelength of water waves emitted when a stone is Challenge 409 ny thrown into a pond?
Ref. 3 Challenge 410 n Challenge 411 n Challenge 412 ny Challenge 413 n
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Yakov Perelman lists the following four problems in his delightful physics problem book. (1) A stone falling into a lake produces circular waves. What is the shape of waves
produced by a stone falling into a river, where the water ows in one direction? (2) It is possible to build a lens for sound, in the same way as it is possible to build
lenses for light. What would such a lens look like? (3) What is the sound heard inside a shell? (4) Light takes about eight minutes to travel from the Sun to the Earth. What con-
sequence does this have for a sunrise?
Challenge 414 n
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Can you describe how a Rubik’s Cube is built? And its generalizations to higher numbers of segments? Is there a limit to the number of segments? ese puzzles are even tougher than the search for a rearrangement of the cube. Similar puzzles can be found in the study of many mechanisms, from robots to textile machines.
Challenge 415 ny
**
Typically, sound produces a pressure variation of − bar on the ear. How is this determined?
e ear is indeed a sensitive device. It is now known that most cases of sea mammals, like whales, swimming onto the shore are due to ear problems: usually some military device (either sonar signals or explosions) has destroyed their ear so that they became deaf and lose orientation.
Ref. 167
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Infrasound, inaudible sound below Hz, is a modern topic of research. In nature, infrasound is emitted by earthquakes, volcanic eruptions, wind, thunder, waterfalls, falling meteorites and the surf. Glacier motion, seaquakes, avalanches and geomagnetic storms also emit infrasound. Human sources include missile launches, tra c, fuel engines and air compressors.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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It is known that high intensities of infrasound lead to vomiting or disturbances of the sense of equilibrium ( dB or more for 2 minutes), and even to death ( dB for 10 minutes). e e ects of lower intensities on human health are not yet known.
Infrasound can travel several times around the world before dying down, as the explosion of the Krakatoa volcano showed in 1883. With modern infrasound detectors, sea surf can be detected hundreds of kilometres away. Infrasound detectors are even used to count meteorites at night. Very rarely, meteorites can be heard with the human ear.
Ref. 168
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e method used to deduce the sine waves contained in a signal, as shown in Figure 101, is called the Fourier transformation. It is of importance throughout science and technology. In the 1980s, an interesting generalization became popular, called the wavelet transformation. In contrast to Fourier transformations, wavelet transformations allow us to localize signals in time. Wavelet transformations are used to compress digitally stored images in an e cient way, to diagnose aeroplane turbine problems, and in many other applications.
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If you like engineering challenges, here is one that is still open. How can one make a Challenge 416 r robust and e cient system that transforms the energy of sea waves into electricity?
Challenge 417 r
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In our description of extended bodies, we assumed that each spot of a body can be followed separately throughout its motion. Is this assumption justi ed? What would happen if it were not?
**
A special type of waves appears in explosions and supersonic ight: shock waves. In a shock wave, the density or pressure of a gas changes abruptly, on distances of a few micrometers. Studying shock waves is a research eld in itself; shock waves determine the
ight of bullets, the snapping of whips and the e ects of detonations.
Ref. 169 Challenge 418 e
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Bats y at night using echolocation. Dolphins also use it. Sonar, used by shing vessels to look for sh, copies the system of dolphins. Less well known is that humans have the same ability. Have you ever tried to echolocate a wall in a completely dark room? You will be surprised at how easily this is possible. Just make a loud hissing or whistling noise that stops abruptly, and listen to the echo. You will be able to locate walls reliably.
D
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We have just discussed the motion of extended bodies in some detail. We have seen that extended bodies show wave motion. But are extended bodies found in nature? Strangely enough, this question has been one of the most intensely discussed questions in physics. Over the centuries, it has reappeared again and again, at each improvement of the description of motion; the answer has alternated between the a rmative and the negative. Many thinkers have been imprisoned, and many still are being persecuted, for giving answers
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n = 1
n = 2
n = 5 n = ∞
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 108 Floors and mountains as fractals (© Paul Martz)
that are not politically correct! In fact, the issue already arises in everyday life.
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Challenge 419 n
Whenever we climb a mountain, we follow the outline of its shape. We usually describe this outline as a curved two-dimensional surface. In everyday life we nd that this is a good approximation. But there are alternative possibilities. e most popular is the idea that mountains are fractal surfaces. A fractal was de ned by Benoit Mandelbrot as a set that is self-similar under a countable but in nite number of magni cation values.* We have already encountered fractal lines. An example of an algorithm for building a (random) fractal surface is shown on the right side of Figure . It produces shapes which look remarkably similar to real mountains. e results are so realistic that they are used in Hollywood movies. If this description were correct, mountains would be extended, but not continuous.
But mountains could also be fractals of a di erent sort, as shown in the le side of Figure . Mountain surfaces could have an in nity of small and smaller holes. In fact, one could also imagine that mountains are described as three-dimensional versions of the le side of the gure. Mountains would then be some sort of mathematical Swiss cheese. Can you devise an experiment to decide whether fractals provide the correct description for mountains? To settle the issue, a chocolate bar can help.
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?
From a drop of water a logician could predict an Atlantic or a Niagara.
“ Arthur Conan Doyle, A Study in Scarlet ” Any child knows how to make a chocolate bar last forever: eat half the remainder every
day. However, this method only works if matter is scale-invariant. In other words, the
* For a de nition of uncountability, see page 649.
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Page 53 Challenge 420 n
method only works if matter is either fractal, as it then would be scale-invariant for a discrete set of zoom factors, or continuous, in which case it would be scale-invariant for any zoom factor. Which case, if either, applies to nature?
We have already encountered a fact making continuity a questionable assumption: continuity would allow us, as Banach and Tarski showed, to multiply food and any other matter by clever cutting and reassembling. Continuity would allow children to eat the same amount of chocolate every day, without ever buying a new bar. Matter is thus not continuous. Now, fractal chocolate is not ruled out in this way; but other experiments settle the question. Indeed, we note that melted materials do not take up much smaller volumes than solid ones. We also nd that even under the highest pressures, materials do not shrink. us matter is not a fractal. What then is its structure?
To get an idea of the structure of matter we can take uid chocolate, or even just some oil – which is the main ingredient of chocolate anyway – and spread it out over a large surface. For example, we can spread a drop of oil onto a pond on a day without rain or wind; it is not di cult to observe which parts of the water are covered by the oil and which are not. A small droplet of oil cannot cover a surface larger than – can you guess the value? Trying to spread the lm further inevitably rips it apart. e child’s method of prolonging chocolate thus does not work for ever: it comes to a sudden end. e oil experiment shows that there is a minimum thickness of oil lms, with a value of about
nm. is simple experiment can even be conducted at home; it shows that there is a smallest size in matter. Matter is made of tiny components. is con rms the observations made by Joseph Loschmidt* in , who was the rst person to measure the size of the components of matter.** In , it was not a surprise that matter was made of small components, as the existence of a smallest size – but not its value – had already been deduced by Galileo, when studying some other simple questions.***
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
* Joseph Loschmidt (b. 1821 Putschirn, d. 1895 Vienna) Austrian chemist and physicist. e oil experiment
was popularized a few decades later, by Kelvin. It is o en claimed that Benjamin Franklin was the rst to
conduct the oil experiment; that is wrong. Franklin did not measure the thickness, and did not even consider
the question of the thickness. He did pour oil on water, but missed the most important conclusion that could
be drawn from it. Even geniuses do not discover everything.
** Loschmidt knew that the (dynamic) viscosity of a gas was given by η = ρlv , where ρ is the density
of the gas, v the average speed of the components and l their mean free path. With Avogadro’s prediction
(made in 1811 without specifying any value) that a volume V of any gas always contains the same number
N of components, one also has l = V πNσ , where σ is the cross-section of the components. ( e cross-
section is the area of the shadow of an object.) Loschmidt then assumed that when the gas is lique ed, the
volume of the liquid is the sum of the volumes of the particles. He then measured all the involved quantities and determined N. e modern value of N, called Avogadro’s number or Loschmidt’s number, is . ë particles in . l of any gas at standard conditions (today called mol).
*** Galileo was brought to trial because of his ideas about atoms, not about the motion of the Earth, as
is o en claimed. To get a clear view of the matters of dispute in the case of Galileo, especially those of
interest to physicists, the best text is the excellent book by P
R
, Galileo eretico, Einaudi,
1983, translated into English as Galileo Heretic, Princeton University Press, 1987. It is also available in many
other languages. Redondi, a renowned historical scholar and colleague of Pierre Costabel, tells the story of
the dispute between Galileo and the reactionary parts of the Catholic Church. He discovered a document
of that time – the anonymous denunciation which started the trial – that allowed him to show that the
condemnation of Galileo to life imprisonment for his views on the Earth’s motion was organized by his
friend the Pope to protect him from a sure condemnation to death over a di erent issue.
e reasons for his arrest, as shown by the denunciation, were not his ideas on astronomy and on the
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Ref. 171 Challenge 421 n
Fleas can jump to heights a hundred times their size, humans only to heights about their own size. In fact, biological studies yield a simple observation: most animals, regardless of their size, achieve about the same jumping height of between . and . m, whether they are humans, cats, grasshoppers, apes, horses or leopards. e explanation of this fact takes only two lines. Can you nd it?
e above observation seems to be an example of scale invariance. But there are some interesting exceptions at both ends of the mass range. At the small end, mites and other small insects do not achieve such heights because, like all small objects, they encounter the problem of air resistance. At the large end, elephants do not jump that high, because doing so would break their bones. But why do bones break at all?
Why are all humans of about the same size? Why are there no giant adults with a height of ten metres? Why aren’t there any land animals larger than elephants? e answer yields the key to understanding the structure of matter. In fact, the materials of which we are made would not allow such changes of scale, as the bones of giants would collapse under the weight they have to sustain. Bones have a nite strength because their constituents stick to each other with a nite attraction. Continuous matter – which exists only in cartoons – could not break at all, and fractal matter would be in nitely fragile. Matter breaks under nite loads because it is composed of small basic constituents.
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F
Challenge 422 ny
e gentle lower slopes of Motion Mountain are covered by trees. Trees are fascinating structures. Take their size. Why do trees have limited size? Already in the sixteenth century, Galileo knew that it is not possible to increase tree height without limits: at some point a tree would not have the strength to support its own weight. He estimated the maximum height to be around m; the actual record, unknown to him at the time, seems to be m, for the Australian tree Eucalyptus regnans. But why does a limit exist at all? e answer is the same as for bones: wood has a nite strength because it is not scale invariant; and it is not scale invariant because it is made of small constituents, namely atoms.*
In fact, the derivation of the precise value of the height limit is more involved. Trees must not break under strong winds. Wind resistance limits the height-to-thickness ratio h d to about for normal-sized trees (for . m < d < m). Can you say why? inner
Ref. 172
motion of the Earth, but his statements on matter. Galileo defended the view that since matter is not scale invariant, it must be made of ‘atoms’ or, as he called them, piccolissimi quanti – smallest quanta. is was and still is a heresy. A true Catholic is still not allowed to believe in atoms. Indeed, the theory of atoms is not compatible with the change of bread and wine into human esh and blood, called transsubstantiation, which is a central tenet of the Catholic faith. In Galileo’s days, church tribunals punished heresy, i.e. deviating personal opinions, by the death sentence. Despite being condemned to prison in his trial, Galileo published his last book, written as an old man under house arrest, on the scaling issue. Today, the Catholic Church still refuses to publish the proceedings and other documents of the trial. Its o cials carefully avoid the subject of atoms, as any statement on this subject would make the Catholic Church into a laughing stock. In fact, quantum theory, named a er the term used by Galileo, has become the most precise description of nature yet. * ere is another important limiting factor: the water columns inside trees must not break. Both factors seem to yield similar limiting heights.
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lamp
•.
three monoatomic steps
eye
F I G U R E 109 Atoms exist: rotating an aluminium rod leads to brightness oscillations
photograph to be included
F I G U R E 110 Atomic steps in broken gallium arsenide crystals can be seen under a light microscope
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Ref. 173 Challenge 423 n
Ref. 174
Challenge 424 ny
trees are limited in height to less than m by the requirement that they return to the vertical a er being bent by the wind.
Such studies of natural constraints also answer the question of why trees are made from wood and not, for example, from steel. You could check for yourself that the maximum height of a column of a given mass is determined by the ratio E ρ between the elastic module and the square of the mass density. Wood is actually the material for which this ratio is highest. Only recently have material scientists managed to engineer slightly better ratios with bre composites.
Why do materials break at all? All observations yield the same answer and con rm Galileo’s reasoning: because there is a smallest size in materials. For example, bodies under stress are torn apart at the position at which their strength is minimal. If a body were completely homogeneous, it could not be torn apart; a crack could not start anywhere. If a body had a fractal Swiss-cheese structure, cracks would have places to start, but they would need only an in nitesimal shock to do so.
A simple experiment that shows that solids have a smallest size is shown in Figure . A cylindrical rod of pure, single crystal aluminium shows a surprising behaviour when it is illuminated from the side: its brightness depends on how the rod is oriented, even though it is completely round. is angular dependence is due to the atomic arrangement of the aluminium atoms in the rod.
It is not di cult to con rm experimentally the existence of smallest size in solids. It is su cient to break a single crystal, such as a gallium arsenide wafer, in two. e breaking surface is either completely at or shows extremely small steps, as shown in Figure .
ese steps are visible under a normal light microscope. (Why?) It turns out that all the step heights are multiples of a smallest height: its value is about . nm. e existence of a smallest height, corresponding to the height of an atom, contradicts all possibilities of scale invariance in matter.
T
Climbing the slopes of Motion Mountain, we arrive in a region of the forest covered with deep snow. We stop for a minute and look around. It is dark; all the animals are asleep; there is no wind and there are no sources of sound. We stand still, without breathing, and
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listen to the silence. (You can have this experience also in a sound studio such as those used for musical recordings, or in a quiet bedroom at night.) In situations of complete silence, the ear automatically becomes more sensitive*; we then have a strange experience. We hear two noises, a lower- and a higher-pitched one, which are obviously generated inside the ear. Experiments show that the higher note is due to the activity of the nerve cells in the inner ear. e lower note is due to pulsating blood streaming through the head. But why do we hear a noise at all?
Many similar experiments con rm that whatever we do, we can never eliminate noise from measurements. is unavoidable type of noise is called shot noise in physics. e statistical properties of this type of noise actually correspond precisely to what would be expected if ows, instead of being motions of continuous matter, were transportation of a large number of equal, small and discrete entities. us, simply listening to noise proves that electric current is made of electrons, that air and liquids are made of molecules, and that light is made of photons. In a sense, the sound of silence is the sound of atoms. Shot noise would not exist in continuous systems.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
L
I prefer knowing the cause of a single thing to being king of Persia.
“ Democritus ” Precise observations show that matter is neither continuous nor a fractal: matter is made
of smallest basic particles. Galileo, who deduced their existence by thinking about giants and trees, called them ‘smallest quanta.’ Today they are called ‘atoms’, in honour of a famous argument of the ancient Greeks. Indeed, years ago, the Greeks asked the following question. If motion and matter are conserved, how can change and transformation exist? e philosophical school of Leucippus and Democritus of Abdera** studied two particular observations in special detail. ey noted that salt dissolves in water. ey also noted that sh can swim in water. In the rst case, the volume of water does not increase when the salt is dissolved. In the second case, when sh advance, they must push water aside. ey deduced that there is only one possible explanation that satis es observations and also reconciles conservation and transformation: nature is made of void and of small, hard, indivisible and conserved particles.*** In this way any example of motion, change or transformation is due to rearrangements of these particles; change and conservation are reconciled.
Challenge 425 d
* e human ear can detect pressure variations at least as small as µPa. ** Leucippus of Elea (Λευκιππος) (c. 490 to c. 430 ), Greek philosopher; Elea was a small town south of Naples. It lies in Italy, but used to belong to the Magna Graecia. Democritus (∆εµοκριτος) of Abdera (c. 460 to c. 356 or 370 ), also a Greek philosopher, was arguably the greatest philosopher who ever lived. Together with his teacher Leucippus, he was the founder of the atomic theory; Democritus was a much admired thinker, and a contemporary of Socrates. e vain Plato never even mentions him, as Democritus was a danger to his own fame. Democritus wrote many books which have been lost; they were not copied during the Middle Ages because of his scienti c and rational world view, which was felt to be a danger by religious zealots who had the monopoly on the copying industry. *** e story is told by Lucrece, or Titus Lucretius Carus, in his famous text De natura rerum, around 50
. Especially if we imagine particles as little balls, we cannot avoid calling this a typically male idea. (What would be the female approach?)
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light detector
laser diode
vertical
piezo
controller
tip
horizontal piezo controller
sample
F I G U R E 111 The principle, and a simple realization, of an atomic force microscope
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Page 240 Ref. 175, Ref. 176
Ref. 177
Ref. 178
Ref. 179
In short, since matter is hard, has a shape and is divisible, Leucippus and Democritus imagined it as being made of atoms. Atoms are particles which are hard, have a shape, but are indivisible. In other words, the Greeks imagined nature as a big Lego set. Lego pieces are rst of all hard or impenetrable, i.e. repulsive at very small distances. ey are attractive at small distances: they remain stuck together. Finally, they have no interaction at large distances. Atoms behave in the same way. (Actually, what the Greeks called ‘atoms’ partly corresponds to what today we call ‘molecules’. e latter term was invented by Amadeo Avogadro in in order to clarify the distinction. But we can forget this detail for the moment.)
Since atoms are so small, it took many years before all scientists were convinced by the experiments showing their existence. In the nineteenth century, the idea of atoms was beautifully veri ed by the discovery of the ‘laws’ of chemistry and those of gas behaviour. Later on, the noise e ects were discovered.
Nowadays, with advances in technology, single atoms can be seen, photographed, hologrammed, counted, touched, moved, li ed, levitated, and thrown around. And indeed, like everyday matter, atoms have mass, size, shape and colour. Single atoms have even been used as lamps and lasers.
Modern researchers in several elds have fun playing with atoms in the same way that children play with Lego. Maybe the most beautiful demonstration of these possibilities is provided by the many applications of the atomic force microscope. If you ever have the opportunity to see one, do not miss it!* It is a simple device which follows the surface of an object with an atomically sharp needle; such needles, usually of tungsten, are easily manufactured with a simple etching method. e changes in the height of the needle along its path over the surface are recorded with the help of a de ected light ray. With a little care, the atoms of the object can be felt and made visible on a computer screen. With special types of such microscopes, the needle can be used to move atoms one by one to speci ed places on the surface. It is also possible to scan a surface, pick up a given atom and throw it towards a mass spectrometer to determine what sort of atom it is.
* A cheap version costs only a few thousand euro, and will allow you to study the di erence between a silicon
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F I G U R E 112 The atoms on the surface of a silicon crystal mapped with an atomic force microscope
F I G U R E 113 The result of moving helium atoms on a metallic surface (© IBM)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Incidentally, the construction of atomic force microscopes is only a small improvement on what nature is building already by the millions; when we use our ears to listen, we are actually detecting changes in eardrum position of about nm. In other words, we all have two ‘atomic force microscopes’ built into our heads.
In summary, matter is not scale invariant: in particular, it is neither smooth nor fractal. Matter is made of atoms. Di erent types of atoms, as well as their various combinations, produce di erent types of substances. Pictures from atomic force microscopes show that the size and arrangement of atoms produce the shape and the extension of objects, con-
rming the Lego model of matter.* As a result, the description of the motion of extended objects can be reduced to the description of the motion of their atoms. Atomic motion will be a major theme in the following pages. One of its consequences is especially important: heat.
C
Before we continue, a few puzzles are due. motion of extended bodies encompasses.
ey indicate the range of phenomena that the
Challenge 426 e
**
How much water is necessary to moisten the air in a room in winter? At °C, the vapour pressure of water is mbar, °C it is mbar. As a result, heating air in the winter gives at most a humidity of 25%. To increase the humidity by 50%, one thus needs about 1 litre of water per m .
**
Page 921
wafer – crystalline – a our wafer – granular-amorphous – and consecrated wafer. * Studying matter in even more detail yields the now well-known idea that matter, at higher and higher magni cations, is made of molecules, atoms, nuclei, protons and neutrons, and nally, quarks. Atoms also contain electrons. A nal type of matter, neutrinos, is observed coming from the Sun and from certain types of radioactive materials. Even though the fundamental bricks have become smaller with time, the basic idea remains: matter is made of smallest entities, nowadays called elementary particles. In the second part of our mountain ascent we will explore this idea in detail. Appendix C lists the measured properties of all known elementary particles.
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You are in a boat on a pond with a stone, a bucket of water and a piece of wood. What happens to the water level of the pond a er you throw the stone in it? A er you throw Challenge 427 n the water into the pond? A er you throw the piece of wood?
Challenge 428 n
**
What is the maximum length of a vertically hanging wire? Could a wire be lowered from a suspended geostationary satellite down to the Earth? is would mean we could realize a space ‘li ’. How long would the cable have to be? How heavy would it be? How would you build such a system? What dangers would it face?
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**
Matter is made of atoms. Over the centuries the stubborn resistance of many people to this idea has lead to the loss of many treasures. For over a thousand years, people thought that genuine pearls could be distinguished from false ones by hitting them with a hammer: only false pearls would break. However, all pearls break. (Also diamonds break in this situation.) As a result, all the most beautiful pearls in the world have been smashed to pieces.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
**
Put a rubber air balloon over the end of a bottle and let it hang inside the bottle. How Challenge 429 e much can you blow up the balloon inside the bottle?
**
Put a small paper ball into the neck of a horizontal bottle and try to blow it into the bottle. Challenge 430 e e paper will y towards you. Why?
**
It is possible to blow an egg from one egg-cup to a second one just behind it. Can you Challenge 431 e perform this trick?
Challenge 432 n
**
In the seventeenth century, engineers who needed to pump water faced a challenge. To pump water from mine sha s to the surface, no water pump managed more than m of height di erence. For twice that height, one always needed two pumps in series, connected by an intermediate reservoir. Why? How then do trees manage to pump water upwards for larger heights?
Challenge 433 e
**
Comic books have di culties with the concept of atoms. Could Asterix really throw Romans into the air using his st? Are Lucky Luke’s precise revolver shots possible? Can Spiderman’s silk support him in his swings from building to building? Can the Roadrunner stop running in three steps? Can the Sun be made to stop in the sky by command? Can space-ships hover using fuel? Take any comic-book hero and ask yourself whether matter made of atoms would allow him the feats he seems capable of. You will nd that most cartoons are comic precisely because they assume that matter is not made of atoms, but continuous! In a sense, atoms make life a serious adventure.
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water
F I G U R E 114 What is your personal stone-skipping record?
Challenge 434 n
**
When hydrogen and oxygen are combined to form water, the amount of hydrogen needed is exactly twice the amount of oxygen, if no gas is to be le over a er the reaction. How does this observation con rm the existence of atoms?
**
Challenge 435 n How are alcohol- lled chocolate pralines made? Note that the alcohol is not injected into them a erwards, because there would be no way to keep the result tight enough.
**
Ref. 180 Challenge 436 r
How o en can a stone jump when it is thrown over the surface of water? e present world record was achieved in 2002: 40 jumps. More information is known about the previous world record, achieved in 1992: a palm-sized, triangular and at stone was thrown with a speed of m s (others say m s) and a rotation speed of about 14 revolutions per second along a river, covering about m with 38 jumps. ( e sequence was lmed with a video recorder from a bridge.)
What would be necessary to increase the number of jumps? Can you build a machine that is a better thrower than yourself?
**
e biggest component of air is nitrogen (about 78 %). Challenge 437 n oxygen (about 21 %). What is the third biggest one?
e second biggest component is
**
Water can ow uphill: Heron’s fountain shows this most clearly. Heron of Alexandria (c. 10 to c. 70) described it 2000 years ago; it is easily built at home, using some plastic Challenge 438 n bottles and a little tubing. How does it work?
Challenge 439 n
**
A light bulb is placed, underwater, in a stable steel cylinder with a diameter of cm. A Fiat Cinquecento ( kg) is placed on a piston pushing onto the water surface. Will the bulb resist?
** Challenge 440 ny What is the most dense gas? e most dense vapour?
** Every year, the Institute of Maritime Systems of the University of Rostock organizes a
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 115 Heron’s fountain
Challenge 441 e
contest. e challenge is to build a paper boat with the highest carrying capacity. e paper boat must weigh at most g; the carrying capacity is measured by pouring lead small shot onto it, until the boat sinks. e 2002 record stands at . kg. Can you achieve
this value? (For more information, see the http://www.paperboat.de website.)
**
Challenge 442 n
A modern version of an old question – already posed by Daniel Colladon (1802–1893) – is the following. A ship of mass m in a river is pulled by horses walking along the riverbank attached by ropes. If the river is of super uid helium, meaning that there is no friction
between ship and river, what energy is necessary to pull the ship upstream along the river until a height h has been gained?
Challenge 443 n
**
e Swiss professor Auguste Piccard (1884–1962) was a famous explorer of the stratosphere. He reached a height of km in his aerostat. Inside the airtight cabin hanging under his balloon, he had normal air pressure. However, he needed to introduce several ropes attached at the balloon into the cabin, in order to be able to pull them, as they controlled his balloon. How did he get the ropes into the cabin while preventing air from leaving the cabin?
Challenge 444 n
**
A human cannot breathe at any depth under water, even if he has a tube going to the surface. At a few metres of depth, trying to do so is inevitably fatal! Even at a depth of
cm only, the human body can only breathe in this way for a few minutes. Why?
**
A human in air falls with a limiting speed of about
Challenge 445 ny long does it take to fall from
m to m?
km h, depending on clothing. How
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
**
Several humans have survived free falls from aeroplanes for a thousand metres or more, Challenge 446 n even though they had no parachute. How was this possible?
**
Liquid pressure depends on height. If the average human blood pressure at the height of Challenge 447 n the heart is . kPa, can you guess what it is inside the feet when standing?
Challenge 448 n
**
e human heart pumps blood at a rate of about . l s. A capillary has the diameter of a red blood cell, around µm, and in it the blood moves at a speed of half a millimetre per second. How many capillaries are there in a human?
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**
Ref. 181
A few drops of tea usually ow along the underside of the spout of a teapot (or fall onto the table). is phenomenon has even been simulated using supercomputer simulations of the motion of liquids, by Kistler and Scriven, using the Navier–Stokes equations. Teapots are still shedding drops, though.
Challenge 449 n
**
e best giant soap bubbles can be made by mixing . l of water, ml of corn syrup and ml of washing-up liquid. Mix everything together and then let it rest for four hours. You can then make the largest bubbles by dipping a metal ring of up to mm diameter into the mixture. But why do soap bubbles burst?
**
Challenge 450 n
Can humans start earthquakes? What would happen if all the 1000 million Indians were to jump at the same time from the kitchen table to the oor?
In fact, several strong earthquakes have been triggered by humans. is has happened when water dams have been lled, or when water has been injected into drilling holes. It has been suggested that the extraction of deep underground water also causes earthquakes. If this is con rmed, a sizeable proportion of all earthquakes could be humantriggered.
**
Challenge 451 n How can a tip of a stalactite be distinguished from a tip of a stalagmite? Does the di erence exist also for icicles?
Challenge 452 ny
**
A drop of water that falls into a pan containing hot oil dances on the surface for a considerable time, if the oil is above °C. Cooks test the temperature of oil in this way. Why does this so-called Leidenfrost e ect* take place?
**
* It is named a er Johann Gottlieb Leidenfrost (1715–1794), German physician.
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h H
F I G U R E 116 Which funnel is faster?
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
How much more weight would your bathroom scales show if you stood on them in a Challenge 453 n vacuum?
** Challenge 454 n Why don’t air molecules fall towards the bottom of the container and stay there?
**
Challenge 455 n Which of the two water funnels in Figure 116 is emptied more rapidly? Apply energy Ref. 182 conservation to the uid’s motion (also called Bernoulli’s ‘law’) to nd the answer.
**
As we have seen, fast ow generates an underpressure. How do sh prevent their eyes Challenge 456 n from popping when they swim rapidly?
Challenge 457 ny
**
Golf balls have dimples for the same reasons that tennis balls are hairy and that shark and dolphin skin is not at: deviations from atness reduce the ow resistance because many small eddies produce less friction than a few large ones. Why?
**
One of the most complex extended bodies is the human body. In modern simulations of the behaviour of humans in car accidents, the most advanced models include ribs, vertebrae, all other bones and the various organs. For each part, its speci c deformation properties are taken into account. With such models and simulations, the protection of passengers and drivers in cars can be optimized.
Challenge 458 n
**
Glass is a solid. Nevertheless, many textbooks say that glass is a liquid. is error has been propagated for about a hundred years, probably originating from a mistranslation of a sentence in a German textbook published in 1933 by Gustav Tamman, Der Glaszustand. Can you give at least three reasons why glass is a solid and not a liquid?
** e recognized record height reached by a helicopter is
m above sea level, though
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 459 n
m has also been claimed. ( e rst height was reached in 1972, the second in 2002, both by French pilots in French helicopters.) Why, then, do people still continue to use their legs in order to reach the top of Mount Sagarmatha, the highest mountain in the world?
**
A loosely knotted sewing thread lies on the surface of a bowl lled with water. Putting a bit of washing-up liquid into the area surrounded by the thread makes it immediately Challenge 460 e become circular. Why?
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Challenge 461 n
**
e deepest hole ever drilled into the Earth is km deep. In 2003, somebody proposed to enlarge such a hole and then to pour millions of tons of liquid iron into it. He claims that the iron would sink towards the centre of the Earth. If a measurement device communication were dropped into the iron, it could send its observations to the surface using sound waves. Can you give some reasons why this would not work?
** Challenge 462 n How can you put a handkerchief under water using a glass, while keeping it dry?
**
Are you able to blow a ping pong ball out of a funnel? What happens if you blow through a funnel towards a burning candle?
**
e economic power of a nation has long been associated with its capacity to produce high-quality steel. Indeed, the Industrial Revolution started with the mass production of steel. Every scientist should know the basics facts about steel. Steel is a combination of iron and carbon to which other elements, mostly metals, may be added as well. One can distinguish three main types of steel, depending on the crystalline structure. Ferritic steels have a body-centred cubic structure, austenitic steels have a face-centred cubic structure, and martensitic steels have a body-centred tetragonal structure. Table 28 gives further details.
Ref. 183 Challenge 463 ny
**
A simple phenomenon which requires a complex explanation is the cracking of a whip. Since the experimental work of Peter Krehl it has been known that the whip cracks when the tip reaches a velocity of twice the speed of sound. Can you imagine why?
**
e fall of a leaf, with its complex path, is still a topic of investigation. We are far from being able to predict the time a leaf will take to reach the ground; the motion of the air around a leaf is not easy to describe. One of the simplest phenomena of hydrodynamics remains one of its most di cult problems.
**
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•.
TA B L E 28 Steel types, properties and uses
F
A
M
‘usual’ steel body centred cubic (bcc) iron and carbon
Examples construction steel
car sheet steel ship steel 12 % Cr stainless ferrite
Properties phases described by the iron-carbon phase diagram
in equilibrium at RT
mechanical properties and grain size depend on heat treatment
hardened by reducing grain size, by forging, by increasing carbon content or by nitration grains of ferrite and paerlite, with cementite (Fe C) ferromagnetic
‘so ’ steel face centred cubic (fcc) iron, chromium, nickel, manganese, carbon
hardened steel, brittle body centred tetragonal (bct) carbon steel and alloys
most stainless (18/8 Cr/Ni) steels
kitchenware
food industry
Cr/V steels for nuclear reactors
knife edges
drill surfaces spring steel, cranksha s
phases described by the Schae er diagram
phases described by the iron-carbon diagram and the TTT (time–temperature transformation) diagram
some alloys in equilibrium at not in equilibrium at RT, but
RT
stable
mechanical properties and grain size depend on thermo-mechanical pre-treatment
mechanical properties and grain size strongly depend on heat treatment
hardened by cold working only
hard anyway – made by laser irradiation, induction heating, etc.
grains of austenite
grains of martensite
not magnetic or weakly magnetic
ferromagnetic
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Ref. 184
Fluids exhibit many interesting e ects. Soap bubbles in air are made of a thin spherical lm of liquid with air on both sides. In 1932, anti-bubbles, thin spherical lms of air with
liquid on both sides, were rst observed. In 2004, the Belgian physicist Stéphane Dorbolo and his team showed that it is possible to produce them in simple experiments, and in particular, in Belgian beer.
**
A bicycle chain is an extended object with no sti ness. However, if it is made to rotate rapidly, it gets dynamical sti ness, and can roll down an inclined plane. is surprising e ect can be seen on the http://www.iwf.de/Navigation/Projekte/LNW/Pohl/index.asp website.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
TA B L E 29 Extensive quantities in nature, i.e. quantities that flow and accumulate
D
E
C
I
-E
R
(
(
)
(
(
)(
)
)
-
)
Rivers
mass m
Gases
volume V
Mechanics momentum p
mass ow m t height
P= hm t
di erence h
volume ow V t pressure p P = pV t
force F = dp dt velocity v P = v F
Rm = ht m [m s kg]
RV = pt V [kg s m ]
Rp = t m [s kg]
angular
torque
momentum L M = dL dt
angular
P =ωM
velocity ω
Chemistry amount of substance n
substance ow chemical P = µ In
In = dn dt
potential µ
ermo- entropy S dynamics
entropy ow IS = dS dt
temperature P = T IS T
Light
like all massless radiation, it can ow but cannot accumulate
RL = t mr [s kg m ]
Rn = µt n [Js mol ]
RS = Tt S [K W]
Electricity charge q
electrical current electrical P = U I
I = dq dt
potential U
R=U I [Ω]
Magnetism no accumulable magnetic sources are found in nature
Nuclear physics
extensive quantities exist, but do not appear in everyday life
Gravitation empty space can move and ow, but the motion is not observed in everyday life
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Ref. 185 Ref. 186
**
Mechanical devices are not covered in this text. ere is a lot of progress in the area even at present. For example, people have built robots that are able to ride a unicycle. But even the physics of human unicycling is not simple.
W
?
Before we continue to the next way to describe motion globally, we will have a look at the possibilities of motion in everyday life. One overview is given in Table . e domains that belong to everyday life – motion of uids, of matter, of matter types, of heat, of light and of charge – are the domains of continuum physics.
Within continuum physics, there are three domains we have not yet studied: the mo-
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•.
Ref. 187
tion of charge and light, called electrodynamics, the motion of heat, called thermodynamics, and the motion of the vacuum. Once we have explored these domains, we will have completed the rst step of our description of motion: continuum physics. In continuum physics, motion and moving entities are described with continuous quantities that can take any value, including arbitrarily small or arbitrarily large values.
But nature is not continuous. We have already seen that matter cannot be inde nitely divided into ever-smaller entities. In fact, we will discover that there are precise experiments that provide limits to the observed values for every domain of continuum physics.
ere is a limit to mass, to speed, to angular momentum, to force, to entropy and to change of charge. e consequences of these discoveries form the second step in our description of motion: quantum theory and relativity. Quantum theory is based on lower limits; relativity is based on upper limits. e third and last step of our description of motion will be formed by the uni cation of quantum theory and general relativity.
Every domain of physics, regardless of which one of the above steps it belongs to, describes change in terms two quantities: energy, and an extensive quantity characteristic of the domain. An observable quantity is called extensive if it increases with system size. Table provides an overview. e intensive and extensive quantities corresponding to what in everyday language is called ‘heat’ are temperature and entropy.
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H
?
Ref. 188
We continue our short stroll through the eld of global descriptions of motion with an overview of heat and the main concepts associated with it. For our purposes we only need to know the basic facts about heat. e main points that are taught in school are almost su cient.
Macroscopic bodies, i.e. bodies made of many atoms, have temperature. e temperature of a macroscopic body is an aspect of its state. It is observed that any two bodies in contact tend towards the same temperature: temperature is contagious. In other words, temperature describes an equilibrium situation. e existence and contagiousness of temperature is o en called the zeroth principle of thermodynamics. Heating is the increase of temperature.
How is temperature measured? e eighteenth century produced the clearest answer: temperature is best de ned and measured by the expansion of gases. For the simplest, socalled ideal gases, the product of pressure p and volume V is proportional to temperature:
pV T .
(88)
Ref. 189 Page 1154
e proportionality constant is xed by the amount of gas used. (More about it shortly.) e ideal gas relation allows us to determine temperature by measuring pressure and volume. is is the way (absolute) temperature has been de ned and measured for about a century. To de ne the unit of temperature, one only has to x the amount of gas used. It is customary to x the amount of gas at mol; for oxygen this is g. e proportionality constant, called the ideal gas constant R, is de ned to be R = . J mol K. is number has been chosen in order to yield the best approximation to the independently de ned Celsius temperature scale. Fixing the ideal gas constant in this way de nes K, or one Kelvin, as the unit of temperature. In simple terms, a temperature increase of one Kelvin
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Challenge 464 ny
is de ned as the temperature increase that makes the volume of an ideal gas increase – keeping the pressure xed – by a fraction of / . or . %.
In general, if one needs to determine the temperature of an object, one takes a mole of gas, puts it in contact with the object, waits a while, and then measures the pressure and the volume of the gas. e ideal gas relation ( ) then gives the temperature. Most importantly, the ideal gas relation shows that there is a lowest temperature in nature, namely that temperature at which an ideal gas would have a vanishing volume. at would happen at T = K, i.e. at − . °C. Obviously, other e ects, like the volume of the atoms themselves, prevent the volume of the gas from ever reaching zero. e third principle of thermodynamics provides another reason why this is impossible.
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T
Ref. 190
e temperature achieved by a civilization can be used as a measure of its technological
achievements. One can de ne the Bronze Age ( . kK,
) , the Iron Age ( . kK,
), the Electric Age ( kK from c. ) and the Atomic Age (several MK, from
) in this way. Taking into account also the quest for lower temperatures, one can
de ne the Quantum Age ( K, starting ).
Heating implies ow of energy. For example, friction heats up and slows down mov-
ing bodies. In the old days, the ‘creation’ of heat by friction was even tested experiment-
ally. It was shown that heat could be generated from friction, just by continuous rubbing,
without any limit; this ‘creation’ implies that heat is not a material uid extracted from
the body – which in this case would be consumed a er a certain time – but something
else. Indeed, today we know that heat, even though it behaves in some ways like a uid, is
due to disordered motion of particles. e conclusion of these studies is simple. Friction
is the transformation of mechanical energy into thermal energy.
To heat kg of water by K by friction, . kJ of mechanical energy must be trans-
formed through friction. e rst to measure this quantity with precision was, in ,
the German physician Julius Robert Mayer ( – ). He regarded his experiment as
proof of the conservation of energy; indeed, he was the rst person to state energy con-
servation! It is something of an embarrassment to modern physics that a medical doctor
was the rst to show the conservation of energy, and furthermore, that he was ridiculed
by most physicists of his time. Worse, conservation of energy was accepted only when
it was repeated many years later by two authorities: Hermann von Helmholtz – himself
also a physician turned physicist – and William omson, who also cited similar, but
later experiments by James Joule.* All of them acknowledged Mayer’s priority. Publicity
by William omson eventually led to the naming of the unit of energy a er Joule.
In short, the sum of mechanical energy and thermal energy is constant. is is usu-
ally called the rst principle of thermodynamics. Equivalently, it is impossible to produce
mechanical energy without paying for it with some other form of energy. is is an im-
portant statement, because among others it means that humanity will stop living one day.
* Hermann von Helmholtz (b. 1821 Potsdam, d. 1894 Berlin), important Prussian scientist. William omson (later William Kelvin) (1824–1907), important Irish physicist. James Prescott Joule (1818–1889), English physicist. Joule is pronounced so that it rhymes with ‘cool’, as his descendants like to stress. ( e pronunciation of the name ‘Joule’ varies from family to family.)
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•.
TA B L E 30 Some temperature values
O
T
Lowest, but unattainable, temperature In the context of lasers, it sometimes makes sense to talk about negative temperature. Temperature a perfect vacuum would have at Earth’s surface
Page 871
Sodium gas in certain laboratory experiments – coldest matter system achieved by man and possibly in the universe Temperature of neutrino background in the universe Temperature of photon gas background (or background radiation) in the universe Liquid helium Oxygen triple point Liquid nitrogen Coldest weather ever measured (Antarctic) Freezing point of water at standard pressure Triple point of water Average temperature of the Earth’s surface Smallest uncomfortable skin temperature Interior of human body Hottest weather measured Boiling point of water at standard pressure Liquid bronze Liquid, pure iron Freezing point of gold Light bulb lament Earth’s centre Sun’s surface Air in lightning bolt Hottest star’s surface (centre of NGC 2240) Space between Earth and Moon (no typo) Sun’s centre Inside the JET fusion tokamak Centre of hottest stars Maximum temperature of systems without electron–positron pair generation Universe when it was s old Hagedorn temperature Heavy ion collisions – highest man-made value Planck temperature – nature’s upper temperature limit
K = − . °C
zK
. nK
c. K .K
.K .K K K = − °C . K = . °C . K = . °C .K K ( K above normal) . . K = . . °C K = °C . K or . °C c. K
K .K . kK kK . kK kK kK up to MK MK MK GK ca. GK
GK . TK up to . TK
K
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Challenge 465 n Page 255
Indeed, we live mostly on energy from the Sun; since the Sun is of nite size, its energy content will eventually be consumed. Can you estimate when this will happen?
ere is also a second (and the mentioned third) principle of thermodynamics, which will be presented later on. e study of these topics is called thermostatics if the systems concerned are at equilibrium, and thermodynamics if they are not. In the latter case, we distinguish situations near equilibrium, when equilibrium concepts such as temperature can still be used, from situations far from equilibrium, such as self-organization, where such concepts o en cannot be applied.
Does it make sense to distinguish between thermal energy and heat? It does. Many older texts use the term ‘heat’ to mean the same as thermal energy. However, this is confusing; in this text, ‘heat’ is used, in accordance with modern approaches, as the everyday term for entropy. Both thermal energy and heat ow from one body to another, and both accumulate. Both have no measurable mass.* Both the amount of thermal energy and the amount of heat inside a body increase with increasing temperature. e precise relation will be given shortly. But heat has many other interesting properties and stories to tell. Of these, two are particularly important: rst, heat is due to particles; and secondly, heat is at the heart of the di erence between past and future. ese two stories are intertwined.
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Ref. 187
– It’s irreversible. – Like my raincoat!
“ Mel Brooks, Spaceballs, ” Every domain of physics describes change in terms of two quantities: energy, and an ex-
tensive quantity characteristic of the domain. Even though heat is related to energy, the quantity physicists usually call heat is not an extensive quantity. Worse, what physicists call heat is not the same as what we call heat in our everyday speech. e extensive quantity corresponding to what we call ‘heat’ in everyday speech is called entropy.** Entropy describes heat in the same way as momentum describes motion. When two objects di ering in temperature are brought into contact, an entropy ow takes place between them, like the ow of momentum that take place when two objects of di erent speeds collide. Let us de ne the concept of entropy more precisely and explore its properties in some more detail.
Entropy measures the degree to which energy is mixed up inside a system, that is, the degree to which energy is spread or shared among the components of a system. erefore, entropy adds up when identical systems are composed into one. When two litre bottles of water at the same temperature are poured together, the entropy of the water adds up.
Like any other extensive quantity, entropy can be accumulated in a body; it can ow into or out of bodies. When water is transformed into steam, the entropy added into the water is indeed contained in the steam. In short, entropy is what is called ‘heat’ in
Page 314
* is might change in future, when mass measurements improve in precision, thus allowing the detection of relativistic e ects. In this case, temperature increase may be detected through its related mass increase. However, such changes are noticeable only with twelve or more digits of precision in mass measurements. ** e term ‘entropy’ was invented by the German physicist Rudolph Clausius (1822–1888) in 1865. He formed it from the Greek ἐν ‘in’ and τρόπος ‘direction’, to make it sound similar to ‘energy’. It has always had the meaning given here.
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TA B L E 31 Some measured entropy values
P
/S
E
Melting of kg of ice Water under standard conditions Boiling of kg of liquid water at . kPa Iron under standard conditions Oxygen under standard conditions
. kJ K kg = . J K mol . J K mol . kJ K= J K mol . J K mol . J K mol
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Challenge 467 ny
everyday speech. In contrast to several other important extensive quantities, entropy is not conserved. e sharing of energy in a system can be increased, for example by heating it. However,
entropy is ‘half conserved’: in closed systems, entropy does not decrease; mixing cannot be undone. What is called equilibrium is simply the result of the highest possible mixing. In short, the entropy in a closed system increases until it reaches the maximum possible value.
When a piece of rock is detached from a mountain, it falls, tumbles into the valley, heating up a bit, and eventually stops. e opposite process, whereby a rock cools and tumbles upwards, is never observed. Why? e opposite motion does not contradict any rule or pattern about motion that we have deduced so far.
Rocks never fall upwards because mountains, valleys and rocks are made of many particles. Motions of many-particle systems, especially in the domain of thermostatics, are called processes. Central to thermostatics is the distinction between reversible processes, such as the ight of a thrown stone, and irreversible processes, such as the aforementioned tumbling rock. Irreversible processes are all those processes in which friction and its generalizations play a role. ey are those which increase the sharing or mixing of energy. ey are important: if there were no friction, shirt buttons and shoelaces would not stay fastened, we could not walk or run, co ee machines would not make co ee, and maybe most importantly of all, we would have no memory.
Irreversible processes, in the sense in which the term is used in thermostatics, transform macroscopic motion into the disorganized motion of all the small microscopic components involved: they increase the sharing and mixing of energy. Irreversible processes are therefore not strictly irreversible – but their reversal is extremely improbable. We can say that entropy measures the ‘amount of irreversibility’: it measures the degree of mixing or decay that a collective motion has undergone.
Entropy is not conserved. Entropy – ‘heat’ – can appear out of nowhere, since energy sharing or mixing can happen by itself. For example, when two di erent liquids of the same temperature are mixed – such as water and sulphuric acid – the nal temperature of the mix can di er. Similarly, when electrical current ows through material at room temperature, the system can heat up or cool down, depending on the material.
e second principle of thermodynamics states that ‘entropy ain’t what it used to be.’ More precisely, the entropy in a closed system tends towards its maximum. Here, a closed system is a system that does not exchange energy or matter with its environment. Can you think of an example?
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Entropy never decreases. Everyday life shows that in a closed system, the disorder increases with time, until it reaches some maximum. To reduce disorder, we need e ort, i.e. work and energy. In other words, in order to reduce the disorder in a system, we need to connect the system to an energy source in some clever way. Refrigerators need electrical current precisely for this reason.
Because entropy never decreases, white colour does not last. Whenever disorder increases, the colour white becomes ‘dirty’, usually grey or brown. Perhaps for this reason white objects, such as white clothes, white houses and white underwear, are valued in our society. White objects defy decay.
Entropy allows to de ne the concept of equilibrium more precisely as the state of maximum entropy, or maximum energy sharing.
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F
We know from daily experience that transport of an extensive quantity always involves friction. Friction implies generation of entropy. In particular, the ow of entropy itself produces additional entropy. For example, when a house is heated, entropy is produced in the wall. Heating means to keep a temperature di erence ∆T between the interior and the exterior of the house. e heat ow J traversing a square metre of wall is given by
J = κ∆T = κ(Ti − Te)
(89)
where κ is a constant characterizing the ability of the wall to conduct heat. While con-
ducting heat, the wall also produces entropy. e entropy production σ is proportional to
the di erence between the interior and the exterior entropy ows. In other words, one
has
σ
=
J Te
−
J Ti
=
κ
(Ti − Te) Ti Te
.
(90)
Note that we have assumed in this calculation that everything is near equilibrium in each slice parallel to the wall, a reasonable assumption in everyday life. A typical case of a good wall has κ = W m K in the temperature range between K and K. With this value, one gets an entropy production of
σ= ë − W m K.
(91)
Challenge 468 ny
Can you compare the amount of entropy that is produced in the ow with the amount that is transported? In comparison, a good goose-feather duvet has κ = . W m K, which in shops is also called tog.*
ere are two other ways, apart from heat conduction, to transport entropy: convection, used for heating houses, and radiation, which is possible also through empty space. For
* at unit is not as bad as the o cial (not a joke) BthU ë h sq cm °F used in some remote provinces of
our galaxy.
e insulation power of materials is usually measured by the constant λ = κd which is independent of the thickness d of the insulating layer. Values in nature range from about W K m for diamond, which is the best conductor of all, down to between . W K m and . W K m for wood, between . W K m and . W K m for wools, cork and foams, and the small value of ë − W K m for krypton gas.
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F I G U R E 117 The basic idea of statistical mechanics about gases
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example, the Earth radiates about . W m K into space, in total thus about . PW K. e entropy is (almost) the same that the Earth receives from the Sun. If more entropy
had to be radiated away than received, the temperature of the surface of the Earth would have to increase. is is called the greenhouse e ect. (It is also called global warming.) Let’s hope that it remains small in the near future.
D
?
Challenge 469 n
In all our discussions so far, we have assumed that we can distinguish the system under investigation from its environment. But do such isolated or closed systems, i.e. systems not interacting with their environment, actually exist? Probably our own human condition was the original model for the concept: we do experience having the possibility to act independently of our environment. An isolated system may be simply de ned as a system not exchanging any energy or matter with its environment. For many centuries, scientists saw no reason to question this de nition.
e concept of an isolated system had to be re ned somewhat with the advent of quantum mechanics. Nevertheless, the concept provides useful and precise descriptions of nature also in that domain. Only in the third part of our walk will the situation change drastically. ere, the investigation of whether the universe is an isolated system will lead to surprising results. (What do you think?)* We’ll take the rst steps towards the answer shortly.
W
?–T
Heat properties are material-dependent. Studying them should therefore enable us to understand something about the constituents of matter. Now, the simplest materials of all are gases.** Gases need space: an amount of gas has pressure and volume. Indeed, it did not take long to show that gases could not be continuous. One of the rst scientists to think about gases as made up of atoms was Daniel Bernoulli.*** Bernoulli reasoned
* A strange hint: your answer is almost surely wrong. ** By the way, the word gas is a modern construct. It was coined by the Brussels alchemist and physician Johan Baptista van Helmont (1579–1644), to sound similar to ‘chaos’. It is one of the few words which have been invented by one person and then adopted all over the world. *** Daniel Bernoulli (b. 1700 Bâle, d. 1782 Bâle), important Swiss mathematician and physicist. His father Johann and his uncle Jakob were famous mathematicians, as were his brothers and some of his nephews. Daniel Bernoulli published many mathematical and physical results. In physics, he studied the separation
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F I G U R E 118 Which balloon wins?
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Challenge 470 ny
that if atoms are small particles, with mass and momentum, he should be able to make quantitative predictions about the behaviour of gases, and check them with experiment. If the particles y around in a gas, then the pressure of a gas in a container is produced by the steady ow of particles hitting the wall. It was then easy to conclude that if the particles are assumed to behave as tiny, hard and perfectly elastic balls, the pressure p, volume V and temperature T must be related by
pV = k NT
(92)
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Challenge 471 ny Challenge 472 n Challenge 473 e
Challenge 474 n
where N is the number of particles contained in the gas. ( e Boltzmann constant k, one of the fundamental constants of nature, is de ned below.) A gas made of particles with such textbook behaviour is called an ideal gas. Relation ( ) has been con rmed by experiments at room and higher temperatures, for all known gases.
Bernoulli thus derived the gas relation, with a speci c prediction for the proportionality constant, from the single assumption that gases are made of small massive constituents. is derivation provides a clear argument for the existence of atoms and for their behaviour as normal, though small objects. (Can you imagine how N might be determined experimentally?)
e ideal gas model helps us to answer questions such as the one illustrated in Figure . Two identical rubber balloons, one lled up to a larger size than the other, are connected via a pipe and a valve. Daniel Bernoulli
e valve is opened. Which one de ates? e ideal gas relation states that hotter gases, at given pressure, need more volume.
e relation thus explains why winds and storms exist, why hot air balloons rise, why car engines work, why the ozone layer is destroyed by certain gases, or why during the extremely hot summer of in the south of Turkey, oxygen maks were necessary to walk outside around noon.
Now you can take up the following challenge: how can you measure the weight of a car or a bicycle with a ruler only?
of compound motion into translation and rotation. In 1738 he published the Hydrodynamique, in which he deduced all results from a single principle, namely the conservation of energy. e so-called Bernoulli’s principle states that (and how) the pressure of a uid decreases when its speed increases. He studied the tides and many complex mechanical problems, and explained the Boyle–Mariotte gas law. For his publications he won the prestigious prize of the French Academy of Sciences – a forerunner of the Nobel Prize – ten times.
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Ref. 192
Ref. 193 Page 844, page 847
e picture of gases as being made of hard constituents without any long-distance interactions breaks down at very low temperatures. However, the ideal gas relation ( ) can be improved to overcome these limitations by taking into account the deviations due to interactions between atoms or molecules. is approach is now standard practice and allows us to measure temperatures even at extremely low values. e e ects observed below K, such as the solidi cation of air, frictionless transport of electrical current, or frictionless ow of liquids, form a fascinating world of their own, the beautiful domain of low-temperature physics; it will be explored later on.
B
Ref. 194 Challenge 475 ny
It is easy to observe, under a microscope, that small particles (such as pollen) in a liquid never come to rest. ey seem to follow a random zigzag movement. In , the English botanist Robert Brown ( – ) showed with a series of experiments that this observation is independent of the type of particle and of the type of liquid. In other words, Brown had discovered a fundamental noise in nature. Around , this motion was attributed to the molecules of the liquid colliding with the particles. In and , Marian von Smoluchowski and, independently, Albert Einstein argued that this theory could be tested experimentally, even though at that time nobody was able to observe molecules directly. e test makes use of the speci c properties of thermal noise.
It had already been clear for a long time that if molecules, i.e. indivisible matter particles, really existed, then heat had to be disordered motion of these constituents and temperature had to be the average energy per degree of freedom of the constituents. Bernoulli’s model of Figure implies that for monoatomic gases the kinetic energy Tkin per particle is given by
Tkin = kT
(93)
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Challenge 476 ny
where T is temperature. e so-called Boltzmann constant k = . ë − J K is the standard conversion factor between temperature and energy.*At a room temperature of K, the kinetic energy is thus zJ.
Using relation ( ) to calculate the speed of air molecules at room temperature yields values of several hundred metres per second. Why then does smoke from a candle take so long to di use through a room? Rudolph Clausius ( – ) answered this question in the mid-nineteenth century: di usion is slowed by collisions with air molecules, in the same way as pollen particles collide with molecules in liquids.
At rst sight, one could guess that the average distance the pollen particle has moved a er n collisions should be zero, because the molecule velocities are random. However, this is wrong, as experiment shows.
* e important Austrian physicist Ludwig Boltzmann (b. 1844 Vienna, d. 1906 Duino) is most famous for his work on thermodynamics, in which he explained all thermodynamic phenomena and observables, including entropy, as results of the behaviour of molecules. Planck named the Boltzmann constant a er his investigations. He was one of the most important physicists of the late nineteenth century and stimulated many developments that led to quantum theory. It is said that Boltzmann committed suicide partly because of the resistance of the scienti c establishment to his ideas. Nowadays, his work is standard textbook material.
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probability density evolution
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F I G U R E 119 Example paths for particles in Brownian motion and their displacement distribution
Challenge 477 ny
An average square displacement, written d , is observed for the pollen particle. It cannot be predicted in which direction the particle will move, but it does move. If the distance the particle moves a er one collision is l, the average square displacement a er n collisions is given, as you should be able to show yourself, by
d = nl .
(94)
For molecules with an average velocity v over time t this gives
d = nl = vlt .
(95)
Ref. 195 Challenge 478 d
In other words, the average square displacement increases proportionally with time. Of course, this is only valid if the liquid is made of separate molecules. Repeatedly measuring the position of a particle should give the distribution shown in Figure for the probability that the particle is found at a given distance from the starting point. is is called the (Gaussian) normal distribution. In , Jean Perrin* performed extensive experiments in order to test this prediction. He found that equation ( ) corresponded completely with observations, thus convincing everybody that Brownian motion is indeed due to collisions with the molecules of the surrounding liquid, as Smoluchowski and Einstein had predicted.** Perrin received the Nobel Prize for these experiments.
Einstein also showed that the same experiment could be used to determine the number of molecules in a litre of water (or equivalently, the Boltzmann constant k). Can you work out how he did this?
Ref. 196 Page 259
* Jean Perrin (1870–1942), important French physicist, devoted most of his career to the experimental proof of the atomic hypothesis and the determination of Avogadro’s number; in pursuit of this aim he perfected the use of emulsions, Brownian motion and oil lms. His Nobel Prize speech (http://nobelprize.org/physics/ laureates/1926/perrin-lecture.html) tells the interesting story of his research. He wrote the in uential book Les atomes and founded the Centre National de la Recherche Scienti que. He was also the rst to speculate, in 1901, that an atom is similar to a small solar system. ** In a delightful piece of research, Pierre Gaspard and his team showed in 1998 that Brownian motion is also chaotic, in the strict physical sense given later on.
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TA B L E 32 Some typical entropy values per particle at standard temperature and pressure as multiples of the Boltzmann constant
M
E
Monoatomic solids Diamond Graphite Lead Monoatomic gases Helium Radon Diatomic gases Polyatomic solids Polyatomic liquids Polyatomic gases Icosane
0.3 k to 10 k 0.29 k 0.68 k 7.79 k 15-25 k 15.2 k 21.2 k 15 k to 30 k 10 k to 60 k 10 k to 80 k 20 k to 60 k 112 k
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Once it had become clear that heat and temperature are due to the motion of microscopic particles, people asked what entropy was microscopically. e answer can be formulated in various ways. e two most extreme answers are:
— Entropy is the expected number of yes-or-no questions, multiplied by k ln , the answers of which would tell us everything about the system, i.e. about its microscopic state.
— Entropy measures the (logarithm of the) number W of possible microscopic states. A given macroscopic state can have many microscopic realizations. e logarithm of this number, multiplied by the Boltzmann constant k, gives the entropy.*
In short, the higher the entropy, the more microstates are possible. rough either of these de nitions, entropy measures the quantity of randomness in a system. In other words, it measures the transformability of energy: higher entropy means lower transformability. Alternatively, entropy measures the freedom in the choice of microstate that a system has. High entropy means high freedom of choice for the microstate. For example, when a molecule of glucose (a type of sugar) is produced by photosynthesis, about bits of entropy are released. is means that a er the glucose is formed, additional yes-or-no questions must be answered in order to determine the full microscopic state of the system. Physicists o en use a macroscopic unit; most systems of interest are large, and thus an entropy of bits is written as J K.**
Challenge 479 ny
* When Max Planck went to Austria to search for the anonymous tomb of Boltzmann in order to get him buried in a proper grave, he inscribed the formula S = k ln W on the tombstone. (Which physicist would
nance the tomb of another, nowadays?) ** is is only approximate. Can you nd the precise value?
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Ref. 197
Ref. 198 Challenge 480 ny
To sum up, entropy is thus a speci c measure for the characterization of disorder of thermal systems. ree points are worth making here. First of all, entropy is not the measure of disorder, but one measure of disorder. It is therefore not correct to use entropy as a synonym for the concept of disorder, as is o en done in the popular literature. Entropy is only de ned for systems that have a temperature, in other words, only for systems that are in or near equilibrium. (For systems far from equilibrium, no measure of disorder has been found yet; probably none is possible.) In fact, the use of the term entropy has degenerated so much that sometimes one has to call it thermodynamic entropy for clarity.
Secondly, entropy is related to information only if information is de ned also as −k ln W. To make this point clear, take a book with a mass of one kilogram. At room temperature, its entropy content is about kJ K. e printed information inside a book, say pages of lines with each containing characters out of possibilities, corresponds to an entropy of ë − J K. In short, what is usually called ‘information’ in everyday life is a negligible fraction of what a physicist calls information. Entropy is de ned using the physical concept of information.
Finally, entropy is also not a measure for what in normal life is called the complexity of a situation. In fact, nobody has yet found a quantity describing this everyday notion.
e task is surprisingly di cult. Have a try! In summary, if you hear the term entropy used with a di erent meaning than S = k ln W, beware. Somebody is trying to get you, probably with some ideology.
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T
–
Before we complete our discussion of thermostatics we must point out in another way the importance of the Boltzmann constant k. We have seen that this constant appears whenever the granularity of matter plays a role; it expresses the fact that matter is made of small basic entities. e most striking way to put this statement is the following: ere is a smallest entropy in nature. Indeed, for all systems, the entropy obeys
S k.
(96)
Ref. 199 Ref. 200
is result is almost years old; it was stated most clearly (with a di erent numerical factor) by the Hungarian–German physicist Leo Szilard. e same point was made by the French physicist Léon Brillouin (again with a di erent numerical factor). e statement can also be taken as the de nition of the Boltzmann constant.
e existence of a smallest entropy in nature is a strong idea. It eliminates the possibility of the continuity of matter and also that of its fractality. A smallest entropy implies that matter is made of a nite number of small components. e limit to entropy expresses the fact that matter is made of particles.* e limit to entropy also shows that Galilean physics cannot be correct: Galilean physics assumes that arbitrarily small quantities do exist. e entropy limit is the rst of several limits to motion that we will encounter until we nish the second part of our ascent. A er we have found all limits, we can start the third and nal part, leading to uni cation.
* e minimum entropy implies that matter is made of tiny spheres; the minimum action, which we will encounter in quantum theory, implies that these spheres are actually small clouds.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
e existence of a smallest quantity implies a limit on the precision of measurement. Measurements cannot have in nite precision. is limitation is usually stated in the form of an indeterminacy relation. Indeed, the existence of a smallest entropy can be rephrased as an indeterminacy relation between the temperature T and the inner energy U of a system:
∆ T ∆U k .
(97)
Ref. 201 Page 1079
Ref. 203 Ref. 200
Page 704
is relation* was given by Niels Bohr; it was discussed by Werner Heisenberg, who called it one of the basic indeterminacy relations of nature. e Boltzmann constant (divided by ) thus xes the smallest possible entropy value in nature. For this reason, Gilles Cohen-Tannoudji calls it the quantum of information and Herbert Zimmermann calls it the quantum of entropy.
e relation ( ) points towards a more general pattern. For every minimum value for an observable, there is a corresponding indeterminacy relation. We will come across this several times in the rest of our adventure, most importantly in the case of the quantum of action and Heisenberg’s indeterminacy relation.
e existence of a smallest entropy has numerous consequences. First of all, it sheds light on the third principle of thermodynamics. A smallest entropy implies that absolute zero temperature is not achievable. Secondly, a smallest entropy explains why entropy values are nite instead of in nite. irdly, it xes the absolute value of entropy for every system; in continuum physics, entropy, like energy, is only de ned up to an additive constant. e entropy limit settles all these issues.
e existence of a minimum value for an observable implies that an indeterminacy relation appears for any two quantities whose product yields that observable. For example, entropy production rate and time are such a pair. Indeed, an indeterminacy relation connects the entropy production rate P = dS dt and the time t:
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∆P ∆t k .
(98)
Ref. 203, Ref. 202 Challenge 481 ny
From this and the previous relation ( ) it is possible to deduce all of statistical physics, i.e., the precise theory of thermostatics and thermodynamics. We will not explore this further here. (Can you show that the zeroth principle follows from the existence of a smallest entropy?) We will limit ourselves to one of the cornerstones of thermodynamics: the second principle.
W
’
?
Page 45
It’s a poor sort of memory which only works backwards.
“ Lewis Carroll, Alice in Wonderland ” When we rst discussed time, we ignored the di erence between past and future. But ob-
viously, a di erence exists, as we do not have the ability to remember the future. is is
Ref. 202 * It seems that the historical value for the right hand side, given by k, has to be corrected to k .
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TA B L E 33 Some minimum flow values found in nature
O
M
Matter ow
one molecule, one atom or one particle
Volume ow
one molecule, one atom or one particle
Momentum ow
Planck’s constant divided by wavelength
Angular momentum ow Planck’s constant
Chemical amount of substance one molecule, one atom or one particle
Entropy ow
minimum entropy
Charge ow
elementary charge
Light ow
Planck’s constant divided by wavelength
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Challenge 482 ny Challenge 483 ny
not a limitation of our brain alone. All the devices we have invented, such as tape recorders, photographic cameras, newspapers and books, only tell us about the past. Is there a way to build a video recorder with a ‘future’ button? Such a device would have to solve a deep problem: how would it distinguish between the near and the far future? It does not take much thought to see that any way to do this would con ict with the second principle of thermodynamics. at is unfortunate, as we would need precisely the same device to show that there is faster-than-light motion. Can you nd the connection?
In summary, the future cannot be remembered because entropy in closed systems tends towards a maximum. Put even more simply, memory exists because the brain is made of many particles, and so the brain is limited to the past. However, for the most simple types of motion, when only a few particles are involved, the di erence between past and future disappears. For few-particle systems, there is no di erence between times gone by and times approaching. We could say that the future di ers from the past only in our brain, or equivalently, only because of friction. erefore the di erence between the past and the future is not mentioned frequently in this walk, even though it is an essential part of our human experience. But the fun of the present adventure is precisely to overcome our limitations.
I
?
Ref. 204 Page 233
A physicist is the atom’s way of knowing about atoms.
“ George Wald ” Historically, the study of statistical mechanics has been of fundamental importance for
physics. It provided the rst demonstration that physical objects are made of interacting particles. e story of this topic is in fact a long chain of arguments showing that all the properties we ascribe to objects, such as size, sti ness, colour, mass density, magnetism, thermal or electrical conductivity, result from the interaction of the many particles they consist of. e discovery that all objects are made of interacting particles has o en been called the main result of modern science.
How was this discovery made? Table listed the main extensive quantities used in physics. Extensive quantities are able to ow. It turns out that all ows in nature are com-
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Ref. 205
Challenge 484 ny Page 1046
posed of elementary processes, as shown in Table . We have seen that the ow of mass,
volume, charge, entropy and substance are composed. Later, quantum theory will show
the same for the ow of linear and angular momentum. All ows are made of particles.
is success of this idea has led many people to generalize it to the statement:
‘Everything we observe is made of parts.’ is approach has been applied with success
to chemistry with molecules, materials science and geology with crystals, electricity with
electrons, atoms with elementary particles, space with points, time with instants, light
with photons, biology with cells, genetics with genes, neurology with neurons, mathem-
atics with sets and relations, logic with elementary propositions, and even to linguist-
ics with morphemes and phonemes. All these sciences have ourished on the idea that
everything is made of related parts. e basic idea seems so self-evident that we nd it
di cult even to formulate an alternative. Just try!
However, in the case of the whole of nature, the idea that nature is a sum of related
parts is incorrect. It turns out to be a prejudice, and a prejudice so entrenched that it
retarded further developments in physics in the latter decades of the twentieth century. In
particular, it does not apply to elementary particles or to space-time. Finding the correct
description for the whole of nature is the biggest challenge of our adventure, as it requires
a complete change in thinking habits. ere is a lot of fun ahead.
“Jede Aussage über Komplexe läßt sich in eine Aussage über deren Bestandteile und in diejenigen Sätze zerlegen, welche die Komplexe
vollständig beschreiben.*
W
Ludwig Wittgenstein, Tractatus, .
,
”
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Page 242
e exploration of temperature yields another interesting result. Researchers rst studied gases, and measured how much energy was needed to heat them by K. e result is simple: all gases share only a few values, when the number of molecules N is taken into account. Monoatomic gases (in a container with constant volume) require N k , diatomic gases (and those with a linear molecule) N k , and almost all other gases N k, where k = . ë − J K is the Boltzmann constant.
e explanation of this result was soon forthcoming: each thermodynamic degree of freedom** contributes the energy kT to the total energy, where T is the temperature. So the number of degrees of freedom in physical bodies is nite. Bodies are not continuous, nor are they fractals: if they were, their speci c thermal energy would be in nite. Matter is indeed made of small basic entities.
All degrees of freedom contribute to the speci c thermal energy. At least, this is what classical physics predicts. Solids, like stones, have thermodynamic degrees of freedom and should show a speci c thermal energy of N k. At high temperatures, this is indeed observed. But measurements of solids at room temperature yield lower values, and the
Ref. 206
* Every statement about complexes can be resolved into a statement about their constituents and into the propositions that describe the complexes completely. ** A thermodynamic degree of freedom is, for each particle in a system, the number of dimensions in which it can move plus the number of dimensions in which it is kept in a potential. Atoms in a solid have six, particles in monoatomic gases have only three; particles in diatomic gases or rigid linear molecules have
ve. e number of degrees of freedom of larger molecules depends on their shape.
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match head
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 120 The fire pump
lower the temperature, the lower the values become. Even gases show values lower than those just mentioned, when the temperature is su ciently low. In other words, molecules and atoms behave di erently at low energies: atoms are not immutable little hard balls.
e deviation of these values is one of the rst hints of quantum theory.
C
Even though heat is disordered motion, it follows simple rules. Some of them are surprising.
**
Compression of air increases its temperature. is is shown directly by the re pump, a variation of a bicycle pump, shown in Figure 120. (For a working example, see the website http://www.tn.tudel .nl/cdd). A match head at the bottom of an air pump made of transparent material is easily ignited by the compression of the air above it. e temperature of the air a er compression is so high that the match head ignites spontaneously.
**
If heat really is disordered motion of atoms, a big problem appears. When two atoms collide head-on, in the instant of smallest distance, neither atom has velocity. Where does the kinetic energy go? Obviously, it is transformed into potential energy. But that implies that atoms can be deformed, that they have internal structure, that they have parts, and thus that they can in principle be split. In short, if heat is disordered atomic motion, atoms are not indivisible! In the nineteenth century this argument was put forward in order to show that heat cannot be atomic motion, but must be some sort of uid. But since we know that heat really is kinetic energy, atoms must indeed be divisible, even though their name means ‘indivisible’. We do not need an expensive experiment to show this.
Challenge 485 n
**
Not only gases, but also most other materials expand when the temperature rises. As a result, the electrical wires supported by pylons hang much lower in summer than in winter. True?
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1
2
3
F I G U R E 121 Can you boil water in this paper cup?
4
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**
Ref. 207 e following is a famous Fermi problem. Given that a human corpse cools down in four Challenge 486 ny hours a er death, what is the minimum number of calories needed per day in our food?
**
e energy contained in thermal motion is not negligible. A g bullet travelling at the speed of sound has a kinetic energy of only . kcal.
Challenge 487 n How does a typical,
** m hot-air balloon work?
Challenge 488 n
**
If you do not like this text, here is a proposal. You can use the paper to make a cup, as shown in Figure 121, and boil water in it over an open ame. However, to succeed, you have to be a little careful. Can you nd out in what way?
**
Mixing kg of water at °C and kg of water at °C gives kg of water at Challenge 489 ny is the result of mixing kg of ice at °C and kg of water at °C?
°C. What
**
Ref. 208 Challenge 490 n
e highest recorded air temperature in which a man has survived is °C. is was tested in 1775 in London, by the secretary of the Royal Society, Charles Blagden, together with a few friends, who remained in a room at that temperature for 45 minutes. Interestingly, the raw steak which he had taken in with him was cooked (‘well done’) when he and his friends le the room. What condition had to be strictly met in order to avoid cooking the people in the same way as the steak?
** Challenge 491 n Why does water boil at . °C instead of °C?
Challenge 492 n Can you ll a bottle precisely with
** − kg of water?
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invisible pulsed laser beam emitting sound
laser cable to amplifier
F I G U R E 122 The invisible loudspeaker
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 493 n
**
One gram of fat, either butter or human fat, contains kJ of chemical energy (or, in ancient units more familiar to nutritionists, kcal). at is the same value as that of petrol. Why are people and butter less dangerous than petrol?
**
In 1992, the Dutch physicist Martin van der Mark invented a loudspeaker which worked the heating of air by heating air with a laser beam. He demonstrated that with the right wavelength and with a suitable modulation of the intensity, a laser beam in air can generate sound, . e e ect at the basis of this device, called the photoacoustic e ect, appears in many materials. e best wavelength for air is in the infrared domain, on one of the few absorption lines of water vapour. In other words, a properly modulated infrared laser beam that shines through the air generates sound. e light can be emitted from a small matchbox-sized semiconductor laser hidden in the ceiling and shining downwards. e sound is emitted in all directions perpendicular to the beam. Since infrared laser light is not visible, Martin van der Mark thus invented an invisible loudspeaker! Unfortunately, the e ciency of present versions is still low, so that the power of the speaker is not yet su cient for practical applications. Progress in laser technology should change this, so that in the future we should be able to hear sound that is emitted from the centre of an otherwise empty room.
**
A famous exam question: How can you measure the height of a building with a barometer, Challenge 494 n a rope and a ruler? Find at least six di erent ways.
**
What is the approximate probability that out of one million throws of a coin you get Challenge 495 ny exactly 500 000 heads and as many tails?
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
You may want to use Stirling’s formula n! πn (n e)n to calculate the result.*
** Challenge 496 n Does it make sense to talk about the entropy of the universe?
** Challenge 497 ny Can a helium balloon li the tank which lled it?
**
All friction processes, such as osmosis, di usion, evaporation, or decay, are slow. ey take a characteristic time. It turns out that any (macroscopic) process with a time-scale is irreversible. is is no real surprise: we know intuitively that undoing things always takes more time than doing them. at is again the second principle of thermodynamics.
Ref. 209
**
It turns out that storing information is possible with negligible entropy generation. However, erasing information requires entropy. is is the main reason why computers, as well as brains, require energy sources and cooling systems, even if their mechanisms would otherwise need no energy at all.
**
When mixing hot rum and cold water, how does the increase in entropy due to the mixing Challenge 498 ny compare with the entropy increase due to the temperature di erence?
**
Why aren’t there any small humans, e.g. mm in size, as in many fairy tales? In fact, Challenge 499 n there are no warm-blooded animals of that size. Why not?
**
Shining a light onto a body and repeatedly switching it on and o produces sound. is is called the photoacoustic e ect, and is due to the thermal expansion of the material. By changing the frequency of the light, and measuring the intensity of the noise, one reveals a characteristic photoacoustic spectrum for the material. is method allows us to detect gas concentrations in air of one part in . It is used, among other methods, to study the gases emitted by plants. Plants emit methane, alcohol and acetaldehyde in small quantities; the photoacoustic e ect can detect these gases and help us to understand the processes behind their emission.
**
What is the rough probability that all oxygen molecules in the air would move away from Challenge 500 ny a given city for a few minutes, killing all inhabitants?
**
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* ere are many improvements to Stirling’s formula. A simple one is n! is πn (n e)ne ( n+ ) < n! < πn (n e)ne ( n).
( n + )π (n e)n. Another
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If you pour a litre of water into the sea, stir thoroughly through all the oceans and then Challenge 501 ny take out a litre of the mixture, how many of the original atoms will you nd?
** Challenge 502 ny How long would you go on breathing in the room you are in if it were airtight?
**
Challenge 503 ny What happens if you put some ash onto a piece of sugar and set re to the whole? (Warning: this is dangerous and not for kids.)
Challenge 504 ny Challenge 505 ny
**
Entropy calculations are o en surprising. For a system of N particles with two states each, there are Wall = N states. For its most probable con guration, with exactly half the particles in one state, and the other half in the other state, we have Wmax = N! ((N )!) . Now, for a macroscopic system of particles, we might typically have N = . at gives Wall Wmax; indeed, the former is times larger than the latter. On the other hand, we
nd that ln Wall and ln Wmax agree for the rst 20 digits! Even though the con guration with exactly half the particles in each state is much more rare than the general case, where
the ratio is allowed to vary, the entropy turns out to be the same. Why?
Challenge 506 ny Challenge 507 ny
**
If heat is due to motion of atoms, our built-in senses of heat and cold are simply detectors of motion. How could they work?
By the way, the senses of smell and taste can also be seen as motion detectors, as they signal the presence of molecules ying around in air or in liquids. Do you agree?
Challenge 508 n
**
e Moon has an atmosphere, although an extremely thin one, consisting of sodium (Na) and potassium (K). is atmosphere has been detected up to nine Moon radii from its surface. e atmosphere of the Moon is generated at the surface by the ultraviolet radiation from the Sun. Can you estimate the Moon’s atmospheric density?
**
Does it make sense to add a line in Table 29 for the quantity of physical action? A column? Challenge 509 ny Why?
Challenge 510 n
**
Di usion provides a length scale. For example, insects take in oxygen through their skin. As a result, the interiors of their bodies cannot be much more distant from the surface than about a centimetre. Can you list some other length scales in nature implied by di usion processes?
**
Rising warm air is the reason why many insects are found in tall clouds in the evening. Many insects, especially that seek out blood in animals, are attracted to warm and humid air.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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cold air
hot air
room temperature air
F I G U R E 123 The Wirbelrohr or Ranque–Hilsch vortex tube
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
**
ermometers based on mercury can reach °C. How is this possible, given that merChallenge 511 n cury boils at °C?
** Challenge 512 n What does a burning candle look like in weightless conditions?
Challenge 513 n
**
It is possible to build a power station by building a large chimney, so that air heated by the Sun ows upwards in it, driving a turbine as it does so. It is also possible to make a power station by building a long vertical tube, and letting a gas such as ammonia rise into it which is then lique ed at the top by the low temperatures in the upper atmosphere; as it falls back down a second tube as a liquid – just like rain – it drives a turbine. Why are such schemes, which are almost completely non-polluting, not used yet?
Challenge 514 n
**
One of the most surprising devices ever invented is the Wirbelrohr or Ranque–Hilsch vortex tube. By blowing compressed air at room temperature into it at its midpoint, two
ows of air are formed at its ends. One is extremely cold, easily as low as − °C, and one extremely hot, up to °C. No moving parts and no heating devices are found inside. How does it work?
**
It is easy to cook an egg in such a way that the white is hard but the yolk remains liquid. Challenge 515 n Can you achieve the opposite?
Ref. 210
**
ermoacoustic engines, pumps and refrigerators provide many strange and fascinating applications of heat. For example, it is possible to use loud sound in closed metal chambers to move heat from a cold place to a hot one. Such devices have few moving parts and are being studied in the hope of nding practical applications in the future.
** Challenge 516 ny Does a closed few-particle system contradict the second principle of thermodynamics?
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Figures to be added
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F I G U R E 124 Examples of self-organization for sand
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**
Challenge 517 ny
What happens to entropy when gravitation is taken into account? We carefully le gravitation out of our discussion. In fact, many problems appear – just try to think about the issue. For example, Jakob Bekenstein has discovered that matter reaches its highest possible entropy when it forms a black hole. Can you con rm this?
Challenge 518 ny
**
e numerical values (but not the units!) of the Boltzmann constant k = . ë − J K and the combination h ce agree in their exponent and in their rst three digits, where h is Planck’s constant and e is the electron charge. Can you dismiss this as mere coincidence?
S-
Ref. 211
Ref. 212 Challenge 519 n
To speak of non-linear physics is like calling zoology the study of non-elephant animals.
“ Stanislaw Ulam ” In our list of global descriptions of motion, the high point is the study of self-organization.
Self-organization is the appearance of order. Order is a term that includes shapes, such as the complex symmetry of snow akes; patterns, such as the stripes of zebras; and cycles, such as the creation of sound when singing. Every example of what we call beauty is a combination of shapes, patterns and cycles. (Do you agree?) Self-organization can thus be called the study of the origin of beauty.
e appearance of order is a general observation across nature. Fluids in particular exhibit many phenomena where order appears and disappears. Examples include the more or less regular ickering of a burning candle, the apping of a ag in the wind, the regular stream of bubbles emerging from small irregularities in the surface of a champagne glass, and the regular or irregular dripping of a water tap.
e appearance of order is found from the cell di erentiation in an embryo inside a woman’s body; the formation of colour patterns on tigers, tropical sh and butter ies; the symmetrical arrangements of ower petals; the formation of biological rhythms; and so on.
All growth processes are self-organization phenomena. Have you ever pondered the
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TA B L E 34 Sand patterns in the sea and on land
P
P
A
sand banks sand waves megaribbles ribbles singing sand
2 to km 100 to m m
cm 95 to Hz
2 to m m .m mm
up to dB
O
tides tides tides waves wind on sand dunes, avalanches making the dune vibrate
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Ref. 213 Page 698 Challenge 520 e
Ref. 214
Ref. 215
incredible way in which teeth grow? A practically inorganic material forms shapes in the upper and the lower rows tting exactly into each other. How this process is controlled is still a topic of research. Also the formation, before and a er birth, of neural networks in the brain is another process of self-organization. Even the physical processes at the basis of thinking, involving changing electrical signals, is to be described in terms of selforganization.
Biological evolution is a special case of growth. Take the evolution of animal shapes. It turns out that snake tongues are forked because that is the most e cient shape for following chemical trails le by prey and other snakes of the same species. (Snakes smell with help of the tongue.) e xed numbers of ngers in human hands or of petals of
owers are also consequences of self-organization. Many problems of self-organization are mechanical problems: for example, the form-
ation of mountain ranges when continents move, the creation of earthquakes, or the creation of regular cloud arrangements in the sky. It can be fascinating to ponder, during an otherwise boring ight, the mechanisms behind the formation of the clouds you see from the aeroplane.
Studies into the conditions required for the appearance or disappearance of order have shown that their description requires only a few common concepts, independently of the details of the physical system. is is best seen looking at a few examples.
All the richness of self-organization reveals itself in the study of plain sand. Why do sand dunes have ripples, as does the sand oor at the bottom of the sea? We can also study how avalanches occur on steep heaps of sand and how sand behaves in hourglasses, in mixers, or in vibrating containers. e results are o en surprising. For example, as recently as Paul Umbanhowar and his colleagues found that when a at container holding tiny bronze balls (around . mm in diameter) is shaken up and down in vacuum at certain frequencies, the surface of this bronze ‘sand’ forms stable heaps. ey are shown in Figure . ese heaps, so-called oscillons, also bob up and down. Oscillons can move and interact with one another.
Oscillons in sand are simple example for a general e ect in nature: discrete systems with nonlinear interactions can exhibit localized excitations. is fascinating topic is just beginning to be researched. It might well be that it will yield results relevant to our understanding of elementary particles.
Sand shows many other pattern-forming processes. A mixture of sand and sugar, when poured onto a heap, forms regular layered structures that in cross section look like zebra
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n = 21 n = 23
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 125 Oscillons formed by shaken bronze balls; horizontal size is about cm
(© Paul Umbanhowar)
time
F I G U R E 126 Magic numbers: 21 spheres, when swirled in a dish, behave differently from non-magic numbers, like 23, of spheres (redrawn from photographs, © Karsten Kötter)
Ref. 216 Ref. 217
Ref. 218
stripes. Horizontally rotating cylinders with binary mixtures inside them separate the mixture out over time. Or take a container with two compartments separated by a cm wall. Fill both halves with sand and rapidly shake the whole container with a machine. Over time, all the sand will spontaneously accumulate in one half of the container. As another example of self-organization in sand, people have studied the various types of sand dunes that ‘sing’ when the wind blows over them. In fact, the behaviour of sand and dust is proving to be such a beautiful and fascinating topic that the prospect of each human returning dust does not look so grim a er all.
Another simple and beautiful example of self-organization is the e ect discovered in by Karsten Kötter and his group. ey found that the behaviour of a set of spheres
swirled in a dish depends on the number of spheres used. Usually, all the spheres get continuously mixed up. But for certain ‘magic’ numbers, such as , stable ring patterns emerge, for which the outside spheres remain outside and the inside ones remain inside.
e rings, best seen by colouring the spheres, are shown in Figure . ese and many other studies of self-organizing systems have changed our understand-
ing of nature in a number of ways. First of all, they have shown that patterns and shapes are similar to cycles: all are due to motion. Without motion, and thus without history, there is no order, neither patterns nor shapes. Every pattern has a history; every pattern is a result of motion.
Secondly, patterns, shapes and cycles are due to the organized motion of large numbers of small constituents. Systems which self-organize are always composite: they are cooperative structures.
irdly, all these systems obey evolution equations which are nonlinear in the con guration variables. Linear systems do not self-organize. Many self-organizing systems also
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show chaotic motion. Fourthly, the appearance and disappearance of order depends on the strength of a
driving force, the so-called order parameter. O en, chaotic motion appears when the driving is increased beyond the value necessary for the appearance of order. An example of chaotic motion is turbulence, which appears when the order parameter, which is proportional to the speed of the uid, is increased to high values.
Moreover, all order and all structure appears when two general types of motion compete with each other, namely a ‘driving’, energy-adding process, and a ‘dissipating’, braking mechanism. ermodynamics plays a role in all self-organization. Self-organizing systems are always dissipative systems, and are always far from equilibrium. When the driving and the dissipation are of the same order of magnitude, and when the key behaviour of the system is not a linear function of the driving action, order may appear.*
All self-organizing systems at the onset of order appearance can be described by equations for the pattern amplitude A of the general form
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∂A(t, ∂t
x)
=
λA −
µ
A
A + κ ∆A + higher orders .
(99)
Challenge 521 ny Challenge 522 ny
Ref. 219
Here, the – possibly complex – observable A is the one that appears when order appears, such as the oscillation amplitude or the pattern amplitude. e rst term λA is the driving term, in which λ is a parameter describing the strength of the driving. e next term is a typical nonlinearity in A, with µ a parameter that describes its strength, and the third term κ ∆A = κ(∂ A ∂x + ∂ A ∂y + ∂ A ∂z ) is a typical dissipative (and di usive) term.
One can distinguish two main situations. In cases where the dissipative term plays no role (κ = ), one nds that when the driving parameter λ increases above zero, a temporal oscillation appears, i.e. a stable cycle with non-vanishing amplitude. In cases where the di usive term does play a role, equation ( ) describes how an amplitude for a spatial oscillation appears when the driving parameter λ becomes positive, as the solution A = then becomes spatially unstable.
In both cases, the onset of order is called a bifurcation, because at this critical value of the driving parameter λ the situation with amplitude zero, i.e. the homogeneous (or unordered) state, becomes unstable, and the ordered state becomes stable. In nonlinear systems, order is stable. is is the main conceptual result of the eld. Equation ( ) and its numerous variations allow us to describe many phenomena, ranging from spirals, waves, hexagonal patterns, and topological defects, to some forms of turbulence. For every physical system under study, the main task is to distil the observable A and the parameters λ, µ and κ from the underlying physical processes.
Self-organization is a vast eld which is yielding new results almost by the week. To
* To describe the ‘mystery’ of human life, terms like ‘ re’, ‘river’ or ‘tree’ are o en used as analogies. ese are all examples of self-organized systems: they have many degrees of freedom, have competing driving and braking forces, depend critically on their initial conditions, show chaos and irregular behaviour, and sometimes show cycles and regular behaviour. Humans and human life resemble them in all these respects; thus there is a solid basis to their use as metaphors. We could even go further and speculate that pure beauty is pure self-organization. e lack of beauty indeed o en results from a disturbed equilibrium between external braking and external driving.
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fixed point
oscillation, limit cycle
quasiperiodic motion
chaotic motion
configuration variables
configuration variables
F I G U R E 127 Examples of different types of motion in configuration space
state value
system 1
system 2 time F I G U R E 128 Sensitivity to initial conditions
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Challenge 523 ny
Ref. 220 Challenge 524 n
discover new topics of study, it is o en su cient to keep one’s eye open; most e ects are
comprehensible without advanced mathematics. Good hunting!
Most systems that show self-organization also show another type of motion. When the
driving parameter of a self-organizing system is increased to higher and higher values,
T oprhdyesricbisetcs,ocmheasomorecamndotimonories
irregular, and in the most irregular
the end one usually nds type of motion.* Chaos can
chaos. be de
For ned
i
independently of self-organization, namely as that motion of systems for which small
changes in initial conditions evolve into large changes of the motion (exponentially with
time), as shown in Figure . More precisely, chaos is irregular motion characterized by a
positive Lyapounov exponent. e weather is such a system, as are dripping water-taps, the
fall of dice, and many other common systems. For example, research on the mechanisms
by which the heart beat is generated has shown that the heart is not an oscillator, but a
chaotic system with irregular cycles. is allows the heart to be continuously ready for
demands for changes in beat rate which arise once the body needs to increase or decrease
its e orts.
Incidentally, can you give a simple argument to show that the so-called butter y e ect
does not exist? is ‘e ect’ is o en cited in newspapers: the claim is that nonlinearities
imply that a small change in initial conditions can lead to large e ects; thus a butter y
wing beat is alleged to be able to induce a tornado. Even though nonlinearities do indeed
lead to growth of disturbances, the butter y e ect has never been observed; it does not
* On the topic of chaos, see the beautiful book by H.-O. P
, H. J
& D. S , Chaos and
Fractals, Springer Verlag, 1992. It includes stunning pictures, the necessary mathematical background, and
some computer programs allowing personal exploration of the topic. ‘Chaos’ is an old word: according to
Greek mythology, the rst goddess, Gaia, i.e. the Earth, emerged from the chaos existing at the beginning.
She then gave birth to the other gods, the animals and the rst humans.
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•.
Challenge 525 ny Ref. 221
Challenge 527 ny
exist. ere is chaotic motion also in machines: chaos appears in the motion of trains on the
rails, in gear mechanisms, and in re- ghter’s hoses. e precise study of the motion in a zippo cigarette lighter will probably also yield an example of chaos. e mathematical description of chaos – simple for some textbook examples, but extremely involved for others – remains an important topic of research.
All the steps from disorder to order, quasiperiodicity and nally to chaos, are examples of self-organization. ese types of motion, illustrated in Figure , are observed in many
uid systems. eir study should lead, one day, to a deeper understanding of the mysteries of turbulence. Despite the fascination of this topic, we will not explore it further, because it does not lead towards the top of Motion Mountain.
But self-organization is of interest also for a more general reason. It is sometimes said that our ability to formulate the patterns or rules of nature from observation does not imply the ability to predict all observations from these rules. According to this view, socalled ‘emergent’ properties exist, i.e. properties appearing in complex systems as something new that cannot be deduced from the properties of their parts and their interactions. ( e ideological backdrop to this view is obvious; it is the last attempt to ght the determinism.) e study of self-organization has de nitely settled this debate. e properties of water molecules do allow us to predict Niagara Falls.* Similarly, the di usion of signal molecules do determine the development of a single cell into a full human being: in particular, cooperative phenomena determine the places where arms and legs are formed; they ensure the (approximate) right–le symmetry of human bodies, prevent mix-ups of connections when the cells in the retina are wired to the brain, and explain the fur patterns on zebras and leopards, to cite only a few examples. Similarly, the mechanisms at the origin of the heart beat and many other cycles have been deciphered.
Self-organization provides general principles which allow us in principle to predict the behaviour of complex systems of any kind. ey are presently being applied to the most complex system in the known universe: the human brain. e details of how it learns to coordinate the motion of the body, and how it extracts information from the images in the eye, are being studied intensely. e ongoing work in this domain is fascinating. If you plan to become a scientist, consider taking this path.
Such studies provide the nal arguments that con rm what J. O rey de la Mettrie in stated and explored in his famous book L’homme machine: humans are complex machines. Indeed, the lack of understanding of complex systems in the past was due mainly to the restrictive teaching of the subject of motion, which usually concentrated – as we do in this walk – on examples of motion in simple systems. Even though the subject of self-organization provides fascinating insights, and will do so for many years to come,
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Ref. 222 Challenge 526 ny
* Already small versions of Niagara Falls, namely dripping water taps, show a large range of cooperative phenomena, including the chaotic, i.e. non-periodic, fall of water drops. is happens when the water ow has the correct value, as you can verify in your own kitchen. Several cooperative uid phenomena have been simulated even on the molecular level.
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water pipe
pearls
λ
finger
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F I G U R E 129 The wavy surface of icicles
F I G U R E 130 Water pearls
F I G U R E 131 A braiding water stream (© Vakhtang Putkaradze)
we now leave it. We continue with our own adventure exploring the basics of motion.*
“Ich sage euch: man muss noch Chaos in sich haben, um einen tanzenden Stern gebären zu können. Ich sage euch: ihr habt noch Chaos in
euch.
C
” Friedrich Nietzsche, Also sprach Zarathustra. -
Every example of a pattern or of beauty contains a physical challenge:
Challenge 528 ny
**
All icicles have a wavy surface, with a crest-to-crest distance of about cm, as shown in Figure 129. e distance is determined by the interplay between water ow and surface cooling. How?
**
When wine is made to swirl in a wine glass, a er the motion has calmed down, the wine owing down the glass walls forms little arcs. Can you explain in a few words what forms
Challenge 529 ny them?
**
Ref. 223 * An important case of self-organization is humour.
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How does the average distance between cars parked along a street change over time, asChallenge 530 d suming a constant rate of cars leaving and arriving?
**
When a ne stream of water leaves a water tap, putting a nger in the stream leads to a Challenge 531 d wavy shape, as shown in Figure 130. Why?
**
Ref. 224 Challenge 532 ny
When water emerges from a oblong opening, the stream forms a braid pattern, as shown in Figure 131. is e ect results from the interplay and competition between inertia and surface tension: inertia tends to widen the stream, while surface tension tends to narrow it. Predicting the distance from one narrow region to the next is still a topic of research.
If the experiment is done in free air, without a plate, one usually observes an additional e ect: there is a chiral braiding at the narrow regions, induced by the asymmetries of the water ow. You can observe this e ect in the toilet! Scienti c curiosity knows no limits: are you a right-turner or a le -turner, or both? On every day?
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**
Ref. 225 Challenge 533 e
Gerhard Müller has discovered a simple but beautiful way to observe self-organization in solids. His system also provides a model for a famous geological process, the formation of hexagonal columns in basalt, such as the Devil’s Staircase in Ireland. Similar formations are found in many other places of the Earth. Just take some rice our or corn starch, mix it with about half the same amount of water, put the mixture into a pan and dry it with a lamp. Hexagonal columns form. e analogy works because the drying of starch and the cooling of lava are di usive processes governed by the same equations, because the boundary conditions are the same, and because both materials respond with a small reduction in volume.
Ref. 226
**
Water ow in pipes can be laminar (smooth) or turbulent (irregular and disordered).
e transition depends on the diameter d of the pipe and the speed v of the water. e
transition usually happens when the so-called Reynolds number – de ned as R = vd η
(η being the kinematic viscosity of the water, around mm s) – becomes greater than
about 2000. However, careful experiments show that with proper handling, laminar ows
can be produced up to R =
. A linear analysis of the equations of motion of the
uid, the Navier–Stokes equations, even predicts stability of laminar ow for all Reynolds
numbers. is riddle was solved only in the years 2003 and 2004. First, a complex math-
ematical analysis showed that the laminar ow is not always stable, and that the transition
to turbulence in a long pipe occurs with travelling waves. en, in 2004, careful experi-
ments showed that these travelling waves indeed appear when water is owing through
a pipe at large Reynolds numbers.
**
For some beautiful pictures on self-organization in uids, see the http://serve.me.nus.edu. sg/limtt website. Among others, it shows that a circular vortex can ‘suck in’ a second one behind it, and that the process can then repeat.
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Ref. 227
**
Also dance is an example of self-organization. Self-organization takes part in the brain. Like for all complex movements, learning then is o en a challenge. Nowadays there are beautiful books that tell how physics can help you improve your dancing skills and the grace of your movements.
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I only know that I know nothing.
“ ” Socrates, as cited by Plato
Socrates’ saying applies also to Galilean physics, despite its general success in engineering and in the description of everyday life. We will now give a short overview of the limitations of the eld.
R
Even though the science of mechanics is now several hundred years old, research into its details is still continuing.
Ref. 228 Ref. 229
— We have already mentioned above the issue of the stability of the solar system. e long-term future of the planets is unknown. In general, the behaviour of few-body systems interacting through gravitation is still a research topic of mathematical physics. Answering the simple question of how long a given set of bodies gravitating around each other will stay together is a formidable challenge. e history of this so-called many-body problem is long and involved. Interesting progress has been achieved, but the nal answer still eludes us.
— Many challenges remain in the elds of self-organization, of nonlinear evolution equations, and of chaotic motion; and they motivate numerous researchers in mathematics, physics, chemistry, biology, medicine and the other sciences.
— Perhaps the toughest of all problems in physics is how to describe turbulence. When the young Werner Heisenberg was asked to continue research on turbulence, he refused – rightly so – saying it was too di cult; he turned to something easier and discovered quantum mechanics instead. Turbulence is such a vast topic, with many of its concepts still not settled, that despite the number and importance of its applications, only now, at the beginning of the twenty- rst century, are its secrets beginning to be unravelled. It is thought that the equations of motion describing uids, the so-called Navier–Stokes equations, are su cient to understand turbulence.* But the mathematics behind them is mind-boggling. ere is even a prize of one million dollars o ered by the Clay Mathematics Institute for the completion of certain steps on the way to solving the equations.
* ey are named a er Claude Navier (b. 1785 Dijon, d. 1836 Paris), important French engineer and bridge builder, and Georges Gabriel Stokes (b. 1819 Skreen, d. 1903 Cambridge), important Irish physicist and mathematician.
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W
?
Ref. 230 Page 77
“Democritus declared that there is a unique sort of motion: that ensuing from collision. Simplicius, Commentary on the Physics of ” Aristotle, ,
Of the questions unanswered by classical physics, the details of contact and collisions are among the most pressing. Indeed, we de ned mass in terms of velocity changes during collisions. But why do objects change their motion in such instances? Why are collisions between two balls made of chewing gum di erent from those between two stainless-steel balls? What happens during those moments of contact?
Contact is related to material properties, which in turn in uence motion in a complex way. e complexity is such that the sciences of material properties developed independently from the rest of physics for a long time; for example, the techniques of metallurgy (o en called the oldest science of all) of chemistry and of cooking were related to the properties of motion only in the twentieth century, a er having been independently pursued for thousands of years. Since material properties determine the essence of contact, we need knowledge about matter and about materials to understand the notion of mass, and thus of motion. e second part of our mountain ascent will reveal these connections.
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P
Appendix B Challenge 534 n Challenge 535 n
When we started climbing Motion Mountain, we stated that to gain height means to in-
crease the precision of our description of nature. To make even this statement itself more
precise, we distinguish between two terms: precision is the degree of reproducibility; ac-
curacy is the degree of correspondence to the actual situation. Both concepts apply to
measurements,* to statements and to physical concepts.
At present, the record number of digits ever measured for a physical quantity is .
Why so few? Classical physics doesn’t provide an answer. What is the maximum number
of digits we can expect in measurements; what determines it; and how can we achieve
it? ese questions are still open at this point in our ascent; they will be covered in the
second part of it.
On the other hand, statements with false accuracy abound. What should we think
of a car company – Ford – who claim that the drag coe cient cw of a certain model is . ? Or of the o cial claim that the world record in fuel consumption for cars is
. km l? Or of the statement that . % of all citizens share a certain opinion?
One lesson we learn from investigations into measurement errors is that we should never
provide more digits for a result than we can put our hand into re for.
Is it possible to draw or produce a rectangle for which the ratio of lengths is a real num-
ber, e.g. of the form .
..., whose digits encode a book? (A
simple method would code a space as , the letter ‘a’ as , ‘b’ as , ‘c’ as , etc. Even
more interestingly, could the number be printed inside its own book?)
In our walk we aim for precision and accuracy, while avoiding false accuracy. ere-
fore, concepts have mainly to be precise, and descriptions have to be accurate. Any in-
* For measurements, both precision and accuracy are best described by their standard deviation, as explained in Appendix B, on page 1164.
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accuracy is a proof of lack of understanding. To put it bluntly, ‘inaccurate’ means wrong. Increasing the accuracy and precision of our description of nature implies leaving behind us all the mistakes we have made so far. at is our aim in the following.
C
?
Page 1049 Challenge 536 e
Let us have some fun with a paradox related to our adventure. If a perfect physics publication describing all of nature existed, it must also describe itself, its own production – including its author – and most important of all, its own contents. Is this possible? Using the concept of information, we can state that such a book should contain all information contained in the universe, including the information in the book itself. Is this possible?
If nature requires an in nitely long book to be fully described, such a publication obviously cannot exist. In this case, only approximate descriptions of nature are possible.
If nature requires a nite amount of information for its description, then the universe cannot contain more information than is already contained in the book. is would imply that the rest of the universe would not add to the information already contained in the book. It seems that the entropy of the book and the entropy of the universe must be similar. is is possible, but seems somewhat unlikely.
We note that the answer to this puzzle also implies the answer to another puzzle: whether a brain can contain a full description of nature. In other words, the question is: can humans understand nature? We do believe so. In other words, we seem to believe something rather unlikely: that the universe does not contain more information than what our brain could contain or even contains already. However, this conclusion is not correct. e terms ‘universe’ and ‘information’ are not used correctly in this reasoning, as you might want to verify. We will solve this puzzle later in our adventure. Until then, do make up your own mind.
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W
?
In the description of gravity given so far, the one that everybody learns – or should learn – at school, acceleration is connected to mass and distance via a = GM r . at’s all. But this simplicity is deceiving. In order to check whether this description is correct, we have to measure lengths and times. However, it is impossible to measure lengths and time intervals with any clock or any ruler based on the gravitational interaction alone! Try to Challenge 537 n conceive such an apparatus and you will be inevitably be disappointed. You always need a non-gravitational method to start and stop the stopwatch. Similarly, when you measure length, e.g. of a table, you have to hold a ruler or some other device near it. e interaction necessary to line up the ruler and the table cannot be gravitational.
A similar limitation applies even to mass measurements. Try to measure mass using Challenge 538 n gravitation alone. Any scale or balance needs other – usually mechanical, electromagnetic
or optical – interactions to achieve its function. Can you con rm that the same applies Challenge 539 n to speed and to angle measurements? In summary, whatever method we use, in order to
measure velocity, length, time, and mass, interactions other than gravity are needed. Our ability to measure shows that gravity is not all there is.
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I
?
Galilean physics does not explain the ability to measure. In fact, it does not even explain the existence of standards. Why do objects have xed lengths? Why do clocks work with regularity? Galilean physics cannot explain these observations.
Galilean physics also makes no clear statements on the universe as a whole. It seems to suggest that it is in nite. Finitude does not t with the Galilean description of motion. Galilean physics is thus limited in its explanations because it disregards the limits of motion.
We also note that the existence of in nite speeds in nature would not allow us to de ne time sequences. Clocks would then be impossible. In other words, a description of nature that allows unlimited speeds is not precise. Precision requires limits. To achieve the highest possible precision, we need to discover all limits to motion. So far, we have discovered only one: there is a smallest entropy. We now turn to another, more striking one: the limit for speed. To understand this limit, we will explore the most rapid motion we know: the motion of light.
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B
138 G
F
, Chi l’ha detto?, Hoepli, . Cited on page .
139 e beautiful story of the south-pointing carriage is told in Appendix B of J
F
& J.D. N
, A Short Course in General Relativity, Springer Verlag, nd edition,
. Such carriages have existed in China, as told by the great sinologist Joseph Needham,
but their construction is unknown. e carriage described by Foster and Nightingale is the
one reconstructed in by George Lancaster, a British engineer. Cited on page .
140 See for example Z. G
, Building blocks of movement, Nature 407, pp. – ,
. Researchers in robot control are also interested in such topics. Cited on page .
141 G. G
, C. F , A. C
& D. F
, Fluid ow up the wall of
a spinning egg, American Journal of Physics 66, pp. – , . Cited on page .
142 A historical account is given in W
Y
Variational Principles in Dynamics and Quantum
and .
&S eory, Dover,
M
,
. Cited on pages
143 M P
, Prinzipe der Mechanik, Walter de Gruyter & Co., . Cited on page .
144 e relations between possible Lagrangians are explained by H
G
sical Mechanics, nd edition, Addison-Wesley, . Cited on page .
, Clas-
145 C.G. G , G. K & V.A. N
, From Maupertius to Schrödinger. Quantization
of classical variational principles, American Journal of Physics 67, pp. – , . Cited
on page .
146 e Hemingway statement is quoted by Marlene Dietrich in A
E. H
Hemingway, Random House, , in part , chapter . Cited on page .
, Papa
147 J.A. M
, An innovation in physics instruction for nonscience majors, American
Journal of Physics 46, pp. – , . Cited on page .
148 See e.g. A P. B , Extrasolar planets, Physics Today 49, pp. – . September . e most recent information can be found at the ‘Extrasolar Planet Encyclopaedia’ main-
tained at http://www.obspm.fr/planets by Jean Schneider at the Observatoire de Paris. Cited on page .
149 A good review article is by D
W. H
, Comets and Asteroids, Contemporary
Physics 35, pp. – , . Cited on page .
150 G.B. W , J.H. B
& B.J. E
, A general model for the origin of allometric
scaling laws in biology, Science 276, pp. – , April , with a comment on page of
the same issue. e rules governing branching properties of blood vessels, of lymph systems
and of vessel systems in plants are explained. For more about plants, see also the paper G.B.
W , J.H. B
& B.J. E
, A general model for the structure and allometry of
plant vascular systems, Nature 400, pp. – , . Cited on page .
151 J.R. B
, A. M
& A. R
, Size and form in e cient transportation
networks, Nature 399, pp. – , . Cited on page .
152 J M
R
, An Introduction to Early Greek Philosophy, Houghton Mu n
, chapter . Cited on page .
153 See e.g. B. B .
, A child’s theory of mind, Science News 144, pp. – . Cited on page
154 e most beautiful book on this topic is the text by B
G
& G.C. S -
, Tilings and Patterns, W.H. Freeman and Company, New York, . It has been trans-
lated into several languages and republished several times. Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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155 U. N
, e maximal kinematical invariance group of the free Schrödinger equation,
Helvetica Physica Acta 45, pp. – , . See also the introduction by O. J & V.V.
S
, e maximal invariance group of Newton’s equations for a free point particle,
http://www.arxiv.org/abs/math-ph/
. Cited on page .
156 e story is told in the interesting biography of Einstein by A. P , ‘Subtle is the Lord...’ – e Science and the Life of Albert Einstein, Oxford University Press, . Cited on page .
157 W. Z & R. W Journal 1, pp. – ,
-S
, Globale Eigenschwingungen der Erde, Physik
. Cited on page .
158 N. G
, What happens to energy and momentum when two oppositely-moving
wave pulses overlap?, American Journal of Physics 71, pp. – , . Cited on page .
159 An informative and modern summary of present research about the ear and the details of its function is http://www.physicsweb.org/article/world/ / / . Cited on page .
160 A.L. H
& A.F. H
, A quantitative description of membrane current and
its application to conduction and excitation in nerve, Journal of Physiology 117, pp. –
, . is famous paper of theoretical biology earned the authors the Nobel Prize in
Medicine in . Cited on page .
161 T. F
, e Versatile Soliton, Springer Verlag, . See also J.S. R
, Report of
the Fourteenth Meeting of the British Association for the Advancement of Science, Murray,
London, , pp. – . Cited on pages and .
162 N.J. Z
& M.D. K
, Interaction of solitons in a collisionless plasma and the
recurrence of initial states, Physical Review Letters 15, pp. – , . Cited on page .
163 O. M
, De kortste knal ter wereld, Nederlands tijdschri voor natuurkunde pp. –
, . Cited on page .
164 E. H
, Freak waves: just bad luck, or avoidable?, Europhysics News pp. – , Septem-
ber/October , downloadable at www.europhysicsnews.org. Cited on page .
165 For more about the ocean sound channel, see the novel by T C
, e Hunt
for Red October. See also the physics script by R.A. M
, Government secrets
of the oceans, atmosphere, and UFOs, http://web.archive.org/web/*/http://muller.lbl.gov/
teaching/Physics /chapters/ -SecretsofUFOs.html . Cited on page .
166 B. W
, R.S. B
& L.M. D , Paci c and Atlantic herring produce burst pulse
sounds, Biology Letters 271, number S , February . Cited on page .
167 See for example the article by G. F , . Cited on page .
, Infraschall, Physik in unserer Zeit 13, pp. –
168 Wavelet transformations were developed by the French mathematicians Alex Grossmann,
Jean Morlet and ierry Paul. e basic paper is A. G
, J. M
& T. P ,
Integral transforms associated to square integrable representations, Journal of Mathematical
Physics 26, pp. – , . For a modern introduction, see S
M
,A
Wavelet Tour of Signal Processing, Academic Press, . Cited on page .
169 J I
, e Velocity of Honey, Viking, o . Cited on page .
170 M. A
, Proceedings of the Royal Society in London A 400, pp. – ,
on page .
. Cited
171 T.A. M M
& J. T
B
, Form und Leben – Konstruktion vom Reißbrett
der Natur, Spektrum Verlag, . Cited on page .
172 G.W. K , S.C. S Nature 428, pp. – ,
, G.M. J
& S.D. D
. Cited on page .
, e limits to tree height,
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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173 A simple article explaining the tallness of trees is A. M
, Trees worthy of Paul
Bunyan, Quantum pp. – , January–February . (Paul Bunyan is a mythical giant lum-
berjack who is the hero of the early frontier pioneers in the United States.) Note that the
transport of liquids in trees sets no limits on their height, since water is pumped up along
tree stems (except in spring, when it is pumped up from the roots) by evaporation from the
leaves. is works almost without limits because water columns, when nucleation is care-
fully avoided, can be put under tensile stresses of over bar, corresponding to m.
See also P. N , Plant Physiology, Academic Press, nd Edition, . Cited on page
.
174 Such information can be taken from the excellent overview article by M.F. A , On the engineering properties of materials, Acta Metallurgica 37, pp. – , . e article explains the various general criteria which determine the selection of materials, and gives numerous tables to guide the selection. Cited on page .
175 For a photograph of a single Barium atom – named Astrid – see H D
, Experi-
ments with an isolated subatomic particle at rest, Reviews of Modern Physics 62, pp. – ,
. For an earlier photograph of a Barium ion, see W. N
, M. H
,
P.E. T
& H. D
, Localized visible Ba+ mono-ion oscillator, Physical Re-
view A 22, pp. – , . Cited on page .
176 Holograms of atoms were rst produced by H -W
F & al., Atomic resolu-
tion in lens-less low-energy electron holography, Physical Review Letters 67, pp. – ,
. Cited on page .
177 A single–atom laser was built in by K. A , J.J. C
, R.R. D
& M.S. F ,
Microlaser: a laser with one atom in an optical resonator, Physical Review Letters 73, p. ,
. Cited on page .
178 See for example C. S
,A.A. K
, T.L. R , C. S
& H.B. E
, Decapitation of tungsten eld emitter tips during sputter sharpening,
Surface Science Letters 339, pp. L –L , . Cited on page .
179 U. W
& J.C.H. S
, An STM with time-of- ight analyzer for atomic spe-
cies identi cation, MSA , Philadelphia, Microscopy and Microanalysis 6, Supplement ,
p. , . Cited on page .
180 L
B
, e physics of stone skipping, American Journal of Physics 17, pp. –
, . e present recod holder is Kurt Steiner, with skips. See http://pastoneskipping.
com/steiner.htm and http://www.stoneskipping.com. e site http://www.yeeha.net/nassa/
guin/g .html is by the a previous world record holder, Jerdome Coleman–McGhee. Cited
on page .
181 S.F. K
& L.E. S
, e teapot e ect: sheetforming ows with de ection, wet-
ting, and hysteresis, Journal of Fluid Mechanics 263, pp. – , . Cited on page .
182 E. H
, Over trechters en zo ..., Nederlands tijdschri voor natuurkunde 68, p. ,
. Cited on page .
183 P. K
, S. E
& D. S
, e puzzle of whip cracking – uncovered
by a correlation of whip-tip kinematics with shock wave emission, Shock Waves 8, pp. – ,
. e authors used high-speed cameras to study the motion of the whip. A new aspect
has been added by A. G
& T. M M
, Shape of a cracking whip, Physical
Review Letters 88, p.
, . is article focuses on the tapered shape of the whip.
However, the neglection of the tu – a piece at the end of the whip which is required to
make it crack – in the latter paper shows that there is more to be discovered still. Cited on
page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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184 S. D
, H. C & N. V
, Fluid instabilities in the birth and death of
antibubbles, New Journal of Physics 5, p. , . Cited on page .
185 Z. S
& K. Y
, Realization of a Human Riding a Unicycle by a Robot, Proceed-
ings of the 1995 IEEE International Conference on Robotics and Automation, Vol. , pp.
– , . Cited on page .
186 On human uncicycling, see J W , e Complete Book of Unicycling, Lodi, , and
S
H
, Einradfahren und die Physik, Reinbeck, . Cited on page .
187 F. H
, Mengenartige Größen im Physikunterricht, Physikalische Blätter 54,
pp. – , September . See also his lecture notes on general introductory physics
on the website http://www-tfp.physik.uni-karlsruhe.de/~didaktik. Cited on pages
and .
188 ermostatics is di cult to learn also because it was not discovered in a systematic way.
See C. T
, e Tragicomical History of ermodynamics 1822–1854, Springer Ver-
lag, . An excellent advanced textbook on thermostatics and thermodynamics is L
R
, A Modern Course in Statistical Physics, Wiley, nd edition, . Cited on page
.
189 Gas expansion was the main method used for the de nition of the o cial temperature scale.
Only in were other methods introduced o cially, such as total radiation thermometry
(in the range K to K), noise thermometry ( K to K and K to K), acoustical
thermometry (around K), magnetic thermometry ( . K to . K) and optical radiation
thermometry (above K). Radiation thermometry is still the central method in the range
from about K to about K. is is explained in detail in R.L. R , R.P. H
,
M. D
, J.F. S
, P.P.M. S
& C.A. S
, e basis of the ITS-
, Metrologia 28, pp. – , . On the water boiling point see also page . Cited on
page .
190 See for example the captivating text by G S , A Matter of Degrees: What Temperature Reveals About the Past and Future of Our Species, Planet and Universe, Viking, New York, . Cited on page .
191 B. P
, What is the best way to lace your shoes?, Nature 420, p. , December .
Cited on page .
192 See for example the article by H. P
-T
, e international temperature scale
of (ITS- ), Metrologia 27, pp. – , , and the errata H. P
-T
,e
international temperature scale of (ITS- ), Metrologia 27, p. , , Cited on page
.
193 For an overview, see C
E &S
H
Physics, Springer, . Cited on page .
, Low-Temperature
194 e famous paper on Brownian motion which contributed so much to Einstein’s fame is A.
E
, Über die von der molekularkinetischen eorie der Wärme geforderte Bewe-
gung von in ruhenden Flüssigkeiten suspendierten Teilchen, Annalen der Physik 17, pp. –
, . In the following years, Einstein wrote a series of further papers elaborating on this
topic. For example, he published his Ph.D. thesis as A. E
, Eine neue Bestim-
mung der Moleküldimensionen, Annalen der Physik 19, pp. – , , and he corrected
a small mistake in A. E
, Berichtigung zu meiner Arbeit: ‘Eine neue Bestimmung
der Moleküldimensionen’, Annalen der Physik 34, pp. – , , where, using new data,
he found the value . ë for Avogadro’s number. Cited on page .
195 e rst tests of the prediction were performed by J. P
, Comptes Rendus de l’Académie
des Sciences 147, pp. – , and pp. – , . He masterfully sums up the whole
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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discussion in J P .
, Les atomes, Librarie Félix Alcan, Paris, . Cited on page
196 P
G
& al., Experimental evidence for microscopic chaos, Nature 394, p. ,
August . Cited on page .
197 ese points are made clearly and forcibly, as is his style, by N.G. K
, Entropie,
Nederlands tijdschri voor natuurkunde 62, pp. – , December . Cited on page
.
198 is is a disappointing result of all e orts so far, as Grégoire Nicolis always stresses in his university courses. Seth Lloyd has compiled a list of proposed de nitions of complexity, containing among others, fractal dimension, grammatical complexity, computational complexity, thermodynamic depth. See, for example, a short summary in Scienti c American p. , June . Cited on page .
199 Minimal entropy is discussed by L. S
, Über die Entropieverminderung in einem
thermodynamischen System bei Eingri en intelligenter Wesen, Zeitschri für Physik 53,
pp. – , . is classic paper can also be found in English translation in his collected
works. Cited on page .
200 G. C
-T
, Les constantes universelles, Pluriel, Hachette, . See also L.
B
, Science and Information eory, Academic Press, . Cited on pages
and .
201 See for example A.E. S
-M
& A.Y . T
, Generalized uncer-
tainty relation in thermodynamics, http://www.arxiv.org/abs/gr-qc/
, or J. U
& J.
L - D , ermodynamic uncertainty relations, Foundations of Physics
29, p. , . Cited on page .
202 H.W. Z
, Particle entropies and entropy quanta IV: the ideal gas, the second
law of thermodynamics, and the P-t uncertainty relation, Zeitschri für physikalische Chemie
217, pp. – , , and H.W. Z
, Particle entropies and entropy quanta V:
the P-t uncertainty relation, Zeitschri für physikalische Chemie 217, pp. – , .
Cited on page .
203 B. L
, Statistical Physics: A Probabilistic Approach, Wiley-Interscience, New York,
. Cited on page .
204 e quote is found in the introduction by George Wald to the text by L
J.
H
, e Fitness of the Environment, Macmillan, New York, , reprinted .
Cited on page .
205 A fascinating introduction to chemistry is the text by J E hibition, Oxford University Press, . Cited on page .
, Molecules at an Ex-
206 An excellent introduction into the physics of heat is the book by L
R
Course in Statistical Physics, Wiley, nd edition, . Cited on page .
, A Modern
207 See V.L. T on page .
, Enrico Fermi in America, Physics Today 55, pp. – , June . Cited
208 K. S
-N
University Press,
, Desert Animals: Physiological Problems of Heat and Water, Oxford . Cited on page .
209 Why entropy is created when information is erased, but not when it is acquired, is explained
in C
H. B
&R L
, Fundamental Limits of Computation,
Scienti c American 253:1, pp. – , . e conclusion: we should pay to throw the news-
paper away, not to buy it. Cited on page .
210 See, for example, G. S , ermoacoustic engines and refrigerators, Physics Today 48, pp. – , July . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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211 Quoted in D. C
, J. C
, J. F
& E. J , Experimental math-
ematics: the role of computation in nonlinear science, Communications of the Association of
Computing Machinery 28, pp. – , . Cited on page .
212 For more about the shapes of snow akes, see the famous book by W.A. B
& W.J.
H
, Snow Crystals, Dover Publications, New York, . is second printing of
the original from shows a large part of the result of Bentley’s lifelong passion, namely
several thousand photographs of snow akes. Cited on page .
213 K. S page .
, Why snakes have forked tongues, Science 263, pp.
–,
. Cited on
214 P.B. U
, F. M & H.L. S
, Localized excitations in a vertically
vibrated granular layer, Nature 382, pp. – , August . Cited on page .
215 D.K. C
, S. F
& Y.S. K
, Localizing energy through nonlinearity
and discreteness, Physics Today 57, pp. – , January . Cited on page .
216 B. A 92, p.
, e song of dunes as a wave-particle mode locking, Physical Review Letters , . Cited on page .
217 K. K
, E. G & M. M
, Shell structures with ‘magic numbers’ of spheres
in a swirled disk, Physical Review E 60, pp. – , . Cited on page .
218 A good introduction is the text by D
W
, Spatiotemporal Pattern Formation,
With Examples in Physics, Chemistry and Materials Science, Springer . Cited on page
.
219 For an overview, see the Ph.D. thesis by J
L , Défauts topologiques associés à la
brisure de l’invariance de translation dans le temps, Université de Nice, . Cited on page
.
220 An idea of the fascinating mechanisms at the basis of the heart beat is given by A. B
-
& A. D
, Is the normal heart a periodic oscillator?, Biological Cybernetics
58, pp. – , . Cited on page .
221 For a short, modern overview of turbulence, see L.P. K
, A model of turbulence,
Physics Today 48, pp. – , September . Cited on page .
222 For a clear introduction, see T. S
& M. M
, A simple mathematical model of
a dripping tap, European Journal of Physics 18, pp. – , . Cited on page .
223 An overview of science humour can be found in the famous anthology compiled by R.L.
W , edited by E. M
, A Random Walk in Science, Institute of Physics, . It
is also available in several expanded translations. Cited on page .
224 K. M
, V. P
Nature 430, p. ,
& P. V . Cited on page .
, Braiding patterns on an inclined plane,
225 G. M
, Starch columns: analog model for basalt columns, Journal of Geophysical Re-
search 103, pp. – , . Cited on page .
226 B. H , C.W.H.
D
, J. W
, F.T.M. N
, H. W ,
R. K
, F. W
, H. F
& B. E
, Experimental observation of
nonlinear traveling waves in turbulent pipe ow, Science 305, pp. – , . Cited
on page .
227 A fascinating book on the topic is K
L &M
S
, Physics and the
Art of Dance: Understanding Movement, Oxford University Press . Cited on page .
228 J.J. L
, Chaotic motion in the solar system, Reviews of Modern Physics 71, pp. –
, . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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229 e present state of our understanding of turbulence is described in ... Cited on page .
230 See J -P D Cited on page .
, Les écoles présocratiques, Folio Essais, Gallimard, , p. .
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C
II
SPECIAL RELATIVIT Y
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ere are limitations on motion that are missed by the Galilean description. e rst limitation we discover is the existence of a maximal speed in nature. e maximum speed implies many fascinating results: it leads to observer-varying time and length intervals, to an intimate relation between mass and energy, and to the existence of event horizons. We explore them now.
.
,
,
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
“ ” Fama nihil est celerius.*
L is indispensable for a precise description of motion. To check whether a ine or a path of motion is straight, we must look along it. In other words, we use ight to de ne straightness. How do we decide whether a plane is at? We look across it,** again using light. How do we measure length to high precision? With light. How do we measure time to high precision? With light: once it was light from the Sun that was Page 1154 used; nowadays it is light from caesium atoms.
In other words, light is important because it is the standard for undisturbed motion. Physics would have evolved much more rapidly if, at some earlier time, light propagation had been recognized as the ideal example of motion.
But is light really a phenomenon of motion? is was already known in ancient Greece, from a simple daily phenomenon, the shadow. Shadows prove that light is a moving entity, emanating from the light source, and moving in straight lines.*** e obvious conclusion that light takes a certain amount of time to travel from the source to the surface
Challenge 540 n
* ‘Nothing is faster than rumour.’ is common sentence is a simpli ed version of Virgil’s phrase: fama, malum qua non aliud velocius ullum. ‘Rumour, the evil faster than all.’ From the Aeneid, book IV, verses 173 and 174. ** Note that looking along the plane from all sides is not su cient for this: a surface that a light beam touches right along its length in all directions does not need to be at. Can you give an example? One needs other methods to check atness with light. Can you specify one? *** Whenever a source produces shadows, the emitted entities are called rays or radiation. Apart from light, other examples of radiation discovered through shadows were infrared rays and ultraviolet rays, which emanate from most light sources together with visible light, and cathode rays, which were found to be to the motion of a new particle, the electron. Shadows also led to the discovery of X-rays, which again turned out to be a version of light, with high frequency. Channel rays were also discovered via their shadows; they turn out to be travelling ionized atoms. e three types of radioactivity, namely α-rays (helium nuclei), β-rays
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•.
,
Jupiter and Io (second measurement)
Earth (second measurement)
Sun
Earth (first measurement)
Jupiter and Io (first measurement)
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F I G U R E 132 Rømer’s method of measuring the speed of light
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 231 Challenge 541 n
Challenge 542 n Page 101 Ref. 232
showing the shadow had already been reached by the Greek thinker Empedocles (c.
to c.
).
We can con rm this result with a di erent, equally simple, but subtle argument. Speed
can be measured. erefore the perfect speed, which is used as the implicit measurement
standard, must have a nite value. An in nite velocity standard would not allow meas-
urements at all. In nature, the lightest entities move with the highest speed. Light, which
is indeed light, is an obvious candidate for motion with perfect but nite speed. We will
con rm this in a minute.
A nite speed of light means that whatever we see is a message from the past. When
we see the stars, the Sun or a loved one, we always see an image of the past. In a sense,
nature prevents us from enjoying the present – we must therefore learn to enjoy the past.
e speed of light is high; therefore it was not measured until , even though many,
including Galileo, had tried to do so earlier. e rst measurement method was worked
out by the Danish astronomer Ole Rømer* when he was studying the orbits of Io and
the other moons of Jupiter. He obtained an incorrect value for the speed of light because
he used the wrong value for their distance from Earth. However, this was quickly correc-
ted by his peers, including Newton himself. You might try to deduce his method from
Figure . Since that time it has been known that light takes a bit more than minutes
to travel from the Sun to the Earth. is was con rmed in a beautiful way y years later,
in , by the astronomer James Bradley. Being English, Bradley thought of the ‘rain
method’ to measure the speed of light.
How can we measure the speed of falling rain? We walk rapidly with an umbrella,
measure the angle α at which the rain appears to fall, and then measure our own velocity
(again electrons), and γ-rays (high-energy X-rays) also produce shadows. All these discoveries were made between 1890 and 1910: those were the ‘ray days’ of physics. * Ole (Olaf) Rømer (1644 Aarhus – 1710 Copenhagen), Danish astronomer. He was the teacher of the Dauphin in Paris, at the time of Louis XIV. e idea of measuring the speed of light in this way was due to the Italian astronomer Givanni Cassini, whose assistant Rømer had been. Rømer continued his measurements until 1681, when Rømer had to leave France, like all protestants (such as Christiaan Huygens), so that his work was interrupted. Back in Denmark, a re destroyed all his measurement notes. As a result, he was not able to continue improving the precision of his method. Later he became an important administrator and reformer of the Danish state.
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,
rain's perspective
,
rain c
v
light's perspective light
c earth v
Sun
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human perspective
human perspective
α c
α
v
Sun
v
F I G U R E 133 The rain method of measuring the speed of light
v. As shown in Figure , the speed c of the rain is then given by
c = v tan α .
(100)
e same measurement can be made for light; we just need to measure the angle at which the light from a star above Earth’s orbit arrives at the Earth. Because the Earth is moving relative to the Sun and thus to the star, the angle is not a right one. is e ect is called the aberration of light; the angle is found most easily by comparing measurements made six months apart. e value of the angle is . ′′; nowadays it can be measured with a precision of ve decimal digits. Given that the speed of the Earth around the Sun is v = πR T = . km s, the speed of light must therefore be c = . ë m s.* is is
Challenge 543 n Challenge 544 n
Challenge 545 n
* Umbrellas were not common in Britain in 1726; they became fashionable later, a er being introduced from China. e umbrella part of the story is made up. In reality, Bradley had his idea while sailing on the ames, when he noted that on a moving ship the apparent wind has a di erent direction from that on land. He had observed 50 stars for many years, notably Gamma Draconis, and during that time he had been puzzled by the sign of the aberration, which was opposite to the e ect he was looking for, namely the star parallax. Both the parallax and the aberration for a star above the ecliptic make them describe a small ellipse in the course of an Earth year, though with di erent rotation senses. Can you see why?
By the way, it follows from special relativity that the formula (100) is wrong, and that the correct formula is c = v sin α; can you see why?
To determine the speed of the Earth, we rst have to determine its distance from the Sun. e simplest method is the one by the Greek thinker Aristarchos of Samos (c. 310 to c. 230 ). We measure the angle between the Moon and the Sun at the moment when the Moon is precisely half full. e cosine of that angle gives the ratio between the distance to the Moon (determined, for example, by the methods of page 117) and the distance to the Sun. e explanation is le as a puzzle for the reader.
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mirror
•.
,
half-silvered mirror
light source
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F I G U R E 134 Fizeau’s set-up to measure the speed of light (© AG Didaktik und Geschichte der Physik, Universität Oldenburg)
Page 57 Challenge 547 n
Ref. 235
Page 560
an astonishing value, especially when compared with the highest speed ever achieved by a man-made object, namely the Voyager satellites, which travel at Mm h = km s, with the growth of children, about nm s, or with the growth of stalagmites in caves, about . pm s. We begin to realize why measurement of the speed of light is a science in its own right.
e rst precise measurement of the speed of light was made in by the French physicist Hippolyte Fizeau ( – ). His value was only % greater than the modern one. He sent a beam of light towards a distant mirror and measured the time the light took to come back. How did Fizeau measure the time without any electric device? In fact, he used the same ideas that are used to measure bullet speeds; part of the answer is given in Figure . (How far away does the mirror have to be?) A modern reconstruction of his experiment by Jan Frercks has achieved a precision of %. Today, the experiment is much simpler; in the chapter on electrodynamics we will discover how to measure the speed of light using two standard UNIX or Linux computers connected by a cable.
e speed of light is so high that it is even di cult to prove that it is nite. Perhaps the most beautiful way to prove this is to photograph a light pulse ying across one’s eld of view, in the same way as one can photograph a car driving by or a bullet ying through
Ref. 233
Page 1167 Challenge 546 n
Ref. 234
e angle in question is almost a right angle (which would yield an in nite distance), and good instruments are needed to measure it with precision, as Hipparchos noted in an extensive discussion of the problem around 130 . Precise measurement of the angle became possible only in the late seventeenth century, when it was found to be . °, giving a distance ratio of about 400. Today, thanks to radar measurements of planets, the distance to the Sun is known with the incredible precision of 30 metres. Moon distance variations can even be measured to the nearest centimetre; can you guess how this is achieved?
Aristarchos also determined the radius of the Sun and of the Moon as multiples of those of the Earth. Aristarchos was a remarkable thinker: he was the rst to propose the heliocentric system, and perhaps the
rst to propose that stars were other, faraway suns. For these ideas, several of his contemporaries proposed that he should be condemned to death for impiety. When the Polish monk and astronomer Nicolaus Copernicus (1473–1543) again proposed the heliocentric system two thousand years later, he did not mention Aristarchus, even though he got the idea from him.
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,
,
red shutter switch beam
path of light pulse
10 mm
F I G U R E 135 A photograph of a light pulse moving from right to left through a bottle with milky water, marked in millimetres (© Tom Mattick)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 136 A consequence of the finiteness of the speed of light
Ref. 236 Challenge 548 n
Challenge 549 n
the air. Figure shows the rst such photograph, produced in with a standard o -the-shelf re ex camera, a very fast shutter invented by the photographers, and, most noteworthy, not a single piece of electronic equipment. (How fast does such a shutter have to be? How would you build such a shutter? And how would you make sure it opened at the right instant?)
A nite speed of light also implies that a rapidly rotating light beam behaves as shown as in Figure . In everyday life, the high speed of light and the slow rotation of lighthouses make the e ect barely noticeable.
In short, light moves extremely rapidly. It is much faster than lightning, as you might like to check yourself. A century of increasingly precise measurements of the speed have culminated in the modern value
c=
m s.
(101)
In fact, this value has now been xed exactly, by de nition, and the metre has been de ned in terms of c. Table gives a summary of what is known today about the motion of light. Two surprising properties were discovered in the late nineteenth century. ey form the
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•.
,
TA B L E 35 Properties of the motion of light O
Light can move through vacuum.
Light transports energy.
Light has momentum: it can hit bodies.
Light has angular momentum: it can rotate bodies.
Light moves across other light undisturbed.
Light in vacuum always moves faster than any material body does.
e speed of light, its true signal speed, is the forerunner speed. Page 579
In vacuum its value is
m s.
e proper speed of light is in nite. Page 297
Shadows can move without any speed limit.
Light moves in a straight line when far from matter.
High-intensity light is a wave.
Light beams are approximations when the wavelength is neglected.
In matter, both the forerunner speed and the energy speed of light are lower than in vacuum.
In matter, the group velocity of light pulses can be zero, positive, negative or in nite.
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Ref. 237 basis of special relativity.
Ref. 238 Challenge 550 n
C
?
Et nihil est celerius annis.*
“ ” Ovid, Metamorphoses.
We all know that in order to throw a stone as far as possible, we run as we throw it; we know instinctively that in that case the stone’s speed with respect to the ground is higher. However, to the initial astonishment of everybody, experiments show that light emitted from a moving lamp has the same speed as light emitted from a resting one. Light (in vacuum) is never faster than light; all light beams have the same speed. Many specially designed experiments have con rmed this result to high precision. e speed of light can be measured with a precision of better than m s; but even for lamp speeds of more than
m s no di erences have been found. (Can you guess what lamps were used?) In everyday life, we know that a stone arrives more rapidly if we run towards it. Again, for light no di erence has been measured. All experiments show that the velocity of light has the same value for all observers, even if they are moving with respect to each other or with respect to the light source. e speed of light is indeed the ideal, perfect measurement standard.**
Page 563
* ‘Nothing is faster than the years.’ Book X, verse 520. ** An equivalent alternative term for the speed of light is ‘radar speed’ or ‘radio speed’; we will see below why this is the case.
e speed of light is also not far from the speed of neutrinos. is was shown most spectacularly by the
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,
,
Ref. 241 Page 536
Page 536
ere is also a second set of experimental evidence for the
constancy of the speed of light. Every electromagnetic device,
such as an electric toothbrush, shows that the speed of light is
constant. We will discover that magnetic elds would not res-
ult from electric currents, as they do every day in every motor
and in every loudspeaker, if the speed of light were not constant.
is was actually how the constancy was rst deduced, by sev-
eral researchers. Only a er understanding this, did the German–
Swiss physicist Albert Einstein* show that the constancy is also
in agreement with the motion of bodies, as we will do in this
section. e connection between electric toothbrushes and re-
lativity will be described in the chapter on electrodynamics.** In simple terms, if the speed of light were not constant, observers
Albert Einstein
would be able to move at the speed of light. Since light is a wave, such observers would
see a wave standing still. However, electromagnetism forbids the such a phenomenon.
erefore, observers cannot reach the speed of light.
In summary, the velocity v of any physical system in nature (i.e., any localized mass or
energy) is bound by
v c.
(102)
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Ref. 244
is relation is the basis of special relativity; in fact, the full theory of special relativity is contained in it. Einstein o en regretted that the theory was called ‘Relativitätstheorie’ or ‘theory of relativity’; he preferred the name ‘Invarianztheorie’ or ‘theory of invariance’, but was not able to change the name.
Challenge 551 n Ref. 239 Ref. 240
Page 318
Ref. 242
Ref. 243
observation of a supernova in 1987, when the ash and the neutrino pulse arrived a 12 seconds apart. (It is not known whether the di erence is due to speed di erences or to a di erent starting point of the two
ashes.) What is the rst digit for which the two speed values could di er, knowing that the supernova was . ë light years away?
Experiments also show that the speed of light is the same in all directions of space, to at least 21 digits of precision. Other data, taken from gamma ray bursts, show that the speed of light is independent of frequency, to at least 20 digits of precision. * Albert Einstein (b. 1879 Ulm, d. 1955 Princeton); one of the greatest physicists ever. He published three important papers in 1905, one about Brownian motion, one about special relativity, and one about the idea of light quanta. Each paper was worth a Nobel Prize, but he was awarded the prize only for the last one. Also in 1905, he proved the famous formula E = mc (published in early 1906), possibly triggered by an idea of Olinto de Pretto. Although Einstein was one of the founders of quantum theory, he later turned against it. His famous discussions with his friend Niels Bohr nevertheless helped to clarify the eld in its most counterintuitive aspects. He explained the Einstein–de Haas e ect which proves that magnetism is due to motion inside materials. In 1915 and 1916, he published his highest achievement: the general theory of relativity, one of the most beautiful and remarkable works of science.
Being Jewish and famous, Einstein was a favourite target of attacks and discrimination by the National Socialist movement; in 1933 he emigrated to the USA. He was not only a great physicist, but also a great thinker; his collection of thoughts about topics outside physics are worth reading.
Anyone interested in emulating Einstein should know that he published many papers, and that many of them were wrong; he would then correct the results in subsequent papers, and then do so again. is happened so frequently that he made fun of himself about it. Einstein realizes the famous de nition of a genius as a person who makes the largest possible number of mistakes in the shortest possible time. ** For information about the in uences of relativity on machine design, see the interesting textbook by Van Bladel.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
e constancy of the speed of light is in complete contrast with Galilean mechanics, and proves that the latter is wrong at high velocities. At low velocities the description remains good, because the error is small. But if we want a description valid at all velocities, we have to discard Galilean mechanics. For example, when we play tennis we use the fact that by hitting the ball in the right way, we can increase or decrease its speed. But with light this is impossible. Even if we take an aeroplane and y a er a light beam, it still moves away with the same speed. Light does not behave like cars. If we accelerate a bus we are driving, the cars on the other side of the road pass by with higher and higher speeds. For light, this is not so: light always passes by with the same speed.*
Why is this result almost unbelievable, even though the measurements show it unambiguously? Take two observers O and Ω (pronounced ‘omega’) moving with relative velocity v, such as two cars on opposite sides of the street. Imagine that at the moment they pass each other, a light ash is emitted by a lamp in O. e light ash moves through positions x(t) for O and through positions ξ(τ) (pronounced ‘xi of tau’) for Ω. Since the speed of light is the same for both, we have
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x t
=c=
ξ τ
.
(103)
Challenge 552 e Ref. 245
Ref. 241
However, in the situation described, we obviously have x ξ. In other words, the constancy of the speed of light implies that t τ, i.e. that time is di erent for observers moving relative to each other. Time is thus not unique. is surprising result, which has been con-
rmed by many experiments, was rst stated clearly in by Albert Einstein. ough many others knew about the invariance of c, only the young Einstein had the courage to say that time is observer-dependent, and to face the consequences. Let us do so as well.
Already in , the discussion of viewpoint invariance had been called the theory of relativity by Henri Poincaré.** Einstein called the description of motion without gravity the theory of special relativity, and the description of motion with gravity the theory of general relativity. Both elds are full of fascinating and counter-intuitive results. In particular, they show that everyday Galilean physics is wrong at high speeds.
e speed of light is a limit speed. We stress that we are not talking of the situation where a particle moves faster than the speed of light in matter, but still slower than the speed of light in vacuum. Moving faster than the speed of light in matter is possible. If the particle is charged, this situation gives rise to the so-called Čerenkov radiation. It corresponds to the V-shaped wave created by a motor boat on the sea or the cone-shaped shock wave around an aeroplane moving faster than the speed of sound. Čerenkov radiation is regularly observed; for example it is the cause of the blue glow of the water in nuclear reactors. Incidentally, the speed of light in matter can be quite low: in the centre of the Sun,
Ref. 239 Ref. 246, Ref. 247
* Indeed, even with the current measurement precision of ë − , we cannot discern any changes of the speed of light with the speed of the observer. ** Henri Poincaré (1854–1912), important French mathematician and physicist. Poincaré was one of the most productive men of his time, advancing relativity, quantum theory, and many parts of mathematics.
e most beautiful and simple introduction to relativity is still that given by Albert Einstein himself, for example in Über die spezielle und allgemeine Relativitätstheorie, Vieweg, 1997, or in e Meaning of Relativity, Methuen, London, 1951. It has taken a century for books almost as beautiful to appear, such as the text by Taylor and Wheeler.
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Ref. 248, Ref. 249
the speed of light is estimated to be only around km year, and even in the laboratory, for some materials, it has been found to be as low as . m s. In the following, when we use the term ‘speed of light’, we mean the speed of light in vacuum. e speed of light in air in smaller than that in vacuum only by a fraction of a percent, so that in most cases, the di erence can be neglected.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
S
Ref. 250
Challenge 553 n Challenge 554 n
e speed of light is constant for all observers. We
can thus deduce all relations between what two different observers measure with the help of Figure . t It shows two observers moving with constant speed against each other in space-time, with the rst send-
first observer or clock
second observer or clock
ing a light ash to the second, from where it is re ected back to the rst. Since light speed is constant, light
k2 T light
is the only way to compare time and space coordin-
ates for two distant observers. Two distant clocks (like two distant metre bars) can only be compared, or
t1 = (k2+1)T/2
t2 = kT
synchronized, using light or radio ashes. Since light
speed is constant, light paths are parallel in such dia-
T
grams.
A constant relative speed between two observers implies that a constant factor k relates the time co-
O
ordinates of events. (Why is the relation linear?) If a
x
ash starts at a time T as measured for the rst ob- F I G U RE 137 A drawing containing server, it arrives at the second at time kT, and then most of special relativity
back again at the rst at time k T. e drawing shows
that
k=
c+v c−v
or
v c
=
k k
− +
.
(104)
Page 284
is factor will appear again in the Doppler e ect.* e gure also shows that the time coordinate t assigned by the rst observer to the
moment in which the light is re ected is di erent from the coordinate t assigned by the second observer. Time is indeed di erent for two observers in relative motion. Figure illustrates the result.
e time dilation factor between the two time coordinates is found from Figure by comparing the values t and t ; it is given by
t t
=
= γ(v) .
−
v c
(105)
Time intervals for a moving observer are shorter by this factor γ; the time dilation factor is always larger than . In other words, moving clocks go slower. For everyday speeds the
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* e explanation of relativity using the factor k is o en called k-calculus.
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one moving watch
first
second
time
time
two fixed watches F I G U R E 138 Moving clocks go slow
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Challenge 555 e
e ect is tiny. at is why we do not detect time di erences in everyday life. Nevertheless, Galilean physics is not correct for speeds near that of light. e same factor γ also appears in the formula E = γmc , which we will deduce below. Expression ( ) or ( ) is the only piece of mathematics needed in special relativity: all other results derive from it.
If a light ash is sent forward starting from the second observer and re ected back, he will make the same statement: for him, the rst clock is moving, and also for him, the moving clock goes slower. Each of the observers observes that the other clock goes slower.
e situation is similar to that of two men comparing the number of steps between two identical ladders that are not parallel. A man on either ladder will always observe that the steps of the other ladder are shorter. For another analogy, take two people moving away from each other: each of them notes that the other gets smaller as their distance increases.
Naturally, many people have tried to nd arguments to avoid the strange conclusion that time di ers from observer to observer. But none have succeeded, and experimental results con rm this conclusion. Let us have a look at some of them.
A
D
Page 567 Challenge 556 n
Ref. 251, Ref. 252
Light can be accelerated. Every mirror does this! We will see in the chapter on electromagnetism that matter also has the power to bend light, and thus to accelerate it. However, it will turn out that all these methods only change the direction of propagation; none has the power to change the speed of light in a vacuum. In short, light is an example of a motion that cannot be stopped. ere are only a few other such examples. Can you name one?
What would happen if we could accelerate light to higher speeds? For this to be possible, light would have to be made of particles with non-vanishing mass. Physicists call such particles massive particles. If light had mass, it would be necessary to distinguish the ‘massless energy speed’ c from the speed of light cL, which would be lower and would depend on the kinetic energy of those massive particles. e speed of light would not be constant, but the massless energy speed would still be so. Massive light particles could be captured, stopped and stored in a box. Such boxes would make electric illumination unnecessary; it would be su cient to store some daylight in them and release the light, slowly, during the following night, maybe a er giving it a push to speed it up.*
Physicists have tested the possibility of massive light in quite some detail. Observations now put any possible mass of light (particles) at less than . ë − kg from terrestrial
* Incidentally, massive light would also have longitudinal polarization modes. is is in contrast to observations, which show that light is polarized exclusively transversally to the propagation direction.
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sender v
y
receiver
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y sender
light signal
θs
v
x
receiver z
θr x
z F I G U R E 139 The set-up for the observation of the Doppler effect
Challenge 557 e
experiments, and at less than ë − kg from astrophysical arguments (which are a bit less strict). In other words, light is not heavy, light is light.
But what happens when light hits a moving mirror? If the speed of light does not change, something else must. e situation is akin to that of a light source moving with respect to the receiver: the receiver will observe a di erent colour from that observed by the sender. is is called the Doppler e ect. Christian Doppler* was the rst to study the frequency shi in the case of sound waves – the well-known change in whistle tone between approaching and departing trains – and to extend the concept to the case of light waves. As we will see later on, light is (also) a wave, and its colour is determined by its frequency, or equivalently, by its wavelength λ. Like the tone change for moving trains, Doppler realized that a moving light source produces a colour at the receiver that is di erent from the colour at the source. Simple geometry, and the conservation of the number of maxima and minima, leads to the result
* Christian Andreas Doppler (b. 1803 Salzburg, d. 1853 Venezia), Austrian physicist. Doppler studied the e ect named a er him for sound and light. In 1842 he predicted (correctly) that one day we would be able to use the e ect to measure the motion of distant stars by looking at their colours.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•.
,
λr λs
=
−v
c
(
−
v c
cos
θr)
=
γ
(
−
v c
cos
θr)
.
(106)
Ref. 253 Challenge 558 n
Page 563
e variables v and θr in this expression are de ned in Figure . Light from an approaching source is thus blue-shi ed, whereas light from a departing source is red-shi ed. e
rst observation of the Doppler e ect for light was made by Johannes Stark* in , who studied the light emitted by moving atoms. All subsequent experiments con rmed the calculated colour shi within measurement errors; the latest checks have found agreement to within two parts per million. In contrast to sound waves, a colour change is also found when the motion is transverse to the light signal. us, a yellow rod in rapid motion across the eld of view will have a blue leading edge and a red trailing edge prior to the closest approach to the observer. e colours result from a combination of the longitudinal ( rst-order) Doppler shi and the transverse (second-order) Doppler shi . At a particular angle θunshifted the colours will be the same. (How does the wavelength change in the purely transverse case? What is the expression for θunshifted in terms of v?)
e colour shi is used in many applications. Almost all solid bodies are mirrors for radio waves. Many buildings have doors that open automatically when one approaches. A little sensor above the door detects the approaching person. It usually does this by measuring the Doppler e ect of radio waves emitted by the sensor and re ected by the approaching person. (We will see later that radio waves and light are manifestations of the same phenomenon.) So the doors open whenever something moves towards them. Police radar also uses the Doppler e ect, this time to measure the speed of cars.**
e Doppler e ect also makes it possible to measure the velocity of light sources. Indeed, it is commonly used to measure the speed of distant stars. In these cases, the Doppler shi is o en characterized by the red-shi number z, de ned with the help of wavelength λ or frequency F by
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z=
∆λ λ
=
fS fR
−
=
c+v c−v
−
.
(107)
Challenge 560 n Challenge 561 n
Page 133 Challenge 562 n
Can you imagine how the number z is determined? Typical values for z for light sources in the sky range from − . to . , but higher values, up to more than , have also been found. Can you determine the corresponding speeds? How can they be so high?
In summary, whenever one tries to change the speed of light, one only manages to change its colour. at is the Doppler e ect.
We know from classical physics that when light passes a large mass, such as a star, it is de ected. Does this de ection lead to a Doppler shi ?
Challenge 559 n
* Johannes Stark (1874–1957), discovered in 1905 the optical Doppler e ect in channel rays, and in 1913 the splitting of spectral lines in electrical elds, nowadays called the Stark e ect. For these two discoveries he received the 1919 Nobel Prize for physics. He le his professorship in 1922 and later turned into a fullblown National Socialist. A member of the NSDAP from 1930 onwards, he became known for aggressively criticizing other people’s statements about nature purely for ideological reasons; he became rightly despised by the academic community all over the world. ** At what speed does a red tra c light appear green?
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T
Ref. 254
Challenge 563 n Page 987 Ref. 255
e Doppler e ect for light is much more important than the Doppler e ect for sound. Even if the speed of light were not yet known to be constant, this e ect alone would prove that time is di erent for observers moving relative to each other. Why? Time is what we read from our watch. In order to determine whether another watch is synchronized with our own one, we look at both watches. In short, we need to use light signals to synchronize clocks. Now, any change in the colour of light moving from one observer to another necessarily implies that their watches run di erently, and thus that time is di erent for the two of them. One way to see this is to note that also a light source is a clock – ‘ticking’ very rapidly. So if two observers see di erent colours from the same source, they measure di erent numbers of oscillations for the same clock. In other words, time is di erent for observers moving against each other. Indeed, equation ( ) implies that the whole of relativity follows from the full Doppler e ect for light. (Can you con rm that the connection between observer-dependent frequencies and observer-dependent time breaks down in the case of the Doppler e ect for sound?)
Why does the behaviour of light imply special relativity, while that of sound in air does not? e answer is that light is a limit for the motion of energy. Experience shows that there are supersonic aeroplanes, but there are no superluminal rockets. In other words, the limit v c is valid only if c is the speed of light, not if c is the speed of sound in air.
However, there is at least one system in nature where the speed of sound is indeed a limit speed for energy: the speed of sound is the limit speed for the motion of dislocations in crystalline solids. (We discuss this in detail later on.) As a result, the theory of special relativity is also valid for such dislocations, provided that the speed of light is replaced everywhere by the speed of sound! Dislocations obey the Lorentz transformations, show length contraction, and obey the famous energy formula E = γmc . In all these e ects the speed of sound c plays the same role for dislocations as the speed of light plays for general physical systems.
If special relativity is based on the statement that nothing can move faster than light, this statement needs to be carefully checked.
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C
’
?
Challenge 564 e Ref. 256
“ ” Quid celerius umbra?*
for Lucky Luke to achieve the feat shown in Figure , his bullet has to move faster than the speed of light. (What about his hand?) In order to emulate Lucky Luke, we could take the largest practical amount of energy available, taking it directly from an electrical power station, and accelerate the lightest ‘bullets’ that can be handled, namely electrons. is experiment is carried out daily in particle accelerators such as the Large Electron Positron ring, the LEP, of km circumference, located partly in France and partly in Switzerland, near Geneva. ere, MW of electrical power (the same amount used by a small city) accelerates electrons and positrons to energies of over nJ ( . GeV) each, and their speed is measured. e result is shown in Figure : even with these impressive means
* ‘What is faster than the shadow?’ A motto o en found on sundials.
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F I G U R E 140 Lucky Luke
Challenge 565 e Page 313
Challenge 566 n
it is impossible to make electrons move more rapidly than light. (Can you imagine a way
to measure energy and speed separately?) e speed–energy relation of Figure is a
consequence of the maximum speed, and is deduced below. ese and many similar ob-
servations thus show that there is a limit to the velocity of objects. Bodies (and radiation)
cannot move at velocities higher than the speed of light.* e accuracy of Galilean mech-
anics was taken for granted for more than three centuries, so that nobody ever thought
of checking it; but when this was nally done, as in Figure , it was found to be wrong.
e people most unhappy with this
limit are computer engineers: if the speed limit were higher, it would be possible v2
TGal
=
1 2
m
v2
to make faster microprocessors and thus faster computers; this would allow, for c2 example, more rapid progress towards the construction of computers that un-
T = m c2 (
1 1 - v2/c 2
– 1)
derstand and use language.
e existence of a limit speed runs
T
counter to Galilean mechanics. In fact, F I G U R E 141 Experimental values (dots) for the it means that for velocities near that of electron velocity v as function of their kinetic
light, say about
km s or more, the energy T, compared with the prediction of Galilean
expression mv is not equal to the kin- physics (blue) and that of special relativity (red)
etic energy T of the particle. In fact, such
high speeds are rather common: many families have an example in their home. Just calcu-
late the speed of electrons inside a television, given that the transformer inside produces
kV.
Ref. 257
* ere are still people who refuse to accept these results, as well as the ensuing theory of relativity. Every physicist should enjoy the experience, at least once in his life, of conversing with one of these men. (Strangely, no woman has yet been reported as belonging to this group of people.) is can be done, for example, via the internet, in the sci.physics.relativity newsgroup. See also the http://www.crank.net website. Crackpots are a fascinating lot, especially since they teach the importance of precision in language and in reasoning, which they all, without exception, neglect. Encounters with several of them provided the inspiration for this chapter.
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,
t
first observer (e.g. Earth)
second observer (e.g. train)
third observer (e.g. stone)
kseT kteT T
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O x
F I G U R E 142 How to deduce the composition of velocities
Challenge 567 d
Ref. 258 Challenge 568 e Challenge 569 r
e observation of speed of light as a limit speed for objects is easily seen to be a consequence of its constancy. Bodies that can be at rest in one frame of reference obviously move more slowly than the maximum velocity (light) in that frame. Now, if something moves more slowly than something else for one observer, it does so for all other observers as well. (Trying to imagine a world in which this would not be so is interesting: funny things would happen, such as things interpenetrating each other.) Since the speed of light is the same for all observers, no object can move faster than light, for every observer.
We follow that the maximum speed is the speed of massless entities. Electromagnetic waves, including light, are the only known entities that can travel at the maximum speed. Gravitational waves are also predicted to achieve maximum speed. ough the speed of neutrinos cannot be distinguished experimentally from the maximum speed, recent experiments suggest that they do have a tiny mass.
Conversely, if a phenomenon exists whose speed is the limit speed for one observer, then this limit speed must necessarily be the same for all observers. Is the connection between limit property and observer invariance generally valid in nature?
T
If the speed of light is a limit, no attempt to exceed it can succeed. is implies that when velocities are composed, as when one throws a stone while running, the values cannot simply be added. If a train is travelling at velocity vte relative to the Earth, and somebody throws a stone inside it with velocity vst relative to the train in the same direction, it is usually assumed as evident that the velocity of the stone relative to the Earth is given by vse = vst + vte. In fact, both reasoning and measurement show a di erent result.
e existence of a maximum speed, together with Figure , implies that the k-factors must satisfy kse = kstkte.* en we only have to insert the relation ( ) between each
* By taking the (natural) logarithm of this equation, one can de ne a quantity, the rapidity, that measures
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,
Challenge 570 e k-factor and the respective speed to get
vse =
vst + vte + vstvte c
.
(108)
Challenge 571 e
Page 313, page 536 Ref. 252
is is called the velocity composition formula. e result is never larger than c and is always smaller than the naive sum of the velocities.* Expression ( ) has been con rmed by all of the millions of cases for which it has been checked. You may check that it reduces to the naive sum for everyday life values.
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O
Ref. 260
Special relativity is built on a simple principle: e maximum speed of energy transport is the same for all observers.
Or, as Hendrik Lorentz** liked to say, the equivalent: e speed v of a physical system is bound by
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vc
(109)
Ref. 261 Ref. 262
Page 536
for all observers, where c is the speed of light.
is independence of the speed of light from the observer was checked with high precision by Michelson and Morley*** in the years from onwards. It has been con rmed in all subsequent experiments; the most precise to date, which achieved a precision of
− is shown in Figure . In fact, special relativity is also con rmed by all the precision experiments that were performed before it was formulated. You can even con rm it yourself at home. e way to do this is shown in the section on electrodynamics.
e existence of a limit speed has several interesting consequences. To explore them, let us keep the rest of Galilean physics intact.**** e limit speed is the speed of light. It is constant for all observers. is constancy implies:
Ref. 259 Page 86
the speed and is additive. * One can also deduce the Lorentz transformation directly from this expression. ** Hendrik Antoon Lorentz (b. 1853 Arnhem, d. 1928 Haarlem) was, together with Boltzmann and Kelvin, one of the most important physicists of his time. He deduced the so-called Lorentz transformation and the Lorentz contraction from Maxwell’s equation for the electrodynamic eld. He was the rst to understand, long before quantum theory con rmed the idea, that Maxwell’s equations for the vacuum also describe matter and all its properties, as long as moving charged point particles – the electrons – are included. He showed this in particular for the dispersion of light, for the Zeeman e ect, for the Hall e ect and for the Faraday e ect. He gave the correct description of the Lorentz force. In 1902, he received the physics Nobel Prize, together with Pieter Zeeman. Outside physics, he was active in the internationalization of scienti c collaborations. He was also instrumental in the creation of the largest human-made structures on Earth: the polders of the Zuyder Zee. *** Albert Abraham Michelson (b. 1852 Strelno, d. 1931 Pasadena), Prussian–Polish–US-American physicist, awarded the Nobel Prize in physics in 1907. Michelson called the set-up he devised an interferometer, a term still in use today. Edward William Morley (1838–1923), US-American chemist, was Michelson’s friend and long-time collaborator. **** is point is essential. For example, Galilean physics states that only relative motion is physical. Galilean physics also excludes various mathematically possible ways to realize a constant light speed that would con-
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beat frequency change [Hz ] angle/3 [deg]
,
,
30 20 10
0 10 20
0
100 200 300 400 500 600 time since begin of rotation [s]
AOM driver
Power servo
PZT Laser 1
T °C Nd: YAG
T °C Laser 2 Nd: YAG PZT
∑
Frequency servo
Frequency servo
∑
FC AOM
AOM FC Local
oscillator Local
oscillator
Frequency counter PD
Fiber DBM
DBM
AOM driver
Power servo
Fiber
FC Res B PD
FC BS
PD PD
PD Res A
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F I G U R E 143 The result, the schematics and the cryostat set-up for the most precise Michelson–Morley experiment performed to date (© Stephan Schiller)
— In a closed free- oating room, there is no way to tell the speed of the room. — ere is no notion of absolute rest (or space): rest (like space) is an observer-dependent
concept.* — Time depends on the observer; time is not absolute.
More interesting and speci c conclusions can be drawn when two additional conditions are assumed. First, we study situations where gravitation can be neglected. (If this not the case, we need general relativity to describe the system.) Secondly, we also assume that the data about the bodies under study – their speed, their position, etc. – can be gathered without disturbing them. (If this not the case, we need quantum theory to describe the system.)
To deduce the precise way in which the di erent time intervals and lengths measured by two observers are related to each other, we take an additional simplifying step. We start
Ref. 244 Challenge 572 n
tradict everyday life. Einstein’s original 1905 paper starts from two principles: the constancy of the speed of light and the
equivalence of all inertial observers. e latter principle had already been stated in 1632 by Galileo; only the constancy of the speed of light was new. Despite this fact, the new theory was named – by Poincaré – a er the old principle, instead of calling in ‘invariance theory’, as Einstein would have preferred. * Can you give the argument leading to this deduction?
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•.
,
observer (greek) v
light c
observer (roman)
F I G U R E 144 Two inertial observers and a beam of light
Galilean physics tτ
special relativity tτ
L
L
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ξ
O, Ω
x, ξ
O, Ω
x
F I G U R E 145 Space-time diagrams for light seen from two different observers using coordinates (t, x) and (τ, ξ)
Challenge 573 e
with a situation where no interaction plays a role. In other words, we start with relativistic kinematics of bodies moving without disturbance.
If an undisturbed body is observed to travel along a straight line with a constant velocity (or to stay at rest), one calls the observer inertial, and the coordinates used by the observer an inertial frame of reference. Every inertial observer is itself in undisturbed motion. Examples of inertial observers (or frames) thus include – in two dimensions – those moving on a frictionless ice surface or on the oor inside a smoothly running train or ship; for a full example – in all three spatial dimensions – we can take a cosmonaut travelling in a space-ship as long as the engine is switched o . Inertial observers in three dimensions might also be called free- oating observers. ey are thus not so common. Non-inertial observers are much more common. Can you con rm this? Inertial observers are the most simple ones, and they form a special set:
— Any two inertial observers move with constant velocity relative to each other (as longs as gravity plays no role, as assumed above).
— All inertial observers are equivalent: they describe the world with the same equations. Because it implies the lack of absolute space and time, this statement was called the principle of relativity by Henri Poincaré. However, the essence of relativity is the existence of a limit speed.
To see how measured length and space intervals change from one observer to the other, we assume two inertial observers, a Roman one using coordinates x, y, z and t, and a Greek one using coordinates ξ, υ, ζ and τ,* that move with velocity v relative to each other. e axes are chosen in such a way that the velocity points in the x-direction. e
* ey are read as ‘xi’, ‘upsilon’, ‘zeta’ and ‘tau’. e names, correspondences and pronunciations of all Greek letters are explained in Appendix A.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
,
,
constancy of the speed of light in any direction for any two observers means that for the motion of light the coordinate di erentials are related by
= (cdt) − (dx) − (dy) − (dz) = (cdτ) − (dξ) − (dυ) − (dζ) . (110)
Challenge 574 e
Assume also that a ash lamp at rest for the Greek observer, thus with dξ = , produces two ashes separated by a time interval dτ. For the Roman observer, the ash lamp moves with speed v, so that dx = vdt. Inserting this into the previous expression, and assuming linearity and speed direction independence for the general case, we nd that intervals are related by
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dt = γ(dτ + vdξ c ) = dτ + vdξ c −v c
dx = γ(dξ + vdτ) = dξ + vdτ −v c
dy = dυ dz = dζ .
with v = dx dt
(111)
ese expressions describe how length and time intervals measured by di erent observ-
ers are related. At relative speeds v that are small compared to the velocity of light, such
as occur in everyday life, the time intervals are essentially equal; the stretch factor or re-
lativistic correction or relativistic contraction γ is then equal to for all practical purposes.
However, for velocities near that of light the measurements of the two observers give
di erent values. In these cases, space and time mix, as shown in Figure .
e expressions ( ) are also strange in another respect. When two observers look
Challenge 575 n at each other, each of them claims to measure shorter intervals than the other. In other
words, special relativity shows that the grass on the other side of the fence is always shorter
– if one rides along beside the fence on a bicycle and if the grass is inclined. We explore
this bizarre result in more detail shortly.
e stretch factor γ is equal to for most practical purposes in everyday life. e largest
value humans have ever achieved is about ë ; the largest observed value in nature is
Challenge 576 n about . Can you imagine where they occur?
Once we know how space and time intervals change, we can easily deduce how coordin-
ates change. Figures and show that the x coordinate of an event L is the sum of
two intervals: the ξ coordinate and the length of the distance between the two origins. In
other words, we have
ξ = γ(x − vt)
and
v
=
dx dt
.
(112)
Using the invariance of the space-time interval, we get
τ = γ(t − xv c ) .
(113)
Henri Poincaré called these two relations the Lorentz transformations of space and time
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Ref. 263 Page 546
Ref. 264
a er their discoverer, the Dutch physicist Hendrik Antoon Lorentz.* In one of the most beautiful discoveries of physics, in and , Lorentz deduced these relations from the equations of electrodynamics, where they had been lying, waiting to be discovered, since .** In that year James Clerk Maxwell had published the equations in order to describe everything electric and magnetic. However, it was Einstein who rst understood that t and τ, as well as x and ξ, are equally correct and thus equally valid descriptions of space and time.
e Lorentz transformation describes the change of viewpoint from one inertial frame to a second, moving one. is change of viewpoint is called a (Lorentz) boost. e formulae ( ) and ( ) for the boost are central to the theories of relativity, both special and general. In fact, the mathematics of special relativity will not get more di cult than that: if you know what a square root is, you can study special relativity in all its beauty.
Many alternative formulae for boosts have been explored, such as expressions in which the relative acceleration of the two observers is included, as well as the relative velocity. However, they had all to be discarded a er comparing their predictions with experimental results. Before we have a look at such experiments, we continue with a few logical deductions from the boost relations.
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W
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Challenge 577 n Ref. 265
“Von Stund’ an sollen Raum für sich und Zeit für sich völlig zu Schatten herabsinken und nur noch eine Art Union der beiden soll Selbstständigkeit bewaren.*** ” Hermann Minkowski.
e Lorentz transformations tell us something important: that space and time are two aspects of the same basic entity. ey ‘mix’ in di erent ways for di erent observers. is fact is commonly expressed by stating that time is the fourth dimension. is makes sense because the common basic entity – called space-time – can be de ned as the set of all events, events being described by four coordinates in time and space, and because the set of all events has the properties of a manifold.**** (Can you con rm this?)
In other words, the existence of a maximum speed in nature forces us to introduce a space-time manifold for the description of nature. In the theory of special relativity, the space-time manifold is characterized by a simple property: the space-time interval di between two nearby events, de ned as
di
= c dt
− dx
− dy
− dz
= c dt (
−
v c
),
(114)
Page 1214
* For information about Hendrik Antoon Lorentz, see page 290. ** e same discovery had been published rst in 1887 by the German physicist Woldemar Voigt (1850– 1919); Voigt – pronounced ‘Fohgt’ – was also the discoverer of the Voigt e ect and the Voigt tensor. Independently, in 1889, the Irishman George F. Fitzgerald also found the result. *** ‘Henceforth space by itself and time by itself shall completely fade into shadows and only a kind of union of the two shall preserve autonomy.’ is famous statement was the starting sentence of Minkowski’s 1908 talk at the meeting of the Gesellscha für Naturforscher und Ärzte. **** e term ‘manifold’ is de ned in Appendix D.
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,
,
Challenge 578 n
is independent of the (inertial) observer. Such a space-time is also called Minkowski space-time, a er Hermann Minkowski* the teacher of Albert Einstein; he was the rst, in , to de ne the concept of space-time and to understand its usefulness and importance.
e space-time interval di of equation ( ) has a simple interpretation. It is the time measured by an observer moving from event (t, x) to event (t + dt, x + dx), the so-called proper time, multiplied by c. If we neglect the factor c, we could simply call it wristwatch time.
We live in a Minkowski space-time, so to speak. Minkowski space-time exists independently of things. And even though coordinate systems can be di erent from observer to observer, the underlying entity, space-time, is still unique, even though space and time by themselves are not.
How does Minkowski space-time di er from Galilean space-time, the combination of everyday space and time? Both space-times are manifolds, i.e. continuum sets of points, both have one temporal and three spatial dimensions, and both manifolds have the topology of the punctured sphere. (Can you con rm this?) Both manifolds are at, i.e. free of curvature. In both cases, space is what is measured with a metre rule or with a light ray, and time is what is read from a clock. In both cases, space-time is fundamental; it is and remains the background and the container of things and events.
e central di erence, in fact the only one, is that Minkowski space-time, in contrast to the Galilean case, mixes space and time, and in particular, does so di erently for observers with di erent speeds, as shown in Figure . at is why time is an observer-dependent concept.
e maximum speed in nature thus forces us to describe motion with space-time. at is interesting, because in space-time, speaking in simple terms, motion does not exist. Motion exists only in space. In space-time, nothing moves. For each point particle, spacetime contains a world-line. In other words, instead of asking why motion exists, we can equivalently ask why space-time is criss-crossed by world-lines. At this point, we are still far from answering either question. What we can do is to explore how motion takes place.
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C
?–T
We know that time is di erent for di erent observers. Does time nevertheless order events in sequences? e answer given by relativity is a clear ‘yes and no’. Certain sets of events are not naturally ordered by time; others sets are. is is best seen in a spacetime diagram.
Clearly, two events can be placed in sequence only if one event is the cause of the other. But this connection can only apply if the events exchange energy (e.g. through a signal). In other words, a relation of cause and e ect between two events implies that energy or signals can travel from one event to the other; therefore, the speed connecting the two events must not be larger than the speed of light. Figure shows that event E at the origin of the coordinate system can only be in uenced by events in quadrant IV (the past light cone, when all space dimensions are included), and can itself in uence only events
* Hermann Minkowski (1864–1909), German mathematician. He had developed similar ideas to Einstein, but the latter was faster. Minkowski then developed the concept of space-time. Minkowski died suddenly at the age of 44.
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time
light cone
II future
III E
elsewhere
T
light cone
I elsewhere
space
IV past
•.
,
t
light cone
future
T
light cone
E
elsewhere
y
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x
past
F I G U R E 146 A space-time diagram for a moving object T seen from an inertial observer O in the case of one and two spatial dimensions
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Challenge 579 n Challenge 580 n
in quadrant II (the future light cone). Events in quadrants I and III neither in uence nor are in uenced by event E. e light cone de nes the boundary between events that can be ordered with respect to their origin – namely those inside the cone – and those that cannot – those outside the cones, happening elsewhere for all observers. (Some people call all the events happening elsewhere the present.) So, time orders events only partially. For example, for two events that are not causally connected, their temporal order (or their simultaneity) depends on the observer!
In particular, the past light cone gives the complete set of events that can in uence what happens at the origin. One says that the origin is causally connected only to the past light cone. is statement re ects the fact that any in uence involves transport of energy, and thus cannot travel faster than the speed of light. Note that causal connection is an invariant concept: all observers agree on whether or not it applies to two given events. Can you con rm this?
A vector inside the light cone is called timelike; one on the light cone is called lightlike or null; and one outside the cone is called spacelike. For example, the world-line of an observer, i.e. the set of all events that make up its past and future history, consists of timelike events only. Time is the fourth dimension; it expands space to space-time and thus ‘completes’ space-time. is is the relevance of the fourth dimension to special relativity, no more and no less.
Special relativity thus teaches us that causality and time can be de ned only because light cones exist. If transport of energy at speeds faster than that of light did exist, time could not be de ned. Causality, i.e. the possibility of (partially) ordering events for all observers, is due to the existence of a maximal speed.
If the speed of light could be surpassed in some way, the future could in uence the past. Can you con rm this? In such situations, one would observe acausal e ects. However, there is an everyday phenomenon which tells that the speed of light is indeed maximal: our memory. If the future could in uence the past, we would also be able to remember the future. To put it in another way, if the future could in uence the past, the second principle
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of thermodynamics would not be valid and our memory would not work.* No other data from everyday life or from experiments provide any evidence that the future can in uence the past. In other words, time travel to the past is impossible. How the situation changes in quantum theory will be revealed later on. Interestingly, time travel to the future is possible, as we will see shortly.
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C
F
:
?
Challenge 581 e
How far away from Earth can we travel, given that the trip should not last more than
a lifetime, say years, and given that we are allowed to use a rocket whose speed can
approach the speed of light as closely as desired? Given the time t we are prepared to
spend in a rocket, given the speed v of the rocket and assuming optimistically that it can
accelerate and decelerate in a negligible amount of time, the distance d we can move away
is given by
d=
vt
.
−v c
(115)
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e distance d is larger than ct already for v . c, and, if v is chosen large enough, it increases beyond all bounds! In other words, relativity does not limit the distance we can travel in a lifetime, and not even the distance we can travel in a single second. We could, in principle, roam the entire universe in less than a second. In situations such as these it makes sense to introduce the concept of proper velocity w, de ned as
w=d t=
v
=γv .
−v c
(116)
As we have just seen, proper velocity is not limited by the speed of light; in fact the proper velocity of light itself is in nite.**
S
–
?
A maximum speed implies that time is is di erent for di erent observers moving relative to each other. So we have to be careful about how we synchronize clocks that are far
apart, even if they are at rest with respect to each other in an inertial reference frame. For example, if we have two similar watches showing the same time, and if we carry one of them for a walk and back, they will show di erent times a erwards. is experiment has
Ref. 266 Challenge 582 e
Ref. 267
* Another related result is slowly becoming common knowledge. Even if space-time had a nontrivial shape, such as a cylindrical topology with closed time-like curves, one still would not be able to travel into the
past, in contrast to what many science ction novels suggest. is is made clear by Stephen Blau in a recent
pedagogical paper.
** Using proper velocity, the relation given in equation (108) for the superposition of two velocities wa = γava
and wb = γbvb simpli es to
ws = γaγb(va + vb ) and ws = wb ,
(117)
where the signs and designate the component in the direction of and the component perpendicular to va, respectively. One can in fact express all of special relativity in terms of ‘proper’ quantities.
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•.
,
first twin
Earth time
first twin
trip of second twin
time comparison and change of rocket
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F I G U R E 147 The twin paradox
Ref. 268, Ref. 269
actually been performed several times and has fully con rmed the prediction of special relativity. e time di erence for a person or a watch in an aeroplane travelling around the Earth once, at about km h, is of the order of ns – not very noticeable in everyday life. In fact, the delay is easily calculated from the expression
t t′
=
γ
.
(118)
Ref. 270
Human bodies are clocks; they show the elapsed time, usually called age, by various changes in their shape, weight, hair colour, etc. If a person goes on a long and fast trip, on her return she will have aged less than a second person who stayed at her (inertial) home.
e most famous illustration of this is the famous twin paradox (or clock paradox). An adventurous twin jumps on a relativistic rocket that leaves Earth and travels for many years. Far from Earth, he jumps on another relativistic rocket going the other way and returns to Earth. e trip is illustrated in Figure . At his arrival, he notes that his twin brother on Earth is much older than himself. Can you explain this result, especially the asymmetry between the two brothers? is result has also been con rmed in many experiments.
Special relativity thus con rms, in a surprising fashion, the well-known observation that those who travel a lot remain younger. e price of the retained youth is, however, that everything around one changes very much more quickly than if one is at rest with the environment.
e twin paradox can also be seen as a con rmation of the possibility of time travel to the future. With the help of a fast rocket that comes back to its starting point, we can arrive at local times that we would never have reached within our lifetime by staying home. Alas, we can never return to the past.*
Ref. 271 * ere are even special books on time travel, such as the well researched text by Nahin. Note that the concept
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Ref. 272 Challenge 583 n Challenge 584 n
Ref. 252 Ref. 273
Challenge 585 n
One of the simplest experiments con rming
the prolonged youth of fast travellers involves the
higher atmosphere
counting of muons. Muons are particles that are
continuously formed in the upper atmosphere
by cosmic radiation. Muons at rest (with respect
to the measuring clock) have a nite half-life of
. µs (or, at the speed of light, m). A er this
amount of time, half of the muons have decayed.
high
is half-life can be measured using simple muon
counter
counters. In addition, there exist special counters
that only count muons travelling within a certain speed range, say from . c to . c. One
decays
can put one of these special counters on top of
a mountain and put another in the valley below,
low
as shown in Figure . e rst time this exper-
counter
iment was performed, the height di erence was F I G U RE 148 More muons than expected . km. Flying . km through the atmosphere at arrive at the ground because fast travel
the mentioned speed takes about . µs. With the keeps them young
half-life just given, a naive calculation nds that
only about % of the muons observed at the top should arrive at the lower site. However,
it is observed that about % of the muons arrive below. e reason for this result is the
relativistic time dilation. Indeed, at the mentioned speed, muons experience a time dif-
ference of only . µs during the travel from the mountain top to the valley. is shorter
time yields a much lower number of lost muons than would be the case without time
dilation; moreover, the measured percentage con rms the value of the predicted time
dilation factor γ within experimental errors, as you may want to check. A similar e ect
is seen when relativistic muons are produced in accelerators.
Half-life dilation has also been found for many other decaying systems, such as pions,
hydrogen atoms, neon atoms and various nuclei, always con rming the predictions of
special relativity. Since all bodies in nature are made of particles, the ‘youth e ect’ of high
speeds (usually called ‘time dilation’) applies to bodies of all sizes; indeed, it has not only
been observed for particles, but also for lasers, radio transmitters and clocks.
If motion leads to time dilation, a clock on the Equator, constantly running around
the Earth, should go slower than one at the poles. However, this prediction, which was
made by Einstein himself, is incorrect. e centrifugal acceleration leads to a reduction in
gravitational acceleration that exactly cancels the increase due to the velocity. is story
serves as a reminder to be careful when applying special relativity in situations involving
gravity. Special relativity is only applicable when space-time is at, not when gravity is
present.
In short, a mother can stay younger than her daughter. We can also conclude that we
cannot synchronize clocks at rest with respect to each other simply by walking, clock in
hand, from one place to another. e correct way to do so is to exchange light signals.
Can you describe how?
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of time travel has to be clearly de ned; otherwise one has no answer to the clerk who calls his o ce chair a time machine, as sitting on it allows him to get to the future.
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observations by the farmer
•.
,
observations by the pilot
farmer time
pilot
time
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plane ends
barn ends
F I G U R E 149 The observations of the pilot and the barn owner
Challenge 586 e
A precise de nition of synchronization allows us to call two distant events simultan-
eous. In addition, special relativity shows that simultaneity depends on the observer. is
is con rmed by all experiments performed so far.
However, the mother’s wish is not easy to ful l. Let us imagine that a mother is acceler-
ated in a spaceship away from Earth at m s for ten years, then decelerates at m s
for another ten years, then accelerates for ten additional years towards the Earth, and -
nally decelerates for ten nal years in order to land safely back on our planet. e mother
has taken years for the trip. She got as far as
light years from Earth. At her
return on Earth,
years have passed. All this seems ne, until we realize that the
necessary amount of fuel, even for the most e cient engine imaginable, is so large that
the mass returning from the trip is only one part in ë . e necessary amount of fuel
does not exist on Earth.
L
Challenge 587 e
e length of an object measured by an observer attached to the object is called its proper length. According to special relativity, the length measured by an inertial observer passing by is always smaller than the proper length. is result follows directly from the Lorentz transformations.
For a Ferrari driving at km h or m s, the length is contracted by . pm: less than the diameter of a proton. Seen from the Sun, the Earth moves at km s; this gives a length contraction of cm. Neither of these e ects has ever been measured. But larger e ects could be. Let us explore some examples.
Imagine a pilot ying through a barn with two doors, one at each end. e plane is slightly longer than the barn, but moves so rapidly that its relativistically contracted length is shorter than the length of the barn. Can the farmer close the barn (at least for a short time) with the plane completely inside? e answer is positive. But why can the
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ski trap
ski h
trap
F I G U R E 150 The observations of the trap digger and of the snowboarder, as (misleadingly) published in the literature
h
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v
g < h F I G U R E 151 Does the conducting glider keep the lamp lit at large speeds?
B
rope
F
v(t)
v(t)
F I G U R E 152 What happens to the rope?
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Challenge 588 n
Ref. 274 Challenge 590 e Challenge 591 n
Ref. 275 Challenge 592 ny
Ref. 276
pilot not say the following: relative to him, the barn is contracted; therefore the plane does not t inside the barn? e answer is shown in Figure . For the farmer, the doors close (and reopen) at the same time. For the pilot, they do not. For the farmer, the pilot is in the dark for a short time; for the pilot, the barn is never dark. ( at is not completely true: can you work out the details?)
We now explore some variations of the general case. Can a rapid snowboarder fall into a hole that is a bit shorter than his board? Imagine him boarding so fast that the length contraction factor γ = d d′ is .* For an observer on the ground, the snowboard is four times shorter, and when it passes over the hole, it will fall into it. However, for the boarder, it is the hole which is four times shorter; it seems that the snowboard cannot fall into it.
More careful analysis shows that, in contrast to the observation of the hole digger, the snowboarder does not experience the board’s shape as xed: while passing over the hole, the boarder observes that the board takes on a parabolic shape and falls into the hole, as shown in Figure . Can you con rm this? In other words, shape is not an observerinvariant concept. (However, rigidity is observer-invariant, if de ned properly; can you con rm this?)
is explanation, though published, is not correct, as Harald van Lintel and Christian Gruber have pointed out. One should not forget to estimate the size of the e ect. At relativistic speeds the time required for the hole to a ect the full thickness of the board cannot be neglected. e snowboarder only sees his board take on a parabolic shape if it is extremely thin and exible. For usual boards moving at relativistic speeds, the snowboarder has no time to fall any appreciable height h or to bend into the hole before passing it. Figure is so exaggerated that it is incorrect. e snowboarder would simply speed over the hole.
e paradoxes around length contraction become even more interesting in the case of a conductive glider that makes electrical contact between two rails, as shown in Figure .
Challenge 589 n * Even the Earth contracts in its direction of motion around the Sun. Is the value measurable?
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•.
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Challenge 593 n
Ref. 277
Challenge 594 n Ref. 278
Challenge 595 n Challenge 596 n
e two rails are parallel, but one rail has a gap that is longer than the glider. Can you work out whether a lamp connected in series stays lit when the glider moves along the rails with relativistic speed? (Make the simplifying and not fully realistic assumption that electrical current ows as long and as soon as the glider touches the rails.) Do you get the same result for all observers? And what happens when the glider is longer than the detour? (Warning: this problem gives rise to heated debates!) What is unrealistic in this experiment?
Another example of length contraction appears when two objects, say two cars, are connected over a distance d by a straight rope, as shown in Figure Imagine that both are at rest at time t = and are accelerated together in exactly the same way. e observer at rest will maintain that the two cars remain the same distance apart. On the other hand,
the rope needs to span a distance d′ = d − v c , and thus has to expand when the two cars are accelerating. In other words, the rope will break. Is this prediction con rmed by observers on each of the two cars?
A funny – but quite unrealistic – example of length contraction is that of a submarine moving horizontally. Imagine that the resting submarine has tuned its weight to oat in water without any tendency to sink or to rise. Now the submarine moves (possibly with relativistic speed). e captain observes the water outside to be Lorentz contracted; thus the water is denser and he concludes that the submarine will rise. A nearby sh sees the submarine to be contracted, thus denser than water, and concludes that the submarine will sink. Who is wrong, and what is the buoyancy force? Alternatively, answer the following question: why is it impossible for a submarine to move at relativistic speed?
In summary, length contraction is can almost never be realistically observed for macroscopic bodies. However, it does play an important role for images.
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R
–
D
We have encountered several ways in which observations change when an observer moves at high speed. First of all, Lorentz contraction and aberration lead to distorted images. Secondly, aberration increases the viewing angle beyond the roughly degrees that humans are used to in everyday life. A relativistic observer who looks in the direction of motion sees light that is invisible for a resting observer, because for the latter, it comes from behind. irdly, the Doppler e ect produces colour-shi ed images. Fourthly, the rapid motion changes the brightness and contrast of the image: the so-called searchlight e ect. Each of these changes depends on the direction of sight; they are shown in Figure .
Modern computers enable us to simulate the observations made by rapid observers with photographic quality, and even to produce simulated lms.* e images of Figure are particularly helpful in allowing us to understand image distortion. ey show the viewing angle, the circle which distinguish objects in front of the observer from those behind the observer, the coordinates of the observer’s feet and the point on the horizon
* See for example images and lms at http://www.anu.edu.au/Physics/Searle/ by Anthony Searle, at http://www.tat.physik.uni-tuebingen.de/~weiskopf/gallery/index.html by Daniel Weiskopf, at http://www. itp.uni-hannover.de/~dragon/stonehenge/stone1.htm by Norbert Dragon and Nicolai Mokros, or at http:// www.tempolimit-lichtgeschwindigkeit.de by Hanns Ruder’s group.
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F I G U R E 153 Flying through twelve vertical columns (shown in the two uppermost images) with 0.9 times the speed of light as visualized by Nicolai Mokros and Norbert Dragon, showing the effect of speed and position on distortions (© Nicolai Mokros)
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,
F I G U R E 154 Flying through three straight and vertical columns with 0.9 times the speed of light as visualized by Daniel Weiskopf: on the left with the original colours; in the middle including the Doppler effect; and on the right including brightness effects, thus showing what an observer would actually see (© Daniel Weiskopf )
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F I G U R E 155 What a researcher standing and one running rapidly through a corridor observe (ignoring colour effects) (© Daniel Weiskopf )
Ref. 279
toward which the observer is moving. Adding these markers in your head when watching other pictures or lms may help you to understand more clearly what they show.
We note that the shape of the image seen by a moving observer is a distorted version of that seen by one at rest at the same point. A moving observer, however, does not see di erent things than a resting one at the same point. Indeed, light cones are independent of observer motion.
e Lorentz contraction is measurable; however, it cannot be photographed. is surprising distinction was discovered only in . Measuring implies simultaneity at the ob-
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Page 277
Challenge 597 n Challenge 598 r
ject’s position; photographing implies simultaneity at the observer’s position. On a photograph, the Lorentz contraction is modi ed by the e ects due to di erent light travel times from the di erent parts of an object; the result is a change in shape that is reminiscent of, but not exactly the same as, a rotation. e total deformation is an angle-dependent aberration. We discussed aberration at the beginning of this section. Aberration transforms circles into circles: such transformations are called conformal.
e images of Figure , produced by Daniel Weiskopf, also include the Doppler e ect and the brightness changes. ey show that these e ects are at least as striking as the distortion due to aberration.
is leads to the ‘pearl necklace paradox’. If the relativistic motion transforms spheres into spheres, and rods into shorter rods, what happens to a pearl necklace moving along its own long axis? Does it get shorter?
ere is much more to be explored using relativistic lms. For example, the author predicts that lms of rapidly rotating spheres in motion will reveal interesting e ects. Also in this case, optical observation and measurement results will di er. For certain combinations of relativistic rotations and relativistic boosts, it is predicted* that the sense of rotation (clockwise or anticlockwise) will di er for di erent observers. is e ect will play an interesting role in the discussion of uni cation.
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W
?
Ref. 277 Challenge 599 e Challenge 600 ny
Let us explore another surprise of special relativity. Imagine two twins inside two identically accelerated cars, one in front of the other, starting from standstill at time t = , as described by an observer at rest with respect to both of them. ( ere is no connecting rope now.) Both cars contain the same amount of fuel. We easily deduce that the acceleration of the two twins stops, when the fuel runs out, at the same time in the frame of the outside observer. In addition, the distance between the cars has remained the same all along for the outside observer, and the two cars continue rolling with an identical constant velocity v, as long as friction is negligible. If we call the events at which the front car and back car engines switch o f and b, their time coordinates in the outside frame are related simply by tf = tb. By using the Lorentz transformations you can deduce for the frame of the freely rolling twins the relation
tb = γ∆x v c + tf ,
(119)
Challenge 601 n
which means that the front twin has aged more than the back twin! us, in accelerated systems, ageing is position-dependent.
For choosing a seat in a bus, though, this result does not help. It is true that the best seat in an accelerating bus is the back one, but in a decelerating bus it is the front one. At the end of a trip, the choice of seat does not matter.
Is it correct to deduce that people on high mountains age faster than people in valleys, so that living in a valley helps postponing grey hair?
* In July 2005.
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average speed: c/2 time
t'
moving judge J
x'
light signal
•.
,
time average speed: c/3
t'
moving judge
J
light signal x'
space
space
F I G U R E 156 For the athlete on the left, the judge moving in the opposite direction sees both feet off the ground at certain times, but not for the athlete on the right
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H
?
Ref. 280
To walk means to move the feet in such a way that at least one of them is on the ground at any time. is is one of the rules athletes have to follow in Olympic walking competitions; they are disquali ed if they break it. A student athlete was thinking about the theoretical maximum speed he could achieve in the Olympics. e ideal would be that each foot accelerates instantly to (almost) the speed of light. e highest walking speed is achieved by taking the second foot o the ground at exactly the same instant at which the rst is put down. By ‘same instant’, the student originally meant ‘as seen by a competition judge at rest with respect to Earth’. e motion of the feet is shown in the le diagram of Figure ; it gives a limit speed for walking of half the speed of light. But then the student noticed that a moving judge will see both feet o the ground and thus disqualify the athlete for running. To avoid disquali cation by any judge, the second foot has to wait for a light signal from the rst. e limit speed for Olympic walking is thus only one third of the speed of light.
I
?
Page 297 Challenge 602 n
Actually, motion faster than light does exist and is even rather common. Special relativity only constrains the motion of mass and energy. However, non-material points or nonenergy-transporting features and images can move faster than light. ere are several simple examples. To be clear, we are not talking about proper velocity, which in these cases cannot be de ned anyway. (Why?)
e following examples show speeds that are genuinely higher than the speed of light in vacuum.
Consider the point marked X in Figure , the point at which scissors cut paper. If
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v X
F I G U R E 157 A simple example of motion that is faster than light
The Beatles
The Beatles The Beatles
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F I G U R E 158 Another example of faster-than-light motion
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the scissors are closed rapidly enough, the point moves faster than light. Similar examples can also be found in every window frame, and in fact in any device that has twisting parts.
Another example of superluminal motion is a music record – an old-fashioned LP – disappearing into its sleeve, as shown in Figure . e point where the edge of the record meets the edge of the sleeve can travel faster than light.
Another example suggests itself when we remember that we live on a spherical planet. Imagine you lie on the oor and stand up. Can you show that the initial speed with which Challenge 603 n the horizon moves away from you can be larger than that of light?
Finally, a standard example is the motion of a spot of light produced by shining a laser beam onto the Moon. If the laser is moved, the spot can easily move faster than light. e same applies to the light spot on the screen of an oscilloscope when a signal of su ciently high frequency is fed to the input.
All these are typical examples of the speed of shadows, sometimes also called the speed of darkness. Both shadows and darkness can indeed move faster than light. In fact, there Challenge 604 n is no limit to their speed. Can you nd another example?
In addition, there is an ever-increasing number of experimental set-ups in which the phase velocity or even the group velocity of light is higher than c. ey regularly make headlines in the newspapers, usually along the lines of ‘light moves faster than light’. We Page 577 will discuss this surprising phenomenon in more detail later on. In fact, these cases can also be seen – with some imagination – as special cases of the ‘speed of shadow’ phenomenon.
For a di erent example, imagine we are standing at the exit of a tunnel of length l. We see a car, whose speed we know to be v, entering the other end of the tunnel and driving towards us. We know that it entered the tunnel because the car is no longer in the Sun or because its headlights were switched on at that moment. At what time t, a er we see it
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•.
,
time
observer emitted or reflected light
tachyon
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
light cone
space
F I G U R E 159 Hypothetical space-time diagram for tachyon observation
entering the tunnel, does it drive past us? Simple reasoning shows that t is given by
t=l v−l c.
(120)
In other words, the approaching car seems to have a velocity vappr of
vappr
=
l t
=
vc c−v
,
(121)
Ref. 281
which is higher than c for any car velocity v higher than c . For cars this does not happen
too o en, but astronomers know a type of bright object in the sky called a quasar (a
contraction of ‘quasi-stellar object’), which sometimes emits high-speed gas jets. If the
emission is in or near the direction of the Earth, its apparent speed – even the purely
transverse component – is higher than c. Such situations are now regularly observed with
telescopes.
Note that to a second observer at the entrance of the tunnel, the apparent speed of the
car moving away is given by
vleav
=
vc c+v
,
(122)
which is never higher than c . In other words, objects are never seen departing with more than half the speed of light.
e story has a nal twist. We have just seen that motion faster than light can be observed in several ways. But could an object moving faster than light be observed at all? Surprisingly, it could be observed only in rather unusual ways. First of all, since such an imaginary object, usually called a tachyon, moves faster than light, we can never see
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R
v
G
u
O
w
F I G U R E 160 If O’s stick is parallel to R’s and R’s is parallel to G’s, then O’s stick and G’s stick are not
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Challenge 605 e Ref. 282
Page 316
it approaching. If it can be seen at all, a tachyon can only be seen departing. Seeing a tachyon would be similar to hearing a supersonic jet. Only a er a tachyon has passed nearby, assuming that it is visible in daylight, could we notice it. We would rst see a
ash of light, corresponding to the bang of a plane passing with supersonic speed. en we would see two images of the tachyon, appearing somewhere in space and departing in opposite directions, as can be deduced from Figure . Even if one of the two images were approaching us, it would be getting fainter and smaller. is is, to say the least, rather unusual behaviour. Moreover, if you wanted to look at a tachyon at night, illuminating it with a torch, you would have to turn your head in the direction opposite to the arm with the torch! is requirement also follows from the space-time diagram: can you see why? Nobody has ever seen such phenomena. Tachyons, if they existed, would be strange objects: they would accelerate when they lose energy, a zero-energy tachyon would be the fastest of all, with in nite speed, and the direction of motion of a tachyon depends on the motion of the observer. No object with these properties has ever been observed. Worse, as we just saw, tachyons would seem to appear from nothing, defying laws of conservation; and note that, just as tachyons cannot be seen in the usual sense, they cannot be touched either, since both processes are due to electromagnetic interactions, as we will see later in our ascent of Motion Mountain. Tachyons therefore cannot be objects in the usual sense. In the second part of our adventure we will show that quantum theory actually rules out the existence of (real) tachyons. However, quantum theory also requires the existence of ‘virtual’ tachyons, as we will discover.
P
–T
Ref. 283
Relativity has strange consequences indeed. Any two observers can keep a stick parallel to the other’s, even if they are in motion with respect to each other. But strangely, given a a chain of sticks for which any two adjacent ones are parallel, the rst and the last sticks will not generally be parallel. In particular, they never will be if the motions of the various observers are in di erent directions, as is the case when the velocity vectors form a loop.
e simplest set-up is shown in Figure . In special relativity, a general concatenation of pure boosts does not give a pure boost, but a boost plus a rotation. As a result, the endpoints of chains of parallel sticks are usually not parallel.
An example of this e ect appears in rotating motion. If we walk in a fast circle holding a stick, always keeping the stick parallel to the direction it had just before, at the end of the circle the stick will have an angle with respect to the original direction. Similarly, the axis
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•.
,
of a rotating body circling a second body will not be pointing in the same direction a er one turn. is e ect is called omas precession, a er Llewellyn omas, who discovered it in , a full years a er the birth of special relativity. It had escaped the attention of dozens of other famous physicists. omas precession is important in the inner working of atoms; we will return to it in a later section of our adventure. ese surprising phenomena are purely relativistic, and are thus measurable only in the case of speeds comparable to that of light.
A
-
–
e literature on temperature is confusing. Albert Einstein and Wolfgang Pauli agreed
on the following result: the temperature T seen by an observer moving with speed v is
related to the temperature T measured by the observer at rest with respect to the heat
bath via
T =T −v c .
(123)
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Ref. 284
A moving observer thus always measures lower values than a resting one. In , Max Planck used this expression, together with the corresponding transform-
ation for heat, to deduce that the entropy is invariant under Lorentz transformations. Being the discoverer of the Boltzmann constant k, Planck proved in this way that the constant is a relativistic invariant.
Not all researchers agree on the expression. Others maintain that T and T should be interchanged in the temperature transformation. Also, powers other than the simple square root have been proposed. e origin of these discrepancies is simple: temperature is only de ned for equilibrium situations, i.e. for baths. But a bath for one observer is not a bath for the other. For low speeds, a moving observer sees a situation that is almost a heat bath; but at higher speeds the issue becomes tricky. Temperature is deduced from the speed of matter particles, such as atoms or molecules. For moving observers, there is no good way to measure temperature. e naively measured temperature value even depends on the energy range of matter particles that is measured! In short, thermal equilibrium is not an observer-invariant concept. erefore, no temperature transformation formula is correct. (With certain additional assumptions, Planck’s expression does seem to hold, however.) In fact, there are not even any experimental observations that would allow such a formula to be checked. Realizing such a measurement is a challenge for future experimenters – but not for relativity itself.
R
Because the speed of light is constant and velocities do not add up, we need to rethink the de nitions of mass, momentum and energy. We thus need to recreate a theory of mechanics from scratch.
M Page 77 In Galilean physics, the mass ratio between two bodies was de ned using collisions; it
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was given by the negative inverse of the velocity change ratio
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
m m
=
−
∆v ∆v
.
(124)
Challenge 606 ny Ref. 285
However, experiments show that the expression must be di erent for speeds near that of light. In fact, experiments are not needed: thinking alone can show this. Can you do so?
ere is only one solution to this problem. e two Galilean conservation theorems i mivi = const for momentum and i mi = const for mass have to be changed into
γi mivi = const
i
(125)
and
γi mi = const .
(126)
i
Challenge 607 n Challenge 608 e
ese expressions, which will remain valid throughout the rest of our ascent of Motion
Mountain, imply, among other things, that teleportation is not possible in nature. (Can
you con rm this?) Obviously, in order to recover Galilean physics, the relativistic correc-
tion (factors) γi have to be almost equal to for everyday velocities, that is, for velocities nowhere near the speed of light.
Even if we do not know the value of the relativistic correction factor, we can deduce it
from the collision shown in Figure .
In the rst frame of reference (A) we have
γv mv = γV MV and γv m + m = γV M. From the ob- Observer A
servations of the second frame of reference (B) we
m
m
deduce that V composed with V gives v, in other before:
v
words, that
v=
V +V c
.
after:
(127)
V M
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When these equations are combined, the relativ- Observer B
istic correction γ is found to depend on the magnitude of the velocity v through
before:
after:
mV
Vm
γv =
. −v c
(128)
M
F I G U R E 161 An inelastic collision of two identical particles seen from two
With this expression, and a generalization of different inertial frames of reference
the situation of Galilean physics, the mass ratio
between two colliding particles is de ned as the ratio
m m
=
−
∆(γ ∆(γ
v v
) )
.
(129)
(We do not give here the generalized mass de nition, mentioned in the chapter on Ga-
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•.
,
before
A
pA
B
A
after pA
rule: ϕ+θ = 90°
θ ϕ
B
F I G U R E 162 A useful rule for playing non-relativistic snooker
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Page 79
lilean mechanics, that is based on acceleration ratios, because it contains some subtleties, which we will discover shortly.) e correction factors γi ensure that the mass de ned by this equation is the same as the one de ned in Galilean mechanics, and that it is the same for all types of collision a body may have.* In this way, mass remains a quantity characterizing the di culty of accelerating a body, and it can still be used for systems of bodies as well.
Following the example of Galilean physics, we call the quantity
p = γmv
(130)
the (linear) relativistic (three-) momentum of a particle. Again, the total momentum is a conserved quantity for any system not subjected to external in uences, and this conservation is a direct consequence of the way mass is de ned.
For low speeds, or γ , relativistic momentum is the same as that of Galilean physics, and is proportional to velocity. But for high speeds, momentum increases faster than velocity, tending to in nity when approaching light speed.
W
Challenge 610 ny
ere is a well-known property of collisions between a moving sphere or particle and a
resting one of the same mass that is important when playing snooker, pool or billiards. A er such a collision, the two spheres will depart at a right angle from each other, as shown in Figure .
However, experiments show that the right angle rule does not apply to relativistic collisions. Indeed, using the conservation of momentum and a bit of dexterity you can cal-
culate that
tan θ tan φ = γ + ,
(131)
where the angles are de ned in Figure . It follows that the sum φ + θ is smaller than a right angle in the relativistic case. Relativistic speeds thus completely change the game
Challenge 609 e * e results below also show that γ = + T mc , where T is the kinetic energy of a particle.
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accelerator beam
target
θ ϕ
detector
F I G U R E 163 The dimensions of detectors in particle accelerators are based on the relativistic snooker angle rule
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Ref. 252 Challenge 611 ny
of snooker. Indeed, every accelerator physicist knows this: for electrons or protons, these angles can easily be deduced from photographs taken in cloud chambers, which show the tracks le by particles when they move through them. All such photographs con rm the above expression. In fact, the shapes of detectors are chosen according to expression ( ), as sketched in Figure . If the formula – and relativity – were wrong, most of these detectors would not work, as they would miss most of the particles a er the collision. In fact, these experiments also prove the formula for the composition of velocities. Can you show this?
M
Let us go back to the collinear and inelastic collision of Figure Challenge 612 n of the nal system? Calculation shows that
. What is the mass M
M m = ( + γv) .
(132)
In other words, the mass of the nal system is larger than the sum of the two original masses m. In contrast to Galilean mechanics, the sum of all masses in a system is not a conserved quantity. Only the sum i γi mi of the corrected masses is conserved.
Relativity provides the solution to this puzzle. Everything falls into place if, for the energy E of an object of mass m and velocity v, we use the expression
E = γmc =
mc
,
−v c
(133)
applying it both to the total system and to each component. e conservation of the corrected mass can then be read as the conservation of energy, simply without the factor c . In the example of the two identical masses sticking to each other, the two particles are thus each described by mass and energy, and the resulting system has an energy E given by the sum of the energies of the two particles. In particular, it follows that the energy E of a body at rest and its mass m are related by
E = mc ,
(134)
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•.
,
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which is perhaps the most beautiful and famous discovery of modern physics. Since c is so large, we can say that mass is concentrated energy. In other words, special relativity says that every mass has energy, and that every form of energy in a system has mass. Increasing the energy of a system increases its mass, and decreasing the energy content decreases the mass. In short, if a bomb explodes inside a closed box, the mass, weight and momentum of the box are the same before and a er the explosion, but the combined mass of the debris inside the box will be smaller than before. All bombs – not only nuclear ones – thus take their energy from a reduction in mass. In addition, every action of a system – such a caress, a smile or a look – takes its energy from a reduction in mass.
e kinetic energy T is thus given by
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T = γmc − mc =
mv +
ë ë
m
v c
+
ë ë
ë ë
v c
+ ...
(135)
Challenge 613 e Challenge 614 n
(using the binomial theorem) which reduces to the Galilean value only for low speeds. e mass–energy equivalence E = γmc implies that taking any energy from matter
results in a mass decrease. When a person plays the piano, thinks or runs, its mass decreases. When a cup of tea cools down or when a star shines, its mass decreases. e mass–energy equivalence pervades all of nature.
By the way, we should be careful to distinguish the transformation of mass into energy from the transformation of matter into energy. e latter is much more rare. Can you give some examples?
e mass–energy relation ( ) means the death of many science ction fantasies. It implies that there are no undiscovered sources of energy on or near Earth. If such sources existed, they would be measurable through their mass. Many experiments have looked for, and are still looking for, such e ects with a negative result. ere is no free energy in nature.*
e mass–energy relation m = E c also implies that one needs about thousand million kJ (or thousand million kcal) to increase one’s weight by one single gram. Of course, dieticians have slightly di erent opinions on this matter! In fact, humans do get their everyday energy from the material they eat, drink and breathe by reducing its combined mass before expelling it again. However, this chemical mass defect appearing when fuel is burned cannot yet be measured by weighing the materials before and a er the reaction: the di erence is too small, because of the large conversion factor involved. Indeed, for any chemical reaction, bond energies are about aJ ( eV) per bond; this gives a weight change of the order of one part in , too small to be measured by weighing people or determining mass di erences between food and excrement. erefore, for everyday chemical processes mass can be taken to be constant, in accordance with Galilean physics.
Modern methods of mass measurement of single molecules have made it possible to measure the chemical mass defect by comparing the mass of a single molecule with that of its constituent atoms. David Pritchard’s group has developed so-called Penning
* ere may be two extremely diluted, yet undiscovered, form of energy, called dark matter and (confusingly) Page 442 dark energy, scattered throughout the universe. ey are deduced from (quite di cult) mass measurements.
e issue has not yet been nally resolved.
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Ref. 286 Challenge 615 ny
traps, which allow masses to be determined from the measurement of frequencies; the attainable precision of these cyclotron resonance experiments is su cient to con rm ∆E = ∆mc for chemical bonds. In the future, increased precision will even allow bond energies to be determined in this way with precision. Since binding energy is o en radiated as light, we can say that these modern techniques make it possible to weigh light.
inking about light and its mass was the basis for Einstein’s rst derivation of the mass–energy relation. When an object emits two equal light beams in opposite directions, its energy decreases by the emitted amount. Since the two light beams are equal in energy and momentum, the body does not move. If we describe the same situation from the viewpoint of a moving observer, we see again that the rest energy of the object is
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E = mc .
(136)
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Challenge 616 e
In summary, all physical processes, including collisions, need relativistic treatment whenever the energy involved is a sizeable fraction of the rest energy.
Every energy increase produces a mass increase. erefore also heating a body makes it heavier. However, this e ect is so small that nobody has measured it up to this day. It is a challenge for experiments of the future to do this one day.
How are energy and momentum related? e de nitions of momentum ( ) and energy ( ) lead to two basic relations. First of all, their magnitudes are related by
m c =E −p c
(137)
for all relativistic systems, be they objects or, as we will see below, radiation. For the momentum vector we get the other important relation
p
=
E c
v
,
(138)
Challenge 617 e which is equally valid for any type of moving energy, be it an object or a beam or a pulse of radiation.* We will use both relations o en in the rest of our ascent of Motion Mountain, including the following discussion.
C
,
We have just seen that in relativistic collisions the conservation of total energy and momentum are intrinsic consequences of the de nition of mass. Let us now have a look at collisions in more detail, using these new concepts. A collision is a process, i.e. a series of events, for which
— the total momentum before the interaction and a er the interaction is the same; — the momentum is exchanged in a small region of space-time; — for small velocities, the Galilean description is valid.
In everyday life an impact, i.e. a short-distance interaction, is the event at which both objects change momentum. But the two colliding objects are located at di erent points
* In 4-vector notation, we can write v c = P P , where P = E c.
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time t
•.
,
τ
object 1
object 2
E'1 p'1 E p
E1 p1
E'2 p'2 E2 p2
object 1
object 2
space x
ξ
F I G U R E 164 Space-time diagram of a collision for two observers
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Ref. 287 Challenge 618 e
Challenge 619 e
when this happens. A collision is therefore described by a space-time diagram such as the le -hand one in Figure , reminiscent of the Orion constellation. It is easy to check that the process described by such a diagram is a collision according to the above de nition.
e right-hand side of Figure shows the same process seen from another, Greek, frame of reference. e Greek observer says that the rst object has changed its momentum before the second one. at would mean that there is a short interval when momentum and energy are not conserved!
e only way to make sense of the situation is to assume that there is an exchange of a third object, drawn with a dotted line. Let us nd out what the properties of this object are. If we give numerical subscripts to the masses, energies and momenta of the two bodies, and give them a prime a er the collision, the unknown mass m obeys
mc
= (E − E′)
− (p
− p′) c
=
mc
−
E E′(
−v c
v′
)
<
.
(139)
is is a strange result, because it means that the unknown mass is an imaginary number!!* On top of that, we also see directly from the second graph that the exchanged object moves faster than light. It is a tachyon, from the Greek ταχύς ‘rapid’. In other words, collisions involve motion that is faster than light! We will see later that collisions are indeed the only processes where tachyons play a role in nature. Since the exchanged objects appear only during collisions, never on their own, they are called virtual objects, to distinguish them from the usual, real objects, which can move freely without restriction.** We will study their properties later on, when we come to discuss quantum theory.
* It is usual to change the mass–energy and mass–momentum relation of tachyons to E = mc v c −
and p = mv v c − ; this amounts to a rede nition of m. A er the rede nition, tachyons have real mass. e energy and momentum relations show that tachyons lose energy and momentum when they get faster. (Provocatively, a single tachyon in a box could provide us with all the energy we need.) Both signs for the energy and momentum relations must be retained, because otherwise the equivalence of all inertial observers would not be generated. Tachyons thus do not have a minimum energy or a minimum momentum. ** More precisely, a virtual particle does not obey the relation m c = E − p c , valid for real particles.
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Challenge 620 ny
Page 733 Challenge 621 n
Page 759
In nature, a tachyon is always a virtual object. Real objects are always bradyons – from the Greek βραδύς ‘slow’ – or objects moving slower than light. Note that tachyons, despite their high velocity, do not allow the transport of energy faster than light; and that they do not violate causality if and only if they are emitted or absorbed with equal probability. Can you con rm all this?
When we study quantum theory, we will also discover that a general contact interaction between objects is described not by the exchange of a single virtual object, but by a continuous stream of virtual particles. For standard collisions of everyday objects, the interaction turns out to be electromagnetic. In this case, the exchanged particles are virtual photons. In other words, when one hand touches another, when it pushes a stone, or when a mountain supports the trees on it, streams of virtual photons are continuously exchanged.
ere is an additional secret hidden in collisions. In the right-hand side of Figure , the tachyon is emitted by the rst object and absorbed by the second one. However, it is easy to imagine an observer for which the opposite happens. In short, the direction of travel of a tachyon depends on the observer! In fact, this is a hint about antimatter. In space-time diagrams, matter and antimatter travel in opposite directions. Also the connection between relativity and antimatter will become more apparent in quantum theory.
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S
–
Ref. 288
Relativity also forces us to eliminate the cherished concept of centre of mass. We can see this already in the simplest example possible: that of two equal objects colliding.
Figure shows that from the viewpoint in which one of two colliding particles is at rest, there are at least three di erent ways to de ne the centre of mass. In other words, the centre of mass is not an observer-invariant concept. We can deduce from the gure that the concept only makes sense for systems whose components move with small velocities relative to each other. For more general systems, centre of mass is not uniquely de nable. Will this hinder us in our ascent? No. We are more interested in the motion of single particles than that of composite objects or systems.
W
?
For most everyday systems, the time intervals measured by two di erent observers are practically equal; only at large relative speeds, typically at more than a few per cent of the speed of light, is there a noticeable di erence. Most such situations are microscopic. We have already mentioned the electrons inside a television tube or inside a particle accelerator. e particles making up cosmic radiation are another example: their high energy has produced many of the mutations that are the basis of evolution of animals and plants on this planet. Later we will discover that the particles involved in radioactivity are also relativistic.
But why don’t we observe any rapid macroscopic bodies? Moving bodies, including observers, with relativistic velocities have a property not found in everyday life: when they are involved in a collision, part of their energy is converted into new matter via E = γmc . In the history of the universe this has happened so many times that practically all the bodies still in relativistic motion are microscopic particles.
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•.
,
A v
transformed CM A
v = 0
CM-0
CM-1 v
B v
B 2v/(1+v2/c2)
geometrical CM
A
CM-2
B
v = 0
v/(1+v2/c2 )
2v/(1+v2/c2)
momentum CM A
v = 0
CM-3
B
v/(1- v2/c2)1/2
2v/(1+v2/c2)
F I G U R E 165 There is no way to define a relativistic centre of mass
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Challenge 622 n Page 450
A second reason for the disappearance of rapid relative motion is radiation damping. Can you imagine what happens to charges during collisions, or in a bath of light?
In short, almost all matter in the universe moves with small velocity relative to other matter. e few known counter-examples are either very old, such as the quasar jets mentioned above, or stop a er a short time. e huge energies necessary for macroscopic relativistic motion are still found in supernova explosions, but they cease to exist a er only a few weeks. In summary, the universe is mainly lled with slow motion because it is old. We will determine its age shortly.
T
–
P
E
Albert Einstein took several months a er his rst paper on special relativity to deduce the expression
E = γmc
(140)
Ref. 241
which is o en called the most famous formula of physics. He published it in a second, separate paper towards the end of . Arguably, the formula could have been discovered thirty years earlier, from the theory of electromagnetism.
In fact, at least one person did deduce the result before Einstein. In and , before Einstein’s rst relativity paper, a little-known Italian engineer, Olinto De Pretto, was the rst to calculate, discuss and publish the formula E = mc .* It might well be that
* Umberto Bartocci, mathematics professor of the University of Perugia in Italy, published the details of
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t
light cone
future
T
light cone
E
elsewhere
y
x past
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F I G U R E 166 The space-time diagram of a moving object T
Ref. 289 Challenge 623 n
Einstein got the idea for the formula from De Pretto, possibly through his friend Michele Besso or other Italian-speaking friends he met when he visited his parents, who were living in Italy at the time. Of course, the value of Einstein’s e orts is not diminished by this.
In fact, a similar formula had also been deduced in by Friedrich Hasenöhrl and published again in Annalen der Physik in , before Einstein, though with an incorrect numerical prefactor, due to a calculation mistake. e formula E = mc is also part of several expressions in two publications in by Henri Poincaré. e real hero in the story might well be Tolver Preston, who discussed the equivalence of mass and energy already in , in his book Physics of the Ether. e mass-energy equivalence was thus indeed oating in the air, only waiting to be discovered.
In the s there was a similar story: a simple relation between the gravitational acceleration and the temperature of the vacuum was discovered. e result had been waiting to be discovered for over years. Indeed, a number of similar, anterior results were found in the libraries. Could other simple relations be hidden in modern physics waiting to be found?
-
To describe motion consistently for all observers, we have to introduce some new quantities. First of all, motion of particles is seen as a sequence of events. To describe events with precision, we use event coordinates, also called -coordinates. ese are written as
X = (ct, x) = (ct, x, y, z) = Xi .
(141)
In this way, an event is a point in four-dimensional space-time, and is described by four coordinates. e coordinates are called the zeroth, namely time X = ct, the rst, usually
this surprising story in several papers. e full account is found in his book U
B
, Albert
Einstein e Olinto De Pretto: la vera storia della formula più famosa del mondo, Ultreja, Padova, 1998.
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,
called X = x, the second, X = y, and the third, X = z. One can then de ne a distance d between events as the length of the di erence vector. In fact, one usually uses the square of the length, to avoid those unwieldy square roots. In special relativity, the magnitude (‘squared length’) of a vector is always de ned through
XX = X − X − X − X = ct − x − y − z = Xa Xa = ηab Xa Xb = ηab Xa Xb .(142)
In this equation we have introduced for the rst time two notations that are useful in relativity. First of all, we automatically sum over repeated indices. us, Xa Xa means the sum of all products Xa Xa as a ranges over all indices. Secondly, for every -vector X we distinguish two ways to write the coordinates, namely coordinates with superscripts and coordinates with subscripts. (In three dimensions, we only use subscripts.) ey are related by the following general relation
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Xa = ηab Xb = (ct, −x, −y, −z) ,
(143)
where we have introduced the so-called metric ηab, an abbreviation of the matrix*
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ηab = ηab =
− −
.
−
(144)
Page 296
Don’t panic: this is all, and it won’t get more di cult! We now go back to physics. e magnitude of a position or distance vector, also called the space-time interval, is
essentially the proper time times c. e proper time is the time shown by a clock moving in a straight line and with constant velocity from the starting point to the end point in space-time. e di erence from the usual -vectors is that the magnitude of the interval can be positive, negative or even zero. For example, if the start and end points in spacetime require motion with the speed of light, the proper time is zero (this is required for null vectors). If the motion is slower than the speed of light, the squared proper time is positive and the distance is timelike. For negative intervals and thus imaginary proper times, the distance is spacelike.** A simpli ed overview is given by Figure .
Now we are ready to calculate and measure motion in four dimensions. e measurements are based on one central idea. We cannot de ne the velocity of a particle as the derivative of its coordinates with respect to time, since time and temporal sequences depend on the observer. e solution is to de ne all observables with respect to the justmentioned proper time τ, which is de ned as the time shown by a clock attached to the object. In relativity, motion and change are always measured with respect to clocks attached to the moving system. In particular, the relativistic velocity or -velocity U of a body is
* Note that 30 % of all physics textbooks use the negative of η as the metric, the so-called spacelike convention, and thus have opposite signs in this de nition. In this text, as in 70 % of all physics texts, we use the timelike convention. ** In the latter case, the negative of the magnitude, which is a positive number, is called the squared proper distance. e proper distance is the length measured by an odometer as the object moves along.
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thus de ned as the rate of change of the event coordinates or -coordinates X = (ct, x) with respect to proper time, i.e. as
U = dX dτ .
(145)
e coordinates X are measured in the coordinate system de ned by the inertial observer
chosen. e value of the velocity U depends on the observer or coordinate system used;
so the velocity depends on the observer, as it does in everyday life. Using dt = γ dτ and
thus
dx dτ
=
dx dt
dt dτ
=
γ
dx dt
, where as usual
γ=
, −v c
(146)
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we get the relation with the -velocity v = dx dt:
u = γc , ui = γvi or U = (γc, γv) .
(147)
Challenge 625 n
For small velocities we have γ , and then the last three components of the -velocity are those of the usual, Galilean -velocity. For the magnitude of the -velocity U we nd UU = UaU a = ηabU aU b = c , which is therefore independent of the magnitude of the -velocity v and makes it a timelike vector, i.e. a vector inside the light cone.*
Note that the magnitude of a -vector can be zero even though all its components are di erent from zero. Such a vector is called null. Which motions have a null velocity vector?
Similarly, the relativistic acceleration or -acceleration B of a body is de ned as
B = dU dτ = d X dτ .
(149)
Using dγ dτ = γdγ dt = γ va c , we get the following relations between the four comRef. 290 ponents of B and the -acceleration a = dv dt:
B
=γ
va c
,
Bi = γ
ai + γ
(va)vi c
.
(150)
e magnitude b of the -acceleration is easily found via BB = ηcd Bc Bd = −γ (a + γ (va) c ) = −γ (a − (v a) c ). Note that it does depend on the value of the
* In general, a 4-vector is de ned as a quantity (h , h , h , h ), which transforms as
h′ = γV (h − h V c) h′ = γV (h − h V c) h′ = h h′ = h
(148)
Challenge 624 n
when changing from one inertial observer to another moving with a relative velocity V in the x direction; the corresponding generalizations for the other coordinates are understood. is relation allows one to deduce the transformation laws for any 3-vector. Can you deduce the velocity composition formula (108) from this de nition, applying it to 4-velocity?
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,
time (E/c , p)
space
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F I G U R E 167 Energy–momentum is tangent to the world line
Challenge 626 n Page 330
-acceleration a. e magnitude of the -acceleration is also called the proper acceleration because B = −a if v = . (What is the connection between -acceleration and
-acceleration for an observer moving with the same speed as the object?) We note
that -acceleration lies outside the light cone, i.e. that it is a spacelike vector, and that BU = ηcd BcU d = , which means that the -acceleration is always perpendicular to the -velocity.* We also note that accelerations, in contrast to velocities, cannot be called re-
lativistic: the di erence between bi and ai, or between their two magnitudes, does not depend on the value of ai, but only on the value of the speed v. In other words, accelerations require relativistic treatment only when the involved velocities are relativistic. If
the velocities involved are low, even the highest accelerations can be treated with Galilean
methods.
When the acceleration a is parallel to the velocity v, we get B = γ a; when a is perpen-
dicular to v, as in circular motion, we get B = γ a. We will use this result below.
-
To describe motion, we also need the concept of momentum. as
P = mU
e -momentum is de ned (153)
* Similarly, the relativistic jerk or 4-jerk J of a body is de ned as J = dB dτ = d U dτ .
Challenge 627 ny For the relation with the 3-jerk j = da dt we then get
J = (J , Ji) =
γ c
(jv
+
a
+
γ
(va) c
),γ
ji
+
γ c
((jv)vi + a vi +
γ
(va) vi + c
(va)ai )
Challenge 628 ny which we will use later on. Surprisingly, J does not vanish when j vanishes. Why not?
(151) (152)
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and is therefore related to the -momentum p by
P = (γmc, γmv) = (E c, p) .
(154)
For this reason -momentum is also called the energy–momentum -vector. In short, the -momentum of a body is given by mass times -displacement per proper time. is is the simplest possible de nition of momentum and energy. e concept was introduced by Max Planck in . e energy–momentum -vector, also called momenergy, like the -velocity, is tangent to the world line of a particle. is connection, shown in Figure , follows directly from the de nition, since
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(E c, p) = (γmc, γmv) = m(γc, γv) = m(dt dτ, dx dτ) .
(155)
e (square of the) length of momenergy, namely PP = ηab Pa Pb, is by de nition the same for all inertial observers; it is found to be
E c −p =m c ,
(156)
Challenge 629 n Ref. 291
Challenge 630 n
thus con rming a result given above. We have already mentioned that energies or situations are called relativistic if the kinetic energy T = E − E is not negligible when compared to the rest energy E = mc . A particle whose kinetic energy is much higher than its rest mass is called ultrarelativistic. Particles in accelerators or in cosmic rays fall into this category. (What is their energy–momentum relation?)
In contrast to Galilean mechanics, relativity implies an absolute zero for the energy. One cannot extract more energy than mc from a system of mass m. In particular, a zero value for potential energy is xed in this way. In short, relativity shows that energy is bounded from below.
Note that by the term ‘mass’ m we always mean what is sometimes called the rest mass. is name derives from the bad habit of many science ction and secondary-school books of calling the product γm the relativistic mass. Workers in the eld usually (but not unanimously) reject this concept, as did Einstein himself, and they also reject the o en-heard expression that ‘(relativistic) mass increases with velocity’. Relativistic mass and energy would then be two words for the same concept: this way to talk is at the level of the tabloid press. Not all Galilean energy contributes to mass. Potential energy in an outside eld does not. Relativity forces us into precise energy bookkeeping. ‘Potential energy’ in relativity is an abbreviation for ‘energy reduction of the outside eld’. Can you show that for two particles with momenta P and P , one has P P = m E = M E = c γv m m , where v is their relative velocity?
e -force K is de ned as
K = dP dτ = mB .
(157)
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erefore force remains equal to mass times acceleration in relativity. From the de nition Ref. 290, Ref. 292 of K we deduce the relation with -force f = dp dt = md(γv) dt, namely*
K = (K
, K i) = (γ
mva
c, γ
mai + γ
vi
mva c
)
=
(
γ c
dE dt
,
γ
dp dt
)
=
(γ
fv c
,
γf)
.
(158)
Challenge 632 e
e -force, like the -acceleration, is orthogonal to the -velocity. e meaning of the zeroth component of the -force can easily be discerned: it is the power required to accelerate the object. One has KU = c dm dτ = γ (dE dt−fv): this is the proper rate at which the internal energy of a system increases. e product KU vanishes only for rest-massconserving forces. Particle collisions that lead to reactions do not belong to this class. In everyday life, the rest mass is preserved, and then one gets the Galilean expression fv = dE dt.
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R
Page 307 Page 337
Challenge 633 e Challenge 634 e
If at night we turn around our own axis while looking at the sky, the stars move with a
velocity much higher than that of light. Most stars are masses, not images. eir speed
should be limited by that of light. How does this t with special relativity?
is example helps to clarify in an-
other way what the limit velocity ac-
C
tually is. Physically speaking, a rotat-
ing sky does not allow superluminal
A
v
energy transport, and thus does not
contradict the concept of a limit speed. Mathematically speaking, the speed of light limits relative velocities
v v'
B
v' D
only between objects that come near F I G U R E 168 On the definition of relative velocity to each other, as shown on the le of
Figure . To compare velocities of distant objects is only possible if all velocities in-
volved are constant in time; this is not the case in the present example. e di erential
version of the Lorentz transformations make this point particularly clear. In many gen-
eral cases, relative velocities of distant objects can be higher than the speed of light. We
encountered one example earlier, when discussing the car in the tunnel, and we will en-
counter a few more examples shortly.
With this clari cation, we can now brie y consider rotation in relativity. e rst ques-
tion is how lengths and times change in a rotating frame of reference. You may want
to check that an observer in a rotating frame agrees with a non-rotating colleague on
the radius of a rotating body; however, both nd that the rotating body, even if it is ri-
gid, has a circumference di erent from the one it had before it started rotating. Sloppily
speaking, the value of π changes for rotating observers. e ratio between the circumfer-
ence c and the radius r turns out to be c r = πγ: it increases with rotation speed. is
Challenge 631 n
* Some authors de ne 3-force as dp dτ; then K looks slightly di erent. In any case, it is important to note that in relativity, 3-force f = dp dt is indeed proportional to 3-acceleration a; however, force and acceleration are not parallel to each other. In fact, for rest-mass-preserving forces one nds f = γma + (fv)v c . In contrast, in relativity 3-momentum is not proportional to 3-velocity, although it is parallel to it.
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Ref. 293 Challenge 635 ny
counter-intuitive result is o en called Ehrenfest’s paradox. Among other things, it shows
that space-time for an observer on a rotating disc is not the Minkowski space-time of
special relativity.
Rotating bodies behave strangely in many ways. For ex-
ample, one gets into trouble when one tries to synchronize
clocks mounted on a rotating circle,as shown in Figure
If one starts synchronizing the clock at O with that at O , and so on, continuing up to clock On, one nds that the last clock is not synchronized with the rst. is result re ects
O3 O2 O1 On On–1
the change in circumference just mentioned. In fact, a care-
ful study shows that the measurements of length and time
intervals lead all observers Ok to conclude that they live in a rotating space-time. Rotating discs can thus be used as an
introduction to general relativity, where this curvature and F I G URE 169 Observers on a
its e ects form the central topic. More about this in the next rotating object
chapter.
Is angular velocity limited? Yes: the tangential speed in an inertial frame of reference
cannot exceed that of light. e limit thus depends on the size of the body in question.
at leads to a neat puzzle: can one see objects rotating very rapidly?
We mention that -angular momentum is de ned naturally as
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l ab = xa pb − xb pa .
(159)
Challenge 636 ny
Challenge 637 ny Challenge 638 ny
Challenge 639 e Challenge 640 ny
In other words, -angular momentum is a tensor, not a vector, as shown by its two indices. Angular momentum is conserved in special relativity. e moment of inertia is naturally de ned as the proportionality factor between angular velocity and angular momentum.
Obviously, for a rotating particle, the rotational energy is part of the rest mass. You may want to calculate the fraction for the Earth and the Sun. It is not large. By the way, how would you determine whether a microscopic particle, too small to be seen, is rotating?
In relativity, rotation and translation combine in strange ways. Imagine a cylinder in uniform rotation along its axis, as seen by an observer at rest. As Max von Laue has discussed, the cylinder will appear twisted to an observer moving along the rotation axis. Can you con rm this?
Here is a last puzzle about rotation. Velocity is relative; this means that the measured value depends on the observer. Is this the case also for angular velocity?
W
In Galilean physics, a wave is described by a wave vector and a frequency. In special relativity, the two are combined in the wave -vector, given by
L
=
λ
(
ω c
,
n)
,
(160)
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•.
,
where λ is the wavelength, ω the wave velocity, and n the normed direction vector. Suppose an observer with -velocity U nds that a wave L has frequency ν. Show that
ν = LU
(161)
Challenge 641 ny must be obeyed. Interestingly, the wave velocity ω transforms in a di erent way than Ref. 246 particle velocity except in the case ω = c. Also the aberration formula for wave motion
Challenge 642 ny di ers from that for particles, except in the case ω = c.
T
–
?
Page 176
If we want to describe relativistic motion of a free particle in terms of an extremal principle, we need a de nition of the action. We already know that physical action is a measure of the change occurring in a system. For an inertially moving or free particle, the only change is the ticking of its proper clock. As a result, the action of a free particle will be proportional to the elapsed proper time. In order to get the standard unit of energy times time, or Js, for the action, the rst guess for the action of a free particle is
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∫τ
S = −mc
dτ ,
τ
(162)
Challenge 643 ny Challenge 644 ny
where τ is the proper time along its path. is is indeed the correct expression. It implies conservation of (relativistic) energy and momentum, as the change in proper time is maximal for straight-line motion with constant velocity. Can you con rm this? Indeed, in nature, all particles move in such a way that their proper time is maximal. In other words, we again nd that in nature things change as little as possible. Nature is like a wise old man: its motions are as slow as possible. If you prefer, every change is maximally e ective. As we mentioned before, Bertrand Russell called this the law of cosmic laziness.
e expression ( ) for the action is due to Max Planck. In , by exploring it in detail, he found that the quantum of action ħ, which he had discovered together with the Boltzmann constant, is a relativistic invariant (like the Boltzmann constant k). Can you imagine how he did this?
e action can also be written in more complex, seemingly more frightening ways. ese equivalent ways to write it are particularly appropriate to prepare for general relativity:
∫ ∫ ∫ ∫ t
τ
s
S = L dt = −mc t γ dt = −mc τ
uaua dτ = −mc
s
ηab
dxa ds
dxb ds
ds ,
(163)
where s is some arbitrary, but monotonically increasing, function of τ, such as τ itself. As
usual, the metric ηαβ of special relativity is
ηab = ηab =
− −
.
−
(164)
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Challenge 645 ny
Challenge 647 ny Page 176
Challenge 648 ny
You can easily con rm the form of the action ( ) by deducing the equation of motion in the usual way.
In short, nature is in not a hurry: every object moves in a such way that its own clock shows the longest delay possible, compared with any alternative motion nearby.* is general principle is also valid for particles under the in uence of gravity, as we will see in the section on general relativity, and for particles under the in uence of electric or magnetic interactions. In fact, it is valid in all cases of (macroscopic) motion found in nature. For the moment, we just note that the longest proper time is realized when the di erence between kinetic and potential energy is minimal. (Can you con rm this?) For the Galilean case, the longest proper time thus implies the smallest average di erence between the two energy types. We thus recover the principle of least action in its Galilean formulation.
Earlier on, we saw that the action measures the change going on in a system. Special relativity shows that nature minimizes change by maximizing proper time. In nature, proper time is always maximal. In other words, things move along paths of maximal ageing. Can you explain why ‘maximal ageing’ and ‘cosmic laziness’ are equivalent?
We thus again nd that nature is the opposite of a Hollywood movie: nature changes in the most economical way possible. e deeper meaning of this result is le to your personal thinking: enjoy it!
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C
–
?
e distinction between space and time in special relativity depends on the observer. On
the other hand, all inertial observers agree on the position, shape and orientation of the
light cone at a point. us, in the theory of relativity, the light cones are the basic physical
‘objects’. Given the importance of light cones, we might ask if inertial observers are the
only ones that observe the same light cones. Interestingly, it turns out that other observers
do as well.
e rst such category of observers are those using units of measurement in which all
time and length intervals are multiplied by a scale factor λ. e transformations among
these points of view are given by
xa λxa
(165)
and are called dilations. A second category of additional observers are found by applying the so-called special
conformal transformations. ese are compositions of an inversion
xa
xa x
(166)
with a translation by a vector ba, namely
xa xa + ba ,
(167)
* If neutrinos were massless, the action (163) would not be applicable for them. Why? Can you nd an Challenge 646 ny alternative for this (admittedly academic) case?
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and a second inversion. us the special conformal transformations are
xa
xa + bax + baxa + b x
or
xa x
xa x
+
ba
.
(168)
ese transformations are called conformal because they do not change angles of (in nChallenge 649 ny itesimally) small shapes, as you may want to check. ey therefore leave the form (of in-
nitesimally small objects) unchanged. For example, they transform in nitesimal circles into in nitesimal circles. ey are called special because the full conformal group includes the dilations and the inhomogeneous Lorentz transformations as well.*
Note that the way in which special conformal transformations leave light cones invariChallenge 651 ny ant is rather subtle.
Since dilations do not commute with time translations, there is no conserved quantity associated with this symmetry. ( e same is true of Lorentz boosts.) In contrast, rotations and spatial translations do commute with time translations and thus do lead to conserved quantities.
In summary, vacuum is conformally invariant – in the special sense just mentioned – and thus also dilation invariant. is is another way to say that vacuum alone is not su cient to de ne lengths, as it does not x a scale factor. As we would expect, matter is necessary to do so. Indeed, (special) conformal transformations are not symmetries of situations containing matter. Only vacuum is conformally invariant; nature as a whole is not.
However, conformal invariance, or the invariance of light cones, is su cient to allow velocity measurements. Conformal invariance is also necessary for velocity measureChallenge 652 ny ments, as you might want to check.
We have seen that conformal invariance implies inversion symmetry: that is, that the large and small scales of a vacuum are related. is suggest that the constancy of the speed of light is related to the existence of inversion symmetry. is mysterious connection gives us a glimpse of the adventures we will encounter in the third part of our ascent of Motion Mountain. Conformal invariance turns out to be an important property that will lead to some incredible insights.**
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Challenge 650 ny Page 1194
Challenge 653 ny
* e set of all special conformal transformations forms a group with four parameters; adding dilations and the inhomogeneous Lorentz transformations one gets een parameters for the full conformal group.
e conformal group is locally isomorphic to SU(2,2) and to the simple group SO(4,2): these concepts are explained in Appendix D. Note that all this is true only for four space-time dimensions; in two dimensions – the other important case, especially in string theory – the conformal group is isomorphic to the group of arbitrary analytic coordinate transformations, and is thus in nite-dimensional. ** e conformal group does not appear only in the kinematics of special relativity: it is the symmetry group of all physical interactions, such as electromagnetism, provided that all the particles involved have zero mass, as is the case for the photon. A eld that has mass cannot be conformally invariant; therefore conformal invariance is not an exact symmetry of all of nature. Can you con rm that a mass term mφ in a Lagrangian is not conformally invariant?
However, since all particles observed up to now have masses that are many orders of magnitude smaller than the Planck mass, it can be said that they have almost vanishing mass; conformal symmetry can then be seen as an approximate symmetry of nature. In this view, all massive particles should be seen as small corrections, or perturbations, of massless, i.e. conformally invariant, elds. erefore, for the construction of a fundamental theory, conformally invariant Lagrangians are o en assumed to provide a good starting
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A
Ref. 294
So far, we have only studied what inertial, or free- ying, observers say to each other when
they talk about the same observation. For example, we saw that moving clocks always
run slow. e story gets even more interesting when one or both of the observers are
accelerating.
One sometimes hears that special relativity cannot be used to describe accelerating
observers. at is wrong, just as it is wrong to say that Galilean physics cannot be used for
accelerating observers. Special relativity’s only limitation is that it cannot be used in non-
at, i.e. curved, space-time. Accelerating bodies do exist in at space-times, and therefore
they can be discussed in special relativity.
As an appetizer, let us see what an acceler-
ating, Greek, observer says about the clock
of an inertial, Roman, one, and vice versa. observer (Greek)
v
Assume that the Greek observer, shown in
Figure , moves along the path x(t), as light observed by the inertial Roman one. In gen-
c
eral, the Roman/Greek clock rate ratio is observer (Roman)
given by ∆τ ∆t = (τ − τ ) (t − t ). Here the Greek coordinates are constructed with a simple procedure: take the two sets of
F I G U R E 170 The simplest situation for an inertial and an accelerated observer
events de ned by t = t and t = t , and let τ and τ be the points where these sets
intersect the time axis of the Greek observer.* We assume that the Greek observer is iner-
tial and moving with velocity v as observed by the Roman one. e clock ratio of a Greek
observer is then given by
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∆τ ∆t
=
dτ dt
=
− v c = γv ,
(169)
Challenge 654 ny Ref. 294
a formula we are now used to. We nd again that moving clocks run slow.
For accelerated motions, the di erential version of the above reasoning is necessary. e Roman/Greek clock rate ratio is again dτ dt, and τ and τ + dτ are calculated in the same way from the times t and t + dt. Assume again that the Greek observer moves along
the path x(t), as measured by the Roman one. We nd directly that
τ = t − x(t)v(t) c
(170)
and thus
τ + dτ = (t + dt) − [x(t) − dtv(t)][v(t) + dta(t)] c .
(171)
Together, these equations yield
‘dτ dt’ = γv( − vv c − xa c ) .
(172)
approximation. * ese sets form what mathematicians call hypersurfaces.
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•.
,
is shows that accelerated clocks can run fast or slow, depending on their position x and the sign of their acceleration a. ere are quotes in the above equation because we can see directly that the Greek observer notes
‘dt dτ’ = γv ,
(173)
which is not the inverse of equation ( ). is di erence becomes most apparent in the simple case of two clocks with the same velocity, one of which has a constant acceleration
towards the origin, whereas the other moves inertially. We then have
‘dτ dt’ = + x c
(174)
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and
‘dt dτ’ = .
(175)
We will discuss this situation shortly. But rst we must clarify the concept of acceleration.
A
Ref. 247
Accelerations behave di erently from velocities under change of viewpoint. Let us rst
take the simple case in which the object and two inertial observers all move along the
x-axis. If the Roman inertial observer measures an acceleration a = dv dt = d x dt ,
and the Greek observer, also inertial, measures an acceleration α = dω dτ = d ξ dτ , we
get
γv a = γωα .
(176)
is relation shows that accelerations are not Lorentz invariant, unless the velocities are small compared to the speed of light. is is in contrast to our everyday experience, where accelerations are independent of the speed of the observer.
Expression ( ) simpli es if the accelerations are measured at a time t at which ω vanishes – i.e. if they are measured by the so-called comoving inertial observer. In that case the acceleration relation is given by
ac = aγv
(177)
Ref. 295
and the acceleration ac = α is also called proper acceleration, as its value describes what the Greek, comoving observer feels: proper acceleration describes the experience of being
pushed into the back of the accelerating seat.
In general, the observer’s speed and the acceleration are not parallel. We can calculate
how the value of -acceleration a measured by a general inertial observer is related to the
value ac measured by the comoving observer using expressions ( ) and ( ). We get the generalization of ( ):
vac = vaγv
(178)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
and
a = γv
ac − (
− γv )(vac)v − γv (vac)v
v
c
.
Squaring yields the relation
(179)
a = γv
ac
−
(acv) c
(180)
Page 321 Challenge 655 e
Page 476
which we know already in a slightly di erent form. It shows (again) that the comoving or proper -acceleration is always larger than the -acceleration measured by an outside inertial observer. e faster the outside inertial observer is moving, the smaller the acceleration he observes. Acceleration is not a relativistic invariant. e expression also shows that whenever the speed is perpendicular to the acceleration, a boost yields a factor γv, whereas a speed parallel to the acceleration gives the already mentioned γv dependence.
We see that acceleration complicates many issues, and it requires a deeper investigation. To keep matters simple, from now on we only study constant accelerations. Interestingly, this situation serves also as a good introduction to black holes and, as we will see shortly, to the universe as a whole.
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A
How do we check whether we live in an inertial frame of reference? Let us rst de ne the term. An inertial frame (of reference) has two de ning properties. First, lengths and distances measured with a ruler are described by Euclidean geometry. In other words, rulers behave as they do in daily life. In particular, distances found by counting how many rulers (rods) have to be laid down end to end to reach from one point to another – the so-called rod distances – behave as in everyday life. For example, they obey Pythagoras’ theorem in the case of right-angled triangles. Secondly, the speed of light is constant. In other words, any two observers in that frame, independent of their time and of the position, make the following observation: the ratio c between twice the rod distance between two points and the time taken by light to travel from one point to the other and back is always the same.
Equivalently, an inertial frame is one for which all clocks always remain synchronized and whose geometry is Euclidean. In particular, in an inertial frame all observers at xed coordinates always remain at rest with respect to each other. is last condition is, however, a more general one. ere are other, non-inertial, situations where this is still the case.
Non-inertial frames, or accelerating frames, are a useful concept in special relativity. In fact, we all live in such a frame. We can use special relativity to describe it in the same way that we used Galilean physics to describe it at the beginning of our journey.
A general frame of reference is a continuous set of observers remaining at rest with respect to each other. Here, ‘at rest with respect to each other’ means that the time for a light signal to go from one observer to another and back again is constant over time, or equivalently, that the rod distance between the two observers is constant. Any frame of reference can therefore also be called a rigid collection of observers. We therefore note that a general frame of reference is not the same as a set of coordinates; the latter is usu-
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•.
,
t
τ
II
future
horizon Ω
ξ
III
O
c2/g
x
IV
past horizon
I
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F I G U R E 171 The hyperbolic motion of an rectilinearly, uniformly accelerating observer Ω
Ref. 297 Challenge 656 ny
ally not rigid. If all the rigidly connected observers have constant coordinate values, we speak of a rigid coordinate system. Obviously, these are the most useful when it comes to describing accelerating frames of reference.*
Note that if two observers both move with a velocity v, as measured in some inertial frame, they observe that they are at rest with respect to each other only if this velocity is constant. Again we nd, as above, that two people tied to each other by a rope, and at a distance such that the rope is under tension, will see the rope break (or hang loose) if they accelerate together to (or decelerate from) relativistic speeds in precisely the same way. Relativistic acceleration requires careful thinking.
An observer who always feels the same force on his body is called uniformly accelerating. More precisely, a uniformly accelerating observer is an observer whose acceleration at every moment, measured by the inertial frame with respect to which the observer is at rest at that moment, always has the same value B. It is important to note that uniform acceleration is not uniformly accelerating when always observed from the same inertial frame. is is an important di erence from the Galilean case.
For uniformly accelerated motion in the sense just de ned, we need
BëB=−
(181)
Ref. 298 where is a constant independent of t. e simplest case is uniformly accelerating motion that is also rectilinear, i.e. for which the acceleration a is parallel to v at one instant of time
Challenge 657 ny and (therefore) for all other times as well. In this case we can write, using -vectors,
Ref. 296 * ere are essentially only two other types of rigid coordinate frames, apart from the inertial frames:
— e frame ds = dx + dy + dz − c dt ( + k xk c ) with arbitrary, but constant, acceleration of the origin. e acceleration is a = −g( + gx c ).
— e uniformly rotating frame ds = dx + dy + dz + ω(−y dx + x dy)dt − ( − r ω c )dt. Here the z-axis is the rotation axis, and r = x + y .
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
γ a=g
or
dγv dt
=
g
.
(182)
Taking the direction we are talking about to be the x-axis, and solving for v(t), we get
v=
t,
+
t c
(183)
Challenge 658 ny
where it was assumed that v( ) = . We note that for small times we get v = t and for large times v = c, both as expected. e momentum of the accelerated observer increases linearly with time, again as expected. Integrating, we nd that the accelerated observer moves along the path
x(t) = c
+
t c
,
(184)
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Ref. 298, Ref. 299
where it is assumed that x( ) = c , in order to keep the expression simple. Because of
this result, visualized in Figure , a rectilinearly and uniformly accelerating observer is
said to undergo hyperbolic motion. For small times, the world-line reduces to the usual
x = t + x , whereas for large times it is x = ct, as expected. e motion is thus
uniformly accelerated only for the moving body itself, not for an outside observer.
e proper time τ of the accelerated observer is related to the time t of the inertial
frame in the usual way by dt = γdτ. Using the expression for the velocity v(t) of equation
( ) we get*
t = c sinh
τ c
and
x= c
cosh
τ c
(185)
Challenge 659 n
for the relationship between proper time τ and the time t and position x measured by the external, inertial Roman observer. We will encounter this relation again during our study of black holes.
Does all this sound boring? Just imagine accelerating on a motorbike at = m s for the proper time τ of years. at would bring you beyond the end of the known universe! Isn’t that worth a try? Unfortunately, neither motorbikes nor missiles that accelerate like this exist, as their fuel tanks would have to be enormous. Can you con rm this?
Ref. 300
* Use your favourite mathematical formula collection – every student should have one – to deduce this. e hyperbolic sine and the hyperbolic cosine are de ned by sinh y = (ey − e−y) and cosh y = (ey + e−y) .
ey imply that ∫ dy y + a = arsinh y a = Arsh y a = ln(y + y + a ).
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•.
,
For uniform acceleration, the coordinates transform as
t
=
(c
+
ξ c
)
sinh
τ c
x = (c
+ ξ) cosh
τ c
y=υ
z=ζ,
(186) Dvipsbugw
where τ now is the time coordinate in the Greek frame. We note also that the space-time interval dσ satis es
dσ = ( + ξ c ) c dτ − dξ − dυ − dζ = c dt − dx − dy − dz , (187)
Ref. 301
Challenge 660 e Ref. 302
and since for dτ = distances are given by Pythagoras’ theorem, the Greek reference
frame is indeed rigid.
A er this forest of formulae, let’s tackle a simple question, shown in Figure . e
inertial, Roman observer O sees the Greek observer Ω departing with acceleration ,
moving further and further away, following equation ( ). What does the Greek observer
say about his Roman colleague? With all the knowledge we have now, that is easy to
answer to answer. At each point of his trajectory Ω sees that O has the coordinate τ =
(can you con rm this?), which means that the distance to the Roman observer, as seen
by Greek one, is the same as the space-time interval OΩ. Using expression ( ), we see
that this is
dOΩ = ξ = x − c t = c ,
(188)
Challenge 661 n Ref. 303
Challenge 662 e
which, surprisingly enough, is constant in time! In other words, the Greek observer will
observe that he stays at a constant distance from the Roman one, in complete contrast to
what the Roman observer says. Take your time to check this strange result in some other
way. We will need it again later on, to explain why the Earth does not explode. (Can you
guess how that is related to this result?)
e composition theorem for accelerations is more complex than for velocities. e
best explanation of this was published by Mishra. If we call anm the acceleration of system n by observer m, we are seeking to express the object acceleration a as function
of the value a measured by the other observer, the relative acceleration a , and the
proper acceleration a of the other observer: see Figure . Here we will only study
one-dimensional situations, where all observers and all objects move along one axis. (For
clarity, we also write v = v and v = u.) In Galilean physics we have the general connec-
tion
a =a −a +a
(189)
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y
a11 : proper acceleration v11 = 0
y Observer 1
a22 : proper acceleration v22 = 0 Observer 2
x
x
v0n : object speed seen by observer n
Object
a0n : object acceleration seen by observer n
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F I G U R E 172 The definitions necessary to deduce the composition behaviour of accelerations
because accelerations behave simply. In special relativity, one gets
Challenge 663 ny Page 312
Challenge 664 ny
a
=a
( −v c ) ( − uv c )
−a
( − u c )( − v c )− ( − uv c )
+a
( − u c )( − v c ) ( − uv c )
(190)
and you might enjoy checking the expression.
Can you state how the acceleration ratio enters into the de nition of mass in special
relativity?
E
ere are many surprising properties of accelerated motion. Of special interest is the trajectory, in the coordinates ξ and τ of the rigidly accelerated frame, of an object located Challenge 665 ny at the departure point x = x = c at all times t. One gets the two relations*
ξ = −c
(
− sech
τ c
)
dξ dτ = −c sech
τ c
tanh
τ c
.
(192)
ese equations are strange. For large times τ the coordinate ξ approaches the limit value −c and dξ dτ approaches zero. e situation is similar to that of a car accelerating away from a woman standing on a long road. Seen from the car, the woman moves away;
* e functions appearing above, the hyperbolic secant and the hyperbolic tangent, are de ned using the expressions from the footnote on page 333:
sech y = cosh y
and
tanh
y=
sinh cosh
y y
.
(191)
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•.
,
t
τ
II
future
horizon Ω
ξ
III
O
c2/g
x
IV
past horizon
I
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F I G U R E 173 Hyperbolic motion and event horizons
Challenge 666 ny
Challenge 667 n Challenge 668 n
however, a er a while, the only thing one notices is that she is slowly approaching the horizon. In Galilean physics, both the car driver and the woman on the road see the other person approaching their horizon; in special relativity, only the accelerated observer makes this observation.
A graph of the situation helps to clarify the result. In Figure we can see that light emitted from any event in regions II and III cannot reach the Greek observer. ose events are hidden from him and cannot be observed. Strangely enough, however, light from the Greek observer can reach region II. e boundary between the part of spacetime that can be observed and the part that cannot is called the event horizon. In relativity, event horizons act like one-way gates for light and other signals. For completeness, the graph also shows the past event horizon. Can you con rm that event horizons are black?
So, not all events observed in an inertial frame of reference can be observed in a uniformly accelerating frame of reference. Uniformly accelerating frames of reference produce event horizons at a distance −c . For example, a person who is standing can never see further than this distance below his feet.
By the way, is it true that a light beam cannot catch up with an observer in hyperbolic motion, if the observer has a su cient headstart?
Here is a more advanced challenge, which prepares us for general relativity. What is the shape of the horizon seen by a uniformly accelerated observer?
A
Ref. 298, Ref. 304
We saw earlier that a moving receiver sees di erent colours from the sender. So far, we discussed this colour shi , or Doppler e ect, for inertial motion only. For accelerating frames the situation is even stranger: sender and receiver do not agree on colours even if they are at rest with respect to each other. Indeed, if light is emitted in the direction of the acceleration, the formula for the space-time interval gives
dσ =
+
x c
c dt
(193)
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Challenge 669 ny
in which is the proper acceleration of an observer located at x = . We can deduce in
a straightforward way that
fr fs
=
−
rh c
=
+
sh c
(194)
Challenge 670 n
where h is the rod distance between the source and the receiver, and where s = ( + xs c ) and r = ( + oxr c ) are the proper accelerations measured at the source
and at the detector. In short, the frequency of light decreases when light moves in the direction of acceleration. By the way, does this have an e ect on the colour of trees along their vertical extension?
e formula usually given, namely
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fr fs
=
−
h c
,
(195)
is only correct to a rst approximation. In accelerated frames of reference, we have to be careful about the meaning of every quantity. For everyday accelerations, however, the Challenge 671 ny di erences between the two formulae are negligible. Can you con rm this?
C
c?
What speed of light does an accelerating observer measure? Using expression ( an accelerated observer deduces that
) above,
vlight = c (
+
h c
)
(196)
Challenge 672 n
which is higher than c for light moving in front of or ‘above’ him, and lower than c for light moving behind or ‘below’ him. is strange result follows from a basic property of any accelerating frame of reference. In such a frame, even though all observers are at rest with respect to each other, clocks do not remain synchronized. is change of the speed of light has also been con rmed by experiment.* us, the speed of light is only constant when it is de ned as c = dx dt, and if dx and dt are measured with a ruler located at a point inside the interval dx and a clock read o during the interval dt. If the speed of light is de ned as ∆x ∆t, or if the ruler de ning distances or the clock measuring times is located away from the propagating light, the speed of light is di erent from c for accelerating observers! is is the same e ect you can experience when you turn around your vertcial axis at night: the star velocities you observe are much higher than the speed of light.
Note that this result does not imply that signals or energy can be moved faster than c. You may want to check this for yourself.
In fact, all these e ects are negligible for distances l that are much less than c a. For an acceleration of . m s (about that of free fall), distances would have to be of the order
Page 410 * e propagation delays to be discussed in the chapter on general relativity can be seen as con rmations of this e ect.
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•.
,
time clock 1
clock 2
t3
t2
t1
space F I G U R E 174 Clocks and the measurement of the speed of light as two-way velocity
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Challenge 673 n
of one light year, or . ë km, in order for any sizable e ects to be observed. In short, c is the speed of light relative to nearby matter only.
By the way, everyday gravity is equivalent to a constant acceleration. So, why then distant objects, such as stars, move faster than light, following expression ( )?
W
?
We have seen that the speed of light, as usually de ned, is given by c only if either the observer is inertial or the observer measures the speed of light passing nearby (rather than light passing at a distance). In short, the speed of light has to be measured locally. But this condition does not eliminate all subtleties.
An additional point is o en forgotten. Usually, length is measured by the time it takes light to travel. In such a case the speed of light will obviously be constant. But how does one check the constancy? One needs to eliminate length measurements. e simplest way to do this is to re ect light from a mirror, as shown in Figure . e constancy of the speed of light implies that if light goes up and down a short straight line, then the clocks at the two ends measure times given by
t − t = (t − t ) .
(197)
Here it is assumed that the clocks have been synchronised according to the prescription on page . If the factor were not exactly two, the speed of light would not be constant. In fact, all experiments so far have yielded a factor of two, within measurement errors.*
Ref. 305
* e subtleties of the one-way and two-way speed of light will remain a point of discussion for a long time. Many experiments are explained and discussed in Ref. 252. Zhang says in his summary on page 171, that the one-way velocity of light is indeed independent of the light source; however, no experiment really shows that it is equal to the two-way velocity. Moreover, most so called ‘one-way’ experiments are in fact still ‘two-way’
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is result is sometimes expressed by saying that it is impossible to measure the oneChallenge 674 n way velocity of light; only the two-way velocity of light is measurable. Do you agree?
L
An everyday solid object breaks when some part of it moves with respect to some other
part with more than the speed of sound c of the material.* For example, when an object
hits the oor and its front end is stopped within a distance d, the object breaks at the
latest when
v c
d l
.
(198)
In this way, we see that we can avoid the breaking of fragile objects by packing them into foam rubber – which increases the stopping distance – of roughly the same thickness as the object’s size. is may explain why boxes containing presents are usually so much larger than their contents!
e fracture limit can also be written in a di erent way. To avoid breaking, the acceleration a of a solid body with length l must obey
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la < c ,
(199)
Ref. 306 Challenge 675 n
where c is the speed of sound, which is the speed limit for the material parts of solids. Let us now repeat the argument in relativity, using the speed of light instead of that of sound. Imagine accelerating the front of a solid body with some proper acceleration a. e back end cannot move with an acceleration α equal or larger than in nity, or if one prefers, it cannot move with more than the speed of light. A quick check shows that therefore the length l of a solid body must obey
lα < c ,
(200)
Challenge 676 ny
where c is now the speed of light. e speed of light thus limits the size of solid bodies. For
example, for . m s , the acceleration of good motorbike, this expression gives a length
limit of . Pm, about a light year. Not a big restriction: most motorbikes are shorter.
However, there are other, more interesting situations. e highest accelerations achiev-
able today are produced in particle accelerators. Atomic nuclei have a size of a few femo-
tometres. Can you deduce at which energies they break when smashed together in an
accelerator? In fact, inside a nucleus, the nucleons move with accelerations of the order
of v r ħ m r
m s ; this is one of the highest values found in nature.
Note that Galilean physics and relativity produce a similar conclusion: a limiting speed,
be it that of sound or that of light, makes it impossible for solid bodies to be rigid. When
we push one end of a body, the other end always moves a little bit later.
What does this mean for the size of elementary particles? Take two electrons a distance
d apart, and call their size l. e acceleration due to electrostatic repulsion then leads to
experiments (see his page 150). * e (longitudinal) speed of sound is about . km s for glass, iron or steel; about . km s for gold; and about km s for lead. Other sound speeds are given on page 206.
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•.
,
Challenge 677 ny an upper limit for their size given by
l<
πε
c e
d
m
.
(201)
e nearer electrons can get, the smaller they must be. e present experimental limit gives a size smaller than − m. Can electrons be exactly point-like? We will come back to this question during our study of general relativity and quantum theory.
S
is section of our ascent of Motion Mountain can be quickly summarized.
— All (free oating) observers nd that there is a unique, perfect velocity in nature, namely a common maximum energy velocity, which is realized by massless radiation such as light or radio signals, but cannot be achieved by material systems.
— erefore, even though space-time is the same for every observer, times and lengths vary from one observer to another, as described by the Lorentz transformations (112) and (113), and as con rmed by experiment.
— Collisions show that a maximum speed implies that mass is concentrated energy, and that the total energy of a body is given by E = γmc , as again con rmed by experiment.
— Applied to accelerated objects, these results lead to numerous counter-intuitive consequences, such as the twin paradox, the appearance of event horizons and the appearance of short-lived tachyons in collisions.
Special relativity shows that motion, though limited in speed, is relative, de ned using the propagation of light, conserved, reversible and deterministic.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
C
?
Challenge 678 n
e speed of massless light is the limit speed. Assuming that all light is indeed massless, could the speed of light still change from place to place, or as time goes by? is tricky question still makes a fool out of many physicists. e rst answer is usually a loud: ‘Yes, of course! Just look at what happens when the value of c is changed in formulae.’ (In fact, there have even been attempts to build ‘variable speed of light theories’.) However, this o en-heard statement is wrong.
Since the speed of light enters into our de nition of time and space, it thus enters, even if we do not notice it, into the construction of all rulers, all measurement standards and all measuring instruments. erefore there is no way to detect whether the value actually varies. No imaginable experiment could detect a variation of the limit speed, as the limit speed is the basis for all measurements. ‘ at is intellectual cruelty!’, you might say. ‘All experiments show that the speed of light is invariant; we had to swallow one counter-intuitive result a er another to accept the constancy of the speed of light, and now we are supposed to admit that there is no other choice?’ Yes, we are. at is the irony of progress in physics. e observer-invariance of the speed of light is counter-intuitive and astonishing when compared to the lack of observer-invariance at everyday, Galilean speeds. But had we taken into account that every speed measurement is – whether we like it or not – a comparison with the speed of light, we would not have been astonished by
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the constancy of the speed of light; rather, we would have been astonished by the strange properties of small speeds.
In short, there is in principle no way to check the invariance of a standard. To put it another way, the truly surprising aspect of relativity is not the invariance of c; it is the disappearance of c from the formulae of everyday motion.
W
?
As one approaches the speed of light, the quantities in the Lorentz transformation diverge. A division by zero is impossible: indeed, neither masses nor observers can move at the speed of light. However, this is only half the story.
No observable actually diverges in nature. Approaching the speed of light as nearly as possible, even special relativity breaks down. At extremely large Lorentz contractions, there is no way to ignore the curvature of space-time; indeed, gravitation has to be taken into account in those cases. Near horizons, there is no way to ignore the uctuations of speed and position; quantum theory has to be taken into account there. e exploration of these two limitations de ne the next two stages of our ascent of Motion Mountain.
At the start of our adventure, during our exploration of Galilean physics, once we had de ned the basic concepts of velocity, space and time, we turned our attention to gravitation. e invariance of the speed of light has forced us to change these basic concepts. We now return to the study of gravitation in the light of this invariance.
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B
231 A D
, On sense and the sensible, section , part ,
. Cited in J -P
, Les écoles présocratiques, Folio Essais, Gallimard, p. , . Cited on page .
232 e history of the measurement of the speed of light can be found in chapter of the text
by F
A. J
&H
E. W , Fundamentals of Optics, McGraw-Hill,
New York, . Cited on page .
233 On the way to perform such measurements, see S
G. B
, Do-it-yourself As-
tronomy, Edinburgh University Press, . Kepler himself never measured the distances of
planets to the Sun, but only ratios of planetary distances. e parallax of the Sun from two
points of the Earth is at most . ′′; it was rst measured in the eighteenth century. Cited
on page .
234 A M Dover,
, On the sizes and the distances of the Sun and the Moon, c.
, in
J. C
, eories of the World From Antiquity to the Copernican Revolution,
. Cited on page .
235 J. F
, Creativity and technology in experimentation: Fizeau’s terrestrial determina-
tion of the speed of light, Centaurus 42, pp. – , . See also the beautiful website on
reconstrutions of historical science experiments at http://www.uni-oldenburg.de/histodid/
forschung/nachbauten. Cited on page .
236 e way to make pictures of light pulses with an ordinary photographic camera, without
any electronics, is described by M.A. D
& A.T. M
, Ultrahigh speed pho-
tography of picosecond light pulses and echoes, Applied Optics 10, pp. – , . e
picture on page is taken from it. Cited on page .
237 You can learn the basics of special relativity with the help of the web, without referring to any book, using the http://physics.syr.edu/research/relativity/RELATIVITY.html web page as a starting point. is page mentions most of the English-language relativity resources available on the web. Links in other languages can be found with search engines. Cited on page .
238 Observations of gamma ray bursts show that the speed of light does not depend on the
lamp speed to within one part in , as shown by K. B
, Bulletin of the American
Physical Society 45, . He assumed that both sides of the burster emit light. e large
speed di erence and the pulse sharpness then yield this result. See also his older paper K.
B
, Is the speed of light independent of the source?, Physics Letters 39, pp. –
, Errata , . Measuring the light speed from rapidly moving stars is another way.
Some of these experiments are not completely watertight, however. ere is a competing
theory of electrodynamics, due to Ritz, which maintains that the speed of light is c only
when measured with respect to the source; e light from stars, however, passes through
the atmosphere, and its speed might thus be reduced to c.
e famous experiment with light emitted from rapid pions at CERN is not subject to this
criticism. It is described in T. A
, J.M. B
, F.J.M. F
, J. K
&
I. W
, Test of the second postulate of relativity in the GeV region, Physics Letters 12,
pp. – , . See also T. A
& al., Velocity of high-energy gamma rays, Arkiv
för Fysik 31, pp. – , .
Another precise experiment at extreme speeds is described by G.R. K
, N.
B
, E.C. F
& J. A
, Experimental comparison of neutrino, anti-
neutrino, and muon velocities, Physical Review Letters 43, pp. – , . Cited on
page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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239 See e.g. C. W , eory and Experiment in Gravitational Physics, Revised edition, Cambridge University Press, . Cited on pages and .
240 B.E. S
, Severe limits on variations of the speed of light with frequency, Physical
Review Letters 82, pp. – , June . Cited on page .
241 e beginning of the modern theory of relativity is the famous paper by A
E-
, Zur Elektrodynamik bewegter Körper, Annalen der Physik 17, pp. – , . It
still well worth reading, and every physicist should have done so. e same can be said of the
famous paper, probably written a er he heard of Olinto De Pretto’s idea, found in A
E
, Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?, Annalen der
Physik 18, pp. – , . See also the review A
E
, Über das Relativität-
sprinzip und die aus demselben gezogenen Folgerungen, Jahrbuch der Radioaktivität und
Elektronik 4, pp. – , . ese papers are now available in many languages. A later,
unpublished review is available in facsimile and with an English translation as A
E
, Hanoch Gutfreund, ed., Einstein’s 1912 Manuscript on the eory of Relativity,
George Braziller, . Cited on pages , , and .
242 A
E
on page .
, Mein Weltbild, edited by C S , Ullstein Verlag, . Cited
243 J
B
, Relativity and Engineering, Springer, . Cited on page .
244 A
F
pages and .
, Albert Einstein – eine Biographie, Suhrkamp p. ,
. Cited on
245 R.J. K
& E.M. T
, Experimental establishment of the relativity of time,
Physical Review 42, pp. – , . See also H.E. I & G.R. S
, An experi-
mental study of the rate of a moving atomic clock, Journal of the Optical Society of America
28, pp. – , , and 31, pp. – , . For a modern, high-precision versions, see
C. B
, H. M
, O. P
, J. M
, A. P
& S. S
, New
tests of relativity using a cryogenic optical resonator, Physical Review Letters 88, p.
,
. e newest result is in P. A
, M. O
, E. G
& S. S
,
Testing the constancy of the speed of light with rotating cryogenic optical resonators, Phys-
ical Review A 71, p.
, , or http://www.arxiv.org/abs/gr-qc/
. Cited on page
.
246 E
F. T
&J
A. W
, Spacetime Physics – Introduction to Special
Relativity, second edition, Freeman, . See also N M.J. W
, Special Re-
lativity, Springer, . Cited on pages and .
247 W Press,
R
, Relativity – Special, General and Cosmological, Oxford University
. A beautiful book by one of the masters of the eld. Cited on pages and .
248 e slowness of the speed of light inside stars is due to the frequent scattering of photons
by the star matter. e most common estimate for the Sun is an escape time of
to
million years, but estimates between
years and million years can be found in the
literature. Cited on page .
249 L. V
H , S.E. H
, Z. D
& C.H. B
, Light speed
reduction to meters per second in an ultracold atomic gas, Nature 397, pp. – , .
See also Ref. . Cited on page .
250 e method of explaining special relativity by drawing a few lines on paper is due to H -
B , Relativity and Common Sense: A New Approach to Einstein, Dover, New
York, . See also D
-E
L
, Relativitätstheorie mit Zirkel und
Lineal, Akademie-Verlag Berlin, . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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251 R S. L , Experimental limits on the photon mass and cosmic vector potential, Phys-
ical Review Letters 80, pp. – , . e speed of light is independent of frequency
within a factor of ë − , as was shown from gamma ray studies by B.E. S
, Severe
limits on variations of the speed of light with frequency, Physical Review Letters 82, pp. –
, . Cited on page .
252 An overview of experimental results is given in Y and its Experimental Foundations, World Scienti c, , and .
Z
Z
, Special Relativity
. Cited on pages , , , ,
253 R.W. M G
& D.M. G
, New measurement of the relativistic Doppler shi
in neon, Physical Review Letters 70, pp. – , . Cited on page .
254 e present record for clock synchronization seems to be ps for two clocks distant km
from each other. See A. V
, G. S
& Y. S , Distant clock synchron-
ization using entangled photon pairs, Applied Physics Letters 85, pp. – , , or
http://www.arxiv.org/abs/quant-ph/
. Cited on page .
255 J. F
& T. K
, Über die eorie der plastischen Verformung, Physikalis-
che Zeitschri der Sowietunion 13, p. , . F.C. F
, On the equations of motion of
crystal dislocations, Proceedings of the Physical Society A 62, pp. – , . J. E
,
Uniformly moving dislocations, Proceedings of the Physical Society A 62, pp. – , .
See also G. L
& H. D
, Zeitschri für Physik 126, p. , . A general in-
troduction can be found in A. S
& P. S
, Kinks in dislocation lines and their
e ects in internal friction in crystals, Physical Acoustics 3A, W.P. M , ed., Academic
Press, . See also the textbooks by F
R.N. N
, eory of Crystal Dislo-
cations, Oxford University Press, , or J.P. H
& J. L
, eory of Dislocations,
McGraw Hill, . Cited on page .
256 is beautiful graph is taken from Z.G.T. G
, G.B. R
, M.R.
Y
, R. G
& J.J. M
, Relative velocity measurements of electrons
and gamma rays at GeV, Physical Review Letters 34, pp. – , . Cited on page
.
257 To nd out more about the best-known crackpots, and their ideas, send an email to majordomo@zikzak.net with the one-line body ‘subscribe psychoceramics’. Cited on page .
258 e speed of neutrinos is the same as that of light to decimal digits. is is explained
by L S
, e speed of light and the speed of neutrinos, Physics Letters B 201,
p. , . An observation of a small mass for the neutrino has been published by the Ja-
panese Super-Kamiokande collaboration, in Y. F
& al., Evidence for oscillation of
atmospheric neutrinos, Physical Review Letters 81, pp. – , . e newer results
published by the Canadian Sudbury Neutrino Observatory, as Q.R. A
& al., Direct
evidence for neutrino avor transformation from neutral-current interactions in the Sud-
bury Neutrino Observatory, Physical Review Letters 89, p.
, , also con rm that
neutrinos have a mass in the eV region. Cited on pages and .
259 B. R
& G. E
, Lorentz transformations directly from the speed of
light, American Journal of Physics 63, p. , . See also the comment by E. K
,
Comment on “Lorentz transformations directly from the speed of light,” by B. Rothenstein
and G. Eckstein, American Journal of Physics 65, p. , . Cited on page .
260 See e.g. the lectures by Lorentz at Caltech, published as H.A. L Modern Physics, edited by H. Bateman, Ginn and Company, page , .
, Problems of . Cited on page
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261 A.A. M
& E.W. M
, On the relative motion of the Earth and the lumini-
ferous ether, American Journal of Science (3rd series) 34, pp. – , . Michelson pub-
lished many other papers on the topic a er this one. Cited on page .
262 S. S
, P. A
& M. O
, A precision test of the isotropy of
the speed of light using rotating cryogenic resonators, http://www.arxiv.org/abs/physics/
. Cited on page .
263 H.A. L
, De relative beweging van de aarde en dem aether, Amst. Versl. 1, p. ,
, and also H.A. L
, Electromagnetic phenomena in a system moving with any
velocity smaller than that of light, Amst. Proc. 6, p. , , or Amst. Versl. 12, p. , .
Cited on page .
264 A general refutation of such proposals is discussed by S.R. M
& G.E. S -
, Accelerated clock principles, Physical Review A 47, pp. – , . Experiments
on muons at CERN in showed that accelerations of up to m s have no e ect, as
explained by D.H. P
, Introduction to High Energy Physics, Addison-Wesley, ,
or by J. B
& al., Il Nuovo Cimento 9A, p. , . Cited on page .
265 W. R
, General relativity before special relativity: an unconventional overview of
relativity theory, American Journal of Physics 62, pp. – , . Cited on page .
266 S
K. B , Would a topology change allow Ms. Bright to travel backward in time?,
American Journal of Physics 66, pp. – , . Cited on page .
267 On the ‘proper’ formulation of relativity, see for example D. H
, Proper particle
mechanics, Journal of Mathematical Physics 15, pp. – , . Cited on page .
268 e simple experiment to take a precise clock on a plane, y it around the world and then
compare it with an identical one le in place was rst performed by J.C. H
& R.E.
K
, Around-the-world atomic clocks: predicted relativistic time gains, Science 177,
pp. – , and Around-the-world atomic clocks: observed relativistic time gains, pp. –
, July . See also Ref. . Cited on page .
269 A readable introduction to the change of time with observers, and to relativity in general,
is R
U. S & H
KS
, Raum-Zeit-Relativität, . Au age,
Vieweg & Sohn, Braunschweig, . Cited on page .
270 Most famous is the result that moving muons stay younger, as shown for example by D.H.
F
& J.B. S
, Measurement of the relativistic time dilation using µ-mesons, Amer-
ican Journal of Physics 31, pp. – , . For a full pedagogical treatment of the twin
paradox, see E. S
, Relativistic twins or sextuplets?, European Journal of Physics 24,
pp. – , . Cited on page .
271 P J. N , Time Machines – Time Travel in Physics, Metaphysics and Science Fiction, Springer Verlag and AIP Press, second edition, . Cited on page .
272 e rst muon experiment was B. R & D.B. H mesotrons with momentum, Physical Review 59, pp. name for muon. Cited on page .
, Variation of the rate of decay of – , . ‘Mesotron’ was the old
273 A. H
& E. S
. Cited on page .
, A small puzzle from , Physics Today, pp. – , March
274 W. R
, Length contraction paradox, American Journal of Physics 29, pp. – ,
. For a variation without gravity, see R. S , Length contraction paradox, American
Journal of Physics 30, p. , . Cited on page .
275 H.
L
& C. G
, e rod and hole paradox re-examined, European Journal
of Physics 26, pp. – , . Cited on page .
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276 is situation is discussed by G.P. S
, Is length contraction paradoxical?, American
Journal of Physics 55, , pp. – . is paper also contains an extensive literature list
covering variants of length contraction paradoxes. Cited on page .
277 S.P. B
, e case of the identically accelerated twins, American Journal of Physics 57,
pp. – , . Cited on pages and .
278 J.M. S
, Relativistic buoyancy, American Journal of Physics 57 1, pp. – , January
. See also G.E.A. M
, Relativistic Arquimedes law for fast moving bodies and the
general-relativistic resolution of the ‘submarine paradox’, Physical Review D 68, p.
,
, or http://www.arxiv.org/abs/gr-qc/
. Cited on page .
279 e distinction was rst published by J. T
, Invisibility of Lorentz contraction, Phys-
ical Review 116, pp. – , , and R. P
, e apparent shape of a relativistic-
ally moving sphere, Proceedings of the Cambridge Philosophical Society 55, pp. – , .
Cited on page .
280 G.R. R
, Speed limit on walking, American Journal of Physics 59, pp. – , .
Cited on page .
281 e rst examples of such astronomical observations were provided by A.R. W
&
al., Quasars revisited: rapid time variations observed via very-long-baseline interferometry,
Science 173, pp. – , , and by M.H. C
& al., e small-scale structure of
radio galaxies and quasi-stellar sources at . centimetres, Astrophysical Journal 170, pp. –
, . See also T.J. P
, S.C. U
, M.H. C
, R.P. L
, A.C.S.
R
, G.A. S
, R.S. S
& R.C. W
, Superluminal expansion
of quasar C , Nature 290, pp. – , . An overview is given in J.A. Z
& T.J.
P
, editors, Superluminal radio sources, Cambridge University Press, . Another
measurement, using very long baseline interferometry with radio waves, was shown on the
cover of Nature: I.F. M
& L.F. R
, A superluminal source in the galaxy,
Nature 371, pp. – , . A more recent example was reported in Science News 152, p. ,
December .
Pedagogical explanations are given by D.C. G
, e use of quasars in teaching
introductory special relativity, American Journal of Physics 55, pp. – , , and by
E
F. T
&J
A. W
, Spacetime Physics – Introduction to Special
Relativity, second edition, Freeman, , pages - . is excellent book was mentioned
already in the text. Cited on page .
282 O.M. B
& E.C. S
, Particles beyond the light barrier, Physics Today
22, pp. – , , and O.M.P. B
, V.K. D
& E.C.G. S
,
‘Meta’ relativity, American Journal of Physics 30, pp. – , . See also E. R
,
editor, Tachyons, Monopoles and Related Topics, North-Holland, Amsterdam, . Cited
on page .
283 J.P. C
, B.H.J. M K
, A.A. R
& G.J. S
,e
omas rotation, American Journal of Physics 69, pp. – , . Cited on page .
284 See for example S.S. C
& G.E.A. M
able at http://www.arxiv.org/abs/gr-qc/
, Temperature and relativity, preprint avail. Cited on page .
285 R.C. T
& G.N. L , e principle of relativity and non-Newtonian mechanics,
Philosophical Magazine 18, pp. – , , and R.C. T
, Non-Newtonian mech-
anics: the mass of a moving body, Philosophical Magazine 23, pp. – , . Cited on
page .
286 is information is due to a private communication by Frank DiFilippo; part of the story
is given in F. D F
, V. N
, K.R. B
& D.E. P
, Accurate
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atomic masses for fundamental metrology, Physical Review Letters 73, pp. – , .
ese measurements were performed with Penning traps; a review of the possibilities they
o er is given by R.C. T
, Precision measurement aspects of ion traps, Measure-
ment Science and Technology 1, pp. – , . e most important experimenters in the
eld of single particle levitation were awarded the Nobel Prize in . One of the Nobel
Prize lectures can be found in W. P , Electromagnetic traps for neutral and charged
particles, Reviews of Modern Physics 62, pp. – , . Cited on page .
287 J.L. S
, Relativity: e Special eory, North-Holland, , pp. – . More about
antiparticles in special relativity can be found in J.P. C
, B.H.J. M K
&
A.A. R
, Classical antiparticles, American Journal of Physics 65, pp. – ,
. See also Ref. . Cited on page .
288 A. P
, Drehimpuls- und Schwerpunktsatz in der relativistischen Mechanik,
Praktika Acad. Athenes 14, p. , , and A. P
, Drehimpuls- und Schwer-
punktsatz in der Diracschen eorie, Praktika Acad. Athenes 15, p. , . See also
M.H.L. P , e mass-centre in the restricted theory of relativity and its connexion
with the quantum theory of elementary particles, Proceedings of the Royal Society in Lon-
don, A 195, pp. – , . Cited on page .
289 e references preceding Einstein’s E = mc are: S. T
P
, Physics of the Ether,
E. & F.N. Spon, , J.H. P
, La théorie de Lorentz et le principe de réaction,
Archives néerlandaises des sciences exactes et naturelles 5, pp. – , , O. D P
,
Ipotesi dell’etere nella vita dell’universo, Reale Istituto Veneto di Scienze, Lettere ed Arti tomo
LXIII, parte , pp. – , Febbraio , F. H
, Berichte der Wiener Akademie
113, p. , , F. H
, Zur eorie der Strahlung in bewegten Körpern, An-
nalen der Physik 15, pp. – , , F. H
, Zur eorie der Strahlung in be-
wegten Körpern – Berichtigung, Annalen der Physik 16, pp. – , . Hasenöhrl died
in , De Pretto in . All these publications were published before the famous paper
by A
E
, Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?,
Annalen der Physik 18, pp. – , . Cited on page .
290 A jewel among the textbooks on special relativity is the booklet by U
E. S
,
Spezielle Relativitätstheorie, Verlag Harri Deutsch, un, . Cited on pages and .
291 A readable article showing a photocopy of a letter by Einstein making this point is L B. O , e concept of mass, Physics Today, pp. – , June . e topic is not without controversy, as the letters by readers following that article show; they are found in Physics Today, pp. – and pp. – , May . e topic is still a source of debates. Cited on page .
292 C
M
, e eory of Relativity, Clarendon Press, , . is standard
text has been translated in several languages. Cited on page .
293 P. E
, Gleichförmige Rotation starrer Körper und Relativitätstheorie, Physikalis-
che Zeitschri 10, pp. – , . Ehrenfest (incorrectly) suggested that this meant that
relativity cannot be correct. A modern summary of the issue can be found in M.L. R -
, e relative space: space measurements on a rotating platform, http://www.arxiv.
org/abs/gr-qc/
. Cited on page .
294 R.J. L , When moving clocks run fast, European Journal of Physics 16, pp. – , . Cited on page .
295 G. S
& C.W. K
, Special Relativity for Physicists, Longmans, Lon-
don, . See also W.N. M
, Relativistic velocity and acceleration transforma-
tions from thought experiments, American Journal of Physics 73, pp. – , . Cited on
page .
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296 e impossibility of de ning rigid coordinate frames for non-uniformly accelerating observ-
ers is discussed by C
M
,K T
& J A. W
, Gravitation,
Freeman, p. , . Cited on page .
297 E.A. D
& R.J. P
, Uniformly accelerated reference frames in special re-
lativity, American Journal of Physics 55, pp. – , . Cited on page .
298 R.H. G , Uniformly accelerated reference frame and twin paradox, American Journal of Physics 50, pp. – , . Cited on pages , , and .
299 J. D
H
, e uniformly accelerated reference frame, American Journal of
Physics 46, pp. – , . Cited on page .
300 e best and cheapest mathematical formula collection remains the one by K. R
,
Mathematische Formelsammlung, BI Hochschultaschenbücher, . Cited on page .
301 C.G. A
& R.W. B
, Relativistic solutions to a falling body in a uniform gravit-
ation eld, American Journal of Physics 59, pp. – , . Cited on page .
302 See for example the excellent lecture notes by D.J. R
, A radically modern ap-
proach to freshman physics, on the http://www.physics.nmt.edu/~raymond/teaching.html
website. Cited on pages and .
303 L. M
, e relativistic acceleration addition theorem, Classical and Quantum Gravity
11, pp. L –L , . Cited on page .
304 E
A. D
, e gravitational red-shi in a uniform eld, American Journal of
Physics 58, pp. – , . Cited on page .
305 One of the latest of these debatable experiments is T.P. K
,L. M
, G.F. L ,
L.E. P
, R.T. L
, J.D. A
& C.M. W , Test of the isotropy of the
one-way speed of light using hydrogen-maser frequency standards, Physical Review D 42,
pp. – , . Cited on page .
306 E
F. T
& A.P. F
, Limitation on proper length in special relativity,
American Journal of Physics 51, pp. – , . Cited on page .
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C
III
GR AVITATION AND REL ATIVIT Y
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G relativity is easy. Nowadays, it can be made as intuitive as universal ravity and its inverse square law – by using the right approach. e main ideas of eneral relativity, like those of special relativity, are accessible to secondary-school students. Black holes, gravitational waves, space-time curvature and the limits of the universe can then be understood as easily as the Doppler e ect or the twins paradox.
We will discover that, just as special relativity is based on a maximum speed c, general relativity is based on a maximum force c G or on a maximum power c G. We rst show that all known experimental data are consistent with these limits. In fact, we nd that the maximum force and the maximum power are achieved only on insurmountable limit surfaces; these limit surfaces are called horizons. We will then be able to deduce the
eld equations of general relativity. In particular, the existence of a maximum for force or power implies that space-time is curved. It explains why the sky is dark at night, and it shows that the universe is of nite size.
We also discuss the main counter-arguments and paradoxes arising from the limits. e resolutions of the paradoxes clarify why the limits have remained dormant for so long, both in experiments and in teaching. A er this introduction, we will study the e ects of relativistic gravity in more detail. In particular, we will study the consequences of space-time curvature for the motions of bodies and of light in our everyday environment. For example, the inverse square law will be modi ed. (Can you explain why this is necessary in view of what we have learned so Challenge 679 n far?) Most fascinating of all, we will discover how to move and bend the vacuum. en we will study the universe at large; nally, we will explore the most extreme form of gravity: black holes.
.
–
We just saw that the theory of special relativity appears when we recognize the speed limit c in nature and take this limit as a basic principle. At the end of the twentieth century it was shown that general relativity can be approached by using a similar basic principle:*
Ref. 307, Ref. 308 Ref. 309
* is principle was published in the year 2000 in this text, and independently in a conference proceedings in 2002 by Gary Gibbons. e present author discovered the maximum force in 1998 when searching for a way to derive the results of chapter XI that would be so simple that it would convince even a secondary-school student.
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•.
ere is in nature a maximum force:
F
c G
=
.
ë
N.
(202)
In nature, no force in any muscle, machine or system can exceed this value. For the curious, the value of the force limit is the energy of a (Schwarzschild) black
hole divided by twice its radius. e force limit can be understood intuitively by noting that (Schwarzschild) black holes are the densest bodies possible for a given mass. Since there is a limit to how much a body can be compressed, forces – whether gravitational, electric, centripetal or of any other type – cannot be arbitrary large.
Alternatively, it is possible to use another, equivalent statement as a basic principle:
ere is a maximum power in nature:
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P
c G
=
.ë
W.
(203)
No power of any lamp, engine or explosion can exceed this value. e maximum power is realized when a (Schwarzschild) black hole is radiated away in the time that light takes to travel along a length corresponding to its diameter. We will see below precisely what black holes are and why they are connected to these limits.
e existence of a maximum force or power implies the full theory of general relativity. In order to prove the correctness and usefulness of this approach, a sequence of arguments is required. e sequence is the same as for the establishment of the limit speed in special relativity. First of all, we have to gather all observational evidence for the claimed limit. Secondly, in order to establish the limit as a principle of nature, we have to show that general relativity follows from it. Finally, we have to show that the limit applies in all possible and imaginable situations. Any apparent paradoxes will need to be resolved.
ese three steps structure this introduction to general relativity. We start the story by explaining the origin of the idea of a limiting value.
T
In the nineteenth and twentieth centuries many physicists took pains to avoid the concept of force. Heinrich Hertz made this a guiding principle of his work, and wrote an in uential textbook on classical mechanics without ever using the concept. e fathers of quantum theory, who all knew this text, then dropped the term ‘force’ completely from the vocabulary of microscopic physics. Meanwhile, the concept of ‘gravitational force’ was eliminated from general relativity by reducing it to a ‘pseudo-force’. Force fell out of fashion.
Nevertheless, the maximum force principle does make sense, provided that we visualize it by means of the useful de nition: force is the ow of momentum per unit time. Momentum cannot be created or destroyed. We use the term ‘ ow’ to remind us that momentum, being a conserved quantity, can only change by in ow or out ow. In other words, change of momentum always takes place through some boundary surface. is fact is of central importance. Whenever we think about force at a point, we mean the
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Ref. 308 Page 1068
Page 368
momentum ‘ owing’ through a surface at that point. General relativity states this idea usually as follows: force keeps bodies from following geodesics. e mechanism underlying a measured force is not important. In order to have a concrete example to guide the discussion it can be helpful to imagine force as electromagnetic in origin. In fact, any type of force is possible.
e maximum force principle thus boils down to the following: if we imagine any physical surface (and cover it with observers), the integral of momentum ow through the surface (measured by all those observers) never exceeds a certain value. It does not matter how the surface is chosen, as long as it is physical, i.e., as long as we can x observers* onto it.
is principle imposes a limit on muscles, the e ect of hammers, the ow of material, the acceleration of massive bodies, and much more. No system can create, measure or experience a force above the limit. No particle, no galaxy and no bulldozer can exceed it.
e existence of a force limit has an appealing consequence. In nature, forces can be measured. Every measurement is a comparison with a standard. e force limit provides a natural unit of force which ts into the system of natural units** that Max Planck derived from c, G and h (or ħ). e maximum force thus provides a standard of force valid in every place and at every instant of time.
e limit value of c G di ers from Planck’s proposed unit in two ways. First, the numerical factor is di erent (Planck had in mind the value c G). Secondly, the force unit is a limiting value. In the this respect, the maximum force plays the same role as the maximum speed. As we will see later on, this limit property is valid for all other Planck units as well, once the numerical factors have been properly corrected. e factor / has no deeper meaning: it is just the value that leads to the correct form of the eld equations of general relativity. e factor / in the limit is also required to recover, in everyday situations, the inverse square law of universal gravitation. When the factor is properly taken into account, the maximum force (or power) is simply given by the (corrected) Planck energy divided by the (corrected) Planck length or Planck time.
e expression for the maximum force involves the speed of light c and the gravitational constant G; it thus quali es as a statement on relativistic gravitation. e fundamental principle of special relativity states that speed v obeys v c for all observers. Analogously, the basic principle of general relativity states that in all cases force F and power P obey F c G and P c G. It does not matter whether the observer measures the force or power while moving with high velocity relative to the system under observation, during free fall, or while being strongly accelerated. However, we will see that it is essential that the observer records values measured at his own location and that the observer is realistic, i.e., made of matter and not separated from the system by a horizon. ese conditions are the same that must be obeyed by observers measuring velocity in special relativity.
Since physical power is force times speed, and since nature provides a speed limit, the force bound and the power bound are equivalent. We have already seen that force
Page 706
* Observers in general relativity, like in special relativity, are massive physical systems that are small enough so that their in uence on the system under observation is negligible. ** When Planck discovered the quantum of action, he had also noticed the possibility to de ne natural units. On a walk with his seven-year-old son in the forest around Berlin, he told him that he had made a discovery as important as the discovery of universal gravity.
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•.
Page 323 and power appear together in the de nition of -force; we can thus say that the upper bound is valid for every component of a force, as well as for its magnitude. e power bound limits the output of car and motorcycle engines, lamps, lasers, stars, gravitational radiation sources and galaxies. It is equivalent to . ë horsepowers. e maximum power principle states that there is no way to move or get rid of energy more quickly than that. e power limit can be understood intuitively by noting that every engine produces exhausts, i.e. some matter or energy that is le behind. For a lamp, a star or an evaporating black hole, the exhausts are the emitted radiation; for a car or jet engine they are hot gases; for a water turbine the exhaust is the slowly moving water leaving the turbine; for a rocket it is the matter ejected at its back end; for a photon rocket or an electric motor it is electromagnetic energy. Whenever the power of an engine gets close to the limit value, the exhausts increase dramatically in mass–energy. For extremely high exhaust masses, the gravitational attraction from these exhausts – even if they are only radiation – prevents further acceleration of the engine with respect to them. e maximum power principle thus expresses that there is a built-in braking mechanism in nature; this braking mechanism is gravity. Yet another, equivalent limit appears when the maximum power is divided by c .
ere is a maximum rate of mass change in nature:
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dm dt
c G
=
.
ë
kg s .
(204)
is bound imposes a limit on pumps, jet engines and fast eaters. Indeed, the rate of ow of water or any other material through tubes is limited. e mass ow limit is obviously equivalent to either the force or the power limit.
e claim of a maximum force, power or mass change in nature seems almost too fantastic to be true. Our rst task is therefore to check it empirically as thoroughly as we can.
T
Like the maximum speed principle, the maximum force principle must rst of all be checked experimentally. Michelson spent a large part of his research life looking for possible changes in the value of the speed of light. No one has yet dedicated so much e ort to testing the maximum force or power. However, it is straightforward to con rm that no experiment, whether microscopic, macroscopic or astronomical, has ever measured force values larger than the stated limit. Many people have claimed to have produced speeds larger than that of light. So far, nobody has ever claimed to have produced a force larger than the limit value.
e large accelerations that particles undergo in collisions inside the Sun, in the most powerful accelerators or in reactions due to cosmic rays correspond to force values much smaller than the force limit. e same is true for neutrons in neutron stars, for quarks inside protons, and for all matter that has been observed to fall towards black holes. Furthermore, the search for space-time singularities, which would allow forces to achieve or exceed the force limit, has been fruitless.
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Page 366 Challenge 680 n
Ref. 310
Page 371
In the astronomical domain, all forces between stars or galaxies are below the limit value, as are the forces in their interior. Not even the interactions between any two halves of the universe exceed the limit, whatever physically sensible division between the two halves is taken. ( e meaning of ‘physically sensible division’ will be de ned below; for divisions that are not sensible, exceptions to the maximum force claim can be constructed. You might enjoy searching for such an exception.)
Astronomers have also failed to nd any region of space-time whose curvature (a concept to be introduced below) is large enough to allow forces to exceed the force limit. Indeed, none of the numerous recent observations of black holes has brought to light forces larger than the limit value or objects smaller than the corresponding black hole radii. Observations have also failed to nd a situation that would allow a rapid observer to observe a force value that exceeds the limit due to the relativistic boost factor.
e power limit can also be checked experimentally. It turns out that the power – or luminosity – of stars, quasars, binary pulsars, gamma ray bursters, galaxies or galaxy clusters can indeed be close to the power limit. However, no violation of the limit has ever been observed. Even the sum of all light output from all stars in the universe does not exceed the limit. Similarly, even the brightest sources of gravitational waves, merging black holes, do not exceed the power limit. Only the brightness of evaporating black holes in their nal phase could equal the limit. But so far, none has ever been observed.
Similarly, all observed mass ow rates are orders of magnitude below the corresponding limit. Even physical systems that are mathematical analogues of black holes – for example, silent acoustical black holes or optical black holes – do not invalidate the force and power limits that hold in the corresponding systems.
e experimental situation is somewhat disappointing. Experiments do not contradict the limit values. But neither do the data do much to con rm them. e reason is the lack of horizons in everyday life and in experimentally accessible systems. e maximum speed at the basis of special relativity is found almost everywhere; maximum force and maximum power are found almost nowhere. Below we will propose some dedicated tests of the limits that could be performed in the future.
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D
*
Page 357
In order to establish the maximum force and power limits as fundamental physical principles, it is not su cient to show that they are consistent with what we observe in nature. It is necessary to show that they imply the complete theory of general relativity. ( is section is only for readers who already know the eld equations of general relativity. Other readers may skip to the next section.)
In order to derive the theory of relativity we need to study those systems that realize the limit under scrutiny. In the case of the special theory of relativity, the main system that realizes the limit speed is light. For this reason, light is central to the exploration of special relativity. In the case of general relativity, the systems that realize the limit are less obvious. We note rst that a maximum force (or power) cannot be realized throughout a volume of space. If this were possible, a simple boost** could transform the force (or
* is section can be skipped at rst reading. ( e mentioned proof dates from December 2003.) ** A boost was de ned in special relativity as a change of viewpoint to a second observer moving in relation to the rst.
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Ref. 308 Page 336
power) to a higher value. erefore, nature can realize maximum force and power only on surfaces, not volumes. In addition, these surfaces must be unattainable. ese unattainable surfaces are basic to general relativity; they are called horizons. Maximum force and power only appear on horizons. We have encountered horizons in special relativity, where they were de ned as surfaces that impose limits to observation. (Note the contrast with everyday life, where a horizon is only a line, not a surface.) e present de nition of a horizon as a surface of maximum force (or power) is equivalent to the de nition as a surface beyond which no signal may be received. In both cases, a horizon is a surface beyond which interaction is impossible.
e connection between horizons and the maximum force is a central point of relativistic gravity. It is as important as the connection between light and the maximum speed in special relativity. In special relativity, we showed that the fact that light speed is the maximum speed in nature implies the Lorentz transformations. In general relativity, we will now prove that the maximum force in nature, which we can call the horizon force, implies the eld equations of general relativity. To achieve this aim, we start with the realization that all horizons have an energy ow across them. e ow depends on the horizon curvature, as we will see. is connection implies that horizons cannot be planes, as an in nitely extended plane would imply an in nite energy ow.
e simplest nite horizon is a static sphere, corresponding to a Schwarzschild black hole. A spherical horizon is characterized by its radius of curvature R, or equivalently, by its surface gravity a; the two quantities are related by aR = c . Now, the energy owing through any horizon is always nite in extension, when measured along the propagation direction. One can thus speak more speci cally of an energy pulse. Any energy pulse through a horizon is thus characterized by an energy E and a proper length L. When the energy pulse ows perpendicularly through a horizon, the rate of momentum change, or force, for an observer at the horizon is
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F
=
E L
.
(205)
Our goal is to show that the existence of a maximum force implies general relativity. Now, maximum force is realized on horizons. We thus need to insert the maximum possible values on both sides of equation ( ) and to show that general relativity follows.
Using the maximum force value and the area πR for a spherical horizon we get
c G
=
E LA
πR
.
(206)
Ref. 311
e fraction E A is the energy per area owing through any area A that is part of a horizon. e insertion of the maximum values is complete when one notes that the length L of the energy pulse is limited by the radius R. e limit L R follows from geometrical considerations: seen from the concave side of the horizon, the pulse must be shorter than the radius of curvature. An independent argument is the following. e length L of an object accelerated by a is limited, by special relativity, by L c a. Special relativity already shows that this limit is related to the appearance of a horizon. Together with relation ( ), the statement that horizons are surfaces of maximum force leads to the
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following important relation for static, spherical horizons:
E=
c πG
a
A
.
(207)
Ref. 312
is horizon equation relates the energy ow E through an area A of a spherical horizon with surface gravity a. It states that the energy owing through a horizon is limited, that this energy is proportional to the area of the horizon, and that the energy ow is proportional to the surface gravity. ( e horizon equation is also called the rst law of black hole mechanics or the rst law of horizon mechanics.)
e above derivation also yields the intermediate result
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E
c πG
A L
.
(208)
is form of the horizon equation states more clearly that no surface other than a horizon can achieve the maximum energy ow, when the area and pulse length (or surface gravity) are given. No other domain of physics makes comparable statements: they are intrinsic to the theory of gravitation.
An alternative derivation of the horizon equation starts with the emphasis on power instead of on force, using P = E T as the initial equation.
It is important to stress that the horizon equations ( ) and ( ) follow from only two assumptions: rst, there is a maximum speed in nature, and secondly, there is a maximum force (or power) in nature. No speci c theory of gravitation is assumed. e horizon equation might even be testable experimentally, as argued below. (We also note that the horizon equation – or, equivalently, the force or power limit – implies a maximum mass change rate in nature given by dm dt c G.)
Next, we have to generalize the horizon equation from static and spherical horizons to general horizons. Since the maximum force is assumed to be valid for all observers, whether inertial or accelerating, the generalization is straightforward. For a horizon that is irregularly curved or time-varying the horizon equation becomes
δE =
c πG
a
δA
.
(209)
Ref. 313
is di erential relation – it might be called the general horizon equation – is valid for any horizon. It can be applied separately for every piece δA of a dynamic or spatially changing horizon. e general horizon equation ( ) has been known to be equivalent to general relativity at least since , when this equivalence was (implicitly) shown by Jacobson. We will show that the di erential horizon equation has the same role for general relativity as the equation dx = c dt has for special relativity. From now on, when we speak of the horizon equation, we mean the general, di erential form ( ) of the relation.
It is instructive to restate the behaviour of energy pulses of length L in a way that holds
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•.
for any surface, even one that is not a horizon. Repeating the above derivation, one gets
δE δA
c πG
L
.
(210)
Ref. 313
Equality is only realized when the surface A is a horizon. In other words, whenever the
value δE δA in a physical system approaches the right-hand side, a horizon starts to form.
is connection will be essential in our discussion of apparent counter-examples to the
limit principles.
If one keeps in mind that on a horizon the pulse length L obeys L c a, it becomes
clear that the general horizon equation is a consequence of the maximum force c G
or the maximum power c G. In addition, the horizon equation takes also into account
maximum speed, which is at the origin of the relation L c a. e horizon equation
thus follows purely from these two limits of nature.
e remaining part of the argument is simply the derivation of general relativity from
the general horizon equation. is derivation was implicitly provided by Jacobson, and
the essential steps are given in the following paragraphs. (Jacobson did not stress that his
derivation was valid also for continuous space-time, or that his argument could also be
used in classical general relativity.) To see the connection between the general horizon
equation ( ) and the eld equations, one only needs to generalize the general horizon
equation to general coordinate systems and to general directions of energy–momentum
ow. is is achieved by introducing tensor notation that is adapted to curved space-time.
To generalize the general horizon equation, one introduces the general surface element
dΣ and the local boost Killing vector eld k that generates the horizon (with suitable
norm). Jacobson uses these two quantities to rewrite the le -hand side of the general
horizon equation ( ) as
∫ δE = Tab kadΣb ,
(211)
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where Tab is the energy–momentum tensor. is expression obviously gives the energy at the horizon for arbitrary coordinate systems and arbitrary energy ow directions.
Jacobson’s main result is that the factor a δA in the right hand side of the general hori-
zon equation ( ) can be rewritten, making use of the (purely geometric) Raychaudhuri
equation, as
∫ a δA = c Rab kadΣb ,
(212)
where Rab is the Ricci tensor describing space-time curvature. is relation describes how the local properties of the horizon depend on the local curvature.
Combining these two steps, the general horizon equation ( ) becomes
∫ ∫ Tab kadΣb =
c πG
Rab kadΣb .
(213)
Jacobson then shows that this equation, together with local conservation of energy (i.e.,
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vanishing divergence of the energy–momentum tensor) can only be satis ed if
Tab =
c πG
Rab − ( R + Λ) ab
,
(214)
Page 478
where R is the Ricci scalar and Λ is a constant of integration the value of which is not determined by the problem. e above equations are the full eld equations of general relativity, including the cosmological constant Λ. e eld equations thus follow from the horizon equation. ey are therefore shown to be valid at horizons.
Since it is possible, by choosing a suitable coordinate transformation, to position a horizon at any desired space-time point, the eld equations must be valid over the whole of space-time. is observation completes Jacobson’s argument. Since the eld equations follow, via the horizon equation, from the maximum force principle, we have also shown that at every space-time point in nature the same maximum force holds: the value of the maximum force is an invariant and a constant of nature.
In other words, the eld equations of general relativity are a direct consequence of the limit on energy ow at horizons, which in turn is due to the existence of a maximum force (or power). In fact, as Jacobson showed, the argument works in both directions. Maximum force (or power), the horizon equation, and general relativity are equivalent.
In short, the maximum force principle is a simple way to state that, on horizons, energy ow is proportional to area and surface gravity. is connection makes it possible to deduce the full theory of general relativity. In particular, a maximum force value is su cient to tell space-time how to curve. We will explore the details of this relation shortly. Note that if no force limit existed in nature, it would be possible to ‘pump’ any desired amount of energy through a given surface, including any horizon. In this case, the energy ow would not be proportional to area, horizons would not have the properties they have, and general relativity would not hold. We thus get an idea how the maximum ow of energy, the maximum ow of momentum and the maximum ow of mass are all connected to horizons. e connection is most obvious for black holes, where the energy, momentum or mass are those falling into the black hole.
By the way, since the derivation of general relativity from the maximum force principle or from the maximum power principle is now established, we can rightly call these limits horizon force and horizon power. Every experimental or theoretical con rmation of the
eld equations indirectly con rms their existence.
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S-
Challenge 681 n
Imagine two observers who start moving parallel to each other and who continue straight ahead. If a er a while they discover that they are not moving parallel to each other any more, then they can deduce that they have moved on a curved surface (try it!) or in a curved space. In particular, this happens near a horizon. e derivation above showed that a nite maximum force implies that all horizons are curved; the curvature of horizons in turn implies the curvature of space-time. If nature had only at horizons, there would be no space-time curvature. e existence of a maximum force implies that space-time is curved.
A horizon so strongly curved that it forms a closed boundary, like the surface of a
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sphere, is called a black hole. We will study black holes in detail below. e main property
of a black hole, like that of any horizon, is that it is impossible to detect what is ‘behind’
the boundary.*
e analogy between special and general relativity can thus be carried further. In spe-
cial relativity, maximum speed implies dx = c dt, and the change of time depends on the
observer. In general relativity, maximum force (or power) implies the horizon equation
δE
=
ecπmG aaxiδmAuamndfothrceeo(bosreprvoawtieorn)
that thus
space-time is has the same
curved. double role
in
general
relativity
as
the maximum speed has in special relativity. In special relativity, the speed of light is the
maximum speed; it is also the proportionality constant that connects space and time, as
the equation dx = c dt makes apparent. In general relativity, the horizon force is the max-
imum force; it also appears (with a factor π) in the eld equations as the proportionality
constant connecting energy and curvature. e maximum force thus describes both the
elasticity of space-time and – if we use the simple image of space-time as a medium – the
maximum tension to which space-time can be subjected. is double role of a material
constant as proportionality factor and as limit value is well known in materials science.
Does this analogy make you think about aether? Do not worry: physics has no need
for the concept of aether, because it is indistinguishable from vacuum. General relativity
does describe the vacuum as a sort of material that can be deformed and move.
Why is the maximum force also the proportionality factor between curvature and en-
ergy? Imagine space-time as an elastic material. e elasticity of a material is described
by a numerical material constant. e simplest de nition of this material constant is the
ratio of stress (force per area) to strain (the proportional change of length). An exact
de nition has to take into account the geometry of the situation. For example, the shear
modulus G (or µ) describes how di cult it is to move two parallel surfaces of a material
against each other. If the force F is needed to move two parallel surfaces of area A and
length l against each other by a distance ∆l, one de nes the shear modulus G by
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F A
=
G
∆l l
.
(215)
e shear modulus for metals and alloys ranges between and GPa. e continuum
theory of solids shows that for any crystalline solid without any defect (a ‘perfect’ solid)
there is a so-called theoretical shear stress: when stresses higher than this value are ap-
plied, the material breaks. e theoretical shear stress, in other words, the maximum
stress in a material, is given by
Gtss =
G π
.
(216)
Ref. 314
e maximum stress is thus essentially given by the shear modulus. is connection is similar to the one we found for the vacuum. Indeed, imagining the vacuum as a material that can be bent is a helpful way to understand general relativity. We will use it regularly in the following.
What happens when the vacuum is stressed with the maximum force? Is it also torn apart like a solid? Yes: in fact, when vacuum is torn apart, particles appear. We will nd
* Analogously, in special relativity it is impossible to detect what moves faster than the light barrier.
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out more about this connection later on: since particles are quantum entities, we need to study quantum theory rst, before we can describe the e ect in the last part of our mountain ascent.
C
Page 306 Ref. 315
Challenge 682 ny
e maximum force value is valid only under certain assumptions. To clarify this point, we can compare it to the maximum speed. e speed of light (in vacuum) is an upper limit for motion of systems with momentum or energy only. It can, however, be exceeded for motions of non-material points. Indeed, the cutting point of a pair of scissors, a laser light spot on the Moon, or the group velocity or phase velocity of wave groups can exceed the speed of light. In addition, the speed of light is a limit only if measured near the moving mass or energy: the Moon moves faster than light if one turns around one’s axis in a second; distant points in a Friedmann universe move apart from each other with speeds larger than the speed of light. Finally, the observer must be realistic: the observer must be made of matter and energy, and thus move more slowly than light, and must be able to observe the system. No system moving at or above the speed of light can be an observer.
e same three conditions apply in general relativity. In particular, relativistic gravity forbids point-like observers and test masses: they are not realistic. Surfaces moving faster than light are also not realistic. In such cases, counter-examples to the maximum force claim can be found. Try and nd one – many are possible, and all are fascinating. We will explore some of the most important ones.
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G
“Wenn eine Idee am Horizonte eben aufgeht, ist gewöhnlich die Temperatur der Seele dabei sehr kalt. Erst allmählich entwickelt die Idee ihre Wärme, und am heissesten ist diese (das heisst sie tut ihre grössten Wirkungen), wenn der Glaube an die Idee schon wieder im Sinken ist. ” Friedrich Nietzsche*
e last, but central, step in our discussion of the force limit is the same as in the discussion of the speed limit. We need to show that any imaginable experiment – not only any real one – satis es the hypothesis. Following a tradition dating back to the early twentieth century, such an imagined experiment is called a Gedanken experiment, from the German Gedankenexperiment, meaning ‘thought experiment’.
In order to dismiss all imaginable attempts to exceed the maximum speed, it is sufcient to study the properties of velocity addition and the divergence of kinetic energy near the speed of light. In the case of maximum force, the task is much more involved. Indeed, stating a maximum force, a maximum power and a maximum mass change easily provokes numerous attempts to contradict them. We will now discuss some of these.
* ‘When an idea is just rising on the horizon, the soul’s temperature with respect to it is usually very cold. Only gradually does the idea develop its warmth, and it is hottest (which is to say, exerting its greatest in-
uence) when belief in the idea is already once again in decline.’ Friedrich Nietzsche (1844–1900), German philosopher and scholar. is is aphorism 207 – Sonnenbahn der Idee – from his text Menschliches Allzumenschliches – Der Wanderer und sein Schatten.
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•.
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**
e brute force approach. e simplest attempt to exceed the force limit is to try to ac-
celerate an object with a force larger than the maximum value. Now, acceleration implies
the transfer of energy. is transfer is limited by the horizon equation (209) or the limit
(210). For any attempt to exceed the force limit, the owing energy results in the appear-
ance of a horizon. But a horizon prevents the force from exceeding the limit, because it
imposes a limit on interaction.
We can explore this limit directly. In special relativity we found that the acceleration
of an object is limited by its length. Indeed, at a distance given by c a in the direction
opposite to the acceleration a, a horizon appears. In other words, an accelerated body
breaks, at the latest, at that point. e force F on a body of mass M and radius R is thus
limited by
F
M R
c
.
(217)
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It is straightforward to add the (usually small) e ects of gravity. To be observable, an accelerated body must remain larger than a black hole; inserting the corresponding radius R = GM c we get the force limit (202). Dynamic attempts to exceed the force limit thus
fail.
**
e rope attempt. We can also try to generate a higher force in a static situation, for example by pulling two ends of a rope in opposite directions. We assume for simplicity that an unbreakable rope exists. To produce a force exceeding the limit value, we need to store large (elastic) energy in the rope. is energy must enter from the ends. When we increase the tension in the rope to higher and higher values, more and more (elastic) energy must be stored in smaller and smaller distances. To exceed the force limit, we would need to add more energy per distance and area than is allowed by the horizon equation. A horizon thus inevitably appears. But there is no way to stretch a rope across a horizon, even if it is unbreakable. A horizon leads either to the breaking of the rope or to its detachment from the pulling system. Horizons thus make it impossible to generate forces larger than the force limit. In fact, the assumption of in nite wire strength is unnecessary: the force limit cannot be exceeded even if the strength of the wire is only nite.
We note that it is not important whether an applied force pulls – as for ropes or wires – or pushes. In the case of pushing two objects against each other, an attempt to increase the force value without end will equally lead to the formation of a horizon, due to the limit provided by the horizon equation. By de nition, this happens precisely at the force limit. As there is no way to use a horizon to push (or pull) on something, the attempt to achieve a higher force ends once a horizon is formed. Static forces cannot exceed the limit value.
Page 339
**
e braking attempt. A force limit provides a maximum momentum change per time. We can thus search for a way to stop a moving physical system so abruptly that the maximum force might be exceeded. e non-existence of rigid bodies in nature, already known from special relativity, makes a completely sudden stop impossible; but special relativity on its
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own provides no lower limit to the stopping time. However, the inclusion of gravity does. Stopping a moving system implies a transfer of energy. e energy ow per area cannot exceed the value given by the horizon equation. us one cannot exceed the force limit by stopping an object.
Similarly, if a rapid system is re ected instead of stopped, a certain amount of energy needs to be transferred and stored for a short time. For example, when a tennis ball is re ected from a large wall its momentum changes and a force is applied. If many such balls are re ected at the same time, surely a force larger than the limit can be realized? It turns out that this is impossible. If one attempted it, the energy ow at the wall would reach the limit given by the horizon equation and thus create a horizon. In that case, no re ection is possible any more. So the limit cannot be exceeded.
**
e classical radiation attempt. Instead of systems that pull, push, stop or re ect matter, we can explore systems where radiation is involved. However, the arguments hold in exactly the same way, whether photons, gravitons or other particles are involved. In particular, mirrors, like walls, are limited in their capabilities.
It is also impossible to create a force larger than the maximum force by concentrating a large amount of light onto a surface. e same situation as for tennis balls arises: when the limit value E A given by the horizon equation (210) is reached, a horizon appears that prevents the limit from being broken.
**
e brick attempt. e force and power limits can also be tested with more concrete Gedanken experiments. We can try to exceed the force limit by stacking weight. But even building an in nitely high brick tower does not generate a su ciently strong force on its foundations: integrating the weight, taking into account its decrease with height, yields a nite value that cannot reach the force limit. If we continually increase the mass density of the bricks, we need to take into account that the tower and the Earth will change into a black hole. And black holes, as mentioned above, do not allow the force limit to be exceeded.
Ref. 316
**
e boost attempt. A boost can apparently be chosen in such a way that a force value F in one frame is transformed into any desired value F′ in another frame. However, this result is not physical. To be more concrete, imagine a massive observer, measuring the value F, at rest with respect to a large mass, and a second observer moving towards the charged mass with relativistic speed, measuring the value F′. Both observers can be thought as being as small as desired. If one transforms the force eld at rest F applying the Lorentz transformations, the force F′ for the moving observer can reach extremely high values, as long as the speed is high enough. However, a force must be measured by an observer located at the speci c point. One has thus to check what happens when the rapid observer moves towards the region where the force is supposed to exceed the force limit. Suppose the observer has a mass m and a radius r. To be an observer, it must be larger than a black hole; in other words, its radius must obey r Gm c , implying that the observer has a non-vanishing size. When the observer dives into the force eld surrounding
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•.
the sphere, there will be an energy ow E towards the observer determined by the transformed eld value and the crossing area of the observer. is interaction energy can be made as small as desired, by choosing a su ciently small observer, but the energy is never zero. When the moving observer approaches the large massive charge, the interaction energy increases. Before the observer arrives at the point where the force was supposed to be higher than the force limit, the interaction energy will reach the horizon limits (209) or (210) for the observer. erefore, a horizon appears and the moving observer is prevented from observing anything at all, in particular any value above the horizon force.
e same limitation appears when electrical or other interactions are studied using a test observer that is charged. In summary, boosts cannot beat the force limit.
**
e divergence attempt. e force on a test mass m at a radial distance d from a SchwarzRef. 310 schild black hole (for Λ = ) is given by
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F = GMm .
d
−
GM dc
(218)
In addition, the inverse square law of universal gravitation states that the force between
two masses m and M is
F
=
GMm d
.
(219)
Both expressions can take any value; this suggest that no maximum force limit exists. A detailed investigation shows that the maximum force still holds. Indeed, the force
in the two situations diverges only for non-physical point-like masses. However, the maximum force implies a minimum approach distance to a mass m given by
dmin =
Gm c
.
(220)
e minimum approach distance – in simple terms, this would be the corresponding black hole radius – makes it impossible to achieve zero distance between two masses or between a horizon and a mass. is implies that a mass can never be point-like, and that there is a (real) minimum approach distance, proportional to the mass. If this minimum approach distance is introduced in equations (218) and (219), one gets
F
=
c G
Mm (M + m)
c
−
M M+m
G
(221)
and
F=
c G
Mm (M + m)
c G
.
(222)
e maximum force value is thus never exceeded, as long as we take into account the size
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of observers.
**
e consistency problem. If observers cannot be point-like, one might question whether it is still correct to apply the original de nition of momentum change or energy change as the integral of values measured by observers attached to a given surface. In general relativity, observers cannot be point-like, but they can be as small as desired. e original de nition thus remains applicable when taken as a limit procedure for ever-decreasing observer size. Obviously, if quantum theory is taken into account, this limit procedure comes to an end at the Planck length. is is not an issue for general relativity, as long as the typical dimensions in the situation are much larger than this value.
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Challenge 683 ny Ref. 308
**
e quantum problem. If quantum e ects are neglected, it is possible to construct surfaces with sharp angles or even fractal shapes that overcome the force limit. However, such surfaces are not physical, as they assume that lengths smaller than the Planck length can be realized or measured. e condition that a surface be physical implies that it must have an intrinsic uncertainty given by the Planck length. A detailed study shows that quantum e ects do not allow the horizon force to be exceeded.
**
e relativistically extreme observer attempt. Any extreme observer, whether in rapid inertial or in accelerated motion, has no chance to beat the limit. In classical physics we are used to thinking that the interaction necessary for a measurement can be made as small as desired. is statement, however, is not valid for all observers; in particular, extreme observers cannot ful l it. For them, the measurement interaction is large. As a result, a horizon forms that prevents the limit from being exceeded.
Ref. 317
**
e microscopic attempt. We can attempt to exceed the force limit by accelerating a small particle as strongly as possible or by colliding it with other particles. High forces do indeed appear when two high energy particles are smashed against each other. However, if the combined energy of the two particles became high enough to challenge the force limit, a horizon would appear before they could get su ciently close.
In fact, quantum theory gives exactly the same result. Quantum theory by itself already provides a limit to acceleration. For a particle of mass m it is given by
a
mc ħ
.
(223)
Page 1078
Here, ħ = . ë − Js is the quantum of action, a fundamental constant of nature. In particular, this acceleration limit is satis ed in particle accelerators, in particle collisions and in pair creation. For example, the spontaneous generation of electron–positron pairs in intense electromagnetic elds or near black hole horizons does respect the limit (223). Inserting the maximum possible mass for an elementary particle, namely the (corrected) Planck mass, we nd that equation (223) then states that the horizon force is the upper
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•.
bound for elementary particles.
Ref. 310
**
e compaction attempt. Are black holes really the most dense form of matter or energy? e study of black hole thermodynamics shows that mass concentrations with higher density than black holes would contradict the principles of thermodynamics. In black hole thermodynamics, surface and entropy are related: reversible processes that reduce entropy could be realized if physical systems could be compressed to smaller values than the black hole radius. As a result, the size of a black hole is the limit size for a mass in nature. Equivalently, the force limit cannot be exceeded in nature.
**
e force addition attempt. In special relativity, composing velocities by a simple vector addition is not possible. Similarly, in the case of forces such a naive sum is incorrect; any attempt to add forces in this way would generate a horizon. If textbooks on relativity had explored the behaviour of force vectors under addition with the same care with which they explored that of velocity vectors, the force bound would have appeared much earlier in the literature. (Obviously, general relativity is required for a proper treatment.)
** Challenge 684 r Can you propose and resolve another attempt to exceed the force or power limit?
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G
Like the force bound, the power bound must be valid for all imaginable systems. Here are some attempts to refute it.
**
e cable-car attempt. Imagine an engine that accelerates a mass with an unbreakable and massless wire (assuming that such a wire could exist). As soon as the engine reached the power bound, either the engine or the exhausts would reach the horizon equation. When a horizon appears, the engine cannot continue to pull the wire, as a wire, even an in nitely strong one, cannot pass a horizon. e power limit thus holds whether the engine is mounted inside the accelerating body or outside, at the end of the wire pulling it.
**
e mountain attempt. It is possible to de ne a surface that is so strangely bent that it passes just below every nucleus of every atom of a mountain, like the surface A in Figure 175. All atoms of the mountain above sea level are then just above the surface, barely touching it. In addition, imagine that this surface is moving upwards with almost the speed of light. It is not di cult to show that the mass ow through this surface is higher than the mass ow limit. Indeed, the mass ow limit c G has a value of about
kg s. In a time of − s, the diameter of a nucleus divided by the speed of light, only kg need to ow through the surface: that is the mass of a mountain.
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mountain
6000 m
nuclei
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surface A
0 m surface B
F I G U R E 175 The mountain attempt to exceed the maximum mass flow value
is surface seems to provide a counter-example to the limit. However, a closer look shows that this is not the case. e problem is the expression ‘just below’. Nuclei are quantum particles and have an indeterminacy in their position; this indeterminacy is essentially the nucleus–nucleus distance. As a result, in order to be sure that the surface of interest has all atoms above it, the shape cannot be that of surface A in Figure 175. It must be a at plane that remains below the whole mountain, like surface B in the gure. However, a at surface beneath a mountain does not allow the mass change limit to be exceeded.
** e multiple atom attempt. One can imagine a number of atoms equal to the number of the atoms of a mountain that all lie with large spacing (roughly) in a single plane. Again, the plane is moving upwards with the speed of light. But also in this case the uncertainty in the atomic positions makes it impossible to say that the mass ow limit has been exceeded.
** e multiple black hole attempt. Black holes are typically large and the uncertainty in their position is thus negligible. e mass limit c G, or power limit c G, corresponds to the ow of a single black hole moving through a plane at the speed of light. Several black holes crossing a plane together at just under the speed of light thus seem to beat the limit. However, the surface has to be physical: an observer must be possible on each of its points. But no observer can cross a black hole. A black hole thus e ectively punctures the plane surface. No black hole can ever be said to cross a plane surface; even less so a multiplicity
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•.
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of black holes. e limit remains valid.
**
e multiple neutron star attempt. e mass limit seems to be in reach when several neutron stars (which are slightly less dense than a black hole of the same mass) cross a plane surface at the same time, at high speed. However, when the speed approaches the speed of light, the crossing time for points far from the neutron stars and for those that actually cross the stars di er by large amounts. Neutron stars that are almost black holes cannot be crossed in a short time in units of a coordinate clock that is located far from the stars. Again, the limit is not exceeded.
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Ref. 310
Page 1003 Challenge 685 n
**
e luminosity attempt. e existence of a maximum luminosity bound has been discussed by astrophysicists. In its full generality, the maximum bound on power, i.e. on energy per time, is valid for any energy ow through any physical surface whatsoever. e physical surface may even run across the whole universe. However, not even bringing together all lamps, all stars and all galaxies of the universe yields a surface which has a larger power output than the proposed limit.
e surface must be physical.* A surface is physical if an observer can be placed on each of its points. In particular, a physical surface may not cross a horizon, or have local detail
ner than a certain minimum length. is minimum length will be introduced later on; it is given by the corrected Planck length. If a surface is not physical, it may provide a counter-example to the power or force limits. However, these counter-examples make no statements about nature. (Ex falso quodlibet.**)
**
e many lamp attempt. An absolute power limit imposes a limit on the rate of energy transport through any imaginable surface. At rst sight, it may seem that the combined power emitted by two radiation sources that each emit 3/4 of the maximum value should give 3/2 times that value. However, two such lamps would be so massive that they would form a black hole. No amount of radiation that exceeds the limit can leave. Again, since the horizon limit (210) is achieved, a horizon appears that swallows the light and prevents the force or power limit from being exceeded.
**
e light concentration attempt. Another approach is to shine a powerful, short and spherical ash of light onto a spherical mass. At rst sight it seems that the force and power limits can be exceeded, because light energy can be concentrated into small volumes. However, a high concentration of light energy forms a black hole or induces the mass to form one. ere is no way to pump energy into a mass at a faster rate than that dictated by the power limit. In fact, it is impossible to group light sources in such a way that their total output is larger than the power limit. Every time the force limit is approached, a horizon appears that prevents the limit from being exceeded.
* It can also be called physically sensible. ** Anything can be deduced from a falsehood.
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**
e black hole attempt. One possible system in nature that actually achieves the power limit is the nal stage of black hole evaporation. However, even in this case the power limit is not exceeded, but only equalled.
**
e water ow attempt. One could try to pump water as rapidly as possible through a large tube of cross-section A. However, when a tube of length L lled with water owing at speed v gets near to the mass ow limit, the gravity of the water waiting to be pumped through the area A will slow down the water that is being pumped through the area. e limit is again reached when the cross-section A turns into a horizon.
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Challenge 686 ny
Checking that no system – from microscopic to astrophysical – ever exceeds the maximum power or maximum mass ow is a further test of general relativity. It may seem easy to nd a counter-example, as the surface may run across the whole universe or envelop any number of elementary particle reactions. However, no such attempt succeeds.
In summary, in all situations where the force, power or mass- ow limit is challenged, whenever the energy ow reaches the black hole mass–energy density in space or the corresponding momentum ow in time, an event horizon appears; this horizon makes it impossible to exceed the limits. All three limits are con rmed both in observation and in theory. Values exceeding the limits can neither be generated nor measured. Gedanken experiments also show that the three bounds are the tightest ones possible. Obviously, all three limits are open to future tests and to further Gedanken experiments. (If you can think of a good one, let me know.)
H
e absence of horizons in everyday life is the rst reason why the maximum force principle remained undiscovered for so long. Experiments in everyday life do not highlight the force or power limits. e second reason why the principle remained hidden is the erroneous belief in point particles. is is a theoretical reason. (Prejudices against the concept of force in general relativity have also been a factor.) e principle of maximum force – or of maximum power – has thus remained hidden for so long because of a ‘conspiracy’ of nature that hid it both from theorists and from experimentalists.
For a thorough understanding of general relativity it is essential to remember that point particles, point masses and point-like observers do not exist. ey are approximations only applicable in Galilean physics or in special relativity. In general relativity, horizons prevent their existence. e habit of believing that the size of a system can be made as small as desired while keeping its mass constant prevents the force or power limit from being noticed.
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A
“Wir leben zwar alle unter dem gleichen Himmel, aber wir haben nicht alle den gleichen Horizont.* ” Konrad Adenauer
e concepts of horizon force and horizon power can be used as the basis for a direct, intuitive approach to general relativity.
**
What is gravity? Of the many possible answers we will encounter, we now have the rst: gravity is the ‘shadow’ of the maximum force. Whenever we experience gravity as weak, we can remember that a di erent observer at the same point and time would experience the maximum force. Searching for the precise properties of that observer is a good exercise. Another way to put it: if there were no maximum force, gravity would not exist.
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**
e maximum force implies universal gravity. To see this, we study a simple planetary
system, i.e., one with small velocities and small forces. A simple planetary system of size
L consists of a (small) satellite circling a central mass M at a radial distance R = L .
Let a be the acceleration of the object. Small velocity implies the condition aL  c ,
deduced from special relativity; small force implies GMa  c , deduced from the
force limit. ese conditions are valid for the system as a whole and for all its components.
Both expressions have the dimensions of speed squared. Since the system has only one
characteristic speed, the two expressions aL = aR and GMa must be proportional,
yielding
a
=
f
GM R
,
(224)
where the numerical factor f must still be determined. To determine it, we study the escape velocity necessary to leave the central body. e escape velocity must be smaller than the speed of light for any body larger than a black hole. e escape velocity, derived from expression (224), from a body of mass M and radius R is given by vesc = f GM R.
e minimum radius R of objects, given by R = GM c , then implies that f = . erefore, for low speeds and low forces, the inverse square law describes the orbit of a satellite around a central mass.
Page 402
**
If empty space-time is elastic, like a piece of metal, it must also be able to oscillate. Any physical system can show oscillations when a deformation brings about a restoring force. We saw above that there is such a force in the vacuum: it is called gravitation. In other words, vacuum must be able to oscillate, and since it is extended, it must also be able to sustain waves. Indeed, gravitational waves are predicted by general relativity, as we will see below.
* ‘We all live under the same sky, but we do not have the same horizon.’ Konrad Adenauer, German chancellor.
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**
If curvature and energy are linked, the maximum speed must also hold for gravitational energy. Indeed, we will nd that gravity has a nite speed of propagation. e inverse square law of everyday life cannot be correct, as it is inconsistent with any speed limit. More about the corrections induced by the maximum speed will become clear shortly. In addition, since gravitational waves are waves of massless energy, we would expect the maximum speed to be their propagation speed. is is indeed the case, as we will see.
**
A body cannot be denser than a (non-rotating) black hole of the same mass. e maximum force and power limits that apply to horizons make it impossible to squeeze mass into smaller horizons. e maximum force limit can therefore be rewritten as a limit for the size L of physical systems of mass m:
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L
Gm c.
(225)
If we call twice the radius of a black hole its ‘size’, we can state that no physical system of mass m is smaller than this value.* e size limit plays an important role in general
relativity. e opposite inequality, m A π c G, which describes the maximum ‘size’ of black holes, is called the Penrose inequality and has been proven for many physic-
Ref. 318, Ref. 319, ally realistic situations. e Penrose inequality can be seen to imply the maximum force Ref. 320 limit, and vice versa. e maximum force principle, or the equivalent minimum size of
matter–energy systems, thus prevents the formation of naked singularities, and implies the validity of the so-called cosmic censorship.
Ref. 310
**
ere is a power limit for all energy sources. In particular, the value c G limits the luminosity of all gravitational sources. Indeed, all formulae for gravitational wave emission imply this value as an upper limit. Furthermore, numerical relativity simulations never exceed it: for example, the power emitted during the simulated merger of two black holes is below the limit.
**
Perfectly plane waves do not exist in nature. Plane waves are of in nite extension. But neither electrodynamic nor gravitational waves can be in nite, since such waves would carry more momentum per time through a plane surface than is allowed by the force limit. e non-existence of plane gravitational waves also precludes the production of singularities when two such waves collide.
**
In nature, there are no in nite forces. ere are thus no naked singularities in nature. Horizons prevent the appearance of naked singularities. In particular, the big bang was
* e maximum value for the mass to size limit is obviously equivalent to the maximum mass change given above.
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•.
not a singularity. e mathematical theorems by Penrose and Hawking that seem to imply the existence of singularities tacitly assume the existence of point masses – o en in the form of ‘dust’ – in contrast to what general relativity implies. Careful re-evaluation of each such proof is necessary.
**
e force limit means that space-time has a limited stability. e limit suggests that spacetime can be torn into pieces. is is indeed the case. However, the way that this happens is not described general relativity. We will study it in the third part of this text.
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Ref. 321
**
e maximum force is the standard of force. is implies that the gravitational constant G is constant in space and time – or at least, that its variations across space and time cannot be detected. Present data support this claim to a high degree of precision.
Ref. 310
**
e maximum force principle implies that gravitational energy – as long as it can be de ned – falls in gravitational elds in the same way as other type of energy. As a result, the maximum force principle predicts that the Nordtvedt e ect vanishes. e Nordtvedt e ect is a hypothetical periodical change in the orbit of the Moon that would appear if the gravitational energy of the Earth–Moon system did not fall, like other mass–energy, in the gravitational eld of the Sun. Lunar range measurements have con rmed the absence of this e ect.
**
If horizons are surfaces, we can ask what their colour is. Page 477 later on.
is question will be explored
Page 1077 Challenge 687 e
**
Later on we will nd that quantum e ects cannot be used to exceed the force or power limit. (Can you guess why?) Quantum theory also provides a limit to motion, namely a lower limit to action; however, this limit is independent of the force or power limit. (A dimensional analysis already shows this: there is no way to de ne an action by combinations of c and G.) erefore, even the combination of quantum theory and general relativity does not help in overcoming the force or power limits.
A
A maximum power is the simplest possible explanation of Olbers’ paradox. Power and luminosity are two names for the same observable. e sum of all luminosities in the universe is nite; the light and all other energy emitted by all stars, taken together, is nite. If one assumes that the universe is homogeneous and isotropic, the power limit P c G must be valid across any plane that divides the universe into two halves. e part of the universe’s luminosity that arrives on Earth is then so small that the sky is dark at night. In fact, the actually measured luminosity is still smaller than this calculation, as a large part of the power is not visible to the human eye (since most of it is matter anyway). In other
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words, the night is dark because of nature’s power limit. is explanation is not in contrast to the usual one, which uses the nite lifetime of stars, their nite density, their nite size, and the nite age and the expansion of the universe. In fact, the combination of all these usual arguments simply implies and repeats in more complex words that the power limit cannot be exceeded. However, this more simple explanation seems to be absent in the literature.
e existence of a maximum force in nature, together with homogeneity and isotropy, implies that the visible universe is of nite size. e opposite case would be an in nitely large, homogeneous and isotropic universe. But in that case, any two halves of the universe would attract each other with a force above the limit (provided the universe were su ciently old). is result can be made quantitative by imagining a sphere whose centre lies at the Earth, which encompasses all the universe, and whose radius decreases with time almost as rapidly as the speed of light. e mass ow dm dt = ρAv is predicted to reach the mass ow limit c G; thus one has
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dm dt
=
ρ
πR
c=
c G
,
(226)
Ref. 322 Challenge 688 ny
a relation also predicted by the Friedmann models. e precision measurements of the cosmic background radiation by the WMAP satellite con rm that the present-day total energy density ρ (including dark matter and dark energy) and the horizon radius R just reach the limit value. e maximum force limit thus predicts the observed size of the universe.
A nite power limit also suggests that a nite age for the universe can be deduced. Can you nd an argument?
E
Ref. 308
e lack of direct tests of the horizon force, power or mass ow is obviously due to the lack of horizons in the environment of all experiments performed so far. Despite the di culties in reaching the limits, their values are observable and falsi able.
In fact, the force limit might be tested with high-precision measurements in binary pulsars or binary black holes. Such systems allow precise determination of the positions of the two stars. e maximum force principle implies a relation between the position error ∆x and the energy error ∆E. For all systems one has
∆E ∆x
c G
.
(227)
For example, a position error of mm gives a mass error of below ë kg. In everyday life, all measurements comply with this relation. Indeed, the le side is so much smaller than the right side that the relation is rarely mentioned. For a direct check, only systems which might achieve direct equality are interesting. Dual black holes or dual pulsars are such systems.
It might be that one day the amount of matter falling into some black hole, such as the one at the centre of the Milky Way, might be measured. e limit dm dt c G could
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•.
then be tested directly. e power limit implies that the highest luminosities are only achieved when systems
emit energy at the speed of light. Indeed, the maximum emitted power is only achieved when all matter is radiated away as rapidly as possible: the emitted power P = Mc (R v) cannot reach the maximum value if the body radius R is larger than that of a black hole (the densest body of a given mass) or the emission speed v is lower than that of light. e sources with highest luminosity must therefore be of maximum density and emit entities without rest mass, such as gravitational waves, electromagnetic waves or (maybe) gluons. Candidates to detect the limit are black holes in formation, in evaporation or undergoing mergers.
A candidate surface that reaches the limit is the night sky. e night sky is a horizon. Provided that light, neutrino, particle and gravitational wave ows are added together, the limit c G is predicted to be reached. If the measured power is smaller than the limit (as it seems to be at present), this might even give a hint about new particles yet to be discovered. If the limit were exceeded or not reached, general relativity would be shown to be incorrect. is might be an interesting future experimental test.
e power limit implies that a wave whose integrated intensity approaches the force limit cannot be plane. e power limit thus implies a limit on the product of intensity I (given as energy per unit time and unit area) and the size (curvature radius) R of the front of a wave moving with the speed of light c:
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πR I
c G
.
(228)
Challenge 689 e
Obviously, this statement is di cult to check experimentally, whatever the frequency and type of wave might be, because the value appearing on the right-hand side is extremely large. Possibly, future experiments with gravitational wave detectors, X-ray detectors, gamma ray detectors, radio receivers or particle detectors might allow us to test relation ( ) with precision. (You might want to predict which of these experiments will con rm the limit rst.)
e lack of direct experimental tests of the force and power limits implies that indirect tests become particularly important. All such tests study the motion of matter or energy and compare it with a famous consequence of the force and power limits: the eld equations of general relativity. is will be our next topic.
A
ere is a simple axiomatic formulation of general relativity: the horizon force c G and the horizon power c G are the highest possible force and power values. No contradicting observation is known. No counter-example has been imagined. General relativity follows from these limits. Moreover, the limits imply the darkness of the night and the
niteness of the size of the universe. e principle of maximum force has obvious applications for the teaching of general re-
lativity. e principle brings general relativity to the level of rst-year university, and possibly to well-prepared secondary school, students: only the concepts of maximum force and horizon are necessary. space-time curvature is a consequence of horizon curvature.
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Challenge 690 ny Page 460
Page 1076
e concept of a maximum force points to an additional aspect of gravitation. e cosmological constant Λ is not xed by the maximum force principle. (However, the principle does x its sign to be positive.) Present measurements give the result Λ − m . A positive cosmological constant implies the existence of a negative energy volume density −Λc G. is value corresponds to a negative pressure, as pressure and energy density have the same dimensions. Multiplication by the (numerically corrected) Planck area Għ c , the smallest area in nature, gives a force value
F = Λħc = . ë − N .
(229)
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is is also the gravitational force between two (numerically corrected) Planck masses
ħc G located at the cosmological distance Λ . If we make the somewhat wishful assumption that expression ( ) is the smallest possible force in nature (the numerical factors are not yet veri ed), we get the fascinating conjecture that the full theory of general relativity, including the cosmological constant, may be de ned by the combination of a Challenge 691 ny maximum and a minimum force in nature. (Can you nd a smaller force?)
Proving the minimum force conjecture is more involved than for the case of the maximum force. So far, only some hints are possible. Like the maximum force, the minimum force must be compatible with gravitation, must not be contradicted by any experiment, and must withstand any Gedanken experiment. A quick check shows that the minimum force, as we have just argued, allows us to deduce gravitation, is an invariant, and is not contradicted by any experiment. ere are also hints that there may be no way to generate or measure a smaller value. For example, the minimum force corresponds to the energy per length contained by a photon with a wavelength of the size of the universe. It is hard – but maybe not impossible – to imagine the production of a still smaller force.
We have seen that the maximum force principle and general relativity fail to x the value of the cosmological constant. Only a uni ed theory can do so. We thus get two requirements for such a theory. First, any uni ed theory must predict the same upper limit to force. Secondly, a uni ed theory must x the cosmological constant. e appearance of ħ in the conjectured expression for the minimum force suggests that the minimum force is determined by a combination of general relativity and quantum theory. e proof of this suggestion and the direct measurement of the minimum force are two important challenges for our ascent beyond general relativity.
We are now ready to explore the consequences of general relativity and its eld equations in more detail. We start by focusing on the concept of space-time curvature in everyday life, and in particular, on its consequences for the observation of motion.
A
e author thanks Steve Carlip, Corrado Massa, Tom Helmond, Gary Gibbons, Heinrich Neumaier and Peter Brown for interesting discussions on these topics.
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B
307 It seems that the rst published statement of the principle was in an earlier edition of this
text, in the chapter on gravitation and relativity. e reference is C
S
,
Motion Mountain – A Hike rough and Beyond Space and Time Following the Concepts of
Modern Physics, found at http://www.motionmountain.net. e approach was discussed in
various usenet discussion groups in the early twenty- rst century. e result is also pub-
lished in C. S
, General Relativity and cosmology derived from principle of max-
imum power or force, International Journal of eoretical Physics 44, pp. – , .
Cited on page .
308 C. S
, Maximum force and minimum distance: physics in limit statements, part
of this text and downloadable at http://www.motionmountain.net/C -LIMI.pdf. Cited on
pages , , , , and .
309 G.W. G
, e maximum tension principle in general relativity, Foundations of Phys-
ics 32, pp. – , , or http://www.arxiv.org/abs/hep-th/
. Gary Gibbons ex-
plains that the maximum force follows from general relativity; he does not make a statement
about the converse. Cited on page .
310 H.C. O
& R. R
, Gravitation and Spacetime, W.W. Norton & Co., New York,
. Cited on pages , , , , , and .
311 See for example W
R
, Relativity – Special, General and Cosmological, Ox-
ford University Press, , p. , or R ’I
Introducing Einstein’s Relativity,
Clarendon Press, , p. . Cited on page .
312 See for example A. A
, S. F
& B. K
, Isolated horizons:
Hamiltonian evolution and the rst law, http://www.arxiv.org/abs/gr-qc/
. Cited
on page .
313 T. J
, ermodynamics of spacetime: the Einstein equation of state, Physical Re-
view Letters 75, pp. – , or http://www.arxiv.org/abs/gr-qc/
. Cited on
pages and .
314 See for example E
K
, Kontinuumstheorie der Versetzungen und Eigenspan-
nungen, Springer, , volume of the series ‘Ergebnisse der angewandten Mathematik’.
Kröner shows the similarity between the equations, methods and results of solid-state con-
tinuum physics and those of general relativity. Cited on page .
315 E
F. T
&J
A. W
, Spacetime Physics – Introduction to Special
Relativity, second edition, Freeman, . Cited on page .
316 is counter-example was suggested by Steve Carlip. Cited on page .
317 E.R. C
, Lettere al Nuovo Cimento 41, p. , . Cited on page .
318 R. P
, Naked singularities, Annals of the New York Academy of Sciences 224, pp. –
, . Cited on page .
319 G. H
& T. I
, e Riemannian Penrose inequality, Int. Math. Res. Not. 59,
pp. – , . Cited on page .
320 S.A. H
, Inequalities relating area, energy, surface gravity and charge of black holes,
Physical Review Letters 81, pp. – , . Cited on page .
321 C. W , Was Einstein Right? – Putting General Relativity to the Test, Oxford University
Press, . See also his paper http://www.arxiv.org/abs/gr-qc/
. Cited on page .
322 e measurement results by the WMAP satellite are summarized on the website http://map.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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gsfc.nasa.gov/m_mm/mr_limits.html; the papers are available at http://lambda.gsfc.nasa. gov/product/map/current/map_bibliography.cfm. Cited on page .
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
,
.
,
Ref. 323, Ref. 324
Sapere aude.
Horace*
“ ” Gravitational in uences do transport energy.** Our description of motion must therefore
be precise enough to imply that this transport can happen at most with the speed of light.
Henri Poincaré stated this requirement as long ago as . e results following from this
principle will be fascinating: we will nd that empty space can move, that the universe has
a nite age and that objects can be in permanent free fall. It will turn out that empty space
can be bent, although it is much sti er than steel. Despite these strange consequences, the
theory and all its predictions have been con rmed by all experiments.
e theory of universal gravitation, which describes motion due to gravity using the
relation a = GM r , allows speeds higher than that of light. Indeed, the speed of a mass
in orbit is not limited. It is also unclear how the values of a and r depend on the observer.
So this theory cannot be correct. In order to reach the correct description, called general
relativity by Albert Einstein, we have to throw quite a few preconceptions overboard.
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Challenge 692 e Ref. 325
e opposite of motion in daily life is a body at rest, such as a child sleeping or a rock defying the waves. A body is at rest whenever it is not disturbed by other bodies. In the Galilean description of the world, rest is the absence of velocity. In special relativity, rest became inertial motion, since no inertially moving observer can distinguish its own motion from rest: nothing disturbs him. Both the rock in the waves and the rapid protons crossing the galaxy as cosmic rays are at rest. e inclusion of gravity leads us to an even more general de nition.
If any body moving inertially is to be considered at rest, then any body in free fall must also be. Nobody knows this better than Joseph Kittinger, the man who in August
stepped out of a balloon capsule at the record height of . km. At that altitude, the air is so thin that during the rst minute of his free fall he felt completely at rest, as if he were oating. Although an experienced parachutist, he was so surprised that he had to turn upwards in order to convince himself that he was indeed moving away from his balloon! Despite his lack of any sensation of movement, he was falling at up to m s or km h with respect to the Earth’s surface. He only started feeling something when he encountered the rst substantial layers of air. at was when his free fall started to be disturbed. Later, a er four and a half minutes of fall, his special parachute opened; and nine minutes later he landed in New Mexico.
Kittinger and all other observers in free fall, such as the cosmonauts circling the Earth or the passengers in parabolic aeroplane ights,*** make the same observation: it is impossible to distinguish anything happening in free fall from what would happen at rest.
is impossibility is called the principle of equivalence; it is one of the starting points of
* ‘Venture to be wise.’ Quintus Horatius Flaccus, Ep. 1, 2, 40. ** e details of this statement are far from simple. ey are discussed on page 402 and page 433. *** Nowadays it is possible to book such ights in specialized travel agents.
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Challenge 693 n
general relativity. It leads to the most precise – and nal – de nition of rest: rest is free fall. Rest is lack of disturbance; so is free fall.
e set of all free-falling observers at a point in space-time generalizes the specialrelativistic notion of the set of the inertial observers at a point. is means that we must describe motion in such a way that not only inertial but also freely falling observers can talk to each other. In addition, a full description of motion must be able to describe gravitation and the motion it produces, and it must be able to describe motion for any observer imaginable. General relativity realizes this aim.
As a rst step, we put the result in simple words: true motion is the opposite of free fall. is statement immediately rises a number of questions: Most trees or mountains are not in free fall, thus they are not at rest. What motion are they undergoing? And if free fall is rest, what is weight? And what then is gravity anyway? Let us start with the last question.
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?–A
Ref. 326 Challenge 695 e
Page 305 Challenge 696 e
In the beginning, we described gravity as the shadow of the maximum force. But there
is a second way to describe it, more related to everyday life. As William Unruh likes to
explain, the constancy of the speed of light for all observers implies a simple conclusion:
gravity is the uneven running of clocks at di erent places.* Of course, this seemingly ab-
surd de nition needs to be checked. e de nition does not talk about a single situation
seen by di erent observers, as we o en did in special relativity. e de nition depends
of the fact that neighbouring, identical clocks, xed against each other, run di erently in
the presence of a gravitational eld when watched by the same observer; moreover, this
di erence is directly related to what we usually call gravity. ere are two ways to check
this connection: by experiment and by reasoning. Let us start with the latter method, as
it is cheaper, faster and more fun.
An observer feels no di erence between
gravity and constant acceleration. We can thus study constant acceleration and use a way of
v(t)=gt
reasoning we have encountered already in the
chapter on special relativity. We assume light is
B
light
F
emitted at the back end of a train of length ∆h
that is accelerating forward with acceleration ,
as shown in Figure . e light arrives at the F I G U R E 176 Inside an accelerating train or
front a er a time t = ∆h c. However, during bus
this time the accelerating train has picked up
some additional velocity, namely ∆v = t = ∆h c. As a result, because of the Doppler
e ect we encountered in our discussion of special relativity, the frequency f of the light
arriving at the front has changed. Using the expression of the Doppler e ect, we thus get*
* Gravity is also the uneven length of metre bars at di erent places, as we will see below. Both e ects are needed to describe it completely; but for daily life on Earth, the clock e ect is su cient, since it is much Challenge 694 n larger than the length e ect, which can usually be neglected. Can you see why?
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∆f f
=
∆h c
.
(230)
Challenge 698 n Ref. 327
Page 198
Challenge 701 e Challenge 702 n
Ref. 328 Page 753
e sign of the frequency change depends on whether the light motion and the train acceleration are in the same or in opposite directions. For actual trains or buses, the frequency change is quite small; nevertheless, it is measurable. Acceleration induces frequency changes in light. Let us compare this e ect of acceleration with the e ects of gravity.
To measure time and space, we use light. What happens to light when gravity is involved? e simplest experiment is to let light fall or rise. In order to deduce what must happen, we add a few details. Imagine a conveyor belt carrying masses around two wheels, a low and a high one, as shown in Figure . e descending, grey masses are slightly larger. Whenever such a larger mass is near the bottom, some mechanism – not shown in the
gure – converts the mass surplus to light, in accordance with the equation E = mc , and sends the light up towards the top.** At the top, one of the lighter, white masses passing by absorbs the light and, because of its added weight, turns the conveyor belt until it reaches the bottom. en the process repeats.***
As the grey masses on the descending side are always heavier, the belt would turn for ever and this system could continuously generate energy. However, since energy conservation is at the basis of our de nition of time, as we saw in the beginning of our walk, the whole process must be impossible. We have to conclude that the light changes its energy when climbing. e only possibility is that the light arrives at the top with a frequency di erent from the one at which it is emitted from the bottom.****
In short, it turns out that rising light is gravitationally red-shi ed. Similarly, the light descending from the top of a tree down to an observer is blue-shi ed; this gives a darker colour to the top in comparison with the bottom of the tree. General relativity thus says that trees have di erent shades of green along their height.***** How big is the e ect?
e result deduced from the drawing is again the one of formula ( ). at is what we would, as light moving in an accelerating train and light moving in gravity are equivalent situations, as you might want to check yourself. e formula gives a relative change of frequency of only . ë − m near the surface of the Earth. For trees, this so-called gravitational red-shi or gravitational Doppler e ect is far too small to be observable, at least using normal light.
In , Einstein proposed an experiment to check the change of frequency with height by measuring the red-shi of light emitted by the Sun, using the famous Fraunhofer lines as colour markers. e results of the rst experiments, by Schwarzschild and others, were unclear or even negative, due to a number of other e ects that induce colour changes at
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Challenge 697 e Challenge 699 n Challenge 700 ny
* e expression v = t is valid only for non-relativistic speeds; nevertheless, the conclusion of this section is not a ected by this approximation. ** As in special relativity, here and in the rest of our mountain ascent, the term ‘mass’ always refers to rest mass. *** Can this process be performed with 100% e ciency? **** e precise relation between energy and frequency of light is described and explained in our discussion on quantum theory, on page 719. But we know already from classical electrodynamics that the energy of light depends on its intensity and on its frequency. ***** How does this argument change if you include the illumination by the Sun?
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Ref. 329
high temperatures. But in and , Grebe and Bachem, and independently Perot,
con rmed the gravitational red-shi with careful experiments. In later years, technolo-
gical advances made the measurements much easier, until it was even possible to measure
the e ect on Earth. In , in a classic experiment using the Mössbauer e ect, Pound and
Rebka con rmed the gravitational red-shi in their university tower using γ radiation.
But our two thought experiments tell us much more. Let
us use the same arguments as in the case of special relativity:
a colour change implies that clocks run di erently at di er-
m
ent heights, just as they run di erently in the front and in
the back of a train. e time di erence ∆τ is predicted to
depend on the height di erence ∆h and the acceleration of gravity according to
m+E/c2
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∆τ τ
=
∆f f
=
∆h c
.
(231)
h
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Challenge 703 ny Ref. 330 Ref. 331
Ref. 332 Challenge 704 ny Challenge 705 e
erefore, in gravity, time is height-dependent. at was ex-
actly what we claimed above. In fact, height makes old. Can
you con rm this conclusion?
In , by ying four precise clocks in an aeroplane
while keeping an identical one on the ground, Hafele and
light
Keating found that clocks indeed run di erently at di er-
ent altitudes according to expression ( ). Subsequently, in
, the team of Vessot et al. shot a precision clock based
on a maser – a precise microwave generator and oscillator –
upwards on a missile. e team compared the maser inside
the missile with an identical maser on the ground and again F I G U RE 177 The necessity of
con rmed the expression. In , Briatore and Leschiutta blue- and red-shift of light:
showed that a clock in Torino indeed ticks more slowly why trees are greener at the than one on the top of the Monte Rosa. ey con rmed the bottom
prediction that on Earth, for every m of height gained,
people age more rapidly by about ns per day. is e ect has been con rmed for all sys-
tems for which experiments have been performed, such as several planets, the Sun and
numerous other stars.
Do these experiments show that time changes or are they simply due to clocks that
function badly? Take some time and try to settle this question. We will give one argument
only: gravity does change the colour of light, and thus really does change time. Clock
precision is not an issue here.
In summary, gravity is indeed the uneven running of clocks at di erent heights. Note
that an observer at the lower position and another observer at the higher position agree
on the result: both nd that the upper clock goes faster. In other words, when gravity is
present, space-time is not described by the Minkowski geometry of special relativity, but
by some more general geometry. To put it mathematically, whenever gravity is present,
the -distance ds between events is di erent from the expression without gravity:
ds c dt − dx − dy − dz .
(232)
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Challenge 706 ny
We will give the correct expression shortly. Is this view of gravity as height-dependent time really reasonable? No. It turns out that
it is not yet strange enough! Since the speed of light is the same for all observers, we can say more. If time changes with height, length must also do so! More precisely, if clocks run di erently at di erent heights, the length of metre bars must also change with height. Can you con rm this for the case of horizontal bars at di erent heights?
If length changes with height, the circumference of a circle around the Earth cannot be given by πr. An analogous discrepancy is also found by an ant measuring the radius and circumference of a circle traced on the surface of a basketball. Indeed, gravity implies that humans are in a situation analogous to that of ants on a basketball, the only di erence being that the circumstances are translated from two to three dimensions. We conclude that wherever gravity plays a role, space is curved.
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Challenge 707 e Challenge 708 n
Challenge 709 ny Page 129 Ref. 333
During his free fall, Kittinger was able to specify an inertial frame for himself. Indeed,
he felt completely at rest. Does this mean that it is impossible to distinguish acceleration
from gravitation? No: distinction is possible. We only have to compare two (or more)
falling observers.
Kittinger could not have found a frame which is also inertial for
a colleague falling on the opposite side of the Earth. Such a common frame does not exist. In general, it is impossible to nd a single iner-
before
tial reference frame describing di erent observers freely falling near
a mass. In fact, it is impossible to nd a common inertial frame even
after
for nearby observers in a gravitational eld. Two nearby observers
observe that during their fall, their relative distance changes. (Why?)
e same happens to orbiting observers.
In a closed room in orbit around the Earth, a person or a mass
at the centre of the room would not feel any force, and in particular
no gravity. But if several particles are located in the room, they will FIGURE 178 Tidal
behave di erently depending on their exact positions in the room. effects: what bodies Only if two particles were on exactly the same orbit would they keep feel when falling
the same relative position. If one particle is in a lower or higher orbit
than the other, they will depart from each other over time. Even more interestingly, if a
particle in orbit is displaced sideways, it will oscillate around the central position. (Can
you con rm this?)
Gravitation leads to changes of relative distance. ese changes evince another e ect,
shown in Figure : an extended body in free fall is slightly squeezed. is e ect also
tells us that it is an essential feature of gravity that free fall is di erent from point to point.
at rings a bell. e squeezing of a body is the same e ect as that which causes the
tides. Indeed, the bulging oceans can be seen as the squeezed Earth in its fall towards the
Moon. Using this result of universal gravity we can now a rm: the essence of gravity is
the observation of tidal e ects.
In other words, gravity is simple only locally. Only locally does it look like acceleration.
Only locally does a falling observer like Kittinger feel at rest. In fact, only a point-like
observer does so! As soon as we take spatial extension into account, we nd tidal e ects.
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Gravity is the presence of tidal e ects. e absence of tidal e ects implies the absence of gravity. Tidal e ects are the everyday consequence of height-dependent time. Isn’t this a beautiful conclusion?
In principle, Kittinger could have felt gravitation during his free fall, even with his eyes closed, had he paid attention to himself. Had he measured the distance change between his two hands, he would have found a tiny decrease which could have told him that he was falling. is tiny decrease would have forced Kittinger to a strange conclusion. Two inertially moving hands should move along two parallel lines, always keeping the same distance. Since the distance changes, he must conclude that in the space around him lines starting out in parallel do not remain so. Kittinger would have concluded that the space around him was similar to the surface of the Earth, where two lines starting out north, parallel to each other, also change distance, until they meet at the North Pole. In other words, Kittinger would have concluded that he was in a curved space.
By studying the change in distance between his hands, Kittinger could even have concluded that the curvature of space changes with height. Physical space di ers from a sphere, which has constant curvature. Physical space is more involved. e e ect is extremely small, and cannot be felt by human senses. Kittinger had no chance to detect anything. Detection requires special high-sensitivity apparatus. However, the conclusion remains valid. Space-time is not described by Minkowski geometry when gravity is present. Tidal e ects imply space-time curvature. Gravity is the curvature of space-time.
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Ref. 334
“Wenn ein Käfer über die Ober äche einer Kugel krabbelt, merkt er wahrscheinlich nicht, daß der Weg, den er zurücklegt, gekrümmt ist. Ich
dagegen hatte das Glück, es zu merken.*
Albert Einstein’s answer to his son Eduard’s
On the th of November
” question about the reason for his fame
, Albert Einstein became world-famous. On that day, an
article in the Times newspaper in London announced the results of a double expedition
to South America under the heading ‘Revolution in science / new theory of the universe
/ Newtonian ideas overthrown’. e expedition had shown unequivocally – though not
for the rst time – that the theory of universal gravity, essentially given by a = GM r ,
was wrong, and that instead space was curved. A worldwide mania started. Einstein was
presented as the greatest of all geniuses. ‘Space warped’ was the most common headline.
Einstein’s papers on general relativity were reprinted in full in popular magazines. People
could read the eld equations of general relativity, in tensor form and with Greek indices,
in Time magazine. Nothing like this has happened to any other physicist before or since.
What was the reason for this excitement?
e expedition to the southern hemisphere had performed an experiment proposed
by Einstein himself. Apart from seeking to verify the change of time with height, Einstein
had also thought about a number of experiments to detect the curvature of space. In the
one that eventually made him famous, Einstein proposed to take a picture of the stars
* ‘When an insect walks over the surface of a sphere it probably does not notice that the path it walks is curved. I, on the other hand, had the luck to notice it.’
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near the Sun, as is possible during a solar eclipse, and compare it with a picture of the same stars at night, when the Sun is far away. Einstein predicted a change in position of . ′ ( . seconds of arc) for star images at the border of the Sun, a value twice as large as that predicted by universal gravity. e prediction, corresponding to about mm on the photographs, was con rmed in , and thus universal gravity was ruled out.
Does this result imply that space is curved? Not by itself. In fact, other explanations could be given for the result of the eclipse experiment, such as a potential di ering from the inverse square form. However, the eclipse results are not the only data. We already know about the change of time with height. Experiments show that two observers at different heights measure the same value for the speed of light c near themselves. But these experiments also show that if an observer measures the speed of light at the position of the other observer, he gets a value di ering from c, since his clock runs di erently. ere is only one possible solution to this dilemma: metre bars, like clocks, also change with height, and in such a way as to yield the same speed of light everywhere.
If the speed of light is constant but clocks and metre bars change with height, the conclusion must be that space is curved near masses. Many physicists in the twentieth century checked whether metre bars really behave di erently in places where gravity is present. And indeed, curvature has been detected around several planets, around all the hundreds of stars where it could be measured, and around dozens of galaxies. Many indirect e ects of curvature around masses, to be described in detail below, have also been observed. All results con rm the curvature of space and space-time around masses, and in addition con rm the curvature values predicted by general relativity. In other words, metre bars near masses do indeed change their size from place to place, and even from orientation to orientation. Figure gives an impression of the situation.
image star
position of star
image of star
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Sun Sun
Mercury
Earth
Earth
F I G U R E 179 The mattress model of space: the path of a light beam and of a satellite near a spherical mass
Ref. 335 Challenge 711 n
But beware: the right-hand gure, although found in many textbooks, can be misleading. It can easily be mistaken fora reproduction of a potential around a body. Indeed, it is impossible to draw a graph showing curvature and potential separately. (Why?) We will see that for small curvatures, it is even possible to explain the change in metre bar length using a potential only. us the gure does not really cheat, at least in the case of weak gravity. But for large and changing values of gravity, a potential cannot be de ned, and thus there is indeed no way to avoid using curved space to describe gravity. In summary, if we imagine space as a sort of generalized mattress in which masses
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produce deformations, we have a reasonable model of space-time. As masses move, the deformation follows them.
e acceleration of a test particle only depends on the curvature of the mattress. It does not depend on the mass of the test particle. So the mattress model explains why all bodies fall in the same way. (In the old days, this was also called the equality of the inertial and gravitational mass.)
Space thus behaves like a frictionless mattress that pervades everything. We live inside the mattress, but we do not feel it in everyday life. Massive objects pull the foam of the mattress towards them, thus deforming the shape of the mattress. More force, more energy or more mass imply a larger deformation. (Does the mattress remind you of the aether? Do not worry: physics eliminated the concept of aether because it is indistinguishable from vacuum.)
If gravity means curved space, then any accelerated observer, such as a man in a departing car, must also observe that space is curved. However, in everyday life we do not notice any such e ect, because for accelerations and sizes of of everyday life the curvature values are too small to be noticed. Could you devise a sensitive experiment to check the prediction?
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Figure shows the curvature of space only, but in fact space-time is curved. We will shortly nd out how to describe both the shape of space and the shape of space-time, and how to measure their curvature.
Let us have a rst attempt to describe nature with the idea of curved space-time. In the case of Figure , the best description of events is with the use of the time t shown by a clock located at spatial in nity; that avoids problems with the uneven running of clocks at di erent distances from the central mass. For the radial coordinate r, the most practical choice to avoid problems with the curvature of space is to use the circumference of a circle around the central body, divided by π. e curved shape of space-time is best described by the behaviour of the space-time distance ds, or by the wristwatch time dτ = ds c, between two neighbouring points with coordinates (t, r) and (t + dt, r + dr). As we saw above, gravity means that in spherical coordinates we have
dτ
=
ds c
dt − dr c − r dφ c .
(233)
e inequality expresses the fact that space-time is curved. Indeed, the experiments on
time change with height con rm that the space-time interval around a spherical mass is
given by
dτ
=
ds c
=
−
GM rc
dt
− c
dr
−
GM r
−
r c
dφ
.
(234)
is expression is called the Schwarzschild metric a er one of its discoverers.* e metric ( ) describes the curved shape of space-time around a spherical non-rotating mass.
* Karl Schwarzschild (1873–1916), important German astronomer; he was one of the rst people to understand general relativity. He published his formula in December 1915, only a few months a er Einstein
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Challenge 713 n It is well approximated by the Earth or the Sun. (Why can their rotation be neglected?) Expression ( ) also shows that gravity’s strength around a body of mass M and radius R is measured by a dimensionless number h de ned as
h=
GM cR
.
(235)
is ratio expresses the gravitational strain with which lengths and the vacuum are deformed from the at situation of special relativity, and thus also determines how much clocks slow down when gravity is present. ( e ratio also reveals how far one is from any possible horizon.) On the surface of the Earth the ratio h has the small value of . ë − ; on the surface of the Sun is has the somewhat larger value of . ë − . e precision of modern clocks allows one to detect such small e ects quite easily. e various consequences and uses of the deformation of space-time will be discussed shortly.
We note that if a mass is highly concentrated, in particular when its radius becomes equal to its so-called Schwarzschild radius
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RS =
GM c
,
(236)
Page 479 Challenge 714 e
Ref. 337
Ref. 338 Challenge 715 ny
the Schwarzschild metric behaves strangely: at that location, time disappears (note that t is time at in nity). At the Schwarzschild radius, the wristwatch time (as shown by a clock at in nity) stops – and a horizon appears. What happens precisely will be explored below. is situation is not common: the Schwarzschild radius for a mass like the Earth is . mm, and for the Sun is . km; you might want to check that the object size for every system in everyday life is larger than its Schwarzschild radius. Bodies which reach this limit are called black holes; we will study them in detail shortly. In fact, general relativity states that no system in nature is smaller than its Schwarzschild size, in other words that the ratio h de ned by expression ( ) is never above unity.
In summary, the results mentioned so far make it clear that mass generates curvature. e mass–energy equivalence we know from special relativity then tells us that as a consequence, space should also be curved by the presence of any type of energy–momentum. Every type of energy curves space-time. For example, light should also curve space-time. However, even the highest-energy beams we can create correspond to extremely small masses, and thus to unmeasurably small curvatures. Even heat curves space-time; but in most systems, heat is only about a fraction of − of total mass; its curvature e ect is thus unmeasurable and negligible. Nevertheless it is still possible to show experimentally that energy curves space. In almost all atoms a sizeable fraction of the mass is due to the electrostatic energy among the positively charged protons. In Kreuzer con rmed that energy curves space with a clever experiment using a oating mass. It is straightforward to imagine that the uneven running of clock is the temporal equivalent of spatial curvature. Taking the two together, we conclude that when gravity is present, space-time is curved.
had published his eld equations. He died prematurely, at the age of 42, much to Einstein’s distress. We will deduce the form of the metric later on, directly from the eld equations of general relativity. e other Ref. 336 discoverer of the metric, unknown to Einstein, was the Dutch physicist J. Droste.
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Let us sum up our chain of thoughts. Energy is equivalent to mass; mass produces gravity; gravity is equivalent to acceleration; acceleration is position-dependent time. Since light speed is constant, we deduce that energy–momentum tells space-time to curve. is statement is the rst half of general relativity.
We will soon nd out how to measure curvature, how to calculate it from energy– momentum and what is found when measurement and calculation are compared. We will also nd out that di erent observers measure di erent curvature values. e set of transformations relating one viewpoint to another in general relativity, the di eomorphism symmetry, will tell us how to relate the measurements of di erent observers.
Since matter moves, we can say even more. Not only is space-time curved near masses, it also bends back when a mass has passed by. In other words, general relativity states that space, as well as space-time, is elastic. However, it is rather sti : quite a lot sti er than steel. To curve a piece of space by % requires an energy density enormously larger than to curve a simple train rail by %. is and other interesting consequences of the elasticity of space-time will occupy us for the remainder of this chapter.
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“Si morior, moror.*
”
We continue on the way towards precision in our understanding of gravitation. All our
theoretical and empirical knowledge about gravity can be summed up in just two general
statements. e rst principle states:
e speed v of a physical system is bounded above:
vc
(237)
for all observers, where c is the speed of light.
e theory following from this rst principle, special relativity, is extended to general relativity by adding a second principle, characterizing gravitation. ere are several equivalent ways to state this principle. Here is one.
For all observers, the force F on a system is limited by
F
c G
,
where G is the universal constant of gravitation.
(238)
In short, there is a maximum force in nature. Gravitation leads to attraction of masses. Challenge 717 e However, this force of attraction is limited. An equivalent statement is:
* ‘If I rest, I die.’ is is the motto of the bird of paradise.
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For all observers, the size L of a system of mass M is limited by
L M
G c
.
(239)
In other words, a massive system cannot be more concentrated than a non-rotating black hole of the same mass. Another way to express the principle of gravitation is the following:
For all systems, the emitted power P is limited by
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P
c G
.
(240)
Page 353 Challenge 718 ny
In short, there is a maximum power in nature. e three limits given above are all equivalent to each other; and no exception is
known or indeed possible. e limits include universal gravity in the non-relativistic case. ey tell us what gravity is, namely curvature, and how exactly it behaves. e limits al-
low us to determine the curvature in all situations, at all space-time events. As we have seen above, the speed limit together with any one of the last three principles imply all of general relativity.*
For example, can you show that the formula describing gravitational red-shi complies with the general limit ( ) on length-to-mass ratios?
We note that any formula that contains the speed of light c is based on special relativity, and if it contains the constant of gravitation G, it relates to universal gravity. If a formula contains both c and G, it is a statement of general relativity. e present chapter frequently underlines this connection.
Our mountain ascent so far has taught us that a precise description of motion requires the speci cation of all allowed viewpoints, their characteristics, their di erences, and the transformations between them. From now on, all viewpoints are allowed, without exception: anybody must be able to talk to anybody else. It makes no di erence whether an observer feels gravity, is in free fall, is accelerated or is in inertial motion. Furthermore, people who exchange le and right, people who exchange up and down or people who say that the Sun turns around the Earth must be able to talk to each other and to us. is gives a much larger set of viewpoint transformations than in the case of special relativity; it makes general relativity both di cult and fascinating. And since all viewpoints are allowed, the resulting description of motion is complete.**
W
E ?–
G
A genius is somebody who makes all possible mistakes in the shortest possible time.
“ ” Anonymous
* is didactic approach is unconventional. It is possible that is has been pioneered by the present author. Ref. 340 e British physicist Gary Gibbons also developed it independently. Earlier references are not known.
** Or it would be, were it not for a small deviation called quantum theory.
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Page 326 Page 56
Challenge 719 ny Challenge 720 ny Challenge 721 ny
In our discussion of special relativity, we saw that inertial or free- oating motion is the motion which connecting two events that requires the longest proper time. In the absence of gravity, the motion ful lling this requirement is straight (rectilinear) motion. On the other hand, we are also used to thinking of light rays as being straight. Indeed, we are all accustomed to check the straightness of an edge by looking along it. Whenever we draw the axes of a physical coordinate system, we imagine either drawing paths of light rays or drawing the motion of freely moving bodies.
In the absence of gravity, object paths and light paths coincide. However, in the presence of gravity, objects do not move along light paths, as every thrown stone shows. Light does not de ne spatial straightness any more. In the presence of gravity, both light and matter paths are bent, though by di erent amounts. But the original statement remains valid: even when gravity is present, bodies follow paths of longest possible proper time. For matter, such paths are called timelike geodesics. For light, such paths are called lightlike or null geodesics.
We note that in space-time, geodesics are the curves with maximal length. is is in contrast with the case of pure space, such as the surface of a sphere, where geodesics are the curves of minimal length. In simple words, stones fall because they follow geodesics. Let us perform a few checks of this statement.
Since stones move by maximizing proper time for inertial observers, they also must do so for freely falling observers, like Kittinger. In fact, they must do so for all observers.
e equivalence of falling paths and geodesics is at least coherent. If falling is seen as a consequence of the Earth’s surface approaching – as we will argue later on – we can deduce directly that falling implies a proper time that is as long as possible. Free fall indeed is motion along geodesics. We saw above that gravitation follows from the existence of a maximum force. e result can be visualized in another way. If the gravitational attraction between a central body and a satellite were stronger than it is, black holes would be smaller than they are; in that case the maximum force limit and the maximum speed could be exceeded by getting close to such a black hole. If, on the other hand, gravitation were weaker than it is, there would be observers for which the two bodies would not interact, thus for which they would not form a physical system. In summary, a maximum force of c G implies universal gravity. ere is no di erence between stating that all bodies attract through gravitation and stating that there is a maximum force with the value c G. But at the same time, the maximum force principle implies that objects move on geodesics. Can you show this? Let us turn to an experimental check. If falling is a consequence of curvature, then the paths of all stones thrown or falling near the Earth must have the same curvature in space-time. Take a stone thrown horizontally, a stone thrown vertically, a stone thrown rapidly, or a stone thrown slowly: it takes only two lines of argument to show that in spacetime all their paths are approximated to high precision by circle segments, as shown in Figure . All paths have the same curvature radius r, given by
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r=c . ë m.
(241)
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, height
slow, steep throw
h d
c · time
rapid, flat throw
throw distance
F I G U R E 180 All paths of flying stones have the same curvature in space-time
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Challenge 722 ny
e large value of the radius, corresponding to a low curvature, explains why we do not notice it in everyday life. e parabolic shape typical of the path of a stone in everyday life is just the projection of the more fundamental path in -dimensional space-time into -dimensional space. e important point is that the value of the curvature does not depend on the details of the throw. In fact, this simple result could have suggested the ideas of general relativity to people a full century before Einstein; what was missing was the recognition of the importance of the speed of light as limit speed. In any case, this simple calculation con rms that falling and curvature are connected. As expected, and as mentioned already above, the curvature diminishes at larger heights, until it vanishes at in nite distance from the Earth. Now, given that the curvature of all paths for free fall is the same, and given that all such paths are paths of least action, it is straightforward that they are also geodesics.
If we describe fall as a consequence of the curvature of space-time, we must show that the description with geodesics reproduces all its features. In particular, we must be able to explain that stones thrown with small speed fall back, and stones thrwon with high speed escape. can you deduce this from space curvature?
In summary, the motion of any particle falling freely ‘in a gravitational eld’ is described by the same variational principle as the motion of a free particle in special relativ-
ity: the path maximizes the proper time ∫ dτ. We rephrase this by saying that any particle
in free fall from point A to point B minimizes the action S given by
∫B
S = −mc dτ .
A
(242)
Page 498 Ref. 341
at is all we need to know about the free fall of objects. As a consequence, any deviation from free fall keeps you young. e larger the deviation, the younger you stay.
As we will see below, the minimum action description of free fall has been tested extremely precisely, and no di erence from experiment has ever been observed. We will also nd out that for free fall, the predictions of general relativity and of universal gravity di er substantially both for particles near the speed of light and for central bodies of high density. So far, all experiments have shown that whenever the two predictions differ, general relativity is right, and universal gravity and other alternative descriptions are wrong.
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Challenge 723 n Ref. 342
All bodies fall along geodesics. is tells us something important. e fall of bodies does not depend on their mass. e geodesics are like ‘rails’ in space-time that tell bodies how to fall. In other words, space-time can indeed be imagined as a single, giant, deformed entity. Space-time is not ‘nothing’; it is an entity of our thinking. e shape of this entity tells objects how to move. Space-time is thus indeed like an intangible mattress; this deformed mattress guides falling objects along its networks of geodesics.
Moreover, bound energy falls in the same way as mass, as is proven by comparing the fall of objects made of di erent materials. ey have di erent percentages of bound energy. (Why?) For example, on the Moon, where there is no air, cosmonauts dropped steel balls and feathers and found that they fell together, alongside each other. e independence on material composition has been checked and con rmed over and over again.
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Page 378 Ref. 343
Page 133
Page 409 Challenge 724 ny
How does radiation fall? Light, like any radiation, is energy without rest mass. It moves like a stream of extremely fast and light objects. erefore deviations from universal gravity become most apparent for light. How does light fall? Light cannot change speed. When light falls vertically, it only changes colour, as we have seen above. But light can also change direction. Long before the ideas of relativity became current, in , the Prussian astronomer Johann Soldner understood that universal gravity implies that light is de ected when passing near a mass. He also calculated how the de ection angle depends on the mass of the body and the distance of passage. However, nobody in the nineteenth century was able to check the result experimentally.
Obviously, light has energy, and energy has weight; the de ection of light by itself is thus not a proof of the curvature of space. General relativity also predicts a de ection angle for light passing masses, but of twice the classical Soldner value, because the curvature of space around large masses adds to the e ect of universal gravity. e de-
ection of light thus only con rms the curvature of space if the value agrees with the one predicted by general relativity. is is the case: observations do coincide with predictions. More details will be given shortly.
Mass is thus not necessary to feel gravity; energy is su cient. is result of the mass– energy equivalence must become second nature when studying general relativity. In particular, light is not light-weight, but heavy. Can you argue that the curvature of light near the Earth must be the same as that of stones, given by expression ( )?
In summary, all experiments show that not only mass, but also energy falls along geodesics, whatever its type (bound or free), and whatever the interaction (be it electromagnetic or nuclear). Moreover, the motion of radiation con rms that space-time is curved.
Since experiments show that all particles fall in the same way, independently of their mass, charge or any other property, we can conclude that the system of all possible trajectories forms an independent structure. is structure is what we call space-time.
We thus nd that space-time tells matter, energy and radiation how to fall. is statement is the second half of general relativity. It complements the rst half, which states that energy tells space-time how to curve. To complete the description of macroscopic motion, we only need to add numbers to these statements, so that they become testable. As usual, we can proceed in two ways: we can deduce the equations of motion directly,
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,
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or we can rst deduce the Lagrangian and then deduce the equations of motion from it. But before we do that, let’s have some fun.
C
“Wenn Sie die Antwort nicht gar zu ernst nehmen und sie nur als eine Art Spaß ansehen, so kann ich Ihnen das so erklären: Früher hat man geglaubt, wenn alle Dinge aus der Welt verschwinden, so bleiben noch Raum und Zeit übrig. Nach der Relativitätstheorie verschwinden aber auch Zeit und Raum mit den Dingen.* ” Albert Einstein in in New York
General relativity is a beautiful topic with numerous interesting aspects.
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**
Challenge 725 n
Take a plastic bottle and make some holes in it near the bottom. Fill the bottle with water, closing the holes with your ngers. If you let the bottle go, no water will leave the bottle during the fall. Can you explain how this experiment con rms the equivalence of rest and free fall?
Challenge 726 n
**
On his 76th birthday, Einstein received a birthday present specially made for him, shown in Figure 181. A rather deep cup is mounted on the top of a broom stick. e cup contains a weak piece of elastic rubber attached to its bottom, to which a ball is attached at the other end. In the starting position, the ball hangs outside the cup. e rubber is too weak to pull the ball into the cup against gravity. What is the most elegant way to get the ball into the cup?
Challenge 727 n
**
Gravity has the same properties in the whole universe – except in the US patent o ce. In 2005, it awarded a patent, Nr 6 960 975, for an antigravity device that works by distorting space-time in such a way that gravity is ‘compensated’ (see http://pat .uspto.gov). Do you know a simpler device?
**
e radius of curvature of space-time at the Earth’s surface is . ë Challenge 728 e this value?
m. Can you con rm
**
Challenge 729 ny A piece of wood oats on water. Does it stick out more or less in a li accelerating upwards?
* ‘If you do not take the answer too seriously and regard it only for amusement, I can explain it to you in the following way: in the past it was thought that if all things were to disappear from the world, space and time would remain. But following relativity theory, space and time would disappear together with the things.’
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Page 305 Challenge 730 ny
**
We saw in special relativity that if two twins are identically accelerated in the same direction, with one twin some distance ahead of the other, then the twin ahead ages more than the twin behind. Does this happen in a gravitational eld as well? And what happens when the eld varies with height, as on Earth?
**
A maximum force and a maximum power also imply a maximum ow of mass. Can you Challenge 731 ny show that no mass ow can exceed . ë kg s?
Challenge 732 ny
**
e experiments of Figure 176 and 177 di er in one point: one happens in at space, the other in curved space. One seems to be connected with energy conservation, the other not. Do these di erences invalidate the equivalence of the observations?
** Challenge 733 n How can cosmonauts weigh themselves to check whether they are eating enough?
Challenge 734 n
**
Is a cosmonaut in orbit really oating freely? No. It turns out that space stations and satellites are accelerated by several small e ects. e important ones are the pressure of the light from the Sun, the friction of the thin air, and the e ects of solar wind. (Micrometeorites can usually be neglected.) ese three e ects all lead to accelerations of the order of
− m s to − m s , depending on the height of the orbit. Can you estimate how long it would take an apple oating in space to hit the wall of a space station, starting from the middle? By the way, what is the magnitude of the tidal accelerations in this situation?
Page 80 Page 868
**
ere is no negative mass in nature, as discussed in the beginning of our walk (even antimatter has positive mass). is means that gravitation cannot be shielded, in contrast to electromagnetic interactions. Since gravitation cannot be shielded, there is no way to make a perfectly isolated system. But such systems form the basis of thermodynamics! We will study the fascinating implications of this later on: for example, we will discover an upper limit for the entropy of physical systems.
Ref. 344 Challenge 735 ny
**
Can curved space be used to travel faster than light? Imagine a space-time in which two points could be connected either by a path leading through a at portion, or by a second path leading through a partially curved portion. Could that curved portion be used to travel between the points faster than through the at one? Mathematically, this is possible; however, such a curved space would need to have a negative energy density. Such a situation is incompatible with the de nition of energy and with the non-existence of negative mass. e statement that this does not happen in nature is also called the weak energy condition. Is it implied by the limit on length-to-mass ratios?
**
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e statement of a length-to-mass limit L M G c invites experiments to try to overcome it. Can you explain what happens when an observer moves so rapidly past a mass Challenge 736 ny that the body’s length contraction reaches the limit?
**
ere is an important mathematical property of R which singles out three dimensional space from all other possibilities. A closed (one-dimensional) curve can form knots only in R : in any higher dimension it can always be unknotted. ( e existence of knots also explains why three is the smallest dimension that allows chaotic particle motion.) However, general relativity does not say why space-time has three plus one dimensions. It is simply based on the fact. is deep and di cult question will be settled only in the third part of our mountain ascent.
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Challenge 737 ny
**
Henri Poincaré, who died in 1912, shortly before the general theory of relativity was nished, thought for a while that curved space was not a necessity, but only a possibility. He imagined that one could continue using Euclidean space provided light was permitted to follow curved paths. Can you explain why such a theory is impossible?
**
Can two hydrogen atoms circle each other, in their mutual gravitational eld? What Challenge 738 n would the size of this ‘molecule’ be?
** Challenge 739 n Can two light pulses circle each other, in their mutual gravitational eld?
Page 86 Page 1161
Ref. 345
**
e various motions of the Earth mentioned in the section on Galilean physics, such as its rotation around its axis or around the Sun, lead to various types of time in physics and astronomy. e time de ned by the best atomic clocks is called terrestrial dynamical time. By inserting leap seconds every now and then to compensate for the bad de nition of the second (an Earth rotation does not take 86 400, but 86 400.002 seconds) and, in minor ways, for the slowing of Earth’s rotation, one gets the universal time coordinate or UTC. en there is the time derived from this one by taking into account all leap seconds. One then has the – di erent – time which would be shown by a non-rotating clock in the centre of the Earth. Finally, there is barycentric dynamical time, which is the time that would be shown by a clock in the centre of mass of the solar system. Only using this latter time can satellites be reliably steered through the solar system. In summary, relativity says goodbye to Greenwich Mean Time, as does British law, in one of the rare cases were the law follows science. (Only the BBC continues to use it.)
**
Space agencies thus have to use general relativity if they want to get arti cial satellites to Mars, Venus, or comets. Without its use, orbits would not be calculated correctly, and satellites would miss their targets and usually even the whole planet. In fact, space agencies play on the safe side: they use a generalization of general relativity, namely the so-
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called parametrized post-Newtonian formalism, which includes a continuous check on whether general relativity is correct. Within measurement errors, no deviation has been found so far.*
Ref. 346
**
General relativity is also used by space agencies around the world to calculate the exact positions of satellites and to tune radios to the frequency of radio emitters on them. In addition, general relativity is essential for the so-called global positioning system, or GPS. is modern navigation tool** consists of 24 satellites equipped with clocks that y around the world. Why does the system need general relativity to operate? Since all the satellites, as well as any person on the surface of the Earth, travel in circles, we have dr = , and we can rewrite the Schwarzschild metric (234) as
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
dτ dt
=
−
GM rc
−
r c
(
dφ dt
)
=
−
GM rc
−
v c
.
Challenge 740 e For the relation between satellite time and Earth time we then get
(244)
dtsat dtEarth
=
− − G M vsat
rsat c
c
.
− − G M
rEarth c
vEarth c
(245)
Challenge 741 n Ref. 347
Can you deduce how many microseconds a satellite clock gains every day, given that the GPS satellites orbit the Earth once every twelve hours? Since only three microseconds would give a position error of one kilometre a er a single day, the clocks in the satellites must be adjusted to run slow by the calculated amount. e necessary adjustments are monitored, and so far have con rmed general relativity every single day, within experimental errors, since the system began operation.
**
Ref. 348 e gravitational constant G does not seem to change with time. e latest experiments limit its rate of change to less than 1 part in per year. Can you imagine how this can
* To give an idea of what this means, the unparametrized post-Newtonian formalism, based on general relativity, writes the equation of motion of a body of mass m near a large mass M as a deviation from the inverse square expression for the acceleration a:
Page 407
a=
GM r
+
f
GM v rc
+f
GM v rc
+f
Gm v rc
+ ëëë
(243)
Here the numerical factors fn are calculated from general relativity and are of order one. e rst two odd terms are missing because of the (approximate) reversibility of general relativistic motion: gravity wave emission, which is irreversible, accounts for the small term f ; note that it contains the small mass m instead of the large mass M. All factors fn up to f have now been calculated. However, in the solar system, only the term f has ever been detected. is situation might change with future high-precision satellite experiments. Higher-order e ects, up to f , have been measured in the binary pulsars, as discussed below.
In a parametrized post-Newtonian formalism, all factors fn, including the uneven ones, are tted through the data coming in; so far all these ts agree with the values predicted by general relativity. ** For more information, see the http://www.gpsworld.com website.
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Challenge 742 d be checked?
**
Could our experience that we live in only three spatial dimensions be due to a limitation Challenge 743 n of our senses? How?
** Challenge 744 ny Can you estimate the e ect of the tides on the colour of the light emitted by an atom?
**
Ref. 349 Challenge 745 ny
e strongest possible gravitational eld is that of a small black hole. e strongest gravitational eld ever observed is somewhat less though. In 1998, Zhang and Lamb used the X-ray data from a double star system to determine that space-time near the km sized neutron star is curved by up to 30 % of the maximum possible value. What is the corresponding gravitational acceleration, assuming that the neutron star has the same mass as the Sun?
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**
Ref. 350 Light de ection changes the angular size δ of a mass M with radius r when observed at Challenge 746 e distance d. e e ect leads to the pretty expression
δ = arcsin( r d
− RS d ) − RS r
where
RS =
GM c.
(246)
Challenge 747 ny What percentage of the surface of the Sun can an observer at in nity see? We will examine Page 486 this issue in more detail shortly.
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Ref. 351 Challenge 748 ny
ere is no way for a single (and point-like) observer to distinguish the e ects of gravity from those of acceleration. is property of nature allows one to make a strange statement: things fall because the surface of the Earth accelerates towards them. erefore, the weight of an object results from the surface of the Earth accelerating upwards and pushing against the object. at is the principle of equivalence applied to everyday life. For the same reason, objects in free fall have no weight.
Let us check the numbers. Obviously, an accelerating surface of the Earth produces a weight for each body resting on it. is weight is proportional to the inertial mass. In other words, the inertial mass of a body is identical to the gravitational mass. is is indeed observed in experiments, and to the highest precision achievable. Roland von Eőtvős* performed many such high-precision experiments throughout his life, without nding any discrepancy. In these experiments, he used the fact that the inertial mass determines centrifugal e ects and the gravitational mass determines free fall. (Can you imagine how he tested the equality?) Recent experiments showed that the two masses agree to one part
* Roland von Eőtvős (b. 1848 Budapest, d. 1919 Budapest), Hungarian physicist. He performed many highprecision gravity experiments; among other discoveries, he discovered the e ect named a er him. e uni-
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 351 Page 77
Page 133
in − . However, the mass equality is not a surprise. Remembering the de nition
of mass ratio as negative inverse acceleration ratio, independently of the origin of the acceleration, we are reminded that mass measurements cannot be used to distinguish between inertial and gravitational mass. As we have seen, the two masses are equal by de nition in Galilean physics, and the whole discussion is a red herring. Weight is an intrinsic e ect of mass.
e equality of acceleration and gravity allows us to imagine the following. Imagine stepping into a li in order to move down a few stories. You push the button. e li is pushed upwards by the accelerating surface of the Earth somewhat less than is the building; the building overtakes the li , which therefore remains behind. Moreover, because of the weaker push, at the beginning everybody inside the li feels a bit lighter. When the contact with the building is restored, the li is accelerated to catch up with the accelerating surface of the Earth. erefore we all feel as if we were in a strongly accelerating car, pushed in the direction opposite to the acceleration: for a short while, we feel heavier, until the li arrives at its destination.
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Challenge 749 ny
W
?
FIG-
Vires acquirit eundo.
Vergilius*
“ ” An accelerating car will soon catch up with an object thrown forward from
U R E 181 A puzzle
it. For the same reason, the surface of the Earth soon catches up with a stone
thrown upwards, because it is continually accelerating upwards. If you enjoy this way of
seeing things, imagine an apple falling from a tree. At the moment when it detaches, it
stops being accelerated upwards by the branch. e apple can now enjoy the calmness of
real rest. Because of our limited human perception, we call this state of rest free fall. Un-
fortunately, the accelerating surface of the Earth approaches mercilessly and, depending
on the time for which the apple stayed at rest, the Earth hits it with a greater or lesser
velocity, leading to more or less severe shape deformation.
Falling apples also teach us not to be disturbed any more by the statement that gravity
is the uneven running of clocks with height. In fact, this statement is equivalent to saying
that the surface of the Earth is accelerating upwards, as the discussion above shows.
Can this reasoning be continued inde nitely? We can go on for quite a while. It is
fun to show how the Earth can be of constant radius even though its surface is acceler-
ating upwards everywhere. We can thus play with the equivalence of acceleration and
gravity. However, this equivalence is only useful in situations involving only one acceler-
ating body. e equivalence between acceleration and gravity ends as soon as two falling
objects are studied. Any study of several bodies inevitably leads to the conclusion that
gravity is not acceleration; gravity is curved space-time.
versity of Budapest is named a er him. * ‘Going it acquires strength.’ Publius Vergilius Maro (b. 70 175.
Andes, d. 19
Brundisium), Aeneis 4,
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–
Many aspects of gravity and curvature can be understood with no or only a little mathematics. e next section will highlight some of the di erences between universal gravity and general relativity, showing that only the latter description agrees with experiment. A er that, a few concepts relating to the measurement of curvature are introduced and applied to the motion of objects and space-time. If the reasoning gets too involved for a
rst reading, skip ahead. In any case, the section on the stars, cosmology and black holes again uses little mathematics.
.
–
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I have the impression that Einstein understands relativity theory very well.
“ Chaim Weitzmann, rst president of Israel ” Before we tackle the details of general relativity, we will explore how the motion of objects
and light di ers from that predicted by universal gravity, and how these di erences can be measured.
W
Gravity is strong near horizons. obey
is happens when the mass M and the distance scale R
GM Rc
.
(247)
erefore, gravity is strong mainly in three situations: near black holes, near the horizon of the universe, and at extremely high particle energies. e rst two cases are explored later on, while the last will be explored in the third part of our mountain ascent. In contrast, in most parts of the universe there are no nearby horizons; in these cases, gravity is a weak e ect. Despite the violence of avalanches or of falling asteroids, in everyday life gravity is much weaker than the maximum force. On the Earth the ratio just mentioned is only about − . In this and all cases of everyday life, gravitation can still be approximated by a eld, despite what was said above. ese weak eld situations are interesting because they are simple to understand; they mainly require for their explanation the different running of clocks at di erent heights. Weak eld situations allow us to mention space-time curvature only in passing, and allow us to continue to think of gravity as a source of acceleration. However, the change of time with height already induces many new and interesting e ects. e only thing we need is a consistent relativistic treatment.
TT
Ref. 352
In , the Austrian physicist Hans irring published two simple and beautiful predictions of motions, one of them with his collaborator Josef Lense. Neither motion appears in universal gravity, but they both appear in general relativity. Figure shows these predictions.
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universal gravity prediction
THIRRING EFFECT relativistic prediction
Moon
a
m
Earth
M
universe or mass shell
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universal gravity prediction Foucault's pendulum or orbiting satellite
THIRRING-LENSE EFFECT relativistic prediction
Earth
universe or mass shell
Earth
F I G U R E 182 The Thirring and the Thirring–Lense effects
Challenge 750 ny
In the rst example, nowadays called the irring e ect, centrifugal accelerations as well as Coriolis accelerations for masses in the interior of a rotating mass shell are predicted. irring showed that if an enclosing mass shell rotates, masses inside it are attracted towards the shell. e e ect is very small; however, this prediction is in stark contrast to that of universal gravity, where a spherical mass shell – rotating or not – has no e ect on masses in its interior. Can you explain this e ect using the gure and the mattress analogy?
e second e ect, the irring–Lense e ect,* is more famous. General relativity predicts that an oscillating Foucault pendulum, or a satellite circling the Earth in a polar orbit, does not stay precisely in a xed plane relative to the rest of the universe, but that the rotation of the Earth drags the plane along a tiny bit. is frame-dragging, as the effect is also called, appears because the Earth in vacuum behaves like a rotating ball in a foamy mattress. When a ball or a shell rotates inside the foam, it partly drags the foam along with it. Similarly, the Earth drags some vacuum with it, and thus turns the plane of the pendulum. For the same reason, the Earth’s rotation turns the plane of an orbiting satellite.
e irring–Lense or frame-dragging e ect is extremely small. It was measured for the rst time in by an Italian group led by Ignazio Ciufolini, and then again by the same group in the years up to . ey followed the motion of two special arti cial satellites – shown in Figure – consisting only of a body of steel and some Cat’s-eyes.
* Even though the order of the authors is Lense and irring, it is customary (but not universal) to stress the idea of Hans irring by placing him rst.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 353 Ref. 354
e group measured the satellite’s motion around the Earth with extremely high precision, making use of re ected laser pulses. is method allowed this low-budget experiment to beat by many years the e orts of much larger but much more sluggish groups.* e results con rm the predictions of general relativity with an error of about %.
Frame dragging e ects have also been measured in binary star systems. is is possible if one of the stars is a pulsar, because such stars send out regular radio pulses, e.g. every millisecond, with extremely high precision. By measuring the exact times when the pulses arrive on Earth, one can deduce the way these stars move and con rm that such subtle e ects as frame dragging do take place.
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**
Ref. 355 Page 536
Frame-dragging and the irring–Lense e ect can be F I G U RE 183 The LAGEOS seen as special cases of gravitomagnetism. (We will show satellites: metal spheres with a the connection below.) is approach to gravity, already diameter of 60 cm, a mass of studied in the nineteenth century by Holzmüller and by 407 kg, and covered with 426 Tisserand, has become popular again in recent years, espe- retroreflectors cially for its didactic advantages. As mentioned above, talking about a gravitational eld is always an approximation. In the case of weak gravity, such as occurs in everyday life, the approximation is very good. Many relativistic e ects can be described in terms of the gravitational eld, without using the concept of space curvature or the metric tensor. Instead of describing the complete space-time mattress, the gravitational- eld model only describes the deviation of the mattress from the at state, by pretending that the deviation is a separate entity, called the gravitational eld. But what is the relativistically correct way to describe the gravitational
eld? We can compare the situation to electromagnetism. In a relativistic description of elec-
trodynamics, the electromagnetic eld has an electric and a magnetic component. e electric eld is responsible for the inverse-square Coulomb force. In the same way, in a relativistic description of (weak) gravity,*** the gravitational eld has an gravitoelectric and a gravitomagnetic component. e gravitoelectric eld is responsible for the inverse square acceleration of gravity; what we call the gravitational eld in everyday life is the gravitoelectric part of the full relativistic gravitational eld.
In nature, all components of energy–momentum tensor produce gravity e ects. In other words, it is not only mass and energy that produce a eld, but also mass or energy currents. is latter case is called gravitomagnetism (or frame dragging). e name is due to the analogy with electrodynamics, where it is not only charge density that produces a
eld (the electric eld), but also charge current (the magnetic eld). In the case of electromagnetism, the distinction between magnetic and electric eld
* One is the so-called Gravity Probe B satellite experiment, which should signi cantly increase the measurement precision; the satellite was put in orbit in 2005, a er 30 years of planning. ** is section can be skipped at rst reading. *** e approximation requires low velocities, weak elds, and localized and stationary mass–energy distributions.
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•.
Ref. 356 Page 523
depends on the observer; each of the two can (partly) be transformed into the other. Grav-
itation is exactly analogous. Electromagnetism provides a good indication as to how the
two types of gravitational elds behave; this intuition can be directly transferred to grav-
ity. In electrodynamics, the motion x(t) of a charged particle is described by the Lorentz
equation
mx¨ = qE − qx˙ B .
(248)
In other words, the change of speed is due to electric elds E, whereas magnetic elds B
give a velocity-dependent change of the direction of velocity, without changing the speed
itself. Both changes depend on the value of the charge q. In the case of gravity this expres-
sion becomes
mx¨ = mG − mx˙ H .
(249)
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e role of charge is taken by mass. In this expression we already know the eld G, given
by
G
=
∇φ
=
∇
GM r
=
−
GMx r
.
(250)
Ref. 357 Page 536
Challenge 751 ny
As usual, the quantity φ is the (scalar) potential. e eld G is the usual gravitational
eld of universal gravity, produced by every mass, and in this context is called the grav-
itoelectric eld; it has the dimension of an acceleration. Masses are the sources of the
gravitoelectric eld. e gravitoelectric eld obeys ∆G = − πGρ, where ρ is the mass
density. A static eld G has no vortices; it obeys ∆ G = .
It is not hard to show that if gravitoelectric elds exist,
gravitomagnetic elds must exist as well; the latter appear
whenever one changes from an observer at rest to a moving one. M
rod
(We will use the same argument in electrodynamics.) A particle
falling perpendicularly towards an in nitely long rod illustrates
v
the point, as shown in Figure . An observer at rest with respect
to the rod can describe the whole situation with gravitoelectric
m
particle
forces alone. A second observer, moving along the rod with con- F I G URE 184 The reality
stant speed, observes that the momentum of the particle along of gravitomagnetism
the rod also increases. is observer will thus not only measure
a gravitoelectric eld; he also measures a gravitomagnetic eld. Indeed, a mass moving
with velocity v produces a gravitomagnetic ( -) acceleration on a test mass m given by
ma = −mv H
(251)
where, almost as in electrodynamics, the static gravitomagnetic eld H obeys
H = ∇ A = πNρv
(252)
where ρ is mass density of the source of the eld and N is a proportionality constant. e
quantity A is called the gravitomagnetic vector potential. In nature, there are no sources for the gravitomagnetic eld; it thus obeys ∇H = . e gravitomagnetic eld has dimension
of inverse time, like an angular velocity.
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Challenge 752 ny
When the situation in Figure is evaluated, we nd that the proportionality constant
N is given by
N
=
G c
=
.
ë
−
m kg ,
(253)
Challenge 753 n
an extremely small value. We thus nd that as in the electrodynamic case, the gravito-
magnetic eld is weaker than the gravitoelectric eld by a factor of c . It is thus hard to
observe. In addition, a second aspect renders the observation of gravitomagnetism even
more di cult. In contrast to electromagnetism, in the case of gravity there is no way to
observe pure gravitomagnetic elds (why?); they are always mixed with the usual, gravito-
electric ones. For these reasons, gravitomagnetic e ects were measured for the rst time
only in the s. We see that universal gravity is the approximation of general relativity
that arises when all gravitomagnetic e ects are neglected.
In summary, if a mass moves, it also produces a gravitomagnetic eld. How can one
imagine gravitomagnetism? Let’s have a look at its e ects. e experiment of Figure
showed that a moving rod has the e ect to slightly accelerate a test mass in the same dir-
ection. In our metaphor of the vacuum as a mattress, it looks as if a moving rod drags the
vacuum along with it, as well as any test mass that happens to be in that region. Gravito-
magnetism can thus be seen as vacuum dragging. Because of a widespread reluctance to
think of the vacuum as a mattress, the expression frame dragging is used instead.
In this description, all frame dragging e ects are gravitomagnetic e ects. In particular,
a gravitomagnetic eld also appears when a large mass rotates, as in the irring–Lense
e ect of Figure . For an angular momentum J the gravitomagnetic eld H is a dipole
eld; it is given by
H=∇ h=∇
−J x r
(254)
exactly as in the electrodynamic case. e gravitomagnetic eld around a spinning mass
has three main e ects.
First of all, as in electromagnetism, a spinning test particle with angular momentum
S feels a torque if it is near a large spinning mass with angular momentum J. is torque
T is given by
T
=
dS dt
=
S
H.
(255)
Challenge 754 ny
e torque leads to the precession of gyroscopes. For the Earth, this e ect is extremely small: at the North Pole, the precession has a conic angle of . milli-arcseconds and a rotation rate of the order of − times that of the Earth.
Since for a torque one has T = Ω˙ S, the dipole eld of a large rotating mass with angular momentum J yields a second e ect. An orbiting mass will experience precession of its orbital plane. Seen from in nity one gets, for an orbit with semimajor axis a and eccentricity e,
Dvipsbugw
Ω˙ = − H
=
−
G c
J x
+
G c
(Jx)x x
=
G c
a
(
J −e
)
(256)
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•.
Ref. 353 Challenge 756 ny
which is the prediction of Lense and irring.* e e ect is extremely small, giving a change of only ′′ per orbit for a satellite near the surface of the Earth. Despite this smallness and a number of larger e ects disturbing it, Ciufolini’s team have managed to con-
rm the result. As a third e ect of gravitomagnetism, a rotating mass leads to the precession of the
periastron. is is a similar e ect to the one produced by space curvature on orbiting masses even if the central body does not rotate. e rotation just reduces the precession due to space-time curvature. is e ect has been fully con rmed for the famous binary pulsar PSR B + , as well as for the ‘real’ double pulsar PSR J - , discovered in .
is latter system shows a periastron precession of . ° a, the largest value observed so far.
e split into gravitoelectric and gravitomagnetic e ects is thus a useful approximation to the description of gravity. It also helps to answer questions such as: How can gravity keep the Earth orbiting around the Sun, if gravity needs minutes to get from the Sun to us? To nd the answer, thinking about the electromagnetic analogy can help. In addition, the split of the gravitational eld into gravitoelectric and gravitomagnetic components allows a simple description of gravitational waves.
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G
One of the most fantastic predictions of physics is the existence of gravitational waves. Gravity waves** prove that empty space itself has the ability to move and vibrate. e basic idea is simple. Since space is elastic, like a large mattress in which we live, space should be able to oscillate in the form of propagating waves, like a mattress or any other elastic medium.
TA B L E 36 The expected spectrum of gravitational waves
F
W
N
E
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< − Hz − Hz– − Hz
Tm Tm– Gm
− Hz– Hz Hz– Hz Hz– Hz Hz
Gm– Mm Mm– km km– m <m
extremely low frequencies
slow binary star systems, supermassive black holes
very low frequencies fast binary star systems, massive black holes, white dwarf vibrations
low frequencies
binary pulsars, medium and light black holes
medium frequencies supernovae, pulsar vibrations
high frequencies
unknown; maybe future human-made sources
maybe unknown cosmological sources
Challenge 755 ny
* A homogeneous spinning sphere has an angular momentum given by J = MωR . ** To be strict, the term ‘gravity wave’ has a special meaning: gravity waves are the surface waves of the sea, where gravity is the restoring force. However, in general relativity, the term is used interchangeably with ‘gravitational wave’.
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Ref. 358
Ref. 359 Challenge 757 ny
Jørgen Kalckar and Ole Ulfbeck have given a simple argument for the necessity of gravitational waves based on the existence of a maximum speed. ey studied two equal masses falling towards each other under the e ect of gravita- F I G U R E 185 A Gedanken experiment showing the tional attraction, and imagined a necessity of gravity waves spring between them. Such a spring will make the masses bounce towards each other again and again. e central spring stores the kinetic energy from the falling masses. e energy value can be measured by determining the length by which the spring is compressed. When the spring expands again and hurls the masses back into space, the gravitational attraction will gradually slow down the masses, until they again fall towards each other, thus starting the same cycle again.
However, the energy stored in the spring must get smaller with each cycle. Whenever a sphere detaches from the spring, it is decelerated by the gravitational pull of the other sphere. Now, the value of this deceleration depends on the distance to the other mass; but since there is a maximal propagation velocity, the e ective deceleration is given by the distance the other mass had when its gravity e ect started out towards the second mass. For two masses departing from each other, the e ective distance is thus somewhat smaller than the actual distance. In short, while departing, the real deceleration is larger than the one calculated without taking the time delay into account.
Similarly, when one mass falls back towards the other, it is accelerated by the other mass according to the distance it had when the gravity e ect started moving towards it.
erefore, while approaching, the acceleration is smaller than the one calculated without time delay.
erefore, the masses arrive with a smaller energy than they departed with. At every bounce, the spring is compressed a little less. e di erence between these two energies is lost by each mass: it is taken away by space-time, in other words, it is radiated away as gravitational radiation. e same thing happens with mattresses. Remember that a mass deforms the space around it as a metal ball on a mattress deforms the surface around it.* If two metal balls repeatedly bang against each other and then depart again, until they come back together, they will send out surface waves on the mattress. Over time, this e ect will reduce the distance that the two balls depart from each other a er each bang. As we will see shortly, a similar e ect has already been measured, where the two masses, instead of being repelled by a spring, were orbiting each other.
A simple mathematical description of gravity waves follows from the split into gravitomagnetic and gravitoelectric e ects. It does not take much e ort to extend gravitomagnetostatics and gravitoelectrostatics to gravitodynamics. Just as electrodynamics can be deduced from Coulomb’s attraction when one switches to other inertial observers, gravitodynamics can be deduced from universal gravity. One gets the four equations
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* However, in contrast to actual mattresses, there is no friction between the ball and the mattress.
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•.
∇ G = − πGρ
,
∇
G
=
−
∂H ∂t
∇H =
,
∇
H=−
πGρv
+
N G
∂G ∂t
.
(257)
Challenge 758 ny Challenge 759 e
We have met two of these equations already. e two other equations are expanded versions of what we have encountered, taking time-dependence into account. Except for a factor of instead of in the last equation, the equations for gravtitodynamics are the same as Maxwell’s equations for electrodynamics.* ese equations have a simple property: in vacuum, one can deduce from them a wave equation for the gravitoelectric and the gravitomagnetic elds G and H. (It is not hard: try!) In other words, gravity can behave like a wave: gravity can radiate. All this follows from the expression of universal gravity when applied to moving observers, with the requirement that neither observers nor energy can move faster than c. Both the above argument involving the spring and the present mathematical argument use the same assumptions and arrive at the same conclusion.
A few manipulations show that the speed of these waves is given by
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Page 563
c=
G N
.
is result corresponds to the electromagnetic expression
(258)
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c= ε µ .
(259)
Ref. 361 Ref. 363
e same letter has been used for the two speeds, as they are identical. Both in uences travel with the speed common to all energy with vanishing rest mass. (We note that this is, strictly speaking, a prediction: the speed of gravitational waves has not yet been measured. A claim from to have done so has turned out to be false.)
How should one imagine these waves? We sloppily said above that a gravitational wave corresponds to a surface wave of a mattress; now we have to do better and imagine that we live inside the mattress. Gravitational waves are thus moving and oscillating deformations of the mattress, i.e., of space. Like mattress waves, it turns out that gravity waves are transverse. us they can be polarized. (Surface waves on mattresses cannot, because in two dimensions there is no polarization.) Gravity waves can be polarized in two independent ways. e e ects of a gravitational wave are shown in Figure , for both linear
Ref. 360
* e additional factor re ects the fact that the ratio between angular momentum and energy (the ‘spin’) of gravity waves is di erent from that of electromagnetic waves. Gravity waves have spin 2, whereas electromagnetic waves have spin 1. Note that since gravity is universal, there can exist only a single kind of spin 2 radiation particle in nature. is is in strong contrast to the spin 1 case, of which there are several examples in nature.
By the way, the spin of radiation is a classical property. e spin of a wave is the ratio E Lω, where E is the energy, L the angular momentum, and ω is the angular frequency. For electromagnetic waves, this ratio is equal to 1; for gravitational waves, it is 2.
Note that due to the approximation at the basis of the equations of gravitodynamics, the equations are neither gauge-invariant nor generally covariant.
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Wave, moving perpendicularly to page
t1
t2
t3
t4
t5
No wave (all times)
linear polarization in + direction linear polarization in x direction
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circular polarization in R sense
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circular polarization in L sense
F I G U R E 186 Effects on a circular or spherical body due to a plane gravitational wave moving in a direction perpendicular to the page
and circular polarization.* We note that the waves are invariant under a rotation by π and that the two linear polarizations di er by an angle π ; this shows that the particles corresponding to the waves, the gravitons, are of spin . (In general, the classical radi-
* A (small amplitude) plane gravity wave travelling in the z-direction is described by a metric given by
=
− + hxx
hx y
hx y
− + hxx
−
(260)
where its two components, whose amplitude ratio determine the polarization, are given by
hab = Bab sin(kz − ωt + φab)
(261)
as in all plane harmonic waves. e amplitudes Bab, the frequency ω and the phase φ are determined by the speci c physical system. e general dispersion relation for the wave number k resulting from the wave
equation is
ω k
=
c
(262)
and shows that the waves move with the speed of light.
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•.
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Challenge 760 ny
Challenge 761 ny Ref. 362
ation eld for a spin S particle is invariant under a rotation by π S. In addition, the two orthogonal linear polarizations of a spin S particle form an angle π S. For the photon, for example, the spin is ; indeed, its invariant rotation angle is π and the angle formed by the two polarizations is π .)
If we image empty space as a mattress that lls space, gravitational waves are wobbling deformations of the mattress. More precisely, Figure shows that a wave of circular polarization has the same properties as a corkscrew advancing through the mattress. We will discover later on why the analogy between a corkscrew and a gravity wave with circular polarization works so well. Indeed, in the third part we will nd a speci c model of the space-time mattress material that automatically incorporates corkscrew waves (instead of the spin waves shown by ordinary latex mattresses).
How does one produce gravitational waves? Obviously, masses must be accelerated. But how exactly? e conservation of energy forbids mass monopoles from varying in strength. We also know from universal gravity that a spherical mass whose radius oscillates would not emit gravitational waves. In addition, the conservation of momentum forbids mass dipoles from changing.
As a result, only changing quadrupoles can emit waves. For example, two masses in orbit around each other will emit gravitational waves. Also, any rotating object that is not cylindrically symmetric around its rotation axis will do so. As a result, rotating an arm leads to gravitational wave emission. Most of these statements also apply to masses in mattresses. Can you point out the di erences?
Einstein found that the amplitude h of waves at a distance r from a source is given, to a good approximation, by the second derivative of the retarded quadrupole moment Q:
Dvipsbugw
hab =
G c
r dtt Qarebt
=
G c
r
dtt
Qab(t
−
r
c) .
(264)
Challenge 762 ny
is expression shows that the amplitude of gravity waves decreases only with r, in contrast to naive expectations. However, this feature is the same as for electromagnetic waves. In addition, the small value of the prefactor, . ë − Wm s, shows that truly gigantic systems are needed to produce quadrupole moment changes that yield any detectable length variations in bodies. To be convinced, just insert a few numbers, keeping in mind that the best present detectors are able to measure length changes down to h = δl l = − . e production of detectable gravitational waves by humans is probably impossible.
Gravitational waves, like all other waves, transport energy.* If we apply the general formula for the emitted power P to the case of two masses m and m in circular orbits
In another gauge, a plane wave can be written as
c ( + φ) A
A
A
=
A A
− + φ hxy
hx y
− + hxx
A
−
(263)
Page 552 Ref. 357
where
φ
and
A
are
the
potentials
such
that
G
=
∇φ
−
∂A c∂t
and
H
=
∇
A.
* Gravitomagnetism and gravitoelectricity allow one to de ne a gravitational Poynting vector. It is as easy
to de ne and use as in the case of electrodynamics.
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Ref. 324 around each other at distance l and get
P
=
−
dE dt
=
G c
Q...raebt Q...raebt =
G mm c m +m
lω
(265)
which, using Kepler’s relation π r T = G(m + m ), becomes
P=
G c
(m m ) (m l
+m ) .
(266)
Ref. 324 Challenge 763 ny
Ref. 364 Challenge 764 ny
Page 394
For elliptical orbits, the rate increases with the ellipticity, as explained by Goenner. Insert-
ing the values for the case of the Earth and the Sun, we get a power of about W, and
a value of W for the Jupiter–Sun system. ese values are so small that their e ect
cannot be detected at all.
For all orbiting systems, the frequency of the waves is twice the orbital frequency, as
you might want to check. ese low frequencies make it even more di cult to detect
them.
As a result, the only observation of e ects
of gravitational waves to date is in binary pulsars. Pulsars are small but extremely dense
time delay
stars; even with a mass equal to that of the Sun, 1975 1980
1990
2000
their diameter is only about km. erefore
they can orbit each other at small distances
year
and high speeds. Indeed, in the most famous
binary pulsar system, PSR + , the two stars
orbit each other in an amazing . h, even
though their semimajor axis is about Mm,
just less than twice the Earth–Moon distance.
Since their orbital speed is up to km s, the
system is noticeably relativistic.
Pulsars have a useful property: because of their rotation, they emit extremely regular ra-
preliminary figure
dio pulses (hence their name), o en in milli- F I G U R E 187 Comparison between measured
second periods. erefore it is easy to follow time delay for the periastron of the binary
their orbit by measuring the change of pulse pulsar PSR 1913+16 and the prediction due to arrival time. In a famous experiment, a team energy loss by gravitational radiation
of astrophysicists led by Joseph Taylor** meas-
ured the speed decrease of the binary pulsar system just mentioned. Eliminating all other
e ects and collecting data for years, they found a decrease in the orbital frequency,
shown in Figure . e slowdown is due to gravity wave emission. e results exactly
t the prediction by general relativity, without any adjustable parameter. (You might want
to check that the e ect must be quadratic in time.) is is the only case so far in which
general relativity has been tested up to (v c) precision. To get an idea of the precision,
consider that this experiment detected a reduction of the orbital diameter of . mm per
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** In 1993 he shared the Nobel Prize in physics for his life’s work.
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•.
L1
mirror
mirror
light source
L2
F I G U R E 188 Detection of gravitational waves
Dvipsbugw
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Ref. 364 Ref. 324
Challenge 766 r Ref. 361
Challenge 767 ny
orbit, or . m per year! e measurements were possible only because the two stars in this system are neutron stars with small size, large velocities and purely gravitational interactions. e pulsar rotation period around its axis, about ms, is known to eleven digits of precision, the orbital time of . h is known to ten digits and the eccentricity of the orbit to six digits.
e direct detection of gravitational waves is one of the aims of experimental general relativity. e race has been on since the s. e basic idea is simple, as shown in Figure : take four bodies, usually four mirrors, for which the line connecting one pair is perpendicular to the line connecting the other pair. en measure the distance changes of each pair. If a gravitational wave comes by, one pair will increase in distance and the other will decrease, at the same time.
Since detectable gravitational waves cannot be produced by humans, wave detection rst of all requires the patience to wait for a strong enough wave to come by. Secondly, a system able to detect length changes of the order of − or better is needed – in other words, a lot of money. Any detection is guaranteed to make the news on television.*
It turns out that even for a body around a black hole, only about % of the rest mass can be radiated away as gravitational waves; furthermore, most of the energy is radiated during the nal fall into the black hole, so that only quite violent processes, such as black hole collisions, are good candidates for detectable gravity wave sources.
Gravitational waves are a fascinating area of study. ey still provide many topics to explore. For example: can you nd a method to measure their speed? A well-publicized but false claim appeared in . Indeed, any correct measurement that does not simply use two spaced detectors of the type of Figure would be a scienti c sensation.
For the time being, another question on gravitational waves remains open: If all change is due to motion of particles, as the Greeks maintained, how do gravity waves t into the picture? If gravitational waves were made of particles, space-time would also have to be. We have to wait until the beginning of the third part of our ascent to say more.
B
As we know from above, gravity also in uences the motion of light. A distant observer measures a changing value for the light speed v near a mass. (Measured at his own location, the speed of light is of course always c.) It turns out that a distant observer measures
Ref. 365 * e topic of gravity waves is full of interesting sidelines. For example, can gravity waves be used to power Challenge 765 ny a rocket? Yes, say Bonnor and Piper. You might ponder the possibility yourself.
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Ref. 366
a lower speed, so that for him, gravity has the same e ects as a dense optical medium. It
takes only a little bit of imagination to see that this e ect will thus increase the bending
of light near masses already deduced in by Soldner for universal gravity.
e following is a simple way to calculate the e ect. As usual, we
use the coordinate system of at space-time at in nity. e idea is to
do all calculations to rst order, as the value of the bending is very
small. e angle of de ection α, to rst order, is simply
α
∫ α = −
∂v ∂x
d
y
,
(267)
where v is the speed of light measured by a distant observer. (Can you Challenge 768 ny con rm this?) e next step is to use the Schwarzschild metric
b M y
Dvipsbugw
dτ
=( −
GM rc
)dt
− (c
dr −
GM r
)
−
r c
dφ
(268)
Challenge 769 ny and transform it into (x, y) coordinates to rst order. at gives
dτ
=( −
GM rc
)dt
−( +
GM rc
)
c
(dx
+ dy )
(269)
x
F I G U R E 189 Calculating the bending of light by a mass
which again to rst order leads to
∂v ∂x
=(
−
GM rc
)c
.
(270)
Challenge 770 ny
is con rms what we know already, namely that distant observers see light slowed down
when passing near a mass. us we can also speak of a height-dependent index of refrac-
tion. In other words, constant local light speed leads to a global slowdown.
Inserting the last result in ( ) and using a clever substitution, we get a deviation
angle α given by
α=
GM cb
(271)
Page 133 Challenge 771 ny
Ref. 367 Page 133
where the distance b is the so-called impact parameter of the approaching light beam. e resulting deviation angle α is twice the result we found for universal gravity. For a
beam just above the surface of the Sun, the result is the famous value of . ′′ which was con rmed by the measurement expedition of . (How did they measure the deviation angle?) is was the experiment that made Einstein famous, as it showed that universal gravity is wrong. In fact, Einstein was lucky. Two earlier expeditions organized to measure the value had failed. In , it was impossible to take data because of rain, and in in Crimea, scientists were arrested (by mistake) as spies, because the world war had just begun. But in , Einstein had already published an incorrect calculation, giving only the Soldner value with half the correct size; only in , when he completed general relativity, did he nd the correct result. erefore Einstein became famous only because of the
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•.
Ref. 346, Ref. 323, Ref. 324
Page 468
Page 417 Challenge 772 ny
failure of the two expeditions that took place before he published his correct calculation. For high-precision experiments around the Sun, it is more e ective to measure the
bending of radio waves, as they encounter fewer problems when they propagate through the solar corona. So far, over a dozen independent experiments have done so, using radio sources in the sky which lie on the path of the Sun. ey have con rmed general relativity’s prediction within a few per cent.
So far, bending of radiation has also been observed near Jupiter, near certain stars, near several galaxies and near galaxy clusters. For the Earth, the angle is at most nrad, too small to be measured yet, even though this may be feasible in the near future. ere is a chance to detect this value if, as Andrew Gould proposes, the data of the satellite Hipparcos, which is taking precision pictures of the night sky, are analysed properly in the future.
Of course, the bending of light also con rms that in a triangle, the sum of the angles does not add up to π, as is predicted later for curved space. (What is the sign of the curvature?)
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T
Ref. 368 Ref. 369 Ref. 370 Challenge 774 ny
e above calculation of the bending of light near masses shows that for a distant observer, light is slowed down near a mass. Constant local light speed leads to a global light speed slowdown. If light were not slowed down near a mass, it would have to go faster than c for an observer near the mass!* In , Irwin Shapiro had the idea to measure this e ect. He proposed two methods. e rst was to send radar pulses to Venus, and measure the time taken for the re ection to get back to Earth. If the signals pass near the Sun, they will be delayed. e second was to use an arti cial satellite communicating with Earth.
e rst measurement was published in , and directly con rmed the prediction of general relativity within experimental errors. All subsequent tests of the same type, such as the one shown in Figure , have also con rmed the prediction within experimental errors, which nowadays are of the order of one part in a thousand. e delay has also been measured in binary pulsars, as there are a few such systems in the sky for which the line of sight lies almost precisely in the orbital plane.
e simple calculations presented here suggest a challenge: Is it also possible to describe full general relativity – thus gravitation in strong elds – as a change of the speed of light with position and time induced by mass and energy?
E
Astronomy allows precise measurements of motions. So, Einstein rst of all tried to apply his results to the motion of planets. He looked for deviations of their motions from the predictions of universal gravity. Einstein found such a deviation: the precession of the perihelion of Mercury. e e ect is shown in Figure . Einstein said later that the moment he found out that his calculation for the precession of Mercury matched observations was one of the happiest moments of his life.
Challenge 773 e
* A nice exercise is to show that the bending of a slow particle gives the Soldner value, whereas with increasing speed, the value of the bending approaches twice that value. In all these considerations, the rotation of the mass has been neglected. As the e ect of frame dragging shows, rotation also changes the deviation angle; however, in all cases studied so far, the in uence is below the detection threshold.
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Earth orbit Sun
10 May 1970 31 March 1970
Mariner 6 orbit
Time delay (µs)
240
180
120
60
0 Jan Feb Mar Apr May Jun 1970
F I G U R E 190 Time delay in radio signals – one of the experiments by Irwin Shapiro
a: semimajor axis
periastron a
M
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F I G U R E 191 The orbit around a central body in general relativity
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Ref. 323, Ref. 324 Page 384
Challenge 775 e
e calculation is not di cult. In universal gravity, orbits are calculated by setting agrav = acentri, in other words, by setting GM r = ω r and xing energy and angular momentum. e mass of the orbiting satellite does not appear explicitly.
In general relativity, the mass of the orbiting satellite is made to disappear by rescaling energy and angular momentum as e = E mc and j = J m. Next, the space curvature needs to be included. We use the Schwarzschild metric ( ) mentioned above to deduce that the initial condition for the energy e, together with its conservation, leads to a relation between proper time τ and time t at in nity:
dt dτ
=
e − GM rc
,
(272)
whereas the initial condition on the angular momentum j and its conservation imply that
dφ dτ
=
j r
.
(273)
ese relations are valid for any particle, whatever its mass m. Inserting all this into the Schwarzschild metric, we nd that the motion of a particle follows
(
dr cdτ
)
+V
(j, r) = e
(274)
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•.
where the e ective potential V is given by
V (J, r) = ( −
GM rc
)(
+
r
j c
).
(275)
Challenge 776 ny e expression di ers slightly from the one in universal gravity, as you might want to Challenge 777 e check. We now need to solve for r(φ). For circular orbits we get two possibilities
r=
GM c
−
(
GM cj
)
(276)
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where the minus sign gives a stable and the plus sign an unstable orbit. If c j GM < , no stable orbit exists; the object will impact the surface or, for a black hole, be swallowed.
ere is a stable circular orbit only if the angular momentum j is larger than GM c. We thus nd that in general relativity, in contrast to universal gravity, there is a smallest
stable circular orbit. e radius of this smallest stable circular orbit is GM c = RS. What is the situation for elliptical orbits? Setting u = r in ( ) and di erentiating,
the equation for u(φ) becomes
u′
+
u
=
GM j
+
G c
M
u
.
(277)
Without the nonlinear correction due to general relativity on the far right, the solutions Challenge 778 e are the famous conic sections
u
(φ) =
GM j
(
+ ε cos φ) ,
(278)
Page 127 Challenge 779 e
i.e. ellipses, parabolas or hyperbolas. e type of conic section depends on the value of the parameter ε, the so-called eccentricity. We know the shapes of these curves from universal gravity. Now, general relativity introduces the nonlinear term on the right-hand side of equation ( ). us the solutions are not conic sections any more; however, as the correction is small, a good approximation is given by
u
(φ) =
GM j
[
+ ε cos(φ −
GM jc
φ)] .
(279)
Challenge 780 e
e hyperbolas and parabolas of universal gravity are thus slightly deformed. Instead of elliptical orbits we get the famous rosetta path shown in Figure . Such a path is above all characterized by a periastron shi . e periastron, or perihelion in the case of the Sun, is the nearest point to the central body reached by an orbiting body. e periastron turns around the central body by an angle
α
GM π a( − ε )c
(280)
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geodesic precession
Lense– Thirring precession
N
start after one S orbit
F I G U R E 192 The geodesic effect
Earth
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Ref. 370 Page 407
for every orbit, where a is the semimajor axis. For Mercury, the value is ′′ per century. Around , this was the only known e ect that was unexplained by universal gravity; when Einstein’s calculation led him to exactly that value, he was over owing with joy for many days.
To be sure about the equality between calculation and experiment, all other e ects leading to rosetta paths must be eliminated. For some time, it was thought that the quadrupole moment of the Sun could be an alternative source of this e ect; later measurements ruled out this possibility.
In the meantime, the perihelion shi has been measured also for the orbits of Icarus, Venus and Mars around the Sun, as well as for several binary star systems. In binary pulsars, the periastron shi can be as large as several degrees per year. In all cases, expression ( ) describes the motion within experimental errors.
We note that even the rosetta orbit itself is not really stable, due to the emission of gravitational waves. But in the solar system, the power lost this way is completely negligible even over thousands of millions of years, as we saw above, so that the rosetta path remains a good description of observations.
T
When a pointed body orbits a central mass m at distance r, the direction of the tip will not be the same a er a full orbit. is e ect exists only in general relativity. e angle α describing the direction change is given by
α= π −
−
Gm rc
πGm rc
.
(281)
Challenge 781 e
e angle change is called the geodesic e ect – ‘geodetic’ in other languages. It is a further consequence of the split into gravitoelectric and gravitomagnetic elds, as you may want to show. Obviously, it does not exist in universal gravity.
In cases where the pointing of the orbiting body is realized by an intrinsic rotation, such as a spinning satellite, the geodesic e ect produces a precession of the axis. us the
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•.
Ref. 371 Ref. 372 Page 309
e ect is comparable to spin–orbit coupling in atomic theory. ( e irring–Lense e ect mentioned above is analogous to spin–spin coupling.)
e geodesic e ect, or geodesic precession, was predicted by Willem de Sitter in ; in particular, he proposed detecting that the Earth–Moon system would change its pointing direction in its fall around the Sun. e e ect is tiny; for the axis of the Moon the precession angle is about . arcsec per year. e e ect was rst detected in by an Italian team for the Earth–Moon system, through a combination of radio-interferometry and lunar ranging, making use of the Cat’s-eyes deposited by Lunakhod and Apollo on the Moon. Experiments to detect it in arti cial satellites are also under way.
At rst sight, geodesic precession is similar to the omas precession found in special relativity. In both cases, a transport along a closed line results in the loss of the original direction. However, a careful investigation shows that omas precession can be added to geodesic precession by applying some additional, non-gravitational interaction, so the analogy is shaky.
is completes our discussion of weak gravity e ects. We now turn to strong gravity, where curvature cannot be neglected and where the fun is even more intense.
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C Here are some questions to think about.
** Is there a static, oscillating gravitational eld?
** Are beams of gravitational waves, analogous to beams of light, possible?
** Would two parallel beams of gravitational waves attract each other?
H
?
Challenge 782 n
We have seen that in the precise description of gravity, motion depends on space-time curvature. In order quantify this idea, we rst of all need to describe curvature itself as accurately as possible. To clarify the issue, we will start the discussion in two dimensions, and then move to three and four dimensions.
Obviously, a at sheet of paper has no curvature. If we roll it into a cone or a cylinder, it gets what is called extrinsic curvature; however, the sheet of paper still looks at for any two-dimensional animal living on it – as approximated by an ant walking over it. In other words, the intrinsic curvature of the sheet of paper is zero even if the sheet as a whole is extrinsically curved. (Can a one-dimensional space have intrinsic curvature? Is a torus intrinsically curved?)
Intrinsic curvature is thus the stronger concept, measuring the curvature which can be observed even by an ant. e surface of the Earth, the surface of an island, or the slopes of a mountain* are intrinsically curved. Whenever we talk about curvature in general
Challenge 783 e * Unless the mountain has the shape of a perfect cone. Can you con rm this?
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a
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F I G U R E 193 Positive, vanishing and negative curvature in two dimensions
Ref. 373 Challenge 784 e
relativity, we always mean intrinsic curvature, since any observer in nature is by de nition in the same situation as an ant on a surface: their experience, their actions and plans always only concern their closest neighbourhood in space and time.
But how can an ant determine whether it lives on an intrinsically curved surface?** One way is shown in Figure . e ant can check whether either the circumference of a circle or its area bears a Euclidean relation to the measured radius. She can even use the di erence between the measured and the Euclidean values as a measure for the local intrinsic curvature, if she takes the limit for vanishingly small circles and if she normalizes the values correctly. In other words, the ant can imagine to cut out a little disc around the point she is on, to iron it at and to check whether the disc would tear or produce folds. Any two-dimensional surface is intrinsically curved whenever ironing is not able to make a at street map out of it. e ‘density’ of folds or tears is related to the curvature.
is means that we can recognize intrinsic curvature also by checking whether two parallel lines stay parallel, approach each other, or depart from each other. In the rst case, such as lines on a paper cylinder, the surface is said to have vanishing intrinsic curvature; a surface with approaching parallels, such as the Earth, is said to have positive curvature, and a surface with diverging parallels, such as a saddle, is said to have negative curvature. In short, positive curvature means that we are more restricted in our movements, negative that we are less restricted. A constant curvature even implies being locked in a nite space. You might want to check this with Figure .
e third way to measure curvature uses triangles. On curved surfaces the sum of angles in a triangle is either larger or smaller than π.
Let us see how we can quantify curvature. First a question of vocabulary: a sphere with radius a is said, by de nition, to have an intrinsic curvature K = a . erefore a plane has zero curvature. You might check that for a circle on a sphere, the measured radius r, circumference C, and area A are related by
C = πr( − K r + ...) and A = πr ( − K r + ...)
(282)
** Note that the answer to this question also tells us how to distinguish real curvature from curved coordinate systems on a at space. is question is o en asked by those approaching general relativity for the rst time.
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direction of minimal curvature
•. point of interest
right angle direction of
maximal curvature
F I G U R E 194 The maximum and minimum curvature of a curved surface
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where the dots imply higher-order terms. is allows one to de ne the intrinsic curvature K, also called the Gaussian curvature, for a general point on a two-dimensional surface in either of the following two equivalent ways:
K=
lim(
r
−
C πr
)
r
or
K=
lim(
r
−
A πr
)r
.
(283)
Challenge 786 e Challenge 787 e
ese expressions allow an ant to measure the intrinsic curvature at each point for any smooth surface.* From now on in this text, curvature will always mean intrinsic curvature. Note that the curvature can be di erent from place to place, and that it can be positive, as for an egg, or negative, as for the part of a torus nearest to the hole. A saddle is another example of the latter case, but, unlike the torus, its curvature changes along all directions. In fact, it is not possible at all to t a two-dimensional surface of constant negative curvature inside three-dimensional space; one needs at least four dimensions, as you can nd out if you try to imagine the situation.
For any surface, at every point, the direction of maximum curvature and the direction of minimum curvature are perpendicular to each other. is relationship, shown in Figure , was discovered by Leonhard Euler in the eighteenth century. You might want to check this with a tea cup, with a sculpture by Henry Moore, or with any other curved object from your surroundings, such as a Volkswagen Beetle. e Gaussian curvature K de ned in ( ) is in fact the product of the two corresponding inverse curvature radii.
us, even though line curvature is not an intrinsic property, this special product is. Gaussian curvature is a measure of the intrinsic curvature. Intrinsic measures of curvature are needed if one is forced to stay inside the surface or space one is exploring. Physicists are thus particularly interested in Gaussian curvature and its higher-dimensional analogues.
For three-dimensional ‘surfaces’, the issue is a bit more involved. First of all, we have di culties imagining the situation. But we can still accept that the curvature of a small disc around a point will depend on its orientation. Let us rst look at the simplest case. If
* If the n-dimensional volume of a sphere is written as Vn = Cnrn and its (n − )-dimensional ‘surface’ as Ref. 374 On = nCn rn− , one can generalize the expressions for curvature to
K=
(n +
) lim( r
− Vn ) Cnrn r
or
K=
n lim( r
− On nCn r n−
) r
,
(284)
Challenge 785 ny as shown by Vermeil. A famous riddle is to determine the number Cn.
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Challenge 788 ny
the curvature at a point is the same in all directions, the point is called isotropic. We can imagine a small sphere around that point. In this special case, in three dimensions, the relation between the measured radius r and the measured surface area A and volume V of the sphere lead to
A = πr ( − K r + ...) and V = π r ( − K r + ...) ,
(285)
where K is the curvature for an isotropic point. is leads to
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K=
lim( −
r
A πr
)r
=
lim r −
r
A r
π =
lim
r
rexcess r
,
(286)
Challenge 789 ny
de ning the excess radius as rexcess = r − A π . We thus nd that for a threedimensional space, the average curvature is six times the excess radius of a small sphere divided by the cube of the radius. A positive curvature is equivalent to a positive excess radius, and similarly for vanishing and negative cases.
Of course, a curvature value de ned in this way is only an average over all possible directions. e precise de nition of curvature involves a disc. For points that are not isotropic, the value yielded by a disc will di er from the value calculated by using a sphere, as it will depend on the orientation of the disc. In fact, there is a relationship between all possible disc curvatures at a given point; taken together, they must form a tensor. (Why?) For a full description of curvature, we thus have to specify, as for any tensor in three dimensions, the main curvature values in three orthogonal directions.*
What are the curvature values in the space around us? Already in , the mathematician and physicist Carl-Friedrich Gauss* is said to have checked whether the three angles formed by three mountain peaks near his place of residence added up to π. Nowadays we know that the deviation δ from the angle π on the surface of a body of mass M and radius r is given by
δ = π − (α + β + γ)
AtriangleK
=
Atriangle
GM rc
.
(287)
* ese three disc values are not independent however, since together, they must yield the just-mentioned average volume curvature K. In total, there are thus three independent scalars describing the curvature in three dimensions (at each point). With the metric tensor ab and the Ricci tensor Rab to be introduced below, one possibility is to take for the three independent numbers the values R = − K, Rab Rab and detR det . * Carl-Friedrich Gauß (b. 1777 Braunschweig, d. 1855 Göttingen), German mathematician. Together with the Leonhard Euler, the most important mathematician of all times. A famous enfant prodige, when he was 19 years old, he constructed the regular heptadecagon with compass and ruler (see http://www.mathworld. wolfram.com/Heptadecagon.html). He was so proud of this result that he put a drawing of the gure on his tomb. Gauss produced many results in number theory, topology, statistics, algebra, complex numbers and di erential geometry which are part of modern mathematics and bear his name. Among his many accomplishments, he produced a theory of curvature and developed non-Euclidean geometry. He also worked on electromagnetism and astronomy.
Gauss was a di cult character, worked always for himself, and did not found a school. He published little, as his motto was: pauca sed matura. As a consequence, when another mathematician published a new result, he regularly produced a notebook in which he had noted the very same result already years before. His notebooks are now available online at http://www.sub.uni-goettingen.de.
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•. a
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F I G U R E 195 Curvature (in two dimensions) and geodesic behaviour
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is expression is typical for hyperbolic geometries. For the case of mathematical negative curvature K, the rst equality was deduced by Johann Lambert ( – ). However, it was Einstein who discovered that the negative curvature K is related to the mass and gravitation of a body. For the case of the Earth and typical mountain distances, the angle δ is of the order of − rad. Gauss had no chance to detect any deviation, and in fact he detected none. Even today, studies with lasers and high-precision apparatus have detected no deviation yet – on Earth. e right-hand factor, which measures the curvature of spacetime on the surface of the Earth, is simply too small. But Gauss did not know, as we do today, that gravity and curvature go hand in hand.
C
-
Challenge 790 e
Notre tête est ronde pour permettre à la pensée de changer de direction.**
“ Francis Picabia ” In nature, with four space-time dimensions, specifying curvature requires a more in-
volved approach. First of all, the use of space-time coordinates automatically introduces the speed of light c as limit speed, which is a central requirement in general relativity. Furthermore, the number of dimensions being four, we expect a value for an average curvature at a point, de ned by comparing the -volume of a -sphere in space-time with the one deduced from the measured radius; then we expect a set of ‘almost average’ curvatures de ned by -volumes of -spheres in various orientations, plus a set of ‘low-level’ curvatures de ned by usual -areas of usual -discs in even more orientations. Obviously, we need to bring some order to bear on this set, and we need to avoid the double counting we encountered in the case of three dimensions.
Fortunately, physics can help to make the mathematics easier. We start by de ning what we mean by curvature of space-time. en we will de ne curvatures for discs of various orientations. To achieve this, we interpret the de nition of curvature in another way, which allows us to generalize it to time as well. Figure illustrates the fact that the curvature K also describes how geodesics diverge. Geodesics are the straightest paths on a surface, i.e. those paths that a tiny car or tricycle would follow if it drove on the surface keeping the steering wheel straight.
If a space is curved, the separation s will increase along the geodesics as
** ‘Our head is round in order to allow our thougths to change direction.’ Francis Picabia (b. 1879 Paris, d. 1953 Paris) French dadaist and surrealist painter.
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ds dl
=
−Ks
+
higher
orders
(288)
where l measures the length along the geodesic, and K is the curvature, in other words,
the inverse squared curvature radius. In space-time, this relation is extended by sub-
stituting proper time (times the speed of light) for proper length. us separation and
curvature are related by
ds dτ
= −Kc
s + higher orders .
(289)
Page 129
Challenge 791 ny Ref. 375
Challenge 792 ny Challenge 793 ny Challenge 794 e
Ref. 376
But this is the de nition of an acceleration. In other words, what in the purely spatial case is described by curvature, in the case of space-time becomes the relative acceleration of two particles freely falling from nearby points. Indeed, we have encountered these accelerations already: they describe tidal e ects. In short, space-time curvature and tidal e ects are precisely the same.
Obviously, the magnitude of tidal e ects, and thus of curvature, will depend on the orientation – more precisely on the orientation of the space-time plane formed by the two particle velocities. e de nition also shows that K is a tensor, so that later on we will have to add indices to it. (How many?) e fun is that we can avoid indices for a while by looking at a special combination of spatial curvatures. If we take three planes in space, all orthogonal to each other and intersecting at a given point, the sum of the three so-called sectional curvature values does not depend on the observer. ( is corresponds to the tensor trace.) Can you con rm this, by using the de nition of the curvature just given?
e sum of the three sectional curvatures de ned for mutually orthogonal planes K( ), K( ) and K( ), is related to the excess radius de ned above. Can you nd out how?
If a surface has constant (intrinsic) curvature, i.e. the same curvature at all locations, geometrical objects can be moved around without deforming them. Can you picture this?
In summary, curvature is not such a di cult concept. It describes the deformation of space-time. If we imagine space (-time) as a big blob of rubber in which we live, the curvature at a point describes how this blob is squeezed at that point. Since we live inside the rubber, we need to use ‘insider’ methods, such as excess radii and sectional curvatures, to describe the deformation. Relativity o en seems di cult to learn because people do not like to think about the vacuum in this way, and even less to explain it in this way. (For a hundred years it was an article of faith for every physicist to say that the vacuum was empty.) Picturing vacuum as a substance can help us in many ways to understand general relativity.
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C
As mentioned above, one half of general relativity is the statement that any object moves along paths of maximum proper time, i.e. along geodesics. e other half is contained in a single expression: for every observer, the sum of all three proper sectional spatial curvatures at a point is given by
K( ) + K(
) + K( ) =
πG c
W
(
)
(290)
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•.
Challenge 795 e
where W( ) is the proper energy density at the point. e lower indices indicate the mixed
curvatures de ned by the three orthogonal directions , and . is is all of general
relativity in one paragraph.
An equivalent expression is easily found using the excess radius de ned above, by
introducing the mass M = V W( ) c . For the surface area A of the volume V containing
the mass, we get
rexcess = r −
A
π=
G c
M.
(291)
Challenge 796 ny
In short, general relativity a rms that for every observer, the excess radius of a small sphere is given by the mass inside the sphere.*
Note that the above expression implies that the average space curvature at a point in empty space vanishes. As we will see shortly, this means that near a spherical mass the negative of the curvature towards the mass is equal to twice the curvature around the mass; the total sum is thus zero.
Curvature will also di er from point to point. In particular, the expression implies that if energy moves, curvature will move with it. In short, both space curvature and, as we will see shortly, space-time curvature change over space and time.
We note in passing that curvature has an annoying e ect: the relative velocity of distant observers is unde ned. Can you provide the argument? In curved space, relative velocity is de ned only for nearby objects – in fact only for objects at no distance at all. Only in
at space are relative velocities of distant objects well de ned. e quantities appearing in expression ( ) are independent of the observer. But o en
people want to use observer-dependent quantities. e relation then gets more involved; the single equation ( ) must be expanded to ten equations, called Einstein’s eld equations. ey will be introduced below. But before we do that, we will check that general relativity makes sense. We will skip the check that it contains special relativity as a limiting case, and go directly to the main test.
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U
Challenge 797 e
“
e only reason which keeps me here is gravity.
” Anonymous
For small velocities and low curvature values, the temporal curvatures K( j) turn out to have a special property. In this case, they can be de ned as the second spatial derivatives
of a single scalar function φ. In other words, we can write
K(
j) =
∂φ ∂(x j)
.
(293)
Ref. 377 * Another, equivalent formulation is that for small radii the area A is given by A = πr ( + r R)
where R is the Ricci scalar, to be introduced later on.
(292)
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In everyday situations, the function φ turns out to be the gravitational potential. Indeed,
universal gravity is the limiting case of general relativity for small speeds and small spatial curvature. ese two limits imply, making use of W( ) = ρc and c , that
K(i j) = and K( ) + K( ) + K( ) = πGρ .
(294)
Challenge 798 ny
In other words, for small speeds, space is at and the potential obeys Poisson’s equation. Universal gravity is thus indeed the low speed and low curvature limit of general relativity.
Can you show that relation ( ) between curvature and energy density indeed means that time near a mass depends on the height, as stated in the beginning of this chapter?
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TS Ref. 375 What is the curvature of space-time near a spherical mass?
– CS – more to be inserted – CS –
Challenge 799 ny
e curvature of the Schwarzschild metric is given by
Krφ
=
Krθ
=
−
G c
M r
and
Kθφ =
GM cr
Ktφ
=
Ktθ
=
G c
M r
and
Ktr = −
GM cr
(295)
Ref. 375 Page 129
Challenge 800 ny
everywhere. e dependence on r follows from the general dependence of all tidal e ects; we have already calculated them in the chapter on universal gravity. e factors G c are due to the maximum force of gravity; only the numerical prefactors need to be calculated from general relativity. e average curvature obviously vanishes, as it does for all vacuum. As expected, the values of the curvatures near the surface of the Earth are exceedingly small.
C Every physicist should have an intuitive understanding of curvature.
Challenge 801 e
**
A y has landed on the outside of a cylindrical glass, cm below its rim. A drop of honey is located halfway around the glass, also on the outside, cm below the rim. What is the shortest distance from the y to the drop? What is the shortest distance if the drop is on the inside of the glass?
** Challenge 802 e Where are the points of highest and lowest Gaussian curvature on an egg?
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•.
A
–
*
“Jeder Straßenjunge in unserem mathematischen Göttingen versteht mehr von vierdimensionaler Geometrie als Einstein. Aber trotzdem hat Einstein die Sache gemacht, und nicht die großen Mathematiker. ” David Hilbert**
Now that we have a feeling for curvature, we want to describe it in a way that allows any observer to talk to any other observer. Unfortunately, this means using formulae with tensors. ese formulae look daunting. e challenge is to see in each of the expressions the essential point (e.g. by forgetting all indices for a while) and not to be distracted by those small letters sprinkled all over them.
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T
-
We mentioned above that a
Il faut suivre sa pente, surtout si elle monte.***
“ ” André Gide
-dimensional space-time is described by -curvature, -
curvature and -curvature. Many texts on general relativity start with -curvature. ese
curvatures describing the distinction between the -volume calculated from a radius and
the actual -volume. ey are described by the Ricci tensor.**** With an argument we
encountered already for the case of geodesic deviation, it turns out that the Ricci tensor
describes how the shape of a spherical cloud of freely falling particles is deformed on its
path.
– CS – to be expanded – CS –
In short, the Ricci tensor is the general-relativistic version of ∆φ, or better, of φ. e most global, but least detailed, de nition of curvature is the one describing the
distinction between the -volume calculated from a measured radius and the actual volume. is is the average curvature at a space-time point and is represented by the socalled Ricci scalar R, de ned as
R = − K = − rcurvature .
(296)
It turns out that the Ricci scalar can be derived from the Ricci tensor by a so-called con-
traction, which is a precise averaging procedure. For tensors of rank two, contraction is
the same as taking the trace:
R = Rλλ = λµRλµ .
(297)
** ‘Every street urchin in our mathematical Göttingen knows more about four-dimensional geometry than Einstein. Nevertheless, it was Einstein who did the work, not the great mathematicians.’ *** ‘One has to follow one’s inclination, especially if it climbs upwards.’ **** Gregorio Ricci-Cubastro (b. 1853 Lugo , d. 1925 Bologna), Italian mathematician. He is the father of absolute di erential calculus, once also called ‘Ricci calculus’. Tullio Levi-Civita was his pupil.
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Challenge 803 ny
e Ricci scalar describes the curvature averaged over space and time. In the image of a falling spherical cloud, the Ricci scalar describes the volume change of the cloud. e Ricci scalar always vanishes in vacuum. is result allows one, on the surface of the Earth, to relate the spatial curvature to the change of time with height.
Now comes an idea discovered by Einstein a er two years of hard work. e important quantity for the description of curvature in nature is not the Ricci tensor Rab, but a tensor built from it. is Einstein tensor Gab is de ned mathematically (for vanishing cosmological constant) as
Gab = Rab − abR .
(298)
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It is not di cult to understand its meaning. e value G is the sum of sectional
curvatures in the planes orthogonal to the direction and thus the sum of all spatial
sectional curvatures:
G = K( ) + K( ) + K( ) .
(299)
Similarly, for each dimension i the diagonal element Gii is the sum (taking into consideration the minus signs of the metric) of sectional curvatures in the planes orthogonal to
the i direction. For example, we have
G = K( ) + K( ) − K( ) .
(300)
Challenge 804 d
e distinction between the Ricci tensor and the Einstein tensor thus lies the way in which the sectional curvatures are combined: discs containing the coordinate in question in one case, discs orthogonal to the coordinate in the other case. Both describe the curvature of space-time equally well, and xing one means xing the other. (What are the trace and the determinant of the Einstein tensor?)
e Einstein tensor is symmetric, which means that it has ten independent components. Most importantly, its divergence vanishes; it therefore describes a conserved quantity. is was the essential property which allowed Einstein to relate it to mass and energy in mathematical language.
T
,
Obviously, for a complete description of gravity, also motions of momentum and energy also need to be quanti ed in such a way that any observer can talk to any other. We have seen that momentum and energy always appear together in relativistic descriptions; the next step is thus to nd out how their motions can be quanti ed for general observers.
First of all, the quantity describing energy, let us call it T, must be de ned using the energy–momentum vector p = mu = (γmc, γmv) of special relativity. Furthermore, T does not describe a single particle, but the way energy–momentum is distributed over space and time. As a consequence, it is most practical to use T to describe a density of energy and momentum. T will thus be a eld, and depend on time and space, a fact usually indicated by the notation T = T(t, x).
Since the energy–momentum density T describes a density over space and time, it de nes, at every space-time point and for every in nitesimal surface dA around that
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•.
point, the ow of energy–momentum dp through that surface. In other words, T is
de ned by the relation
dp = T dA .
(301)
e surface is assumed to be characterized by its normal vector dA. Since the energy– momentum density is a proportionality factor between two vectors, T is a tensor. Of course, we are talking about - ows and -surfaces here. erefore the energy– momentum density tensor can be split in the following way:
wS S S
T=
S S
t t
t t
t t
St t t
energy
energy ow or
=
density energy ow or
momentum density momentum
momentum density
ow density
(302)
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where w = T is a -scalar, S a -vector and t a -tensor. e total quantity T is called the energy–momentum (density) tensor. It has two essential properties: it is symmetric and its divergence vanishes.
e vanishing divergence of the tensor T, o en written as
∂a T ab = or abbreviated T ab , a = ,
(303)
Challenge 805 ny
expresses the fact that the tensor describes a conserved quantity. In every volume, energy can change only via ow through its boundary surface. Can you con rm that the description of energy–momentum with this tensor satis es the requirement that any two observers, di ering in position, orientation, speed and acceleration, can communicate their results to each other?
e energy–momentum density tensor gives a full description of the distribution of energy, momentum and mass over space and time. As an example, let us determine the energy–momentum density for a moving liquid. For a liquid of density ρ, a pressure p and a -velocity u, we have
T ab = (ρ + p)uaub − p ab
(304)
where ρ is the density measured in the comoving frame, the so-called proper density.* Obviously, ρ, ρ and p depend on space and time.
Of course, for a particular material uid, we need to know how pressure p and density ρ are related. A full material characterization thus requires the knowledge of the relation
p = p(ρ) .
(306)
* In the comoving frame we thus have
ρc
Tab =
p p
.
p
(305)
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is relation is a material property and thus cannot be determined from relativity. It has to be derived from the constituents of matter or radiation and their interactions. e simplest possible case is dust, i.e. matter made of point particles** with no interactions at all. Its energy–momentum tensor is given by
Tab = ρ uaub .
(307)
Challenge 806 ny Challenge 807 ny
Can you explain the di erence from the liquid case? e divergence of the energy–momentum tensor vanishes for all times and positions,
as you may want to check. is property is the same as for the Einstein tensor presented above. But before we elaborate on this issue, a short remark. We did not take into account gravitational energy. It turns out that gravitational energy cannot be de ned in general. Gravity is not an interaction and does not have an associated energy.***
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H
’
–
?
When Einstein discussed his work with David Hilbert, Hilbert found a way to do in a
few weeks what had taken years for Einstein. Hilbert understood that general relativity
in empty space could be described by an action integral, like all other physical systems.
us Hilbert set out to nd the measure of change, as this is what an action describes,
for motion due to gravity. Obviously, the measure must be observer-invariant; in particu-
lar, it must be invariant under all possible changes of viewpoints.
Motion due to gravity is determined by curvature. Any curvature measure independ-
ent of the observer must be a combination of the Ricci scalar R and the cosmological
constant Λ. It thus makes sense to expect that the change of space-time is described by
an action S given by
∫ S =
c πG
(R + Λ) dV .
(308)
e volume element dV must be speci ed to use this expression in calculations. e cosmological constant Λ (added some years a er Hilbert’s work) appears as a mathematical possibility to describe the most general action that is di eomorphism-invariant. We will see below that its value in nature, though small, seems to be di erent from zero.
e Hilbert action of a chunk of space-time is thus the integral of the Ricci scalar plus twice the cosmological constant over that chunk. e principle of least action states that space-time moves in such a way that this integral changes as little as possible.
– CS – to be nished – CS –
Page 471 Challenge 808 ny
** Even though general relativity expressly forbids the existence of point particles, the approximation is useful in cases when the particle distances are large compared to their own size. *** In certain special circumstances, such as weak elds, slow motion, or an asymptotically at space-time, we can de ne the integral of the G component of the Einstein tensor as negative gravitational energy. Gravitational energy is thus only de ned approximately, and only for our everyday environment. Nevertheless, this approximation leads to the famous speculation that the total energy of the universe is zero. Do you agree?
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In summary, the question ‘how do things move?’ is answered by general relativity in the same way as by special relativity: things follow the path of maximal ageing. Challenge 809 ny Can you show that the Hilbert action follows from the maximum force?
T e main symmetry of the Lagrangian of general relativity is called di eomorphism in-
variance.
– CS – to be written – CS –
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Ref. 378 Ref. 379
e eld equations for empty space-time also show scale symmetry. is is the invariance of the equations a er multiplication of all coordinates by a common numerical factor. In , Torre and Anderson showed that di eomorphism symmetry and trivial scale symmetry are the only symmetries of the vacuum eld equations.
Apart from di eomorphism symmetry, full general relativity, including mass–energy, has an additional symmetry which is not yet fully elucidated. is symmetry connects the various possible initial conditions of the eld equations; the symmetry is extremely complex and is still a topic of research. ese fascinating investigations should give new insights into the classical description of the big bang.
E
’
“[Einstein’s general theory of relativity] cloaked the ghastly appearance of atheism. A witch hunter from Boston, around
Do you believe in god? Prepaid reply words. Subsequent telegram by another to his hero Albert Einstein
Page 353
I believe in Spinoza’s god, who reveals himself in the orderly harmony of what exists, not in a god who concerns himself with fates and actions of human beings.
” Albert Einstein’s answer
Einstein’s famous eld equations were the basis of many religious worries. ey contain the full description of general relativity. As explained above, they follow from the maximum force – or equivalently, from Hilbert’s action – and are given by
Rab −
Gab = −κ Tab or
ab R − Λ ab = −κ T ab .
(309)
e constant κ, called the gravitational coupling constant, has been measured to be
κ=
πG c
=
.ë
−
N
(310)
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Page 460
and its small value – π divided by the maximum force c G – re ects the weakness of gravity in everyday life, or better, the di culty of bending space-time. e constant Λ, the so-called cosmological constant, corresponds to a vacuum energy volume density, or pressure Λ κ. Its low value is quite hard to measure. e currently favoured value is
Λ − m or Λ κ . nJ m = . nPa .
(311)
Ref. 380 Challenge 810 ny
Current measurements and simulations suggest that this parameter, even though it is numerically near to the square of the present radius of the universe, is a constant of nature that does not vary with time.
In summary, the eld equations state that the curvature at a point is equal to the ow of energy–momentum through that point, taking into account the vacuum energy density. In other words, energy–momentum tells space-time how to curve.*
e eld equations of general relativity can be simpli ed for the case in which speeds are small. In that case T = ρc and all other components of T vanish. Using the de nition of the constant κ and setting φ = (c )h in ab = ηab + hab, we nd
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∇ φ = πρ
and
dx dt
=
−∇φ
(312)
which we know well, since it can be restated as follows: a body of mass m near a body of
mass M is accelerated by
a
=
G
M r
,
(313)
a value which is independent of the mass m of the falling body. And indeed, as noted already by Galileo, all bodies fall with the same acceleration, independently of their size,
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Ref. 381 Page 676
Page 474
* Einstein arrived at his eld equations using a number of intellectual guidelines that are called principles in the literature. Today, many of them are not seen as central any more. Nevertheless, we give a short overview.
- Principle of general relativity: all observers are equivalent; this principle, even though o en stated, is probably empty of any physical content.
- Principle of general covariance: the equations of physics must be stated in tensorial form; even though it is known today that all equations can be written with tensors, even universal gravity, in many cases they require unphysical ‘absolute’ elements, i.e. quantities which a ect others but are not a ected themselves.
is unphysical idea is in contrast with the idea of interaction, as explained above. - Principle of minimal coupling: the eld equations of gravity are found from those of special relativity by taking the simplest possible generalization. Of course, now that the equations are known and tested experimentally, this principle is only of historical interest. - Equivalence principle: acceleration is locally indistinguishable from gravitation; we used it to argue that space-time is semi-Riemannian, and that gravity is its curvature. - Mach’s principle: inertia is due to the interaction with the rest of the universe; this principle is correct, even though it is o en maintained that it is not ful lled in general relativity. In any case, it is not the essence of general relativity. - Identity of gravitational and inertial mass: this is included in the de nition of mass from the outset, but restated ad nauseam in general relativity texts; it is implicitly used in the de nition of the Riemann tensor. - Correspondence principle: a new, more general theory, such as general relativity, must reduce to previous theories, in this case universal gravity or special relativity, when restricted to the domains in which those are valid.
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Challenge 811 e
their mass, their colour, etc. In general relativity also, gravitation is completely democratic.* e independence of free fall from the mass of the falling body follows from the description of space-time as a bent mattress. Objects moving on a mattress also move in the same way, independently of the mass value.
To get a feeling for the complete eld equations, we will take a short walk through their main properties. First of all, all motion due to space-time curvature is reversible, di erentiable and thus deterministic. Note that only the complete motion, of space-time and matter and energy, has these properties. For particle motion only, motion is in fact irreversible, since some gravitational radiation is usually emitted.
By contracting the eld equations we nd, for vanishing cosmological constant, the following expression for the Ricci scalar:
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R = −κT .
(318)
Challenge 812 ny Ref. 382
Challenge 813 ny
is result also implies the relation between the excess radius and the mass inside a sphere.
e eld equations are nonlinear in the metric , meaning that sums of solutions usually are not solutions. at makes the search for solutions rather di cult. For a complete solution of the eld equations, initial and boundary conditions should be speci ed. e ways to do this form a specialized part of mathematical physics; it is not explored here.
Albert Einstein used to say that general relativity only provides the understanding of one side of the eld equations ( ), but not of the other. Can you see which side he meant?
What can we do of interest with these equations? In fact, to be honest, not much that we have not done already. Very few processes require the use of the full equations. Many textbooks on relativity even stop a er writing them down! However, studying them is worthwhile. For example, one can show that the Schwarzschild solution is the only spherically symmetric solution. Similarly, in , Birkho showed that every rotationally symmetric vacuum solution is static. is is the case even if masses themselves move, as for example during the collapse of a star.
Maybe the most beautiful applications of the eld equations are the various lms made of relativistic processes. e worldwide web hosts several of these; they allow one to see
* Here is another way to show that general relativity ts with universal gravity. From the de nition of the Riemann tensor we know that relative acceleration ba and speed of nearby particles are related by
∇e ba = Rcedavcvd .
(314)
From the symmetries of R we know there is a φ such that ba = −∇a φ. at means that
∇e ba = ∇e ∇a φ = Rcaed vc vd
(315)
which implies that
∆φ
=
∇a ∇a φ
=
R
a cad
v
c
v
d
=
Rcd vc vd
=
κ(Tcd vc vd
−
T
)
(316)
Introducing Tab = ρvavb we get as we wanted to show.
∆φ = πGρ
(317)
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what happens when two black holes collide, what happens when an observer falls into a black hole, etc. To generate these lms, the eld equations usually need to be solved directly, without approximations.*
– CS – more to be added – CS –
Another area of application concerns gravitational waves. e full eld equations show that waves are not harmonic, but nonlinear. Sine waves exist only approximately, for small amplitudes. Even more interestingly, if two waves collide, in many cases singularities are predicted to appear. is whole theme is still a research topic and might provide new insights for the quantization of general relativity in the coming years.
We end this section with a side note. Usually, the eld equations are read in one sense only, as stating that energy–momentum produces curvature. One can also read them in the other way, calculating the energy–momentum needed to produces a given curvature. When one does this, one discovers that not all curved space-times are possible, as some would lead to negative energy (or mass) densities. Such solutions would contradict the mentioned limit on length-to-mass ratios for physical systems.
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Ref. 383 Ref. 323, Ref. 324
When the cosmological constant is taken into the picture, the maximum force principle requires a second look. In the case of a non-vanishing cosmological constant, the force limit makes sense only if the constant Λ is positive; this is the case for the currently measured value, which is Λ − m . Indeed, the radius–mass relation of black holes
GM = Rc ( − ΛR )
(319)
Page 373
implies that a radius-independent maximum force is valid only for positive or zero cosmological constant. For a negative cosmological constant the force limit would only be valid for in nitely small black holes. In the following, we take a pragmatic approach and note that a maximum force limit can be seen to imply a vanishing or positive cosmological constant. Obviously, the force limit does not specify the value of the constant; to achieve this, a second principle needs to be added. A straightforward formulation, using the additional principle of a minimum force in nature, was proposed above.
One might ask also whether rotating or charged black holes change the argument that leads from maximum force to the derivation of general relativity. However, the derivation using the Raychaudhuri equation does not change. In fact, the only change of the argument appears with the inclusion of torsion, which changes the Raychaudhuri equation itself. As long as torsion plays no role, the derivation given above remains valid. e inclusion of torsion is still an open research issue.
Another question is how maximum force relates to scalar–tensor theories of gravity, such as the proposal by Brans and Dicke or its generalizations. If a particular scalar-tensor theory obeys the general horizon equation ( ) then it must also imply a maximum force.
e general horizon equation must be obeyed both for static and for dynamic horizons.
* See for example the http://www.photon.at/~werner/black-earth website.
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If that were the case, the speci c scalar–tensor theory would be equivalent to general relativity, as it would allow one, using the argument of Jacobson, to deduce the usual
eld equations. is case can appear if the scalar eld behaves like matter, i.e., if it has mass–energy like matter and curves space-time like matter. On the other hand, if in the particular scalar–tensor theory the general horizon equation ( ) is not obeyed for all moving horizons – which is the general case, as scalar–tensor theories have more de ning constants than general relativity – then the maximum force does not appear and the theory is not equivalent to general relativity. is connection also shows that an experimental test of the horizon equation for static horizons only is not su cient to con rm general relativity; such a test rules out only some, but not all, scalar–tensor theories.
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For the maximum force limit to be considered a basic physical principle, all properties of gravitation, including the full theory of general relativity, must be deduced from it. To make this argument easier to follow, we will split it into several steps. First of all, we show that the force limit implies that in everyday life the inverse square law of universal gravity holds. en we show that it implies the main ideas of general relativity. Finally, we show that the full theory of general relativity follows.
In other words, from this point onwards the force limit is assumed to be valid. We explore its consequences and compare them with the known properties of nature. e maximum force limit can also be visualized in another way. If the gravitational attraction between a central body and a satellite were stronger than it is, black holes would be smaller than they are; in that case the maximum force limit and the maximum speed could be exceeded. If, on the other hand, gravitation were weaker than it is, a fast and accelerating observer would not be able to determine that the two bodies interact. In summary, a maximum force of c G implies universal gravity. ere is no di erence between stating that all bodies attract through gravitation and stating that there is a maximum force with the value c G.
D
Ref. 323, Ref. 324
e next logical step is to show that a maximum force also implies general relativity. e naive approach is to repeat, step by step, the standard approach to general relativity.
Space-time curvature is a consequence of the fact that the speed of light is the maximum speed for all observers, even if they are located in a gravitational eld. e gravitational red shi shows that in gravitational elds, clocks change their rate with height; that change, together with the constancy of the speed of light, implies space-time curvature. Gravity thus implies space-time curvature. e value of the curvature in the case of weak gravitational elds is completely xed by the inverse square law of gravity. Since universal gravity follows from the maximum force, we deduce that maximum force implies spacetime curvature.
Apart from curvature, we must also check the other basic ideas of general relativity. e principle of general relativity states that all observers are equivalent; since the maximum force principle applies to all observers, the principle of general relativity is contained in it. e equivalence principle states that, locally, gravitation can be transformed away by changing to a suitable observer. is is also the case for the maximum force
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Ref. 384
principle, which is claimed for all observers, thus also for observers that locally eliminate gravitation. Mach’s principle, whose precise formulation varies, states that only relative quantities should play a role in the description of nature. Since the maximum force is a relative quantity – in particular, the relation of mass and curvature remains – Mach’s principle is also satis ed.
Free bodies in at space move with constant speed. By the equivalence principle, this statement generalizes to the statement that freely falling bodies move along geodesics.
e maximum force principle keeps intact the statement that space-time tells matter how to move.
e curvature of space-time for weak gravitational elds is xed by the inverse square law of gravity. Space curvature is thus present in the right amount around each mass. As Richard Feynman explains, by extending this result to all possible observers, we can deduce all low-curvature e ects of gravitation. In particular, this implies the existence of linear (low-amplitude) gravitational waves and of the irring–Lense e ect. Linearized general relativity thus follows from the maximum force principle.
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H
e other half of general relativity states that bodies fall along geodesics. All orbits are geodesics, thus curves with the longest proper time. It is thus useful to be able to calculate these trajectories.* To start, one needs to know the shape of space-time, the notion of ‘shape’ being generalized from its familiar two-dimensional meaning. For a being living on the surface, it is usually described by the metric ab, which de nes the distances between neighbouring points through
ds = dxa dxa = ab(x) dxa dxb .
(320)
It is a famous exercise of calculus to show from this expression that a curve xa(s) depending on a well behaved (a ne) parameter s is a timelike or spacelike (metric) geodesic, Challenge 814 ny i.e. the longest possible path between the two events,** only if
d ds
(
a
d
dxd ds
)
=
∂ bc ∂xa
dxb ds
dx c ds
,
(321)
as long as ds is di erent from zero along the path.*** All bodies in free fall follow such Page 388 geodesics. We showed above that the geodesic property implies that a stone thrown in the
air falls back, unless if it is thrown with a speed larger than the escape velocity. Expression
* is is a short section for the more curious; it can be skipped at rst reading.
** We remember that in space in everyday life, geodesics are the shortest possible paths; however, in space-
time in general relativity, geodesics are the longest possible paths. In both cases, they are the ‘straightest’
possible paths.
*** is is o en written as
d xa ds
+
Γbac
dx b ds
dx c ds
=
(322)
where the condition
dx a ab ds
dx b ds
=
(323)
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Ref. 385 Challenge 815 ny
( ) thus replaces both the expression d x dt = −∇φ valid for falling bodies and the expression d x dt = valid for freely oating bodies in special relativity.
e path does not depend on the mass or on the material of the body. erefore antimatter also falls along geodesics. In other words, antimatter and matter do not repel; they also attract each other. Interestingly, even experiments performed with normal matter can show this, if they are carefully evaluated. Can you nd out how?
For completeness, we mention that light follows lightlike or null geodesics. In other words, there is an a ne parameter u such that the geodesics follow
d xa du
+
Γ
a
bc
dxb du
dx c du
=
(325)
with the di erent condition
dx a ab du
dxb du
=
.
(326)
Given all these de nitions of various types of geodesics, what are the lines drawn in Challenge 816 ny Figure on page ?
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Challenge 817 ny
e di eomorphism-invariance of general relativity makes life quite interesting. We will see that it allows us to say that we live on the inside of a hollow sphere, and that it does not allow us to say where energy is actually located. If energy cannot be located, what about mass? It soon becomes clear that mass, like energy, can be localized only if distant space-time is known to be at. It is then possible to de ne a localized mass value by making precise an intuitive idea: the mass is measured by the time a probe takes to orbit the unknown body.*
e intuitive mass de nition requires at space-time at in nity; it cannot be extended to other situations. In short, mass can only be localized if total mass can be de ned. And
must be ful lled, thus simply requiring that all the tangent vectors are unit vectors, and that ds all along the path. e symbols Γ appearing above are given by
Γabc =
a bc
=
ad (∂b dc + ∂c db − ∂d bc) ,
(324)
and are called Christo el symbols of the second kind or simply the metric connection. * is de nition was formalized by Arnowitt, Deser and Misner, and since then has o en been called the ADM mass. e idea is to use the metric i j and to take the integral
Ref. 386
∫ m =
π
( i j,i ν j − ii, j ν j)dA
SR
(327)
where SR is the coordinate sphere of radius R, ν is the unit vector normal to the sphere and dA is the area element on the sphere. e limit exists if space-time is asymptotically at and if the mass distribution is su -
ciently concentrated. Mathematical physicists have also shown that for any manifold whose metric changes
at in nity as
i j = ( + f r + O( r ))δi j
(328)
the total mass is given by M = f .
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total mass is de ned only for asymptotically at space-time. e only other notion of mass that is precise in general relativity is the local mass density at a point. In contrast, it is not well understood how to de ne the mass contained in a region larger than a point but smaller than the entirety of space-time.
Now that we can go on talking about mass without (too much of) a bad conscience, we turn to the equations of motion.
I
?
Page 676 Challenge 818 ny
We tend to answer this question a rmatively, as in Galilean physics gravity was seen as an in uence on the motion of bodies. In fact, we described it by a potential, implying that gravity produces motion. But let us be careful. A force or an interaction is what changes the motion of objects. However, we just saw that when two bodies attract each other through gravitation, both always remain in free fall. For example, the Moon circles the Earth because it continuously falls around it. Since any freely falling observer continuously remains at rest, the statement that gravity changes the motion of bodies is not correct for all observers. Indeed, we will soon discover that in a sense to be discussed shortly, the Moon and the Earth both follow ‘straight’ paths.
Is this correction of our idea of gravity only a question of words? Not at all. Since gravity is not an interaction, it is not due to a eld and there is no potential.
Let us check this strange result in yet another way. e most fundamental de nition of ‘interaction’ is as the di erence between the whole and the sum of its parts. In the case of gravity, an observer in free fall could claim that nothing special is going on, independently of whether the other body is present or not, and could claim that gravity is not an interaction.
However, that is going too far. An interaction transports energy between systems. We have indeed seen that gravity can be said to transport energy only approximately. Gravitation is thus an interaction only approximately. But that is not a su cient reason to abandon this characterization. e concept of energy is not useful for gravity beyond the domain of everyday life. For the general case, namely for a general observer, gravity is thus fundamentally di erent from electricity or magnetism.
Another way to look at the issue is the following. Take a satellite orbiting Jupiter with energy–momentum p = mu. If we calculate the energy–momentum change along its path s, we get
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dp ds
=
m
du ds
=
m(ea
dua ds
+
dea ds
ua
)
=
mea
(
dua ds
+ Γabdubuc)
=
(329)
Challenge 819 ny
Ref. 387 Challenge 820 ny
Challenge 821 n
where e describes the unit vector along a coordinate axis. e energy–momentum change vanishes along any geodesic, as you might check. erefore, the energy–momentum of this motion is conserved. In other words, no force is acting on the satellite. One could reply that in equation ( ) the second term alone is the gravitational force. But this term can be made to vanish along the entirety of any given world line. In short, nothing changes between two bodies in free fall around each other: gravity could be said not to be an interaction. e properties of energy con rm this argument.
Of course, the conclusion that gravity is not an interaction is somewhat academic, as
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it contradicts our experience of daily life. But we will need it for the full understanding of motion later on. e behaviour of radiation con rms the deduction. In vacuum, radiation is always moving freely. In a sense, we can say that radiation always is in free fall. Strangely, since we called free fall the same as rest, we should conclude that radiation always is at rest. is is not wrong! We have already seen that light cannot be accelerated.* We even saw that gravitational bending is not an acceleration, since light follows straight paths in space-time in this case as well. Even though light seems to slow down near masses for distant observers, it always moves at the speed of light locally. In short, even gravitation doesn’t manage to move light.
ere is another way to show that light is always at rest. A clock for an observer trying to reach the speed of light goes slower and slower. For light, in a sense, time stops: or, if you prefer, light does not move.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Ref. 388 Ref. 389
If a maximum power or force appearing on horizons is the basis for general relativity, one can ask whether physical systems other than space-time can also be described in this way.
For special relativity, we found that all its main e ects – such as a limit speed, Lorentz contraction or energy–mass equivalence – are also found for dislocations in solids. Do systems analogous to general relativity exist? Attempts to nd such systems have only been partially successful. Several equations and ideas of general relativity are applicable to deformations of solids, since general relativity describes the deformation of the space-time mattress. Kröner has studied this analogy in great detail. Other systems with horizons, and thus with observables analogous to curvature, are found in certain liquids – where vortices play the role of black holes – and in certain quantum uids for the propagation of light. Exploring such systems has become a research topic in its own right. A full analogy of general relativity in a macroscopic system was discovered only a few years ago. is analogy will be presented in the third part of our adventure; we will need an additional ingredient that is not visible at this point of our ascent.
R
Challenge 822 e
Most books introduce curvature the hard way, namely historically,* using the Riemann curvature tensor. is is a short summary, so that you can understand that old stu when you come across it.
We saw above that curvature is best described by a tensor. In dimensions, this curvature tensor, usually called R, must be a quantity which allows us to calculate, among other things, the area for any orientation of a -disc in space-time. Now, in dimensions, orientations of a disc are de ned in terms of two -vectors; let us call them p and q. And instead of a disc, we take the parallelogram spanned by p and q. ere are several possible de nitions.
* Refraction, the slowdown of light inside matter, is not a counter-example. Strictly speaking, light inside matter is constantly being absorbed and re-emitted. In between these processes, light still propagates with the speed of light in vacuum. e whole process only looks like a slowdown in the macroscopic limit. e same applies to di raction and to re ection. A list of apparent ways to bend light can be found on page 567; details of the quantum-mechanical processes at their basis can be found on page 728. * is is a short section for the more curious; it can be skipped at rst reading.
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e Riemann-Christo el curvature tensor R is then de ned as a quantity which allows
us to calculate the curvature K(p, q) for the surface spanned by p and q, with area A,
through
K(p, q)
=
R A
pqpq (p, q)
=
(
αδ
Rabcd paqb pc qd βγ − αγ βδ)pα qβ pγ qδ
(330)
Challenge 823 e Challenge 824 ny
where, as usual, Latin indices a, b, c, d, etc. run from to , as do Greek indices here, and a summation is implied when an index name appears twice. Obviously R is a tensor, of rank . is tensor thus describes only the intrinsic curvature of a space-time. In contrast, the metric describes the complete shape of the surface, not only the curvature.
e curvature is thus the physical quantity of relevance locally, and physical descriptions therefore use only the Riemann** tensor R or quantities derived from it.***
But we can forget the just-mentioned de nition of curvature. ere is a second, more physical way to look at the Riemann tensor. We know that curvature means gravity. As we said above, gravity means that when two nearby particles move freely with the same velocity and the same direction, the distance between them changes. In other words, the local e ect of gravity is relative acceleration of nearby particles.
It turns out that the tensor R describes precisely this relative acceleration, i.e. what we called the tidal e ects earlier on. Obviously, the relative acceleration b increases with the separation d and the square (why?) of the speed u of the two particles. erefore we can also de ne R as a (generalized) proportionality factor among these quantities:
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
b = R u u d or, more clearly, ba = Ra bcd ub uc dd .
(333)
Challenge 825 ny
e components of the Riemann curvature tensor have the dimensions of inverse square length. Since it contains all information about intrinsic curvature, we conclude that if R vanishes in a region, space-time in that region is at. is connection is easily deduced from this second de nition.*
** Bernhard Riemann (b. 1826 Breselenz, d. 1866 Selasca), important German mathematician.
*** We showed above that space-time is curved by noting changes in clock rates, in metre bar lengths and
in light propagation. Such experiments are the easiest way to determine the metric . We know that space-
time is described by a 4-dimensional manifold M with a metric ab that locally, at each space-time point, is a Minkowski metric. Such a manifold is called a Riemannian manifold. Only such a metric allows one to
de ne a local inertial system, i.e. a local Minkowski space-time at every space-time point. In particular, we
have
ab =
ab and
b a
=
a b
=
δba
.
(331)
How are curvature and metric related? e solution to this question usually occupies a large number of
pages in relativity books; just for information, the relation is
Ref. 390
Ra bcd
=
∂Γa bd ∂xc
−
∂Γa bc ∂xd
+ ΓaecΓebd
− Γa fdΓf bc
.
(332)
e curvature tensor is built from the second derivatives of the metric. On the other hand, we can also determine the metric if the curvature is known. An approximate relation is given below. * is second de nition is also called the de nition through geodesic deviation. It is of course not evident that it coincides with the rst. For an explicit proof, see the literature. ere is also a third way to picture the tensor R, a more mathematical one, namely the original way Riemann introduced it. If one paralleltransports a vector w around a parallelogram formed by two vectors u and v, each of length ε, the vector w
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A nal way to de ne the tensor R is the following. For a free-falling observer, the metric ab is given by the metric ηab from special relativity. In its neighbourhood, we have
ab = ηab + Racbd xc xd + O(x ) = (∂c ∂d ab)xcxd + O(x ) .
(335)
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
e curvature term thus describes the departure of the space-time metric from that of at space-time. e curvature tensor R is a large beast; it has = components at each point of space-time; however, its symmetry properties reduce them to independent numbers.** e actual number of importance in physical problems is still smaller, namely only . ese are the components of the Ricci tensor, which can be de ned with the help of the Riemann tensor by contraction, i.e. by setting
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Rbc = Rabac .
(338)
Challenge 828 e Challenge 829 ny
Its components, like those of the Riemann tensor, are inverse square lengths. e values of the tensor Rbc, or those of Rbacd, are independent of the sign convention used in the Minkowski metric, in contrast to Rabcd .
Can you con rm the relation Rabcd Rabcd = m r for the Schwarzschild solution?
C ere are new results in general relativity every year.
**
For a long time, people have speculated why the Pioneer 10 and 11 arti cial satellites, which are now over 70 astronomical units away from the Sun, are subject to a constant deceleration of ë − m s (towards the Sun) since thy passed the orbit of Saturn. is e ect is called the Pioneer anomaly. e origin is not clear and still a subject of research.
is changed to w + δw. One then has
δw = −ε R u v w + higher-order terms .
(334)
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More can be learned about the geodesic deviation by studying the behaviour of the famous south-pointing
carriage. is device, common in China before the compass was discovered, only works if the world is at.
Indeed, on a curved surface, a er following a large closed path, it will show a di erent direction than at the
start of the trip. Can you explain why?
** e free-fall de nition shows that the Riemann tensor is symmetric in certain indices and antisymmetric
in others:
Rabcd = Rcdab , Rabcd = −Rbacd = −Rabdc .
(336)
ese relations also imply that many components vanish. Of importance also is the relation
Rabcd + Radbc + Racdb = .
(337)
Note that the order of the indices is not standardized in the literature. e list of invariants which can be constructed from R is long. We mention that εabcd Rcd e f Rabe f , namely the product R R of the Riemann
tensor with its dual, is the invariant characterizing the irring–Lense e ect.
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But several investigations have shown that the reason is not a deviation from the inverse Ref. 391 square dependence of gravitation, as is sometimes proposed. In other words, the e ect Challenge 830 r must be electromagnetic. Finding it is one of the challenges of modern astrophysics.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
“Zwei Dinge erfüllen das Gemüt mit immer neuer und zunehmender Bewunderung und Ehrfurcht, je ö er und anhaltender sich das Nachdenken damit beschä igt: der bestirnte Himmel über mir und das moralische Gesetz in mir.* ” Immanuel Kant ( – )
On clear nights, between two and ve thousand stars are visible with the naked eye. Several hundred of them have names. Indeed, in all parts of the world, the stars and the constellations they form are seen as memories of ancient events, and stories are told about them.* But the simple fact that we can see the stars is the basis for a story much more fantastic than all myths. It touches almost all aspects of modern physics.
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Ref. 393
“Democritus says [about the Milky Way] that it is a region of light emanating from numerous stars small and near to each other, of which the grouping produces the brightness of the whole. ” Aetius, Opinions.
e stars we see on a clear night are mainly the brightest of our nearest neighbours in the surrounding region of the Milky Way. ey lie at distances between four and a few thousand light years from us. Roughly speaking, in our environment there is a star about every cubic light years.
Almost all visible stars are from our own galaxy. e only extragalactic object constantly visible to the naked eye in the northern hemisphere is the so-called Andromeda nebula, shown enlarged in Figure . It is a whole galaxy like our own, as Immanuel Kant had already conjectured in . Several extragalactic objects are visible with the naked eye in the southern hemisphere: the Tarantula nebula, as well as the large and the small Magellanic clouds. e Magellanic clouds are neighbour galaxies to our own. Other, temporary exceptions are the rare novae, exploding stars which can be seen if they appear in nearby galaxies, or the still rarer supernovae, which can o en be seen even in faraway galaxies.
Ref. 392
* ‘Two things ll the mind with ever new and increasing admiration and awe, the more o en and persistently
thought considers them: the starred sky above me and the moral law inside me.’
* About the myths around the stars and the constellations, see e.g. the text by G. F
, Sternbilder und
ihre Mythen, Springer Verlag, 1993. On the internet there are also the beautiful http://www.astro.wisc.edu/
~dolan/constellations/constellations.html and http://www.astro.uiuc.edu/~kaler/sow/sow.html websites.
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F I G U R E 196 The Andromeda nebula M31, our neighbour galaxy (and the 31st member of the Messier object listing) (NASA)
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F I G U R E 197 How our galaxy looks in the infrared (NASA)
Challenge 831 ny
In fact, the visible stars are special in other respects also. For example, telescopes show that about half of them are in fact double: they consist of two stars circling around each other, as in the case of Sirius. Measuring the orbits they follow around each other allows one to determine their masses. Can you explain how?
Is the universe di erent from our Milky Way? Yes, it is. ere are several arguments to demonstrate this. First of all, our galaxy – th word galaxy is just the original Greek term for ‘Milky Way’ – is attened, because of its rotation. If the galaxy rotates, there must be other masses which determine the background with respect to which this rotation takes place. In fact, there is a huge number of other galaxies – about – in the universe, a discovery dating only from the twentieth century.
Why did our understanding of the place of our galaxy in the universe happen so late? Well, people had the same di culty as they had when trying to determine the shape of the Earth. ey had to understand that the galaxy is not only a milky strip seen on clear nights, but an actual physical system, made of about stars gravitating around each other.* Like the Earth, the galaxy was found to have a three-dimensional shape; it is
* e Milky Way, or galaxy in Greek, was said to have originated when Zeus, the main Greek god, tried to let his son Heracles feed at Hera’s breast in order to make him immortal; the young Heracles, in a sign showing
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F I G U R E 198 The elliptical galaxy NGC 205 (the 205th member of the New Galactic Catalogue) (NASA)
Challenge 832 ny Ref. 394
shown in Figure . Our galaxy is a at and circular structure, with a diameter of
light years; in the centre, it has a spherical bulge. It rotates about once every to
million years. (Can you guess how this is measured?) e rotation is quite slow: since the
Sun was formed, it has made only about to full turns around the centre.
It is even possible to measure the mass of our galaxy. e trick is to use a binary pulsar
on its outskirts. If it is observed for many years, one can deduce its acceleration around
the galactic centre, as the pulsar reacts with a frequency shi which can be measured
on Earth. Many decades of observation are needed and many spurious e ects have to
be eliminated. Nevertheless, such measurements are ongoing. Present estimates put the
mass of our galaxy at
kg.
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Astrophysics leads to a strange conclusion about matter, quite di erent from how we are used to thinking in classical physics: the matter observed in the sky is found in clouds. Clouds are systems in which the matter density diminishes with the distance from the centre, with no de nite border and with no de nite size. Most astrophysical objects are best described as clouds.
e Earth is also a cloud, if we take its atmosphere, its magnetosphere and the dust ring around it as part of it. e Sun is a cloud. It is a gas ball to start with, but is even more a cloud if we take into consideration its protuberances, its heliosphere, the solar wind it generates and its magnetosphere. e solar system is a cloud if we consider its
his future strength, sucked so forcefully that the milk splashed all over the sky.
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•. F I G U R E 199 The colliding galaxies M51 and M110 (NASA)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 200 The X-rays in the night sky, between 1 and MeV (NASA)
Ref. 395
comet cloud, its asteroid belt and its local interstellar gas cloud. e galaxy is a cloud if we remember its matter distribution and the cloud of cosmic radiation it is surrounded by. In fact, even people can be seen as clouds, as every person is surrounded by gases, little dust particles from skin, vapour, etc.
In the universe, almost all clouds are plasma clouds. A plasma is an ionized gas, such as re, lightning, the inside of neon tubes, or the Sun. At least . % of all matter in the universe is in the form of plasma clouds. Only a very small percentage exists in solid or liquid form, such as toasters, subways or their users.
Clouds in the universe have certain common properties. First, clouds seen in the universe, when undisturbed by collisions or other interactions from neighbouring objects,
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F I G U R E 201 Rotating clouds emitting jets along their axis; top row: a composite image (visible and infrared) of the galaxy 0313-192, the galaxy 3C296, and the Vela pulsar; bottom row: the star in formation HH30, the star in formation DG Tauri B, and a black hole jet from the galaxy M87 (NASA)
Challenge 833 r Ref. 396
are rotating. Most clouds are therefore attened and are in shape of discs. Secondly, in many rotating clouds, matter is falling towards the centre: most clouds are accretion discs. Finally, undisturbed accretion discs usually emit something along the rotation axis: they possess jets. is basic cloud structure has been observed for young stars, for pulsars, for galaxies, for quasars and for many other systems. Figure gives some examples. (Does the Sun have a jet? Does the Milky Way have a jet? So far, none has been detected – there is still room for discovery.)
In summary, at night we see mostly rotating, attened plasma clouds emitting jets along their axes. A large part of astronomy and astrophysics collects information about them. An overview about the observations is given in Table .*
TA B L E 37 Some observations about the universe
A
M
V
Phenomena galaxy formation
galactic collisions star formation
observed by Hubble trigger event momentum cloud collapse
several times unknown
to kg m s form stars between . and solar masses
* An overview of optical observations is given by the Sloan Digital Sky Survey at http://skyserver.sdss.org.
More details about the universe can be found in the beautiful text by W.J. K
& R.A. F
,
Universe, h edition, W.H. Freeman & Co., 1999. e most recent discoveries are best followed on the
http://sci.esa.int and http://hubble.nasa.gov websites.
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V
novae
supernovae
hypernovae gamma-ray bursts
radio sources X-ray sources cosmic rays gravitational lensing comets meteorites
frequency
between and solar masses per year per galaxy; around solar mass in the Milky Way
new luminous stars, L < W
ejecting bubble
R tëc
new bright stars,
L< W
rate
to per galaxy per a
optical bursts
L
W
luminosity
L up to W, about one per cent of the whole visible universe’s luminosity
energy
c. J
duration
c. . to s
observed number
c. per day
radio emission
to W
X-ray emission
to W
energy
from eV to eV
light bending
angles down to − ′′
recurrence, evaporation typ. period a, typ. visibility lifetime ka, typ. lifetime ka
age
up to . ë a
Observed components
intergalactic space quasars
mass density red-shi luminosity
galaxy superclusters our own local supercluster galaxy groups
our local group galaxies
number of galaxies number of galaxies size number of galaxies number of galaxies size number containing containing containing
our galaxy
diameter mass containing
c. − kg m up to z = L = W, about the same as one galaxy c. inside our horizon about
Zm between a dozen and
. to Zm c. inside horizon
to globular clusters typically stars each typically one supermassive and several intermediate-mass black holes . ( . ) Zm
kg or ë solar masses Ref. 397 globular clusters each with million stars
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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A
M
V
speed globular clusters (e.g. M ) containing
nebulae, clouds our local interstellar cloud
star systems
age composition size composition types
our solar system our solar system stars
size speed mass
giants and supergiants large size
main sequence stars
brown dwarfs
low mass
low temperature
L dwarfs
low temperature
T dwarfs
low temperature
white dwarfs
small radius
high temperature
neutron stars
nuclear mass density
small size
jet sources
central compact
objects
emitters of X-ray
X-ray emission
bursts
pulsars
periodic radio emission
mass
magnetars
high magnetic elds
(so gamma repeaters, anomalous X-ray pulsars)
mass
black holes
horizon radius
km s towards Hydra-Centaurus
thousands of stars, one intermediate-mass black hole
up to Ga (oldest known objects) dust, oxygen, hydrogen
light years atomic hydrogen at K
orbiting double stars, over stars orbited by brown dwarfs, several planetary systems
light years (Oort cloud)
km s from Aquarius towards Leo up to solar masses (except when stars fuse) Ref. 398
up to Tm
below . solar masses
below
K Ref. 399
to K
to K
r
km
cools from
to K
ρ
kg m
r km
up to around solar masses up to T and higher Ref. 400
above solar masses Ref. 401 r = GM c , observed mass range from to million solar masses
General properties
cosmic horizon expansion ‘age’ of the universe
distance Hubble’s constant
c. m = Ym ( ) km s− Mpc− or . ( ) ë − s− . ( ) Ga
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
vacuum large-scale shape
dimensions matter
baryons dark matter dark energy photons
neutrinos average temperature perturbations
ionization optical depth decoupling
energy density
space curvature topology
number
density
density
density
density number density
energy density energy density photons neutrinos photon anisotropy density amplitude spectral index tensor-to-scalar ratio
. nJ m or ΩΛ = . for k = no evidence for time-dependence
k ΩK = Page 455 simple in our galactic environment, unknown at large scales
for space, for time, at low and moderate energies
to ë − kg m or to hydrogen atoms per cubic metre
ΩM = . Ωb = . , one sixth of the previous (included in ΩM) ΩDM = . (included in ΩM), unknown
ΩDM = . , unknown to ë m
= . to . ë − kg m ΩR = . ë − Ων unknown . ( )K
not measured, predicted value is K ∆T T = ë −
A= . ( )
n= . ( )
r < . with % con dence
τ= . ( )
z=
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But while we are speaking of what we see in the sky, we need to clarify a general issue.
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“I’m astounded by people who want to ‘know’ the universe when it’s hard enough to nd your way around Chinatown. ” Woody Allen
e term universe implies turning. e universe is what turns around us at night. For a physicist, at least three de nitions are possible for the term ‘universe’:
— e (visible) universe is the totality of all observable mass and energy. is includes everything inside the cosmological horizon. Since the horizon is moving away from us, the amount of observable mass and energy is constantly increasing. e content of the term ‘visible universe’ is thus not xed in time. (What is the origin of this increase?
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F I G U R E 202 The universe is full of galaxies – this photograph shows the Perseus cluster (NASA)
We will come back to this issue later on.) — e (believed) universe is the totality of all mass and energy, including any that is not
visible. Numerous books on general relativity state that there de nitely exists matter
or energy beyond the observation boundaries. We will explain the origin of this belief
below. — e (full) universe is the sum of matter and energy as well as space-time itself.
Challenge 834 ny Challenge 835 ny Challenge 836 ny
ese de nitions are o en mixed up in physical and philosophical discussions. ere is no generally accepted consensus on the terms, so one has to be careful. In this text, when we use the term ‘universe’, we imply the last de nition only. We will discover repeatedly that without clear distinction between the de nitions the complete ascent of Motion Mountain becomes impossible. (For example: Is the amount of matter and energy in the full universe the same as in the visible universe?)
Note that the ‘size’ of the visible universe, or better, the distance to its horizon, is a quantity which can be imagined. e value of m is not beyond imagination. If one took all the iron from the Earth’s core and made it into a wire reaching to the edge of the visible universe, how thick would it be? e answer might surprise you. Also, the content of the universe is clearly nite. ere are about as many visible galaxies in the universe as there are grains in a cubic metre of sand. To expand on the comparison, can you deduce how much space you would need to contain all the our you would get if every little speck represented one star?
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F I G U R E 203 An atlas of our cosmic environment: illustrations at scales up to 12.5, 50, 250, 5 000, 50 000, 500 000, 5 million, 100 million, 1 000 million and 14 000 million light years (© Richard Powell, http://www.anzwers.org/free/universe)
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Ref. 402 Ref. 403
“᾽Η τοι µὲν πρώτιστα Ξάος γένετ΄ ... * Hesiod,
”eogony.
Obviously, the universe is full of motion. To get to know the universe a bit, it is useful to
measure the speed and position of as many objects in it as possible. In the twentieth cen-
tury, a large number of such observations were obtained from stars and galaxies. (Can you
imagine how distance and velocity are determined?) is wealth of data can be summed
up in two points.
First of all, on large scales, i.e. averaged over about ve hundred million light years,
the matter density in the universe is homogeneous and isotropic. Obviously, at smaller
scales inhomogeneities exist, such as galaxies or cheesecakes. Our galaxy for example is
neither isotropic nor homogeneous. But at large scales the di erences average out. is
large-scale homogeneity of matter distribution is o en called the cosmological principle.
e second point about the universe is even more important. In the s, independ-
ently, Carl Wirtz, Knut Lundmark and Gustaf Stromberg showed that on the whole, galax-
ies move away from the Earth, and the more so, the more they were distant. ere are a
few exceptions for nearby galaxies, such as the Andromeda nebula itself; but in general,
the speed of ight v of an object increases with distance d. In , the US-American as-
tronomer Edwin Hubble** published the rst measurement of the relation between speed
and distance. Despite his use of incorrect length scales he found a relation
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
v=Hd,
(339)
Challenge 838 ny
where the proportionality constant H is today called the Hubble constant. A graph of the relation is given in Figure . e Hubble constant is known today to have a value around km s− Mpc− . (Hubble’s own value was so far from this value that it is not cited any more.) For example, a star at a distance of Mpc* is moving away from Earth with a speed between of around km s, and proportionally more for stars further away.
In fact, the discovery by Wirtz, Lundmark and Stromberg implies that every galaxy moves away from all the others. (Why?) In other words, the matter in the universe is expanding. e scale of this expansion and the enormous dimensions involved are amazing.
e motion of all the thousand million galaxy groups in the sky is described by the single equation ( )! Some deviations are observed for nearby galaxies, as mentioned above, and for faraway galaxies, as we will see.
e cosmological principle and the expansion taken together imply that the universe cannot have existed before time when it was of vanishing size; the universe thus has a
nite age. Together with the evolution equations, as explained in more detail below, the
Page 1167
* ‘Verily, at rst chaos came to be ...’. e eogony, attributed to the probably mythical Hesiodos, was nalized around 700 . It can be read in English and Greek on the http://www.perseus.tu s.edu website. e famous quotation here is from verse 117. ** Edwin Powell Hubble (1889–1953), important US-American astronomer. A er being an athlete and taking a law degree, he returned to his childhood passion of the stars; he nally proved Immanuel Kant’s 1755 conjecture that the Andromeda nebula was a galaxy like our own. He thus showed that the Milky Way is only a tiny part of the universe. * A megaparsec or Mpc is a distance of . Zm.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 204 The relation between star distance and star velocity
Ref. 404 Challenge 839 ny
Ref. 405 Ref. 406
Hubble constant points to an age value of around
million years. e expansion also
means that the universe has a horizon, i.e. a nite maximum distance for sources whose
signals can arrive on Earth. Signals from sources beyond the horizon cannot reach us.
Since the universe is expanding, in the past it has been much smaller and thus much
denser than it is now. It turns out that it has also been hotter. George Gamow* predicted
in that since hot objects radiate light, the sky cannot be completely black at night,
but must be lled with black-body radiation emitted when it was ‘in heat’. at radiation,
called the background radiation, must have cooled down due to the expansion of the uni-
verse. (Can you con rm this?) Despite various similar predictions by other authors, in
one of the most famous cases of missed scienti c communication, the radiation was
found only much later, by two researchers completely unaware of all this work. A fam-
ous paper in by Doroshkevich and Novikov had even stated that the antenna used
by the (unaware) later discoverers was the best device to search for the radiation! In any
case, only in did Arno Penzias and Robert Wilson discover the radiation. It was in
one of the most beautiful discoveries of science, for which both later received the Nobel
Prize for physics. e radiation turns out to be described by the black-body radiation for
a body with a temperature of . K; it follows the black-body dependence to a precision
of about part in .
But apart from expansion and cooling, the past fourteen thousand million years have
also produced a few other memorable events.
* George Gamow (b. 1904 Odessa, d. 1968 St. Boulder), Russian-American physicist; he explained alpha decay as a tunnelling e ect and predicted the microwave background. He wrote the rst successful popular science texts, such as 1, 2, 3, in nity and the Mr. ompkins series, which were later imitated by many others.
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F I G U R E 205 The Hertzsprung–Russell diagram (© Richard Powell)
D
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Don’t the stars shine beautifully? I am the only person in the world who knows why they do.
“ Friedrich (Fritz) Houtermans ( – ) ” Stars seem to be there for ever. In fact, every now and then a new star appears in the sky:
a nova. e name is Latin and means ‘new’. Especially bright novae are called supernovae. Novae and similar phenomena remind us that stars usually live much longer than humans, but that like us, they are born and die.
It turns out that one can plot all stars on the so-called Hertzsprung–Russell diagram. is diagram, central to every book on astronomy, is shown in Figure . It is a beautiful example of a standard method used by astrophysicists: collecting statistics over many examples of a type of object, one can deduce the life cycle of the object, even though their lifetime is much longer than that of a human. For example, it is possible, by clever use of the diagram, to estimate the age of stellar clusters, and thus arrive at a minimum age of the universe. e result is around thirteen thousand million years. One conclusion is basic: since stars shine, they also die. Stars can only be seen if they
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Page 922
are born but not yet dead at the moment of light emission. is leads to restrictions on their visibility, especially for high red-shi s. Indeed, modern telescope can look at places (and times) so far from now that they contained no stars at all. At those distances one only observers quasars; these are not stars, but much more massive and bright systems.
eir precise structure are still being studied by astrophysicists. On the other hand, since the stars shine, they were also formed somehow. e fascinating details of their birth from dust clouds are a central part of astrophysics but will not be explored here. Yet we do not have the full answer to our question. Why do stars shine at all? Clearly, they shine because they are hot. ey are hot because of nuclear reactions in their interior. We will discuss these processes in more detail in the chapter on the nucleus.
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A
Ref. 407 Ref. 408
“Anima scintilla stellaris essentiae.* Heraclitus of Ephesus (c. to c.
”)
e adventures the universe has experienced, or better, the adventures the matter and radi-
ation inside it have experienced, are summarized in Table . e steps not yet discussed
will be studied in quantum theory. is history table has applications no theoretical phys-
icist would have imagined. e sequence is so beautiful and impressive that nowadays it
is used in certain psychotherapies to point out to people the story behind their existence
and to remind them of their own worth. Enjoy.
TA B L E 38 A short history of the universe
T
T
E
T
-
a
ëa ëa
b
tPl b
Time, space, matter and initial conditions
indeterminate
c. tPl −s
Distinction of space-time from matter and radiation, initial conditions determinate
− s to −s
In ation & GUT epoch starts; strong and electroweak interactions diverge
−s
Antiquarks annihilate; electromagnetic and weak
interaction separate
ë −s
Quarks get con ned into hadrons; universe is a plasma
Positrons annihilate
.s
Universe becomes transparent for neutrinos
a few seconds Nucleosynthesis: D, He, He and Li nuclei form; radiation still dominates
a
Matter domination starts; density perturbations
magnify
K TPl K ëK K K
K K
K
* ‘ e soul is a spark of the substance of the stars.’
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T
a
z=
T
b
a
z = to z= z= z=
z=
a
ëa ëa
.ë a
.ë a .ë a .ë a
.ë a
.ë a
.ë a
ëa ëa
to ëa ( )ë a
E
T
-
Recombination: during these latter stages of the big
K
bang, H, He and Li atoms form, and the universe
becomes ‘transparent’ for light, as matter and
radiation decouple, i.e. as they acquire di erent
temperatures; the ‘night’ sky starts to get darker and
darker
Sky is almost black except for black-body radiation Tγ = To( + z)
Galaxy formation
Oldest object seen so far
Galaxy clusters form
First generation of stars (population II) is formed, starting hydrogen fusion; helium fusion produces carbon, silicon and oxygen
First stars explode as supernovaec; iron is produced
Second generation of stars (population I) appears, and subsequent supernova explosions of the ageing stars form the trace elements (Fe, Se, etc.) we are made of and blow them into the galaxy
Primitive cloud, made from such explosion remnants, collapses; Sun forms
Earth and other planet formation: Azoicum starts
Craters form on the planets
Moon forms from material ejected during the collision of a large asteroid with the still-liquid Earth
Archean eon (Archaeozoicum) starts: bombardment from space stops; Earth’s crust solidi es; oldest minerals form; water condenses
Unicellular (microscopic) life appears; stromatolites form
Proterozoic eon (‘age of rst life’) starts: atmosphere becomes rich in oxygen thanks to the activity of microorganisms Ref. 409
Macroscopic, multicellular life appears
Earth is completely covered with ice for the rst time (reason still unknown) Ref. 410
Earth is completely covered with ice for the last time
Paleozoic era (Palaeozoicum, ‘age of old life’) starts, a er a gigantic ice age: animals appear, oldest fossils (with ( ) start of Cambrian, ( ) Ordovician,
( ) Silurian, ( ) Devonian, ( ) Carboniferous and ( ) Permian periods)
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•.
T
T
a
b
ëa ëa ( )ë a
ëa
ëa ( )ë a ëa .ë a
ëa ( )ë a ëa
a a a a a a a a a a to a
E
T
Land plants appear
Wooden trees appear
Mesozoic era (Mesozoicum, ‘age of middle life’, formerly called Secondary) starts: most insects and other life forms are exterminated; mammals appear (with ( ) start of Triassic, ( ) Jurassic and
( ) Cretaceous periods)
Continent Pangaea splits into Laurasia and Gondwana
e star cluster of the Pleiades forms
Birds appear
Golden time of dinosaurs (Cretaceous) starts
Start of formation of Alps, Andes and Rocky Mountains
Cenozoic era (Caenozoicum, ‘age of new life’) starts: dinosaurs become extinct a er an asteroid hits the Earth in the Yucatan; primates appear (with . start of Tertiary, consisting of Paleogene period with Paleocene, . Eocene and . Oligocene epoch, and of Neogene period, with . Miocene and . Pliocene epoch; then . Quaternary period with Pleistocene (or Diluvium) and . Holocene (or Alluvium) epoch)
Large mammals appear
Hominids appears
Supernova explodes, with following consequences: more intense cosmic radiation, higher formation rate of clouds, Earth cools down drastically, high evolutionary pressure on the hominids and as a result, Homo appears Ref. 411
Formation of youngest stars in galaxy
Homo sapiens appears
Beginning of last ice age
Homo sapiens sapiens appears
End of last ice age, start of Holocene
First written texts
Physics starts
Use of co ee, pencil and modern physics starts
Electricity use begins
Einstein publishes
You are a unicellular being
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T
T
E
T
-
a
present
future
b
c. ë a
You are reading this
You enjoy life; for details and reasons, see page
Tγ = . K, Tν . K, Tb K
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a. e time coordinate used here is the one given by the coordinate system de ned by the microwave background radiation, as explained on page . A year is abbreviated ‘a’ (Latin ‘annus’). Errors in the last digits are given between parentheses. b. is quantity is not exactly de ned since the big bang is not a space-time event. More on this issue on page . c. e history of the atoms on Earth shows that we are made from the le overs of a supernova. We truly are made of stardust.
e geological time scale is the one of the International Commission on Stratigraphy; the times are measured through radioactive dating.
Despite its length and its interest, this table has its limitations. For example, what happened elsewhere in the last few thousand million years? ere is still a story to be written of which next to nothing is known. For obvious reasons, investigations have been rather Earth-centred.
Research in astrophysics is directed at discovering and understanding all phenomena observed in the skies. Here we skip most of this fascinating topic, since as usual, we want to focus on motion. Interestingly, general relativity allows us to explain many of the general observations about motion in the universe.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
T
-
Challenge 840 n Ref. 412
“A number of rabbits run away from a central point in various directions, all with the same speed. While running, one rabbit turns its head, and makes a startling observation. What does it ”see?
e data showing that the universe is sprinkled with stars all over lead to a simple conclusion: the universe cannot be static. Gravity always changes the distances between bodies; the only exceptions are circular orbits. Gravity also changes the average distances between bodies: gravity always tries to collapse clouds. e biggest cloud of all, the one formed by all the matter in the universe, must therefore either be collapsing, or still be in expansion.
e rst to dare to draw this conclusion was Aleksander Friedmann.* In he de-
Page 463, page 464
* Aleksander Aleksandrowitsch Friedmann (1888–1925), Russian physicist who predicted the expansion of the universe. Following his early death from typhus, his work remained almost unknown until Georges A. Lemaître (b. 1894 Charleroi, d. 1966 Leuven), Belgian priest and cosmologist, took it up and expanded it in 1927, focusing, as his job required, on solutions with an initial singularity. Lemaître was one of the propagators of the (erroneous!) idea that the big bang was an ‘event’ of ‘creation’ and convinced his whole organization of it. e Friedmann–Lemaître solutions are o en erroneously called a er two other physicists, who studied them again much later, in 1935 and 1936, namely H.P. Robertson and A.G. Walker.
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•.
Challenge 841 ny
duced the detailed evolution of the universe in the case of homogeneous, isotropic mass
distribution. His calculation is a classic example of simple but powerful reasoning. For a
universe which is homogeneous and isotropic for every point, the line element is given
by
ds = c dt − a (t)(dx + dy + dz )
(340)
and matter is described by a density ρM and a pressure pM. Inserting all this into the eld equations, we get two equations
a˙ a
+k a
=
πG ρM + Λ
and
a¨ = − πG (ρM + pM) a + Λ a
(341) (342)
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which imply
ρ˙M = −
a˙ a
(ρM
+
pM)
.
(343)
Challenge 842 ny
At the present time t , the pressure of matter is negligible. (In the following, the index refers to the present time.) In this case, the expression ρMa is constant in time.
Equations ( ) and ( ) depend on only two constants of nature: the gravitational constant G, related to the maximum force or power in nature, and the cosmological constant Λ, describing the energy density of the vacuum, or, if one prefers, the smallest force in nature.
Before we discuss the equations, rst a few points of vocabulary. It is customary to relate all mass densities to the so-called critical mass density ρc given by
ρc =
H πG
(
) ë − kg m
(344)
corresponding to about , give or take , hydrogen atoms per cubic metre. On Earth, one
would call this value an extremely good vacuum. Such are the di erences between every-
day life and the universe as a whole. In any case, the critical density characterizes a matter
distribution leading to an evolution of the universe just between never-ending expansion
and collapse. In fact, this density is the critical one, leading to a so-called marginal evolu-
tion, only in the case of vanishing cosmological constant. Despite this restriction, the term
is now used for this expression in all other cases as well. One thus speaks of dimensionless
mass densities ΩM de ned as
ΩM = ρ ρc .
(345)
e cosmological constant can also be related to this critical density by setting
ΩΛ
=
ρΛ ρc
=
Λc πGρc
=
Λc H
.
(346)
A third dimensionless parameter ΩK describes the curvature of space. It is de ned in
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clofslaeotdpen universe
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terms of the present-day radius of the universe R
, − , as
ΩK
=
−k RH
and the curvature constant k = (347)
Challenge 843 ny
and its sign is opposite to the one of the curvature k; ΩK vanishes for vanishing curvature. Note that a positively curved universe, when homogeneous and isotropic, is necessarily
closed and of nite volume. A at or negatively curved universe with the same matter
distribution can be open, i.e. of in nite volume, but does not need to be so. It could be
simply or multiply connected. In these cases the topology is not completely xed by the
curvature.
e present-time Hubble parameter is de ned
by H = a˙ a . From equation ( ) we then get the central relation
no big bang
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ΩM + ΩΛ + ΩK = .
(348)
2
In the past, when data were lacking, physicists were divided into two camps: the claustrophobics believing that ΩK and the agoraphobics believing that ΩK < . More details about the measured values of these parameters will be given shortly.
e diagram of Figure shows the most interesting ranges of parameters together with the corresponding behaviours of the universe.
For the Hubble parameter, the most modern measurements give a value of
1 ΩΛ
0
-1
0
experimental
values adcecceeelteleerrarnatetaedldeexexxpppaaannnssisioioonnn collapse
too young
1
2
3
H=
km sMpc = . ë − s (349)
ΩM
F I G U R E 206 The ranges for the Ω
which corresponds to an age of the universe of parameters and their consequences . thousand million years. In other words,
the age deduced from the history of space-time agrees with the age, given above, deduced from the history of stars.
To get a feeling of how the universe evolves, it is customary to use the so-called deceleration parameter q . It is de ned as
q
=
−
a
a¨ H
=
ΩM − ΩΛ .
(350)
e parameter q is positive if the expansion is slowing down, and negative if the expansion is accelerating. ese possibilities are also shown in the diagram.
An even clearer way to picture the expansion of the universe for vanishing pressure is
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•.
scale factor a
l Pl
quantum effects
c t a(t)
t Pl
time t
scale factor a
present
experimental uncertainties
present
time t
F I G U R E 207 The evolution of the universe’s scale a for different values of its mass density
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to rewrite equation ( ) using τ = t H and x(τ) = a(t) a(t ), yielding
(
dx dτ
)
+ U(x) = ΩK
with U(x) = −ΩΛx − ΩΛx
(351)
is looks like the evolution equation for the motion of a particle with mass , with total energy ΩK in a potential U(x). e resulting evolutions are easily deduced.
For vanishing ΩΛ, the universe either expands for ever, or recollapses, depending on the value of the mass–energy density.
For non-vanishing (positive) ΩΛ, the potential has exactly one maximum; if the particle has enough energy to get over the maximum, it will accelerate continuously. at is the situation the universe seems to be in today.
For a certain time range, the result is shown in Figure . ere are two points to be noted: rst the set of possible curves is described by two parameters, not one. In addition, lines cannot be drawn down to zero size. ere are two main reasons: we do not yet understand the behaviour of matter at very high energy, and we do not understand the behaviour of space-time at very high energy. We return to this important issue later on.
e main conclusion to be drawn from Friedmann’s work is that a homogeneous and isotropic universe is not static: it either expands or contracts. In either case, it has a nite age. is profound idea took many years to spread around the cosmology community; even Einstein took a long time to get accustomed to it.
Note that due to its isotropic expansion, the universe has a preferred reference frame: the frame de ned by average matter. e time measured in that frame is the time listed in Table and is the one we assume when we talk about the age of the universe.
An overview of the possibilities for the long time evolution is given in Figure .
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k = –1
k = 0 Λ < Λc
scale factor k = +1
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Λ > 0 scale factor
time t scale factor
time t Λ = Λc scale factor
Λ > Λc scale factor
Λ = 0 scale factor
Λ < 0 scale factor
time t
time t
scale factor
scale factor
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time t
time t
scale factor
scale factor
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time t
time t
time t
time t
F I G U R E 208 The long-term evolution of the universe’s scale factor a for various parameters
time t
Challenge 844 ny
e evolution can have various outcomes. In the early twentieth century, people decided among them by personal preference. Albert Einstein rst preferred the solution k = and Λ = a− = πGρM. It is the unstable solution found when x(τ) remains at the top of the potential U(x).
In , the Dutch physicist Willem de Sitter had found, much to Einstein’s personal dismay, that an empty universe with ρM = pM = and k = is also possible. is type of universe expands for large times. e de Sitter universe shows that in special cases, matter is not needed for space-time to exist.
Lemaître had found expanding universes for positive mass, and his results were also contested by Einstein at rst. When later the rst measurements con rmed the calculations, the idea of a massive and expanding universe became popular. It became the standard model in textbooks. However, in a sort of collective blindness that lasted from around to , almost everybody believed that Λ = .* Only towards the end of the twentieth century did experimental progress allow one to make statements based on evidence rather than beliefs or personal preferences, as we will nd out shortly. But rst of all we will settle an old issue.
Challenge 845 ny * In this case, for ΩM , the age of the universe follows t vanishing mass density one has t = Ho.
( H ), where the limits correspond. For
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F I G U R E 209 The fluctuations of the cosmic background radiation (WMAP/NASA)
W
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Ref. 413 Challenge 846 n
“In der Nacht hat ein Mensch nur ein Nachthemd an, und darunter kommt gleich der Charakter.** ” Rober Musil
First of all, the sky is not black at night. It has the same intrinsic colour as during the day, as any long-exposure photograph shows. (See, for example, Figure .) But that colour, like the colour of the sky during the day, is not due to the temperature of the sky, but to scattered light from the stars. If we look for the real colour of the sky, we need to look for its thermal radiation. Indeed, measurements show that even the empty sky is not completely cold or black at night. It is lled with radiation of around GHz; more precise measurements show that the radiation corresponds to the thermal emission of a body at . K. is background radiation is the thermal radiation le over from the big bang.
e universe is indeed colder than the stars. But why is this so? If the universe were homogeneous on large scales and in nitely large, it would have an in nite number of stars. Looking in any direction, we would see the surface of a star. e night sky would be as bright as the surface of the Sun! Can you convince your grandmother about this?
In a deep forest, one sees a tree in every direction. Similarly, in a ‘deep’ universe, we would see a star in every direction. Now, the average star has a surface temperature of about K. If we lived in a deep and old universe, we would e ectively live inside an oven with a temperature of around K, making it impossible to enjoy ice cream.
is paradox was most clearly formulated in by the astronomer Wilhelm Olbers.*** As he extensively discussed the question, it is also called Olbers’ paradox. Today
Page 101
** ‘At night, a person is dressed only with a nightgown, and directly under it there is the character.’ Robert Musil (b. 1880 Klagenfurt, d. 1942 Geneva), German writer. *** Heinrich Wilhelm Matthäus Olbers (b. 1758 Arbergen, d. 1840 Bremen), astronomer. He discovered two planetoids, Pallas and Vesta, and ve comets; he developed the method of calculating parabolic orbits for comets which is still in use today. Olbers also actively supported the mathematician and astronomer Friedrich Wilhelm Bessel in his career choice. e paradox is named a er Olbers, though others had made
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Ref. 414
Ref. 415 Ref. 413 Challenge 848 ny Ref. 416
we know that even if all matter in the universe were converted into radiation, the universe would still not be as bright as just calculated. In other words, the power and lifetime of stars are much too low to produce the oven brightness just mentioned. So something is wrong.
In fact, two main e ects can be invoked to avoid the contradiction. First, since the universe is nite in age, distant stars are shining for less time. We see them in a younger stage or even during their formation, when they were darker. As a result, the share of brightness of distant stars is smaller than that of nearby stars, so that the average temperature of the sky is reduced.* Secondly, we could imagine that the radiation of distant stars is red-shi ed and that the volume the radiation must ll is increasing continuously, so that the average temperature of the sky is also reduced.
Calculations are necessary to decide which e ect is the greater one. is issue has been studied in great detail by Paul Wesson; he explains that the rst e ect is larger than the second by a factor of about three. We may thus state correctly that the sky is dark at night mostly because the universe has a nite age. We can add that the sky would be somewhat brighter if the universe were not expanding.
In addition, the darkness of the sky is possible only because the speed of light is nite. Can you con rm this?
Finally, the darkness of the sky also tells us that the universe has a large (but nite) age. Indeed, the . K background radiation is that cold, despite having been emitted at K, because it is red-shi ed, thanks to the Doppler e ect. Under reasonable assumptions, the temperature T of this radiation changes with the scale factor R(t) of the universe as
T R(t) .
(352)
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Ref. 414 Challenge 849 ny Challenge 850 ny
In a young universe, we would thus not be able to see the stars, even if they existed. From the brightness of the sky at night, measured to be about ë − times that of an
average star like the Sun, we can deduce something interesting: the density of stars in the universe must be much smaller than in our galaxy. e density of stars in the galaxy can be deduced by counting the stars we see at night. But the average star density in the galaxy would lead to much higher values for the night brightness if it were constant throughout the universe. We can thus deduce that the galaxy is much smaller than the universe simply by measuring the brightness of the night sky and by counting the stars in the sky! Can you make the explicit calculation?
In summary, the sky is black at night because space-time and matter are of nite, but old age. As a side issue, here is a quiz: is there an Olbers’ paradox also for gravitation?
Challenge 847 ny
similar points before, such as the Swiss astronomer Jean Philippe Loÿs de Cheseaux in 1744 and Johannes Kepler in 1610. * Can you explain that the sky is not black just because it is painted black or made of black chocolate? Or more generally, that the sky is not made of and does not contain any dark and cold substance, as Olbers himself suggested, and as John Herschel refuted in 1848?
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•.
I
,
?
Ref. 417 Page 468
Ref. 418
“- Doesn’t the vastness of the universe make you feel small? - I can feel small without any help from the
universe.
” Anonymous
Sometimes the history of the universe is summed up in two words: bang!...crunch. But
will the universe indeed recollapse, or will it expand for ever? Or is it in an intermediate,
marginal situation? e parameters deciding its fate are the mass density and cosmolo-
gical constant.
e main news of the last decade of twentieth-century astrophysics are the experi-
mental results allowing one to determine all these parameters. Several methods are being
used. e rst method is obvious: determine the speed and distance of distant stars. For
large distances, this is di cult, since the stars are so faint. But it has now become possible
to search the sky for supernovae, the bright exploding stars, and to determine their dis-
tance from their brightness. is is presently being done with the help of computerized
searches of the sky, using the largest available telescopes.
A second method is the measurement of the anisotropy of the cosmic microwave back-
ground. From the observed power spectrum as a function of the angle, the curvature of
space-time can be deduced.
A third method is the determination of the mass density using the gravitational lensing
e ect for the light of distant quasars bent around galaxies or galaxy clusters.
A fourth method is the determination of the mass density using galaxy clusters. All
these measurements are expected to improve greatly in the years to come.
At present, these four completely independent sets of measurements provide the
values
(ΩM, ΩΛ, ΩK) ( . , . , . )
(353)
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where the errors are of the order of . or less. e values imply that the universe is spatially at, its expansion is accelerating and there will be no big crunch. However, no de nite
statement on the topology is possible. We will return to this last issue shortly. In particular, the data show that the density of matter, including all dark matter, is only
about one third of the critical value.* Two thirds are given by the cosmological term. For the cosmological constant Λ one gets the value
Λ = ΩΛ
H c
− m.
(354)
is value has important implications for quantum theory, since it corresponds to a va-
* e di erence between the total matter density and the separately measurable baryonic matter density, only about one sixth of the former value, is also explained yet. It might even be that the universe contains matter of a type unknown so far. is issue is called the dark matter problem; it is one of the important unsolved questions of cosmology.
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cuum energy density
ρΛc
=
Λc πG
. nJ m
− (GeV) (ħc)
.
(355)
But the cosmological term also implies a negative vacuum pressure pΛ = −ρΛc . Inserting Page 427 this result into the relation for the potential of universal gravity deduced from relativity
∆φ = πG(ρ + p c )
(356)
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Ref. 419 we get
∆φ = πG(ρM − ρΛ) .
(357)
Challenge 851 ny us the gravitational acceleration is
a
=
GM r
−
Λc
r
=
GM r
− ΩΛH
r,
(358)
Challenge 852 ny
which shows that a positive vacuum energy indeed leads to a repulsive gravitational e ect. Inserting the mentioned value ( ) for the cosmological constant Λ we nd that the repulsive e ect is small even for the distance between the Earth and the Sun. In fact, the order of magnitude of the repulsive e ect is so much smaller than that of attraction that one cannot hope for a direct experimental con rmation of this deviation from universal gravity at all. Probably astrophysical determinations will remain the only possible ones. A positive gravitational constant manifests itself through a positive component in the expansion rate, as we will see shortly.
But the situation is puzzling. e origin of this cosmological constant is not explained by general relativity. is mystery will be solved only with the help of quantum theory. In any case, the cosmological constant is the rst local and quantum aspect of nature detected by astrophysical means.
W
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Ref. 420 Challenge 853 ny
Page 939
Could the universe be lled with water, which is transparent, as maintained by some
popular books in order to explain rain? No. Even if it were lled with air, the total mass
would never have allowed the universe to reach the present size; it would have recollapsed
much earlier and we would not exist.
e universe is thus transparent because it is mostly empty. But why is it so empty? First
of all, in the times when the size of the universe was small, all antimatter annihilated with
the corresponding amount of matter. Only a tiny fraction of matter, which originally was
slightly more abundant than antimatter, was le over. is − fraction is the matter we
see now. As a consequence, there are as many photons in the universe as electrons or
quarks.
In addition,
years a er antimatter annihilation, all available nuclei and elec-
trons recombined, forming atoms, and their aggregates, like stars and people. No free
charges interacting with photons were lurking around any more, so that from that period
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onwards light could travel through space as it does today, being a ected only when it hits a star or dust particle.
If we remember that the average density of the universe is − kg m and that most of the matter is lumped by gravity in galaxies, we can imagine what an excellent vacuum lies in between. As a result, light can travel along large distances without noticeable hindrance.
But why is the vacuum transparent? at is a deeper question. Vacuum is transparent because it contains no electric charges and no horizons: charges or horizons are indispensable in order to absorb light. In fact, quantum theory shows that vacuum does contain so-called virtual charges. However, virtual charges have no e ects on the transmission of light.
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“Μελέτη θανάτου.
Learn to die. Plato, Phaedo,
”a.
Above all, the big bang model, which is deduced from the colour of the stars and galaxies,
states that about fourteen thousand million years ago the whole universe was extremely
small. is fact gave the big bang its name. e term was created (with a sarcastic un-
dertone) in by Fred Hoyle, who by the way never believed that it applies to nature.
Nevertheless, the term caught on. Since the past smallness of the universe be checked
directly, we need to look for other, veri able consequences. e central ones are the fol-
lowing:
— all matter moves away from all other matter; — the mass of the universe is made up of about 75% hydrogen and 23% helium; — there is thermal background radiation of about . K; — the maximal age for any system in the universe is around fourteen thousand million
years; — there are background neutrinos with a temperature of about K;* — for non-vanishing cosmological constant, Newtonian gravity is slightly reduced.
Ref. 421 Ref. 422
All predictions except the last two have been con rmed by observations. Technology will probably not allow us to check the last two in the foreseeable future; however, there is no evidence against them.
Competing descriptions of the universe have not been so successful in matching observations. In addition, theoretical arguments state that with matter distributions such as the observed one, and some rather weak general assumptions, there is no way to avoid a period in the nite past in which the universe was extremely small. erefore it is worth having a close look at the situation.
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Was it a kind of explosion? is description implies that some material transforms internal energy into motion of its parts. ere was no such process in the early history
* e theory states that Tν Tγ ( ) . ese neutrinos appeared about . s a er the big bang.
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of the universe. In fact, a better description is that space-time is expanding, rather than matter moving. e mechanism and the origin of the expansion is unknown at this point of our mountain ascent. Because of the importance of spatial expansion, the whole phenomenon cannot be called an explosion at all. And obviously there neither was nor is any sound carrying medium in interstellar space, so that one cannot speak of a ‘bang’ in any sense of the term.
Was it big? e visible universe was rather small about fourteen thousand million years ago, much smaller than an atom. In summary, the big bang was neither big nor a bang; but the rest is correct.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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e big bang theory is a description of what happened in the whole of space-time. Despite what is o en written in careless newspaper articles, at every moment of the expansion space has been of non-vanishing size: space was never a single point. People who pretend it was are making ostensibly plausible, but false statements. e big bang theory is a description of the expansion of space-time, not of its beginning. Following the motion of matter back in time, general relativity cannot deduce the existence of an initial singularity.
e issue of measurement errors is probably not a hindrance; however, the e ect of the nonlinearities in general relativity at situations of high energy densities is not clear.
Most importantly, quantum theory shows that the big bang was not a true singularity, as no physical observable, neither density nor temperature, ever reaches an in nitely large (or in nitely small) value. Such values cannot exist in nature.* In any case, there is a general agreement that arguments based on pure general relativity alone cannot make correct statements about the big bang. Nevertheless, most statements in newspaper articles are of this sort.
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Asking what was before the big bang is like asking what is north of the North Pole. Just as nothing is north of the North Pole, so nothing ‘was’ before the big bang. is analogy could be misinterpreted to imply that the big bang took its start at a single point in time, which of course is incorrect, as just explained. But the analogy is better than it looks: in fact, there is no precise North Pole, since quantum theory shows that there is a fundamental indeterminacy as to its position. ere is also a corresponding indeterminacy for the big bang.
In fact, it does not take more than three lines to show with quantum theory that time and space are not de ned either at or near the big bang. We will give this simple argument in the rst chapter of the third part of our mountain ascent. e big bang therefore cannot be called a ‘beginning’ of the universe. ere never was a time when the scale factor a(t) of the universe was zero.
is conceptual mistake is frequently encountered. In fact, quantum theory shows that near the big bang, events can neither be ordered nor even be de ned. More bluntly, there is no beginning; there has never been an initial event or singularity.
* Many physicists are still wary of making such strong statements on this point. e rst sections of the third Page 996 part of our mountain ascent give the precise arguments leading to them.
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Ref. 423 Page 49 Page 153
Ref. 424
Obviously the concept of time is not de ned ‘outside’ or ‘before’ the existence of the universe; this fact was already clear to thinkers over a thousand years ago. It is then tempting to conclude that time must have started. But as we saw, that is a logical mistake as well:
rst of all, there is no starting event, and secondly, time does not ow, as clari ed already in the beginning of our walk.
A similar mistake lies behind the idea that the universe had certain ‘initial conditions.’ Initial conditions by de nition make sense only for objects or elds, i.e. for entities which can be observed from the outside, i.e. for entities which have an environment. e universe does not comply with this requirement; it thus cannot have initial conditions. Nevertheless, many people still insist on thinking about this issue; interestingly, Stephen Hawking sold millions of copies of a book explaining that a description without initial conditions is the most appealing, overlooking the fact that there is no other possibility anyway.*
In summary, the big bang is not a beginning, nor does it imply one. We will uncover the correct way to think about it in the third part of our mountain ascent.
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Challenge 854 ny Page 51
[ e general theory of relativity produces] universal doubt about god and his creation.
“ A witch hunter ” Creation, i.e. the appearance of something out of nothing, needs an existing concept of
space and time to make sense. e concept of ‘appearance’ makes no sense otherwise. But whatever the description of the big bang, be it classical, as in this chapter, or quantum mechanical, as in later ones, this condition is never ful lled. Even in the present, classical description of the big bang, which gave rise to its name, there is no appearance of matter, nor of energy, nor of anything else. And this situation does not change in any later, improved description, as time or space are never de ned before the appearance of matter.
In fact, all properties of a creation are missing: there is no ‘moment’ of creation, no appearance from nothing, no possible choice of any ‘initial’ conditions out of some set of possibilities, and as we will see in more detail later on, not even any choice of particular physical ‘laws’ from any set of possibilities.
In summary, the big bang does not imply nor harbour a creation process. e big bang was not an event, not a beginning and not a case of creation. It is impossible to continue the ascent of Motion Mountain if one cannot accept each of these three conclusions. To deny them is to continue in the domain of beliefs and prejudices, thus e ectively giving up on the mountain ascent.
Note that this requirement is not new. In fact, it was already contained in equation ( ) at the start of our walk, as well as in all the following ones. It appears even more clearly at this point. But what then is the big bang? We’ll nd out in the third part. We now return to the discussion of what the stars can tell us about nature.
* is statement will still provoke strong reactions among physicists; it will be discussed in more detail in the section on quantum theory.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 210 The absorption of the atmosphere (NASA)
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First of all, the Sun is visible because air is transparent. It is not self-evident that air is transparent; in fact it is transparent only to visible light and to a few selected other frequencies. Infrared and ultraviolet radiation are mostly absorbed. e reasons lie in the behaviour of the molecules the air consists of, namely mainly nitrogen, oxygen and a few other transparent gases. Several moons and planets in the solar system have opaque atmospheres: we are indeed lucky to be able to see the stars at all.
In fact, even air is not completely transparent; air molecules scatter light a little bit. at is why the sky and distant mountains appear blue and sunsets red,* and stars are invisible during daylight. At many frequencies far from the visible spectrum the atmosphere is even opaque, as Figure shows.
Secondly, we can see the Sun because the Sun, like all hot bodies, emits light. We describe the details of incandescence, as this e ect is called, below.
irdly, we can see the Sun because we and our environment and the Sun’s environment are colder than the Sun. In fact, incandescent bodies can be distinguished from their background only if the background is colder. is is a consequence of the properties of incandescent light emission, usually called black-body radiation. e radiation is materialindependent, so that for an environment with the same temperature as the body, nothing can be seen at all. Just have a look at the photograph on page as a proof.
Finally, we can see the Sun because it is not a black hole. If it were, it would emit (almost) no light.
Obviously, each of these conditions applies to stars as well. For example, we can only see them because the night sky is black. But then, how to explain the multicoloured sky?
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Stars are visible because they emit visible light. We have encountered several important e ects which determine colours: the diverse temperatures among the stars, the Doppler shi due to a relative speed with respect to the observer, and the gravitational red-shi .
* Air scattering makes the sky blue also at night, as can be proven by long-exposure photographs. (See, for example, Figure 61.) However, our eyes are not able to perceive this, and the low levels of light make it appear black to us.
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TA B L E 39 The colour of the stars
C
T
-E
L
C
O
kK
Mintaka
δ Orionis
blue-violet
O
( ) kK Alnitak
ζ Orionis
blue-violet
B
( ) kK Bellatrix
γ Orionis
blue
B
kK
Saiph
κ Orionis
blue-white
B
kK
Rigel
β Orionis
blue-white
B
kK
Alnilam
ε Orionis
blue-white
B
( ) kK Regulus
α Leonis
blue-white
A
. kK
Sirius
α Canis Majoris blue-white
A
. kK
Megrez
δ Ursae Majoris white
A
. ( ) kK Altair
α Aquilae
yellow-white
F
. ( ) kK Canopus
α Carinae
yellow-white
F
. kK
Procyon
α Canis Minoris yellow-white
G
. kK
Sun
ecliptic
yellow
K
. ( ) kK Aldebaran
α Tauri
orange
M
. ( ) kK Betelgeuse
α Orionis
red
D
< kK
–
–
all
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Note. White dwarfs, or class-D stars, are remnants of imploded stars, with a size of only a few tens of kilometres. Not all are white; they can be yellow or red. ey comprise 5% of all stars. None is visible with
the naked eye. Temperature uncertainties in the last digit are given between parentheses. e size of all other stars is an independent variable and is sometimes added as roman numerals at the end of the spectral type. (Sirius is an A1V star, Arcturus a K2III star.) Giants and supergiants exist in all classes
from O to M. To accommodate brown dwarfs, two new star classes, L and T, have been proposed.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 855 ny
Not all stars are good approximations to black bodies, so that the black-body radiation law does not always accurately describe their colour. However, most stars are reasonable approximations of black bodies. e temperature of a star depends mainly on its size, its mass, its composition and its age, as astrophysicists are happy to explain. Orion is a good example of a coloured constellation: each star has a di erent colour. Long-exposure photographs beautifully show this.
e basic colour determined by temperature is changed by two e ects. e rst, the Doppler red-shi z, depends on the speed v between source and observer as
z=
∆λ λ
=
fS fO
−
=
c+v c−v
−
.
(359)
Such shi s play a signi cant role only for remote, and thus faint, stars visible through the telescope. With the naked eye, Doppler shi s cannot be seen. But Doppler shi s can make distant stars shine in the infrared instead of in the visible domain. Indeed, the highest Doppler shi s observed for luminous objects are larger than . , corresponding to a re-
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Challenge 856 ny cessional speed of more than % of the speed of light. Note that in the universe, the red-shi is also related to the scale factor R(t) by
z
=
R(t ) R(temission)
−
.
(360)
Light at a red-shi of . was thus emitted when the universe was one sixth of its present
age.
e other colour-changing e ect, the gravitational red-shi zg, depends on the matter density of the source and is given by
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zg =
∆λ λ
=
fS f
−
=
−.
−
GM cR
(361)
Challenge 857 ny It is usually quite a bit smaller than the Doppler shi . Can you con rm this? No other red-shi processes are known; moreover, such processes would contradict
Page 476 all the known properties of nature. But the colour issue leads to the next question.
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It could be that some stars are not seen because they are dark. is could be one explanation for the large amount of dark matter seen in the recent measurements of the background radiation. is issue is currently of great interest and hotly debated. It is known that objects more massive than Jupiter but less massive than the Sun can exist in states which emit hardly any light. ey are called brown dwarfs. It is unclear at present how many such objects exist. Many of the so-called extrasolar ‘planets’ are probably brown dwarfs. e issue is not yet settled.
Another possibility for dark stars are black holes. ese are discussed in detail below.
A
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Ref. 426 Ref. 427
“Per aspera ad astra.*
”
Are we sure that at night, two stars are really di erent? e answer is no. Recently, it
was shown that two ‘stars’ were actually two images of the same object. is was found
by comparing the icker of the two images. It was found that the icker of one image
was exactly the same as the other, just shi ed by days. is result was found by the
Estonian astrophysicist Jaan Pelt and his research group while observing two images of
quasars in the system Q + .
e two images are the result of gravitational lensing, as shown in Figure . Indeed, a
large galaxy can be seen between the two images, much nearer to the Earth. is e ect had
been already considered by Einstein; however he did not believe that it was observable.
e real father of gravitational lensing is Fritz Zwicky, who predicted in that the e ect
* ‘ rough hardship to the stars.’ A famous Latin motto. O en incorrectly given as ‘per ardua at astra’.
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first image GRAVITATIONAL LENSING
•. first image
TOPOLOGICAL EFFECT
star
star
galaxy
Earth
second image
second image
F I G U R E 211 How one star can lead to several images
Earth
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F I G U R E 212 The Zwicky-Einstein ring B1938+666, seen in the radio spectrum (left) and in the optical domain (right) (NASA)
Challenge 858 ny
would be quite common and easy to observe, if lined-up galaxies instead of lined-up stars were considered, as indeed turned out to be the case.
Interestingly, when the time delay is known, astronomers are able to determine the size of the universe from this observation. Can you imagine how?
In fact, if the two observed objects are lined up exactly behind each other, the more distant one is seen as ring around the nearer one. Such rings have indeed been observed, and the galaxy image around a central foreground galaxy at B + , shown in Figure , is one of the most beautiful examples. In , several cases of gravitational lensing by stars were also discovered. More interestingly, three events where one of the two stars has a Earth-mass planet have also been observed. e coming years will surely lead to many additional observations, helped by the sky observation programme in the southern hemisphere that checks the brightness of about million stars every night.
Generally speaking, images of nearby stars are truly unique, but for the distant stars the problem is tricky. For single stars, the issue is not so important, seen overall. Reassuringly, only about multiple star images have been identi ed so far. But when whole galaxies are seen as several images at once (and several dozens are known so far) we might start to get nervous. In the case of the galaxy cluster CL + , shown in Figure , seven
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F I G U R E 213 Multiple blue images of a galaxy formed by the yellow cluster CL0024+1654 (NASA)
thin, elongated, blue images of the same distant galaxy are seen around the yellow, nearer, elliptical galaxies.
But multiple images can be created not only by gravitational lenses; the shape of the universe could also play some tricks.
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ere is a popular analogy which avoids some of the just-mentioned problems. e universe in its evolution is similar to the surface of an ever-expanding sphere: the surface is
nite, but it has no boundary. e universe simply has an additional dimension; therefore its volume is also ever increasing, nite, but without boundary. is statement presupposes that the universe has the same topology, the same ‘shape’ as that of a sphere with an additional dimension.
But what is the experimental evidence for this statement? ere is none. Nothing is yet known about the shape of the universe. It is extremely hard to determine it, simply because of its sheer size.
What do experiments say? In the nearby region of the universe, say within a few million light years, the topology is simply connected. But for large distances, almost nothing is certain. Maybe research into gamma-ray bursts will tell us something about the topology, as these bursts o en originate from the dawn of time.* Maybe even the study of
uctuations of the cosmic background radiation can tell us something. All this research is still in its infancy.
Since little is known, we can ask about the range of possible answers. As just mentioned, in the standard model with k = , space-time is usually assumed to be a product of linear time, with the topology R of the real line, and a sphere S for space. at is the simplest possible shape, corresponding to a simply-connected universe. For k = , the simplest
* e story is told from the mathematical point of view by B O
, Poetry of the Universe, 1996.
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topology of space is three-dimensional real space R , and for k = − it is a hyperbolic manifold H .
In addition, Figure showed that depending on the value of the cosmological constant, space could be nite and bounded, or in nite and unbounded. In Friedmann– Lemaître calculations, simple-connectedness is usually tacitly assumed, even though it is not at all required.
It could well be that space-time is multiply connected, like a higher-dimensional version of a torus. It could also have even more complex topologies.** In these cases, it could even be that the actual number of galaxies is much smaller than the observed number.
is situation would correspond to a kaleidoscope, where a few beads produce a large number of images. In addition, topological surprises could also be hidden behind the horizon.
In fact, the range of possibilities is not limited to the simply and multiply connected cases suggested by classical physics. An additional and completely unexpected twist will appear in the third part of our walk, when quantum theory is included in the investigations.
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“
e universe is a big place; perhaps the biggest.
” Kilgore Trout, Venus on the Half Shell.
e horizon is a tricky entity. In fact, all cosmological models show that it moves rapidly
away from us. A detailed investigation shows that for a matter-dominated universe the
horizon moves away from us with a velocity
vhorizon = c .
(362)
Challenge 860 ny
A pretty result, isn’t it? Obviously, since the horizon does not transport any signal, this is not a contradiction of relativity. But what is behind the horizon?
If the universe is open or marginal, the matter we see at night is predicted by naively applied general relativity to be a – literally – in nitely small part of all matter existing. Indeed, an open or marginal universe implies that there is an in nite amount of matter behind the horizon. Is such a statement veri able?
In a closed universe, matter is still predicted to exist behind the horizon; however, in this case it is only a nite amount.
In short, the standard model of cosmology states that there is a lot of matter behind the horizon. Like most cosmologists, we sweep the issue under the rug and take it up only later in our walk. A precise description of the topic is provided by the hypothesis of in ation.
** e Friedmann–Lemaître metric is also valid for any quotient of the just-mentioned simple topologies by a group of isometries, leading to dihedral spaces and lens spaces in the case k = , to tori in the case k = , Ref. 429 and to any hyperbolic manifold in the case k = − .
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Ref. 431
What were the initial conditions of matter? Matter was distributed in a constant density over space expanding with great speed. How could this happen? e person who has explored this question most thoroughly is Alan Guth. So far, we have based our studies of the night sky, cosmology, on two observational principles: the isotropy and the homogeneity of the universe. In addition, the universe is (almost) at. In ation is an attempt to understand the origin of these observations. Flatness at the present instant of time is strange: the at state is an unstable solution of the Friedmann equations. Since the universe is still at a er fourteen thousand million years, it must have been even atter near the big bang.
Guth argued that the precise atness, the homogeneity and the isotropy could follow if in the rst second of its history, the universe had gone through a short phase of exponential size increase, which he called in ation. is exponential size increase, by a factor of about , would homogenize the universe. is extremely short evolution would be driven by a still-unknown eld, the in aton eld. In ation also seems to describe correctly the growth of inhomogeneities in the cosmic background radiation.
However, so far, in ation poses as many questions as it solves. Twenty years a er his initial proposal, Guth himself is sceptical on whether it is a conceptual step forward. e
nal word on the issue has not been said yet.
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Die Energie der Welt ist constant. Die Entropie der Welt strebt einem Maximum zu.*
“ Rudolph Clausius ” e matter–energy density of the universe is near the critical one. In ation, described in
the previous section, is the favourite explanation for this connection. is implies that the actual number of stars is given by the behaviour of matter at extremely high temperatures, and by the energy density le over at lower temperature. e precise connection is still the topic of intense research. But this issue also raises a question about the quotation above. Was the creator of the term ‘entropy’, Rudolph Clausius, right when he made this famous statement? Let us have a look at what general relativity has to say about all this. In general relativity, a total energy can indeed be de ned, in contrast to localized energy, which cannot. e total energy of all matter and radiation is indeed a constant of motion. It is given by the sum of the baryonic, luminous and neutrino parts:
E = Eb + Eγ + Eν
cM T
+ ... + ...
c G
+ ...
.
(363)
is value is constant only when integrated over the whole universe, not when just the inside of the horizon is taken.*
Many people also add a gravitational energy term. If one tries to do so, one is obliged to de ne it in such a way that it is exactly the negative of the previous term. is value
* ‘ e energy of the universe is constant. Its entropy tends towards a maximum.’ * Except for the case when pressure can be neglected.
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Ref. 432
for the gravitational energy leads to the popular speculation that the total energy of the universe might be zero. In other words, the number of stars could also be limited by this relation.
However, the discussion of entropy puts a strong question mark behind all these seemingly obvious statements. Many people have tried to give values for the entropy of the universe. Some have checked whether the relation
S
=
kc Għ
A
=
kG ħc
πM
,
(364)
Challenge 861 ny
Page 39 Challenge 862 ny
which is correct for black holes, also applies to the universe. is assumes that all the matter and all the radiation of the universe can be described by some average temperature.
ey argue that the entropy of the universe is surprisingly low, so that there must be some ordering principle behind it. Others even speculate over where the entropy of the universe comes from, and whether the horizon is the source for it.
But let us be careful. Clausius assumes, without the slightest doubt, that the universe is a closed system, and thus deduces the statement quoted above. Let us check this assumption. Entropy describes the maximum energy that can be extracted from a hot object. A er the discovery of the particle structure of matter, it became clear that entropy is also given by the number of microstates that can make up a speci c macrostate. But neither de nition makes any sense if applied to the universe as a whole. ere is no way to extract energy from it, and no way to say how many microstates of the universe would look like the macrostate.
e basic reason is the impossibility of applying the concept of state to the universe. We rst de ned the state as all those properties of a system which allow one to distinguish it from other systems with the same intrinsic properties, or which di er from one observer to another. You might want to check for yourself that for the universe, such state properties do not exist at all.
We can speak of the state of space-time and we can speak of the state of matter and energy. But we cannot speak of the state of the universe, because the concept makes no sense. If there is no state of the universe, there is no entropy for it. And neither is there an energy value. is is in fact the only correct conclusion one can draw about the issue.
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We are able to see the stars because the universe consists mainly of empty space, in other words, because stars are small and far apart. But why is this the case? Cosmic expansion was deduced and calculated using a homogeneous mass distribution. So why did matter lump together?
It turns out that homogeneous mass distributions are unstable. If for any reason the density uctuates, regions of higher density will attract more matter than regions of lower density. Gravitation will thus cause the denser regions to increase in density and the regions of lower density to be depleted. Can you con rm the instability, simply by assuming a space lled with dust and a = GM r ? In summary, even a tiny quantum uctuation in the mass density will lead, a er a certain time, to lumped matter.
But how did the rst inhomogeneities form? at is one of the big problems of mod-
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Ref. 433
ern physics and astrophysics, and there is no accepted answer yet. Several modern experiments are measuring the variations of the cosmic background radiation spectrum with angular position and with polarization; these results, which will be available in the coming years, might provide some information on the way to settle the issue.
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Given that the matter density is around the critical one, the size of stars, which contain most of the matter, is a result of the interaction of the elementary particles composing them. Below we will show that general relativity (alone) cannot explain any size appearing in nature. e discussion of this issue is a theme of quantum theory.
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Challenge 864 ny Ref. 434
A
?
Can we distinguish between space expanding and galaxies moving apart? Yes, we can. Can you nd an argument or devise an experiment to do so?
e expansion of the universe does not apply to the space on the Earth. e expansion is calculated for a homogeneous and isotropic mass distribution. Matter is neither homogeneous nor isotropic inside the galaxy; the approximation of the cosmological principle is not valid down here. It has even been checked experimentally, by studying atomic spectra in various places in the solar system, that there is no Hubble expansion taking place around us.
I
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Challenge 865 ny Page 812
e existence of ‘several’ universes might be an option when we study the question whether we see all the stars. But you can check that neither de nition of ‘universe’ given above, be it ‘all matter-energy’ or ‘all matter–energy and all space-time’, allows us to answer the question positively.
ere is no way to de ne a plural for universe: either the universe is everything, and then it is unique, or it is not everything, and then it is not the universe. We will discover that quantum theory does not change this conclusion, despite recurring reports to the contrary.
Whoever speaks of many universes is talking ghibberish.
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Ref. 435
Si les astres étaient immobiles, le temps et l’espace n’existeraient plus.
“ Maurice Maeterlink. ” e two arms possessed by humans have played an important role in discussions about
motion, and especially in the development of relativity. Looking at the stars at night, we can make a simple observation, if we keep our arms relaxed. Standing still, our arms hang down. en we turn rapidly. Our arms li up. In fact they do so whenever we see the stars turning. Some people have spent a large part of their lives studying this phenomenon. Why?
Stars and arms prove that motion is relative, not absolute.* is observation leads to
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•.
two possible formulations of what Einstein called Mach’s principle.
— Inertial frames are determined by the rest of the matter in the universe.
is idea is indeed realized in general relativity. No question about it.
— Inertia is due to the interaction with the rest of the universe.
Ref. 436 Ref. 437 Challenge 866 ny
Ref. 437 Challenge 867 n
is formulation is more controversial. Many interpret it as meaning that the mass of an object depends on the distribution of mass in the rest of the universe. at would mean that one needs to investigate whether mass is anisotropic when a large body is nearby. Of course, this question has been studied experimentally; one simply needs to measure whether a particle has the same mass values when accelerated in di erent directions. Unsurprisingly, to a high degree of precision, no such anisotropy has been found. Many therefore conclude that Mach’s principle is wrong. Others conclude with some pain in their stomach that the whole topic is not yet settled.
But in fact it is easy to see that Mach cannot have meant a mass variation at all: one then would also have to conclude that mass is distance-dependent, even in Galilean physics. But this is known to be false; nobody in his right mind has ever had any doubts about it.
e whole debate is due to a misunderstanding of what is meant by ‘inertia’: one can interpret it as inertial mass or as inertial motion (like the moving arms under the stars). ere is no evidence that Mach believed either in anisotropic mass or in distancedependent mass; the whole discussion is an example people taking pride in not making a mistake which is incorrectly imputed to another, supposedly more stupid, person.**
Obviously, inertial e ects do depend on the distribution of mass in the rest of the universe. Mach’s principle is correct. Mach made some blunders in his life (he is infamous for opposing the idea of atoms until he died, against experimental evidence) but his principle is not one of them. Unfortunately it is to be expected that the myth about the incorrectness of Mach’s principle will persist, like that of the derision of Columbus.
In fact, Mach’s principle is valuable. As an example, take our galaxy. Experiments show that it is attened and rotating. e Sun turns around its centre in about million years. Indeed, if the Sun did not turn around the galaxy’s centre, we would fall into it in about million years. As the physicist Dennis Sciama pointed out, from the shape of our galaxy we can draw a powerful conclusion: there must be a lot of other matter, i.e. a lot of other stars and galaxies in the universe. Can you con rm his reasoning?
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A
ere is no preferred frame in special relativity, no absolute space. Is the same true in the actual universe? No; there is a preferred frame. Indeed, in the standard big-bang cosmology, the average galaxy is at rest. Even though we talk about the big bang, any average
* e original reasoning by Newton and many others used a bucket and the surface of the water in it; but the arguments are the same. ** For another example, at school one usually hears that Columbus was derided because he thought the Earth to be spherical. But he was not derided at all for this reason; there were only disagreements on the size of the Earth, and in fact it turned out that his critics were right, and that he was wrong in his own, much too small, estimate of the radius.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 868 ny Ref. 438
galaxy can rightly maintain that it is at rest. Each one is in free fall. An even better realization of this privileged frame of reference is provided by the background radiation.
In other words, the night sky is black because we move with almost no speed through background radiation. If the Earth had a large velocity relative to the background radiation, the sky would be bright even at night, thanks to the Doppler e ect for the background radiation. In other words, the fact that the night sky is dark in all directions is a consequence of our slow motion against the background radiation.
is ‘slow’ motion has a speed of km s. ( is is the value of the motion of the Sun; there are variations due to addition of the motion of the Earth.) the value is large in comparison to everyday life, but small compared to the speed of light. More detailed studies do not change this conclusion. Even the motion of the Milky Way and that of the local group against the cosmic background radiation is of the order of km s; that is still much slower than the speed of light. e reasons why the galaxy and the solar system move with these ‘low’ speeds across the universe have already been studied in our walk. Can you give a summary?
By the way, is the term ‘universe’ correct? Does the universe rotate, as its name implies? If by universe one means the whole of experience, the question does not make sense, because rotation is only de ned for bodies, i.e. for parts of the universe. However, if by universe one only means ‘all matter’, the answer can be determined by experiments. It turns out that the rotation is extremely small, if there is any: measurements of the cosmic background radiation show that in the lifetime of the universe, it cannot have rotated by more than a hundredth of a millionth of a turn! In short, ‘universe’ is a misnomer.
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Ref. 439 Challenge 869 ny
Another reason why we can see stars is that their light reaches us. But why are travelling light rays not disturbed by each other’s gravitation? We know that light is energy and that any energy attracts other energy through gravitation. In particular, light is electromagnetic energy, and experiments have shown that all electromagnetic energy is subject to gravitation. Could two light beams that are advancing with a small angle between them converge, because of mutual gravitational attraction? at could have measurable and possibly interesting e ects on the light observed from distant stars.
e simplest way to explore the issue is to study the following question: Do parallel light beams remain parallel? Interestingly, a precise calculation shows that mutual gravitation does not alter the path of two parallel light beams, even though it does alter the path of antiparallel light beams.* e reason is that for parallel beams moving at light speed, the gravitomagnetic component exactly cancels the gravitoelectric component.
Since light does not attract light moving along, light is not disturbed by its own gravity during the millions of years that it takes to reach us from distant stars. Light does not attract or disturb light moving alongside. So far, all known quantum-mechanical e ects also con rm this conclusion.
* Antiparallel beams are parallel beams travelling in opposite directions.
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Challenge 870 ny Ref. 440
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In the section on quantum theory we will encounter experiments showing that light is made of particles. It is plausible that these photons might decay into some other particle, as yet unknown, or into lower-frequency photons. If that actually happened, we would not be able to see distant stars.
But any decay would also mean that light would change its direction (why?) and thus produce blurred images for remote objects. However, no blurring is observed. In addition, the Soviet physicist Matvey Bronstein demonstrated in the s that any light decay process would have a larger rate for smaller frequencies. When people checked the shi of radio waves, in particular the famous cm line, and compared it with the shi of light from the same source, no di erence was found for any of the galaxies tested.
People even checked that Sommerfeld’s ne-structure constant, which determines the colour of objects, does not change over time. Despite an erroneous claim in recent years, no change could be detected over thousands of millions of years.
Of course, instead of decaying, light could also be hit by some hitherto unknown entity. But this possibility is excluded by the same arguments. ese investigations also show that there is no additional red-shi mechanism in nature apart from the Doppler and gravitational red-shi s.
e visibility of the stars at night has indeed shed light on numerous properties of nature. We now continue our mountain ascent with a more general issue, nearer to our quest for the fundamentals of motion.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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W
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Ref. 337 Ref. 442 Ref. 443
Qui iacet in terra non habet unde cadat.**
“ ” Alanus de Insulis
Black holes are the most extreme gravitational phenomena. ey realize nature’s limit of length-to-mass ratios. ey produce the highest force value possible in nature; as a result, they produce high space-time curvatures. erefore, black holes cannot be studied without general relativity. In addition, the study of black holes is a major stepping stone towards uni cation and the nal description of motion.
‘Black hole’ is shorthand for ‘gravitationally completely collapsed object’. For many years it was unclear whether or not they exist. But the available experimental data have now led most experts to conclude that there is a black hole at the centre of most galaxies, including our own. Black holes are also suspected at the heart of quasars and of gamma ray bursters. It seems that the evolution of galaxies is strongly tied to the evolution of black holes. In addition, half a dozen smaller black holes have been identi ed elsewhere in our galaxy. For these and many other reasons, black holes, the most impressive, the most powerful and the most relativistic systems in nature, are a fascinating subject of study.
** ‘He who lies on the ground cannot fall down from it.’ e author’s original name is Alain de Lille (c. 1128– 1203 ).
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Ref. 444
Ref. 337 Challenge 872 ny
e escape velocity is the speed needed to launch an projectile in such a way that it never falls back down. e escape velocity depends on the mass and the size of the planet from which the launch takes place. What happens when a planet or star has an escape velocity that is larger than the speed of light c? Such objects were rst imagined by the British geologist John Michell in , and independently by the French mathematician Pierre Laplace in , long before general relativity was developed. Michell and Laplace realized something fundamental: even if an object with such a high escape velocity were a hot star, it would appear to be completely black. e object would not allow any light to leave it; in addition, it would block all light coming from behind it. In , John Wheeler* coined the now standard term black hole.
It only takes a short calculation to show that light cannot escape from a body of mass M whenever the radius is smaller than a critical value given by
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RS =
GM c
(365)
Challenge 873 ny
called the Schwarzschild radius. e formula is valid both in universal gravity and in gen-
eral relativity, provided that in general relativity we take the radius as meaning the cir-
cumference divided by π. Such a body realizes the limit value for length-to-mass ratios
in nature. For this and other reasons to be given shortly, we will call RS also the size of the black hole of mass M. (But note that it is only half the diameter. In addition, the term
‘size’ has to be taken with some grain of salt.) In principle, it is possible to imagine an ob-
ject with a smaller length-to-mass ratio; but nobody has yet observed one. In fact, we will
discover that there is no way to observe an object smaller than the Schwarzschild radius,
just as an object moving faster than the speed of light cannot be observed. However, we
can observe black holes – the limit case – just as we can observe entities moving at the
speed of light.
When a test mass approaches the critical radius RS, two things happen. First, the local proper acceleration for (imaginary) point masses increases without bound. For realistic
objects of nite size, the black hole realizes the highest force possible in nature. Something
that falls into a black hole cannot be pulled back out. A black hole thus swallows all matter
that falls into it. It acts like a cosmic vacuum cleaner.
At the surface of a black hole, the red-shi factor for a distant observer also increases
without bound. e ratio between the two quantities is called the surface gravity of a black
hole. It is given by
surf
=
GM RS
=
c GM
=
c RS
.
(366)
A black hole thus does not allow any light to leave it.
* John Archibald Wheeler (1911–), US-American physicist, important expert on general relativity and au-
thor of several excellent textbooks, among them the beautiful J A. W
, A Journey into Gravity
and Spacetime, Scienti c American Library & Freeman, 1990, in which he explains general relativity with
passion and in detail, but without any mathematics.
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Page 335
A surface that realizes the force limit and an in-
nite red-shi makes it is impossible to send light,
matter, energy or signals of any kind to the outside
world. A black hole is thus surrounded by a hori-
zon. We know that a horizon is a limit surface. In
event horizon
fact, a horizon is a limit in two ways. First, a hori-
zon is a limit to communication: nothing can com-
municate across it. Secondly, a horizon is a surface
of maximum force and power. ese properties are
su cient to answer all questions about the e ects
of horizons. For example: What happens when a
light beam is sent upwards from the horizon? And
from slightly above the horizon? Black holes, regarded as astronomical objects,
are thus di erent from planets. During the formation of planets, matter clumps together; as soon
F I G U R E 214 The light cones in the equatorial plane around a non-rotating black hole, seen from above
as it cannot be compressed any further, an equilibrium is reached, which determines the
radius of the planet. at is the same mechanism as when a stone is thrown towards the
Earth: it stops falling when it hits the ground. A ‘ground’ is formed whenever matter hits
other matter. In the case of a black hole, there is no ground; everything continues falling.
at is why, in Russian, black holes used to be called collapsars.
is continuous falling takes place when the concentration of matter is so high that
it overcomes all those interactions which make matter impenetrable in daily life. In ,
Robert Oppenheimer* and Hartland Snyder showed theoretically that a black hole forms
whenever a star of su cient mass stops burning. When a star of su cient mass stops
burning, the interactions that form the ‘ oor’ disappear, and everything continues falling
without end.
A black hole is matter in permanent free fall. Nevertheless, its radius for an outside
observer remains constant! But that is not all. Furthermore, because of this permanent
free fall, black holes are the only state of matter in thermodynamic equilibrium! In a sense,
oors and all other every-day states of matter are metastable: these forms are not as stable
as black holes.
e characterizing property of a black hole is thus its horizon. e rst time we en-
countered horizons was in special relativity, in the section on accelerated observers. e
horizons due to gravitation are similar in all their properties; the section on the maximum
force and power gave a rst impression. e only di erence we have found is due to the
neglect of gravitation in special relativity. As a result, horizons in nature cannot be planar,
in contrast to what is suggested by the observations of the imagined point-like observers
assumed to exist in special relativity.
Both the maximum force principle and the eld equations imply that the space-time
around a rotationally symmetric (thus non-rotating) and electrically neutral mass is de-
* Robert Oppenheimer (1904–1967), important US-American physicist. He can be called the father of theoretical physics in the USA. He worked on quantum theory and atomic physics. He then headed the team that developed the nuclear bomb during the Second World War. He was also the most prominent (innocent) victim of one of the greatest witch-hunts ever organized in his home country. See also the http://www.nap. edu/readingroom/books/biomems/joppenheimer.html website.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Page 385 scribed by
di
=( −
GM rc
)dt
−
dr
−
GM rc
−r
dφ
c
.
(367)
is is the so-called Schwarzschild metric. As mentioned above, r is the circumference divided by π; t is the time measured at in nity. No outside observer will ever receive any signal emitted from a radius value r = GM c or smaller. Indeed, as the proper time i of an observer at radius r is related to the time t of an observer at in nity through
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di =
−
GM rc
dt
,
(368)
we nd that an observer at the horizon would have vanishing proper time. In other words, at the horizon the red-shi is in nite. (In fact, the surface of in nite red-shi and the horizon coincide only for non-rotating black holes. For rotating black holes, the two surfaces are distinct.) Everything happening at the horizon goes on in nitely slowly, as observed by a distant observer. In other words, for a distant observer observing what is going on at the horizon itself, nothing at all ever happens.
In the same way that observers cannot reach the speed of light, observers cannot reach a horizon. For a second observer, it can only happen that the rst is moving almost as fast as light; in the same way, for a second observer, it can only happen that the rst has almost reached the horizon. In addition, a traveller cannot feel how much he is near the speed of light for another, and experiences light speed as unattainable; in the same way, a traveller (into a large black hole) cannot feel how much he is near a horizon and experiences the horizon as unattainable.
In general relativity, horizons of any kind are predicted to be black. Since light cannot escape from them, classical horizons are completely dark surfaces. In fact, horizons are the darkest entities imaginable: nothing in nature is darker. Nonetheless, we will later discover that physical horizons are not completely black.
O
Ref. 440 Challenge 876 ny
Since black holes curve space-time strongly, a body moving near a black hole behaves in
more complicated ways than predicted by universal gravity. In universal gravity, paths
are either ellipses, parabolas, or hyperbolas; all these are plane curves. It turns out that
paths lie in a plane only near non-rotating black holes.*
Around non-rotating black holes, also called Schwarzschild black holes, circular paths
are impossible for radii less than RS (can you show why?) and are unstable to perturbations from there up to a radius of RS. Only at larger radii are circular orbits stable. Around black holes, there are no elliptic paths; the corresponding rosetta path is shown
* For such paths, Kepler’s rule connecting the average distance and the time of orbit
GMt ( π)
=r
Challenge 875 ny still holds, provided the proper time and the radius measured by a distant observer are used.
(369)
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impact parameter
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 215 Motions of uncharged objects around a non-rotating black hole – for different impact parameters and initial velocities
Challenge 877 ny Challenge 878 ny
in Figure . Such a path shows the famous periastron shi in all its glory. Note that the potential around a black hole is not appreciably di erent from r for
distances above about een Schwarzschild radii. For a black hole of the mass of the Sun, that would be km from its centre; therefore, we would not be able to note any di erence for the path of the Earth around the Sun.
We have mentioned several times in our adventure that gravitation is characterized by its tidal e ects. Black holes show extreme properties in this respect. If a cloud of dust falls into a black hole, the size of the cloud increases as it falls, until the cloud envelops the whole horizon. In fact, the result is valid for any extended body. is property of black holes will be of importance later on, when we will discuss the size of elementary particles.
For falling bodies coming from in nity, the situation near black holes is even more interesting. Of course there are no hyperbolic paths, only trajectories similar to hyperbolas for bodies passing far enough away. But for small, but not too small impact parameters, a body will make a number of turns around the black hole, before leaving again. e number of turns increases beyond all bounds with decreasing impact parameter, until a value is reached at which the body is captured into an orbit at a radius of R, as shown in Figure . In other words, this orbit captures incoming bodies if they approach it below a certain critical angle. For comparison, remember that in universal gravity, capture is never possible. At still smaller impact parameters, the black hole swallows the incoming mass. In both cases, capture and de ection, a body can make several turns around the black hole, whereas in universal gravity it is impossible to make more than half a turn around a body.
e most absurd-looking orbits, though, are those corresponding to the parabolic case of universal gravity. ( ese are of purely academic interest, as they occur with probability
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limit orbit
limit orbit
the photon sphere
the photon sphere
F I G U R E 216 Motions of light passing near a non-rotating black hole
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Challenge 879 ny Challenge 880 ny Challenge 881 ny
zero.) In summary, relativity changes the motions due to gravity quite drastically. Around rotating black holes, the orbits of point masses are even more complex than
those shown in Figure ; for bound motion, for example, the ellipses do not stay in one plane – thanks to the irring–Lense e ect – leading to extremely involved orbits in three dimensions lling the space around the black hole.
For light passing a black hole, the paths are equally interesting, as shown in Figure . ere are no qualitative di erences with the case of rapid particles. For a non-rotating black hole, the path obviously lies in a single plane. Of course, if light passes su ciently nearby, it can be strongly bent, as well as captured. Again, light can also make one or several turns around the black hole before leaving or being captured. e limit between the two cases is the path in which light moves in a circle around a black hole, at R . If we were located on that orbit, we would see the back of our head by looking forward! However, this orbit is unstable. e surface containing all orbits inside the circular one is called the photon sphere. e photon sphere thus divides paths leading to capture from those leading to in nity. Note that there is no stable orbit for light around a black hole. Are there any rosetta paths for light around a black hole? For light around a rotating black hole, paths are much more complex. Already in the equatorial plane there are two possible circular light paths: a smaller one in the direction of the rotation, and a larger one in the opposite direction. For charged black holes, the orbits for falling charged particles are even more complex. e electrical eld lines need to be taken into account. Several fascinating e ects appear which have no correspondence in usual electromagnetism, such as e ects similar to electrical versions of the Meissner e ect. e behaviour of such orbits is still an active area of research in general relativity.
H
How is a black hole characterized? It turns out that all properties of black holes follow from a few basic quantities characterizing them, namely their mass M, their angular momentum J, and their electric charge Q.* All other properties – such as size, shape, colour,
* e existence of three basic characteristics is reminiscent of particles. We will nd out more about the connection between black holes and particles in the third part of our mountain ascent.
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Challenge 882 ny
magnetic eld – are uniquely determined by these.** It is as though, to use Wheeler’s colourful analogy, one could deduce every characteristic of a woman from her size, her waist and her height. Physicists also say that black holes ‘have no hair,’ meaning that (classical) black holes have no other degrees of freedom. is expression was also introduced by Wheeler.*** is fact was proved by Israel, Carter, Robinson and Mazur; they showed that for a given mass, angular momentum and charge, there is only one possible black hole. (However, the uniqueness theorem is not valid any more if the black hole carries nuclear quantum numbers, such as weak or strong charges.)
In other words, a black hole is independent of how it has formed, and of the materials used when forming it. Black holes all have the same composition, or better, they have no composition at all (at least classically).
e mass M of a black hole is not restricted by general relativity. It may be as small as that of a microscopic particle and as large as many million solar masses. But for their angular momentum J and electric charge Q, the situation is di erent. A rotating black hole has a maximum possible angular momentum and a maximum possible electric (and magnetic) charge.**** e limit on the angular momentum appears because its perimeter may not move faster than light. e electric charge is also limited. e two limits are not independent: they are related by
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J cM
+
GQ πε c
GM c
.
(370)
Challenge 883 ny is follows from the limit on length-to-mass ratios at the basis of general relativity. Rotating black holes realizing the limit ( ) are called extremal black holes. e limit ( ) implies that the horizon radius of a general black hole is given by
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
rh
=
GM c
+
−
J M
c G
−
Q πε GM
(371)
Ref. 450
For example, for a black hole with the mass and half the angular momentum of the Sun, namely ë kg and . ë kg m s, the charge limit is about . ë C.
How does one distinguish rotating from non-rotating black holes? First of all by the shape. Non-rotating black holes must be spherical (any non-sphericity is radiated away as gravitational waves) and rotating black holes have a slightly attened shape, uniquely determined by their angular momentum. Because of their rotation, their surface of in nite
Ref. 446
Ref. 447 Ref. 337
** Mainly for marketing reasons, non-rotating and electrically neutral black holes are o en called Schwarzschild black holes; uncharged and rotating ones are o en called Kerr black holes, a er Roy Kerr, who discovered the corresponding solution of Einstein’s eld equations in 1963. Electrically charged but nonrotating black holes are o en called Reissner–Nordström black holes, a er the German physicist Hans Reissner and the Finnish physicist Gunnar Nordström. e general case, charged and rotating, is sometimes named a er Kerr and Newman. *** Wheeler claims that he was inspired by the di culty of distinguishing between bald men; however, Feynman, Ru ni and others had a clear anatomical image in mind when they stated that ‘black holes, in contrast to their surroundings, have no hair.’ **** More about the still hypothetical magnetic charge later on. In black holes, it enters like an additional type of charge into all expressions in which electric charge appears.
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Challenge 884 ny
gravity or in nite red-shi , called the static limit, is di erent from their (outer) horizon.
e region in between is called the ergosphere; this is a misnomer as it is not a sphere.
(It is so called because, as we will see shortly, it can be used to extract energy from the
black hole.) e motion of bodies within the ergosphere can be quite complex. It su ces
to mention that rotating black holes drag any in-falling body into an orbit around them;
this is in contrast to non-rotating black holes, which swallow in-falling bodies. In other
words, rotating black holes are not really ‘holes’ at all, but rather vortices.
e distinction between rotating and non-rotating
black holes also appears in the horizon surface area. e (horizon) surface area A of a non-rotating and uncharged
rotation axis
black hole is obviously related to its mass M by
event horizon
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A=
πG c
M
.
(372)
e relation between surface area and mass for a rotating and charged black hole is more complex: it is given by
A=
πG c
M
+
− Jc − Q M G πε GM
(373)
ergosphere
static limit
F I G U R E 217 The ergosphere of a rotating black hole
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where J is the angular momentum and Q the charge. In fact, the relation
A=
πG c
Mrh
(374)
is valid for all black holes. Obviously, in the case of an electrically charged black hole, the rotation also produces a magnetic eld around it. is is in contrast with non-rotating black holes, which cannot have a magnetic eld.
B
Ref. 451
Can one extract energy from a black hole? Roger Penrose has discovered that this is possible for rotating black holes. A rocket orbiting a rotating black hole in its ergosphere could switch its engines on and would then get hurled into outer space at tremendous velocity, much greater than what the engines could have produced by themselves. In fact, the same e ect is used by rockets on the Earth, and is the reason why all satellites orbit the Earth in the same direction; it would require much more fuel to make them turn the other way.*
e energy gained by the rocket would be lost by the black hole, which would thus slow down and lose some mass; on the other hand, there is a mass increases due to the
Challenge 885 ny
* And it would be much more dangerous, since any small object would hit such an against-the-stream satellite at about . km s, thus transforming the object into a dangerous projectile. In fact, any power wanting to destroy satellites of the enemy would simply have to load a satellite with nuts or bolts, send it into space the wrong way, and distribute the bolts into a cloud. It would make satellites impossible for many decades to come.
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Page 875 Ref. 452
exhaust gases falling into the black hole. is increase always is larger than, or at best
equal to, the loss due to rotation slowdown. e best one can do is to turn the engines on
exactly at the horizon; then the horizon area of the black hole stays constant, and only its
rotation is slowed down.**
As a result, for a neutral black hole rotating with its maximum possible angular mo-
mentum, −
= . % of its total energy can be extracted through the Penrose
process. For black holes rotating more slowly, the percentage is obviously smaller.
For charged black holes, such irreversible energy extraction processes are also possible.
Can you think of a way? Using expression ( ), we nd that up to % of the mass
of a non-rotating black hole can be due to its charge. In fact, in the second part of our
mountain ascent we will encounter an energy extraction process which nature seems to
use quite frequently.
e Penrose process allows one to determine how angular momentum and charge
increase the mass of a black hole. e result is the famous mass–energy relation
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
M
=
E c
= (mirr +
Q πε Gmirr
)
+
Jc mirr G
= (mirr +
π
Q ε
ρirr
)
+J ρirr c
(375)
which shows how the electrostatic and the rotational energy enter the mass of a black hole. In the expression, mirr is the irreducible mass de ned as
mirr
=
A(M ,
Q
= π
,
J
=
)c G
=
ρirr
c G
(376)
Ref. 453 Challenge 889 ny
and ρirr is the irreducible radius. Detailed investigations show that there is no process which decreases the horizon area,
and thus the irreducible mass or radius, of the black hole. People have checked this in all
ways possible and imaginable. For example, when two black holes merge, the total area
increases. One calls processes which keep the area and energy of the black hole constant
reversible, and all others irreversible. In fact, the area of black holes behaves like the en-
tropy of a closed system: it never decreases. at the area in fact is an entropy was rst
stated in by Jakob Bekenstein. He deduced that only when an entropy is ascribed to
a black hole, is it possible to understand where the entropy of all the material falling into
it is collected.
e black hole entropy is a function only of the mass, the angular momentum and
the charge of the black hole. You might want to con rm Bekenstein’s deduction that the
entropy S is proportional to the horizon area. Later it was found, using quantum theory,
that
S=
A kc ħG
=
Ak lPl
.
(377)
is famous relation cannot be deduced without quantum theory, as the absolute value of entropy, as for any other observable, is never xed by classical physics alone. We will Page 877 discuss this expression later on in our mountain ascent.
** It is also possible to extract energy from rotational black holes through gravitational radiation.
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If black holes have an entropy, they also must have a temperature. If they have a temperature, they must shine. Black holes thus cannot be black! is was proven by Stephen Hawking in with extremely involved calculations. However, it could have been deduced in the s, with a simple Gedanken experiment which we will present later on. You might want to think about the issue, asking and investigating what strange consequences would appear if black holes had no entropy. Black hole radiation is a further, though tiny (quantum) mechanism for energy extraction, and is applicable even to nonrotating, uncharged black holes. e interesting connections between black holes, thermodynamics, and quantum theory will be presented in the second part of our mountain ascent. Can you imagine other mechanisms that make black holes shine?
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Tiens, les trous noirs. C’est troublant.*
“ ” Anonymous
Black holes have many counter-intuitive properties. We will rst have a look at the classical e ects, leaving the quantum e ects for later on.
Challenge 891 ny
**
Following universal gravity, light could climb upwards from the surface of a black hole and then fall back down. In general relativity, a black hole does not allow light to climb up at all; it can only fall. Can you con rm this?
**
Challenge 892 ny Challenge 893 ny
What happens to a person falling into a black hole? An outside observer gives a clear answer: the falling person never arrives there since she needs an in nite time to reach the horizon. Can you con rm this result? e falling person, however, reaches the horizon in a nite amount of her own time. Can you calculate it?
is result is surprising, as it means that for an outside observer in a universe with nite age, black holes cannot have formed yet! At best, we can only observe systems that are busy forming black holes. In a sense, it might be correct to say that black holes do not exist. Black holes could have existed right from the start in the fabric of space-time. On the other hand, we will nd out later why this is impossible. In short, it is important to keep in mind that the idea of black hole is a limit concept but that usually, limit concepts (like baths or temperature) are useful descriptions of nature. Independently of this last issue, we can con rm that in nature, the length-to-mass ratio always satis es
L M
G c
.
(378)
**
Interestingly, the size of a person falling into a black hole is experienced in vastly di erent ways by the falling person and a person staying outside. If the black hole is large, the infalling observer feels almost nothing, as the tidal e ects are small. e outside observer
* No translation possible.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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observer
dense star
F I G U R E 218 Motion of some light rays from a dense body to an observer
Challenge 894 ny
makes a startling observation: he sees the falling person spread all over the horizon of the black hole. In-falling, extended bodies cover the whole horizon. Can you explain this fact, for example by using the limit on length-to-mass ratios?
is strange result will be of importance later on in our exploration, and lead to important results about the size of point particles.
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Challenge 895 ny Page 395
An observer near a (non-rotating) black hole, or in fact near any object smaller than times its gravitational radius, can even see the complete back side of the object, as shown in Figure 218. Can you imagine what the image looks like? Note that in addition to the paths shown in Figure 218, light can also turn several times around the black hole before reaching the observer! erefore, such an observer sees an in nite number of images of the black hole. e resulting formula for the angular size of the innermost image was given above.
In fact, the e ect of gravity means that it is possible to observe more than half the surface of any spherical object. In everyday life, however, the e ect is small: for example, light bending allows us to see about 50.0002 % of the surface of the Sun.
Ref. 454 Challenge 896 ny
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A mass point inside the smallest circular path of light around a black hole, at R , cannot stay in a circle, because in that region, something strange happens. A body which circles another in everyday life always feels a tendency to be pushed outwards; this centrifugal e ect is due to the inertia of the body. But at values below R , a circulating body is pushed inwards by its inertia. ere are several ways to explain this paradoxical e ect. e simplest is to note that near a black hole, the weight increases faster than the centrifugal force, as you may want to check yourself. Only a rocket with engines switched on and pushing towards the sky can orbit a black hole at R .
**
By the way, how can gravity, or an electrical eld, come out of a black hole, if no signal Challenge 897 n and no energy can leave it?
**
Do white holes exist, i.e. time-inverted black holes, in which everything ows out of, inChallenge 898 ny stead of into, some bounded region?
**
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 899 ny Show that a cosmological constant Λ leads to the following metric for a black hole:
dτ
=
ds c
=
−
GM rc
−
Λr
dt
− c
−
dr
GM r
−
Λc
r
−
r c
dφ
.
(379)
Note that this metric does not turn into the Minkowski metric for large values of r. However, in the case that Λ is small, the metric is almost at for values of r that satisfy
Λ r Gm c . As a result, the inverse square law is also modi ed:
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F
=
−
Gm r
+
c
Λr
.
(380)
With the known values of the cosmological constant, the second term is negligible inside the solar system.
**
In quantum theory, the gyromagnetic ratio is an important quantity for any rotating Challenge 900 ny charged system. What is the gyromagnetic ratio for rotating black holes?
**
Challenge 901 n
A large black hole is, as the name implies, black. Still, it can be seen. If we were to travel towards it in a spaceship, we would note that the black hole is surrounded by a bright circle, like a thin halo. e ring at the radial distance of the photon sphere is due to those photons which come from other luminous objects, then circle the hole, and nally, a er one or several turns, end up in our eye. Can you con rm this result?
Challenge 902 ny
**
Do moving black holes Lorentz-contract? Black holes do shine a little bit. It is true that the images they form are complex, as light can turn around them a few times before reaching the observer. In addition, the observer has to be far away, so that the e ects of curvature are small. All these e ects can be taken into account; nevertheless, the question remains subtle. e reason is that the concept of Lorentz contraction makes no sense in general relativity, as the comparison with the uncontracted situation is di cult to de ne precisely.
Challenge 903 ny
**
Can you con rm that black holes imply a limit to power? Power is energy change over time. General relativity limits power to P c G. In other words, no engine in nature can provide more than . ë W or . ë horsepower.
F
How might black holes form? At present, at least three possible mechanisms have been distinguished; the question is still a hot subject of research. First of all, black holes could Ref. 455 have formed during the early stages of the universe. ese primordial black holes might
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Ref. 456
Ref. 457 Ref. 458 Ref. 442 Ref. 442
grow through accretion, i.e. through the swallowing of nearby matter and radiation, or disappear through one of the mechanisms to be studied later on.
Of the observed black holes, the so-called supermassive black holes are found at the centre of every galaxy studied so far. ey have masses in the range from to solar masses and contain about . % of the mass of a galaxy. ey are conjectured to exist at the centre of all galaxies, and seem to be related to the formation of galaxies themselves. Supermassive black holes are supposed to have formed through the collapse of large dust clouds, and to have grown through subsequent accretion of matter. e latest ideas imply that these black holes accrete a lot of matter in their early stage; the matter falling in emits lots of radiation, which would explain the brightness of quasars. Later on, the rate of accretion slows, and the less spectacular Seyfert galaxies form. It may even be that the supermassive black hole at the centre of the galaxy triggers the formation of stars. Still later, these supermassive black holes become almost dormant, like the one at the centre of the Milky Way.
On the other hand, black holes can form when old massive stars collapse. It is estimated that when stars with at least three solar masses burn out their fuel, part of the matter remaining will collapse into a black hole. Such stellar black holes have a mass between one and a hundred solar masses; they can also continue growing through subsequent accretion. is situation provided the rst ever candidate for a black hole, Cygnus X- , which was discovered in .
Recent measurements suggest also the existence of intermediate black holes, with masses around a thousand solar masses or more; the mechanisms and conditions for their formation are still unknown.
e search for black holes is a popular sport among astrophysicists. Conceptually, the simplest way to search for them is to look for strong gravitational elds. But only double stars allow one to measure gravitational elds directly, and the strongest ever measured is % of the theoretical maximum value. Another way is to look for strong gravitational lenses, and try to get a mass-to-size ratio pointing to a black hole. Still another way is to look at the dynamics of stars near the centre of galaxies. Measuring their motion, one can deduce the mass of the body they orbit. e most favoured method to search for black holes is to look for extremely intense X-ray emission from point sources, using space-based satellites or balloon-based detectors. If the distance to the object is known, its absolute brightness can be deduced; if it is above a certain limit, it must be a black hole, since normal matter cannot produce an unlimited amount of light. is method is being perfected with the aim of directly observing of energy disappearing into a horizon. is may in fact have been observed.
To sum up the experimental situation, measurements show that in all galaxies studied so far – more than a dozen – a supermassive black hole seems to be located at their centre.
e masses vary: the black hole at the centre of our own galaxy has about . million solar masses, while the central black hole of the galaxy M has million solar masses.
About a dozen stellar black holes of between and solar masses are known in the rest of our own galaxy; all have been discovered since , when Cygnus X- was found. In the year , intermediate-mass black holes were found. Astronomers are also studying how large numbers of black holes in star clusters behave, how o en they collide, and what sort of measurable gravitational waves these collisions produce. e list of discoveries is expected to expand dramatically in the coming years.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Ref. 459
Challenge 904 ny Ref. 460
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Solving the equations of general relativity for various initial conditions, one nds that a cloud of dust usually collapses to a singularity, i.e. to a point of in nite density. e same conclusion appears when one follows the evolution of the universe backwards in time. In fact, Roger Penrose and Stephen Hawking have proved several mathematical theorems on the necessity of singularities for many classical matter distributions. ese theorems assume only the continuity of space-time and a few rather weak conditions on the matter in it. e theorems state that in expanding systems such as the universe itself, or in collapsing systems such as black holes in formation, events with in nite matter density should exist somewhere in the past, or in the future, respectively. is result is usually summarized by saying that there is a mathematical proof that the universe started in a singularity.
In fact, the derivation of the initial singularities makes a hidden, but strong assumption about matter: that dust particles have no proper size. In other words, it is assumed that dust particles are singularities. Only with this assumption can one deduce the existence of initial singularities. However, we have seen that the maximum force principle can be reformulated as a minimum size principle for matter. e argument that there must have been an initial singularity of the universe is thus awed. e experimental situation is clear: there is overwhelming evidence for an early state of the universe that was extremely hot and dense; but there is no evidence for in nite temperature or density.
Mathematically inclined researchers distinguish two types of singularities: those with and without a horizon. e latter ones, the so-called naked singularities, are especially strange: for example, a toothbrush can fall into a naked singularity and disappear without leaving any trace. Since the eld equations are time invariant, we could thus expect that every now and then, naked singularities emit toothbrushes. (Can you explain why ‘dressed’ singularities are less dangerous?)
To avoid the spontaneous appearance of toothbrushes, over the years many people have tried to discover some theoretical principles forbidding the existence of naked singularities. It turns out that there are two such principles. e rst is the maximum force or maximum power principle we encountered above. e maximum force implies that no in nite force values appear in nature; in other words, there are no naked singularities in nature. is statement is o en called cosmic censorship. Obviously, if general relativity were not the correct description of nature, naked singularities could still appear. Cosmic censorship is thus still discussed in research articles. e experimental search for naked singularities has not yielded any success; in fact, there is not even a candidate observation for the less abstruse dressed singularities. But the theoretical case for ‘dressed’ singularities is also weak. Since there is no way to interact with anything behind a horizon, it is futile to discuss what happens there. ere is no way to prove that behind a horizon a singularity exists. Dressed singularities are articles of faith, not of physics.
In fact, there is another principle preventing singularities, namely quantum theory. Whenever we encounter a prediction of an in nite value, we have extended our description of nature to a domain for which it was not conceived. To speak about singularities, one must assume the applicability of pure general relativity to very small distances and very high energies. As will become clear in the next two parts of this book, nature does not allow this: the combination of general relativity and quantum theory shows that it
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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makes no sense to talk about ‘singularities’, nor about what happens ‘inside’ a black hole Page 1001 horizon. e reason is that time and space are not continuous at very small scales. *
A
–
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Could it be that we live inside a black hole? Both the universe and black holes have horizons. Interestingly, the horizon distance r of the universe is about
r ct ë m
(381)
and its matter content is about
m
π ρor
whence
Gm c
=
πGρ ct = ë
m
for a density of ë − kg m . us we have
(382)
r
Gm c
,
(383)
Challenge 905 ny which is similar to the black hole relation rS = Gm c . Is this a coincidence? No, it is not: all systems with high curvature more or less obey this relation. But are we nevertheless
Challenge 906 ny falling into a large black hole? You can answer that question by yourself.
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“Tempori parce.*
”Seneca
People in a bad mood say that time is our master. Nobody says that of space. Time and
space are obviously di erent in everyday life. But what is the precise di erence between
them in general relativity? And do we need them at all? In general relativity it is assumed
that we live in a (pseudo-Riemannian) space-time of variable curvature. e curvature is
an observable and is related to the distribution and motion of matter and energy in the
way described by the eld equations.
However, there is a fundamental problem. e equations of general relativity are in-
variant under numerous transformations which mix the coordinates x , x , x and x .
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* Many physicists are still wary of making such strong statements at this point; and there are still some who claim that space and time are continuous even down to the smallest distances. Our discussion of quantum theory, and the rst sections of the third part of our mountain ascent, will give the precise arguments leading to the opposite conclusion. * ‘Care about time.’ Lucius Annaeus Seneca (c. 4 –65), Epistolae 88, 39.
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For example, the viewpoint transformation
x′ = x + x x′ = −x + x x′ = x x′ = x
(384)
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 907 ny
Ref. 416 Page 451, page 616
is allowed in general relativity, and leaves the eld equations invariant. You might want to search for other examples.
is has a consequence that is clearly in sharp contrast with everyday life: di eomorphism invariance makes it impossible to distinguish space from time inside general relativity. More explicitly, the coordinate x cannot simply be identi ed with the physical time t, as implicitly done up to now. is identi cation is only possible in special relativity. In special relativity the invariance under Lorentz (or Poincaré) transformations of space and time singles out energy, linear momentum and angular momentum as the fundamental observables. In general relativity, there is no (non-trivial) metric isometry group; consequently, there are no basic physical observables singled out by their characteristic of being conserved. But invariant quantities are necessary for communication! In fact, we can talk to each other only because we live in an approximately at space-time: if the angles of a triangle did not add up to degrees, there would be no invariant quantities and we would not be able to communicate.
How have we managed to sweep this problem under the rug so far? We have done so in several ways. e simplest was to always require that in some part of the situation under consideration space-time was our usual at Minkowski space-time, where x can be identi ed with t. We can ful l this requirement either at in nity, as we did around spherical masses, or in zeroth approximation, as we did for gravitational radiation and for all other perturbation calculations. In this way, we eliminate the free mixing of coordinates and the otherwise missing invariant quantities appear as expected. is pragmatic approach is the usual way out of the problem. In fact, it is used in some otherwise excellent texts on general relativity that preclude any deeper questioning of the issue.
A common variation of this trick is to let the distinction ‘sneak’ into the calculations by the introduction of matter and its properties, or by the introduction of radiation. e material properties of matter, for example their thermodynamic state equations, always distinguish between space and time. Radiation does the same, by its propagation. Obviously this is true also for those special combinations of matter and radiation called clocks and metre bars. In fact, the method of introducing matter is the same as the method of introducing Minkowski space-time, if one looks closely: properties of matter are always de ned using space-time descriptions of special relativity.*
Another variation of the pragmatic approach is the use of the cosmological time coordinate. An isotropic and homogeneous universe does have a preferred time coordinate, namely the one used in all the tables on the past and the future of the universe. is method is in fact a combination of the previous two.
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* We note something astonishing here: the inclusion of some condition at small distances (matter) has the Challenge 908 ny same e ect as the inclusion of some condition at in nity. Is this just coincidence? We will come back to this
Page 1051 issue in the third part of our mountain ascent.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•.
But we are on a special quest here. We want to understand motion in principle, not only to calculate it in practice. We want a fundamental answer, not a pragmatic one. And for this we need to know how the positions xi and time t are connected, and how we can de ne invariant quantities. e question also prepares us for the task of combining gravity with quantum theory, which will be the goal of the third part of our mountain ascent.
A fundamental solution requires a description of clocks together with the system under consideration, and a deduction of how the reading t of a clock relates to the behaviour of the system in space-time. But we know that any description of a system requires measurements: for example, in order to determine the initial conditions. And initial conditions require space and time. We thus enter a vicious circle: that is precisely what we wanted to avoid in the rst place.
A suspicion arises. Is there in fact a fundamental di erence between space and time? Let us take a tour of various ways to investigate this question.
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Page 265 Ref. 461
In order to distinguish between space and time in general relativity, we must be able to measure them. But already in the section on universal gravity we have mentioned the impossibility of measuring lengths, times and masses with gravitational e ects alone. Does this situation change in general relativity? Lengths and times are connected by the speed of light, and in addition lengths and masses are connected by the gravitational constant. Despite this additional connection, it takes only a moment to convince oneself that the problem persists.
In fact, we need electrodynamics to solve it. It is only be using the elementary charge e that can we form length scales, of which the simplest one is
lemscale =
e πε
G c
. ë − m.
(385)
Page 523
Here, ε the permittivity of free space. Alternatively, we can argue that quantum mechan-
ics provides a length scale, since one can use the quantum of action ħ to de ne the length
scale
lqtscale =
ħG c
. ë − m,
(386)
which is called the Planck length or Planck’s natural length unit. However, this does not change the argument, because one needs electrodynamics to measure the value of ħ. e equivalence of the two arguments is shown by rewriting the elementary charge e as a combination of nature’s fundamental constants:
e = πε cħα .
(387)
Here, α
. is the ne-structure constant that characterizes the strength of electro-
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magnetism. In terms of α, expression ( ) becomes
lscale =
αħG c
=
α lPl .
(388)
Challenge 909 e
Summing up, every length measurement is based on the electromagnetic coupling constant α and on the Planck length. Of course, the same is true for time and mass measurements. ere is thus no way to de ne or measure lengths, times and masses using gravitation or general relativity only.*
Given this sobering result, we can ask whether in general relativity space and time are really required at all.
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Ref. 463
Robert Geroch answers this question in a beautiful ve-page article. He explains how to formulate the general theory of relativity without the use of space and time, by taking as starting point the physical observables only.
He starts with the set of all observables. Among them there is one, called v, which stands out. It is the only observable which allows one to say that for any two observables a , a there is a third one a , for which
(a − v) = (a − v) + (a − v) .
(389)
Such an observable is called the vacuum. Geroch shows how to use such an observable to construct the derivatives of observables. en the so-called Einstein algebra can be built, which comprises the whole of general relativity.
Usually one describes motion by deducing space-time from matter observables, by calculating the evolution of space-time, and then by deducing the motion of matter following from it. Geroch’s description shows that the middle step, and thus the use of space and time, is not necessary.
Indirectly, the principle of maximum force makes the same statement. General relativity can be derived from the existence of limit values for force or power. Space and time are only tools needed to translate this principle into consequences for real-life observers.
us, it is possible to formulate general relativity without the use of space and time. Since both are unnecessary, it seems unlikely that there should be a fundamental di erence between them. Nevertheless, on di erence is well-known.
D
?
Ref. 422
Is it possible that the time coordinate behaves, at least in some regions, like a torus? When we walk, we can return to the point of departure. Is it possible, to come back in time to where we have started? e question has been studied in great detail. e standard reference is the text by Hawking and Ellis; they list the required properties of space-time,
Ref. 462 * In the past, John Wheeler used to state that his geometrodynamic clock, a device which measures time by bouncing light back and forth between two parallel mirrors, was a counter-example; that is not correct,
Challenge 910 n however. Can you con rm this?
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F I G U R E 219 A ‘hole’ in space
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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explaining which are mutually compatible or exclusive. ey nd, for example, that spacetimes which are smooth, globally hyperbolic, oriented and time-oriented do not contain any such curves. It is usually assumed that the observed universe has these properties, so that observation of closed timelike curves is unlikely. Indeed, no candidate has ever been suggested. Later on, we will show that searches for such curves at the microscopic scale have also failed to nd any.
e impossibility of closed timelike curves seems to point to a di erence between space and time. But in fact, this di erence is only apparent. All these investigations are based on the behaviour of matter. us these arguments assume a speci c distinction between space and time right from the start. In short, this line of enquiry cannot help us to decide whether space and time di er. Let us look at the issue in another way.
I
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Ref. 464
When Albert Einstein developed general relativity, he had quite some trouble with di eomorphism invariance. Most startling is his famous hole argument, better called the hole paradox. Take the situation shown in Figure , in which a mass deforms the space-time around it. Einstein imagined a small region of the vacuum, the hole, which is shown as a small ellipse. What happens if we somehow change the curvature inside the hole while leaving the situation outside it unchanged, as shown in the inset of the picture?
On the one hand, the new situation is obviously physically di erent from the original one, as the curvature inside the hole is di erent. is di erence thus implies that the curvature outside a region does not determine the curvature inside it. at is extremely unsatisfactory. Worse, if we generalize this operation to the time domain, we seem to get the biggest nightmare possible in physics: determinism is lost.
On the other hand, general relativity is di eomorphism invariant. e deformation shown in the gure is a di eomorphism; so the new situation must be physically equivalent to the original situation.
Which argument is correct? Einstein rst favoured the rst point of view, and therefore dropped the whole idea of di eomorphism invariance for about a year. Only later did he understand that the second assessment is correct, and that the rst statement makes a fundamental mistake: it assumes an independent existence of the coordinate axes x and y,
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 220 A model of the hollow Earth theory (© Helmut Diehl)
as shown in the gure. But during that deformation, the coordinates x and y automatically change as well, so that there is no physical di erence between the two situations.
e moral of the story is that there is no di erence between space-time and gravitational eld. Space-time is a quality of the eld, as Einstein put it, and not an entity with a separate existence, as suggested by the graph. Coordinates have no physical meaning; only distances (intervals) in space and time have one. In particular, di eomorphism invariance proves that there is no ow of time. Time, like space, is only a relational entity: time and space are relative; they are not absolute.
e relativity of space and time has practical consequences. For example, it turns out that many problems in general relativity are equivalent to the Schwarzschild situation, even though they appear completely di erent at rst sight. As a result, researchers have ‘discovered’ the Schwarzschild solution (of course with di erent coordinate systems) over twenty times, o en thinking that they had found a new, unknown solution. We will now discuss a startling consequence of this insight.
I
E
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Page 56
e hollow Earth hypothesis, i.e. the conjecture that we live on the inside of a sphere, was popular in paranormal circles around the year , and still remains so among certain eccentrics today, especially in Britain, Germany and the US. ey maintain, as illustrated in Figure , that the solid Earth encloses the sky, together with the Moon, the Sun and the stars. Most of us are fooled by education into another description, because we are brought up to believe that light travels in straight lines. Get rid of this wrong belief, they say, and the hollow Earth appears in all its glory.
Interestingly, the reasoning is correct. ere is no way to disprove this sort of de-
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Ref. 465 Challenge 911 e Challenge 912 n
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scription of the universe. In fact, as the great Austrian physicist Roman Sexl used to explain, the di eomorphism invariance of general relativity even proclaims the equivalence between the two views. e fun starts when either of the two camps wants to tell the other that only its own description can be correct. You might check that any such argument is wrong; it is fun to slip into the shoes of such an eccentric and to defend the hollow Earth hypothesis against your friends. It is easy to explain the appearance of day and night, of the horizon, and of the satellite images of the Earth. It is easy to explain what happened during the ight to the Moon. You can drive many bad physicists crazy in this way. e usual description and the hollow Earth description are exactly equivalent. Can you con rm that even quantum theory, with its introduction of length scales into nature, does not change this situation?
Such investigations show that di eomorphism invariance is not an easy symmetry to swallow. But it is best to get used to it now, as the rest of our adventure will throw up even more surprises. Indeed, in the third part of our walk we will discover that there is an even larger symmetry of nature that is similar to the change in viewpoint from the hollow Earth view to the standard view. is symmetry, space-time duality, is valid not only for distances measured from the centre of the Earth, but for distances measured from any point in nature.
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,
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Page 1013 Page 157
Page 433
We can conclude from this short discussion that there is no fundamental distinction between space and time in general relativity. Pragmatic distinctions, using matter, radiation or space-time at in nity, are the only possible ones.
In the third part of our adventure we will discover that even the inclusion of quantum theory is consistent with this view. We will show explicitly that no distinction is possible in principle. We will discover that mass and space-time are on an equal footing, and that, in a sense, particles and vacuum are made of the same substance. Distinctions between space and time turn out to be possible only at low, everyday energies.
In the beginning of our mountain ascent we found that we needed matter to de ne space and time. Now we have found that we even need matter to distinguish between space and time. Similarly, in the beginning we found that space and time are required to de ne matter; now we have found that we even need at space-time to de ne it.
In summary, general relativity does not provide a way out of the circular reasoning we discovered in Galilean physics. Indeed, it makes the issue even less clear than before. Continuing the mountain ascent is really worth the e ort.
.
–
“ ” Sapientia felicitas.*
* ‘Wisdom is happiness.’ is is the motto of Oxford University.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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General relativity is the nal description of paths of motion, or if one prefers, of macroscopic motion. General relativity describes how the observations of motion of any two observers are related to each other; it also describes motion due to gravity. In fact, general relativity is based on the following observations:
— All observers agree that there is a ‘perfect’ speed in nature, namely a common maximum energy speed relative to matter. is speed is realized by massless radiation, such as light or radio signals.
— All observers agree that there is a ‘perfect’ force in nature, a common maximum force that can be realized or measured by realistic observers. is force is realized on event horizons.
ese two statements contain the full theory of relativity. From them we deduce:
— Space-time consists of events in + continuous dimensions, with a variable curvature. e curvature can be deduced from distance measurements among events or from
tidal e ects. We thus live in a pseudo-Riemannian space-time. Measured times, lengths and curvatures vary from observer to observer. — Space-time and space are curved near mass and energy. e curvature at a point is determined by the energy–momentum density at that point, and described by the eld equations. When matter and energy move, the space curvature moves along with them. A built-in delay in this movement renders faster-than-light transport of energy impossible. e proportionality constant between energy and curvature is so small that the curvature is not observed in everyday life; only its indirect manifestation, namely gravity, is observed. — Space is also elastic: it prefers being at. Being elastic, it can oscillate independently of matter; one then speaks of gravitational radiation or of gravity waves. — Freely falling matter moves along geodesics, i.e. along paths of maximal length in curved space-time; in space this means that light bends when it passes near large masses by twice the amount predicted by universal gravity. — To describe gravitation one needs curved space-time, i.e. general relativity, at the latest whenever distances are of the order of the Schwarzschild radius rS = Gm c . When distances are much larger than this value, the relativistic description with gravity and gravitomagnetism (frame-dragging) is su cient. When distances are even larger, the description by universal gravity, namely a = Gm r , together with at Minkowski space-time, will do as a rst approximation. — Space and time are not distinguished globally, but only locally. Matter is required to make the distinction.
In addition, all the matter and energy we observe in the sky lead us to the following conclusions:
— On the cosmological scale, everything moves away from everything else: the universe is expanding. is expansion of space-time is described by the eld equations.
— e universe has a nite age; this is the reason for the darkness of the sky at night. A horizon limits the measurable space-time intervals to about fourteen thousand million years.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•.
T
Ref. 466 Ref. 467
Ref. 466, Ref. 467 Challenge 913 ny
Ref. 466 Page 407
Ref. 467, Ref. 470 Ref. 466
Was general relativity worth the e ort? e discussion of its accuracy is most conveniently
split into two sets of experiments. e rst set consists of measurements of how matter
moves. Do objects really follow geodesics? As summarized in Table , all experiments
agree with the theory to within measurement errors, i.e. at least within part in . In
short, the way matter falls is indeed well described by general relativity.
e second set of measurements concerns the dynamics of space-time itself. Does
space-time move following the eld equations of general relativity? In other words, is
space-time really bent by matter in the way the theory predicts? Many experiments have
been performed, near to and far from Earth, in both weak and strong elds. All agree
with the predictions to within measurement errors. However, the best measurements so
far have only about signi cant digits. Note that even though numerous experiments
have been performed, there are only few types of tests, as Table shows. e discovery
of a new type of experiment almost guarantees fame and riches. Most sought a er, of
course, is the direct detection of gravitational waves.
Another comment on Table is in order. A er many decades in which all measured
e ects were only of the order v c , several so-called strong eld e ects in pulsars allowed
us to reach the order v c . Soon a few e ects of this order should also be detected even
inside the solar system, using high-precision satellite experiments. e present crown of
all measurements, the gravity wave emission delay, is the only v c e ect measured so
far.
e di culty of achieving high precision for space-time curvature measurements is
the reason why mass is measured with balances, always (indirectly) using the prototype
kilogram in Paris, instead of de ning some standard curvature and xing the value of
G. Indeed, no useful terrestrial curvature experiment has ever been carried out. A break-
through in this domain would make the news. e terrestrial curvature methods cur-
rently available would not even allow one to de ne a kilogram of gold or of oranges with
a precision of a single kilogram!
Another way to check general relativity is to search for alternative descriptions of
gravitation. Quite a number of alternative theories of gravity have been formulated and
studied, but so far, only general relativity is in agreement with all experiments.
In summary, as ibault Damour likes to explain, general relativity is at least
.
% correct concerning the motion of matter and energy, and at least
. % correct about the way matter and energy curve and move space-time. No excep-
tions, no anti-gravity and no unclear experimental data are known. All motion on Earth
and in the skies is described by general relativity. e importance of Albert Einstein’s
achievement cannot be understated.
We note that general relativity has not been tested for microscopic motion. In this
context, microscopic motion is any motion for which the action is around the quantum of
action, namely − Js. is issue is central to the third and last part of our adventure.
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R Ref. 471 Research in general relativity is more intense than ever.*
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TA B L E 40 Types of tests of general relativity M
C- T -
R
-
Equivalence principle
−
r dependence (dimensionality of space-time) −
Time independence of G
−s
Red-shi (light and microwaves on Sun, Earth, −
Sirius)
Perihelion shi (four planets, Icarus, pulsars) −
Light de ection (light, radio waves around Sun, − stars, galaxies)
Time delay (radio signals near Sun, near pulsars) −
Gravitomagnetism (Earth, pulsar)
−
Geodesic e ect (Moon, pulsars)
−
Gravity wave emission delay (pulsars)
−
motion of matter Ref. 351,
Ref. 466, Ref. 468
motion of matter Ref. 469 motion of matter Ref. 466 space-time curvature Ref. 329,
Ref. 328, Ref. 466
space-time curvature Ref. 466 space-time curvature Ref. 466
space-time curvature Ref. 466 space-time curvature Ref. 353 space-time curvature Ref. 372,
Ref. 466
space-time curvature Ref. 466
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
**
e most interesting experimental studies at present are those of double pulsars, the search for gravitational waves and various dedicated satellites; among others a special satellite will capture all possible pulsars of the galaxy. All these experiments will allow experimental tests in domains that have not been accessible up to now.
Ref. 472
**
e description of collisions and many-body problems, invloving stars, neutron stars and black holes helps astrophysicists to improve their understanding of the rich behaviour they observe in their telescopes.
Ref. 473
**
e study of the early universe and of elementary particle properties, with phenomena such as in ation, a short period of accelerated expansion during the rst few seconds, is still an important topic of investigation.
Ref. 474
**
e study of chaos in the eld equations is of fundamental interest in the study of the early universe, and may be related to the problem of galaxy formation, one of the biggest open problems in physics.
* ere is even a free and excellent internet-based research journal, called Living Reviews in Relativity, to be found at the http://www.livingreviews.org website.
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•.
Ref. 475
**
Gathering data about galaxy formation is the main aim of many satellite systems and purpose-build telescopes. e main focus is the search for localized cosmic microwave background anisotropies due to protogalaxies.
**
e determination of the cosmological parameters, such as the matter density, the Ref. 418 curvature and the vacuum density, is a central e ort of modern astrophysics.
Ref. 476 Page 875
**
Astrophysicists regularly discover new phenomena in the skies. For example, the various types of gamma-ray bursts, X-ray bursts and optical bursts are still not completely understood. Gamma-ray bursts, for example, can be as bright as sun-like stars combined; however, they last only a few seconds. More details on this research are given later on.
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**
A computer database of all solutions of the eld equations is being built. Among other Ref. 477 things, researchers are checking whether they really are all di erent from each other.
Ref. 478 Ref. 479
**
e inclusion of torsion in the eld equations, a possible extension of the theory, is one of the more promising attempts to include particle spin in general relativity. e inclusion of torsion in general relativity does not require new fundamental constants; indeed, the absence of torsion was assumed in the Raychaudhuri equation used by Jacobson. e use of the extended Raychaudhuri equation, which includes torsion, should allow one to deduce the full Einstein–Cartan theory from the maximum force principle. is issue is a topic for future research.
**
Solutions with non-trivial topology, such as wormholes and particle-like solutions, conRef. 480 stitue a fascinating eld of enquiry, related to string theory.
Ref. 481
**
Other formulations of general relativity, describing space-time with quantities other than the metric, are continuously being developed, in the hope of clarifying its relationship to the quantum world. e so-called Ashtekar variables are such a modern description.
**
e uni cation of quantum physics and general relativity, the topic of the third part of Ref. 482 this mountain ascent, will occupy researchers for many years to come.
Ref. 483
**
Finally, the teaching of general relativity, which for many decades has been hidden behind Greek indices, di erential forms and other antididactic methods, will bene t greatly from future improvements focusing more on the physics and less on the formalism.
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In short, general relativity is still an extremely interesting eld of research and important discoveries are still expected.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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?
Page 340 Challenge 914 ny
Page 499
e constant of gravitation provides a limit for the density and the acceleration of objects, as well as for the power of engines. We based all our deductions on its invariance. Is it possible that the constant of gravitation G changes from place to place or that it changes with time? e question is tricky. At rst sight, the answer is a loud: ‘Yes, of course! Just see what happens when the value of G is changed in formulae.’ However, this answer is wrong, as it was wrong for the speed of light c.
Since the constant of gravitation enters into our de nition of gravity and acceleration, and thus, even if we do not notice it, into the construction of all rulers, all measurement standards and all measuring set-ups, there is no way to detect whether its value actually varies. No imaginable experiment could detect a variation. Every measurement of force is, whether we like it or not, a comparison with the limit force. ere is no way, in principle, to check the invariance of a standard. is is even more astonishing because measurements of this type are regularly reported, as in Table . But the result of any such experiment is easy to predict: no change will ever be found.
Could the number of space dimension be di erent from ? is issue is quite involved. For example, three is the smallest number of dimensions for which a vanishing Ricci tensor is compatible with non-vanishing curvature. On the other hand, more than three dimensions give deviations from the inverse square ‘law’ of gravitation. So far, there are no data pointing in this direction.
Could the equations of general relativity be di erent? Despite their excellent t with experiment, there is one issue that still troubles some people. e rotation speed of matter far from the centre of galaxies does not seem to be consistent with the inverse square dependence. ere could be many reasons for this e ect, and a change in the equations for large distances might be one of them. is issue is still open.
eoreticians have explored many alternative theories, such as scalar-tensor theories, theories with torsion, or theories which break Lorentz invariance. However, none of the alternative theories proposed so far seem to t experimental data.
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T
Challenge 915 e
Despite its successes, the description of motion presented so far is unsatisfactory; maybe you already have some gut feeling about certain unresolved issues.
First of all, even though the speed of light is the starting point of the whole theory, we still do not know what light actually is. is will be our next topic.
Secondly, we have seen that everything falls along geodesics. But a mountain does not fall. Somehow the matter below prevents it from falling. How? And where does mass come from anyway? What is mass? What is matter? General relativity does not provide an answer; in fact, it does not describe matter at all. Einstein used to say that the le -hand side of the eld equations, describing the curvature of space-time, was granite, while the right-hand side, describing matter, was sand. Indeed, at this point we still do not know what matter and mass are. As already remarked, to change the sand into rock we rst need quantum theory and then, in a further step, its uni cation with relativity. is is the
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Page 996 Page 840
programme for the rest of our adventure. We have also seen that matter is necessary to clearly distinguish between space and
time, and in particular, to understand the working of clocks, metre bars and balances. In particular, one question remains: why are there units of mass, length and time in nature at all? is deep question will also be addressed in the following chapter.
Finally, we know little about the vacuum. We need to understand the magnitude of the cosmological constant and the number of space-time dimensions. Only then can we answer the simple question: Why is the sky so far away? General relativity does not help here. Worse, the smallness of the cosmological constant contradicts the simplest version of quantum theory; this is one of the reasons why we still have quite some height to scale before we reach the top of Motion Mountain.
In short, to describe motion well, we need a more precise description of light, of matter and of the vacuum. In other words, we need to know more about everything we know. Otherwise we cannot hope to answer questions about mountains, clocks and stars. In a sense, it seems that we have not achieved much. Fortunately, this is not true. We have learned so much that for the following topic we are forced to go backwards, to situations without gravity, i.e. back to the framework of special relativity. at is the next, middle section of our mountain ascent. Despite this simpli cation to at space-time, a lot of fun awaits us there.
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“It’s a good thing we have gravity, or else when birds died they’d just stay right up there. Hunters would be all confused. ” Steven Wright, comedian.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
B
A man will turn over half a library to make one book.
“ ” Samuel Johnson*
323 e simplest historical source is A
E
, Sitzungsberichte der Preussischen
Akademie der Wissenscha en II pp. – , . It is the rst explanation of the general
theory of relativity, in only three pages. e theory is then explained in detail in the famous
article A
E
, Die Grundlage der allgemeinen Relativitätstheorie, Annalen
der Physik 49, pp. – , . e historic references can be found in German and Eng-
lish in J S
, ed., e Collected Papers of Albert Einstein, Volumes – , Princeton
University Press, – .
Below is a selection of English-language textbooks for deeper study, in ascending order
of depth and di culty:
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— An entertaining book without any formulae, but nevertheless accurate and detailed, is
the paperback by I N
, Black Holes and the Universe, Cambridge University
Press, 1990.
— Almost no formulae, but loads of insight, are found in the enthusiastic text by J A.
W
, A Journey into Gravity and Spacetime, W.H. Freeman, 1990.
— An excellent didactical presentation is E
F. T
& J A. W
, Ex-
ploring Black Holes: Introduction to General Relativity, Addison Wesley Longman, 2000.
— Beauty, simplicity and shortness are the characteristics of M
L
, Gen-
eral Relativity, a Geometric Approach, Cambridge University Press, 1999.
— Good explanation is the strength of B
S
, Gravity From the Ground Up,
Cambridge University Press, 2003.
— A good overview of experiments and theory is given in J
F
& J.D. N -
, A Short Course in General Relativity, Springer Verlag, 2nd edition, 1998.
— A pretty text is S L
, Discovering Relativity for Yourself, Cambridge University
Press, 1981.
— A modern text is by R ’I
Introducing Einstein’s Relativity, Clarendon Press,
1992. It includes an extended description of black holes and gravitational radiation, and
regularly refers to present research.
— A beautiful, informative and highly recommended text is H C. O
&R
R
, Gravitation and Spacetime, W.W. Norton & Co., 1994.
— A well written and modern book, with emphasis on the theory, by one of the great mas-
ters of the eld is W
R
, Relativity – Special, General and Cosmological,
Oxford University Press, 2001.
— A classic is S
W
, Gravitation and Cosmology, Wiley, 1972.
— e passion of general relativity can be experienced also in J K
, ed., Magic
without Magic: John Archibald Wheeler – A Collection of Essays in Honour of His Sixtieth
Birthday, W.H. Freeman & Co., 1972.
— An extensive text is K S. T
, Black Holes and Time Warps – Einstein’s Outrageous
Legacy, W.W. Norton, 1994.
— e most mathematical – and toughest – text is R
M. W , General Relativity,
University of Chicago Press, 1984.
* Samuel Johnson (1709–1784), famous English poet and intellectual.
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— Much information about general relativity is available on the internet. As a good starting point for US-American material, see the http://math.ucr.edu/home/baez/relativity.html website.
ere is still a need for a large and modern textbook on general relativity, with colour material, that combines experimental and theoretical aspects.
For texts in other languages, see the next reference. Cited on pages , , , , and .
324 A beautiful German teaching text is the classic G. F & W. R
, Mechanik, Relativ-
ität, Gravitation – ein Lehrbuch, Springer Verlag, third edition, .
A practical and elegant booklet is U
E. S
, Gravitation – Einführung in
die allgemeine Relativitätstheorie, Verlag Harri Deutsch, Frankfurt am Main, .
A modern reference is T
F
, Allgemeine Relativitätstheorie,
Akademischer Spektrum Verlag, .
Excellent is H
G
, Einführung in die spezielle und allgemeine Relativität-
stheorie, Akademischer Spektrum Verlag, .
In Italian, there is the beautiful, informative, but expensive H C. O
&
RR
, Gravitazione e spazio-tempo, Zanichelli, . It is highly recommended.
Cited on pages , , , , , , , and .
325 P. M
& J.H. S , High altitude free fall, American Journal of Physics 64,
pp. – , . As a note, due to a technical failure Kittinger had his hand in (near)
vacuum during his ascent, without incurring any permanent damage. On the consequences
of human exposure to vacuum, see the http://www.s .net/people/geo rey.landis/vacuum.
html website. Cited on page .
326 is story is told e.g. by W.G. U
, Time, gravity, and quantum mechanics, preprint
available at http://www.arxiv.org/abs/gr-qc/
. Cited on page .
327 H. B , Gravitation, European Journal of Physics 14, pp. – , . Cited on page .
328 J.W. B
, Princeton University Ph.D. thesis, . See also J.L. S
view Letters 28, pp. – , , and for the star Sirius see J.L. G
trophysical Journal 169, p. , . Cited on pages and .
, Physical Re& al., As-
329 e famous paper is R.V. P
& G.A. R
, Apparent weight of photons, Physical Re-
view Letters 4, pp. – , . A higher-precision version was published by R.V. P
& J.L. S
, Physical Review Letters 13, p. , , and R.V. P
& J.L. S
,
Physical Review B 140, p. , . Cited on pages and .
330 J.C. H
&R
E. K
lativistic time gains, Science 177, pp.
served relativistic time gains, pp. –
, Around-the-world atomic clocks: predicted re– , and Around-the-world atomic clocks: ob, July . Cited on page .
331 R.F.C. V
& al., Test of relativistic gravitation with a space-borne hydrogen maser,
Physical Review Letters 45, pp. – , . e experiment was performed in ;
there are more than a dozen co-authors involved in this work, which involved shooting a
maser into space with a scout missile to a height of c.
km. Cited on page .
332 L. B
& S. L
, Evidence for Earth gravitational shi by direct atomic-
time-scale comparison, Il Nuovo Cimento 37B, pp. – , . Cited on page .
333 More information about tides can be found in E.P. C York, . Cited on page .
, e Tides, Doubleday, New
334 e expeditions had gone to two small islands, namely to Sobral, north of Brazil, and to Principe, in the gulf of Guinea. e results of the expedition appeared in e Times before they
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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appeared in a scienti c journal. Today this would be seen as a gross violation of scienti c
honesty. e results were published as F.W. D , A.S. E
& C. D
,
Philosophical Transactions of the Royal Society (London) 220A, p. , , and Memoirs of
the Royal Astronomical Society 62, p. , . Cited on page .
335 A good source for images of space-time is the text by G.F.R. E & R. W and Curved Space-times, Clarendon Press, Oxford, . Cited on page .
, Flat
336 J. D
, Het veld van een enkel centrum in Einstein’s theorie der zwaartekracht, en de
beweging van een sto elijk punt, Verslag gew. Vergad. Wiss. Amsterdam 25, pp. – , .
Cited on page .
337 e name black hole was introduced in at a pulsar conference, as described in his autobi-
ography by J A. W
, Geons, Black Holes, and Quantum Foam: A Life in Physics,
W.W. Norton, , pp. – : ‘In my talk, I argued that we should consider the possibil-
ity that at the center of a pulsar is a gravitationally completely collapsed object. I remarked
that one couldn’t keep saying “gravitationally completely collapsed object” over and over.
One needed a shorter descriptive phrase. “How about black hole?” asked someone in the
audience. I had been searching for just the right term for months, mulling it over in bed, in
the bathtub, in my car, whenever I had quiet moments. Suddenly, this name seemed exactly
right. When I gave a more formal ... lecture ... a few weeks later on, on December , ,
I used the term, and then included it into the written version of the lecture published in
the spring of ... I decided to be casual about the term ”black hole”, dropping it into the
lecture and the written version as if it were an old familiar friend. Would it catch on? Indeed
it did. By now every schoolchild has heard the term.’
e widespread use of the term began with the article by R. R
& J.A. W
,
Introducing the black hole, Physics Today 24, pp. – , January .
In his autobiography, Wheeler also writes that the expression ‘black hole has no hair’ was
criticized as ‘obscene’ by Feynman. An interesting comment by a physicist who used to write
his papers in topless bars. Cited on pages , , , and .
338 L.B. K
, Experimental measurement of the equivalence of active and passive gravit-
ational mass, Physical Review 169, pp. – , . With a clever experiment, he showed
that the gravitational masses of uorine and of bromine are equal. Cited on page .
339 A good and accessible book on the topic is D
B
&G
on a cosmic sea, Allen & Unwin, . Cited on page .
MN
, Ripples
340 G.W. G
, e maximum tension principle in general relativity, Foundations of Phys-
ics 32, pp. – , , or http://www.arxiv.org/abs/hep-th/
. Cited on page .
341 at bodies fall along geodesics has been checked by ... Cited on page .
342 So far, the experiments con rm that electrostatic and (strong) nuclear energy fall like matter to within one part in , and weak (nuclear) energy to within a few per cent. is is summarized in Ref. . Cited on page .
343 J. S .
, Berliner Astronomisches Jahrbuch auf das Jahr 1804, , p. . Cited on page
344 See for example K.D. O , Superluminal travel requires negative energies, Physical Re-
view Letters 81, pp. – , , or M. A
, e warp drive: hyper-fast travel
within general relativity, Classical and Quantum Gravity 11, pp. L –L , . See also
C VDB
, A warp drive with more reasonable total energy requirements,
Classical and Quantum Gravity 16, pp. – , . Cited on page .
345 See the Astronomical Almanac, and its Explanatory Supplement, H.M. Printing O ce, London and U.S. Government Printing O ce, Washington, . For the information about
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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various time coordinates used in the world, such as barycentric coordinate time, the time at the barycentre of the solar system, see also the http://tycho.usno.navy.mil/systime.html web page. It also contains a good bibliography. Cited on page .
346 An overview is given in C. W , eory and Experiment in Gravitational Physics, chapter
. , Cambridge University Press, revised edition, . (Despite being a standard reference,
his view the role of tides and the role of gravitational energy within the principle of equi-
valence has been criticised by other researchers.) See also C. W , Was Einstein Right? –
Putting General Relativity to the Test, Oxford University Press, . See also his paper http://
www.arxiv.org/abs/gr-qc/
. Cited on pages , , and .
347 e calculation omits several smaller e ects, such as rotation of the Earth and red-shi . For
the main e ect, see E
F. T
, ‘ e boundaries of nature: special and general
relativity and quantum mechanics, a second course in physics’ – Edwin F. Taylor’s accept-
ance speech for the Oersted Medal presented by the American Association of Physics
Teachers, January , American Journal of Physics 66, pp. – , . Cited on page
.
348 A.G. L
, Did Popper solve Hume’s problem?, Nature 366, pp. – , November
, Cited on page .
349 e measurement is presented in the Astrophysical Journal, in or . Some beautiful graphics at the http://www.physics.uiuc.edu/groups/tastro/movies/spm/ website show the models of this star system. Cited on page .
350 R.J. N
, Visual distortions near a black hole and a neutron star, American Journal
of Physics 61, pp. – , . Cited on page .
351 e equality was rst tested with precision by R
E
, Annalen der Physik
& Chemie 59, p. , , and by R.
E
, V. P
& E. F
, Beiträge
zum Gesetz der Proportionalität von Trägheit und Gravität, Annalen der Physik 4, Leipzig
68, pp. – , . Eőtvős found agreement to parts in . More experiments were per-
formed by P.G. R , R. K
& R.H. D , e equivalence of inertial and
passive gravitational mass, Annals of Physics (NY) 26, pp. – , , one of the most
interesting and entertaining research articles in experimental physics, and by V.B. B -
& V.I. P
, Soviet Physics – JETP 34, pp. – , . Modern results, with
errors less than one part in , are by Y. S & al., New tests of the universality of free fall,
Physical Review D50, pp. – , . Several experiments have been proposed to test
the equality in space to less than one part in . Cited on pages , , and .
352 e irring e ect was predicted in H. T
, Über die Wirkung rotierender ferner
Massen in der Einsteinschen Gravitationstheorie, Physikalische Zeitschri 19, pp. – ,
, and in H. T
, Berichtigung zu meiner Arbeit: “Über die Wirkung rotierender
Massen in der Einsteinschen Gravitationstheorie”, Physikalische Zeitschri 22, p. , .
e irring–Lense e ect was predicted in J. L
& H. T
, Über den Ein uß
der Eigenrotation der Zentralkörper auf die Bewegung der Planeten und Monde nach der
Einsteinschen Gravitationstheorie, Physikalische Zeitschri 19, pp. – , . See also
Ref. . Cited on page .
353 e feat used the LAGEOS and LAGEOS II satellites and is told in I
C
,e
– measurements of the irring–Lense e ect using laser-ranged satellites, Classical
and Quantum Gravity 17, pp. – , . See also I. C
& E.C. P
,A
con rmation of the general relativistic prediction of the Lense– irring e ect, Nature 431,
pp. – , . Cited on pages , , and .
354 e detection of the irring–Lense e ect in binary pulsars is presented in R.D. B
-
, Lense– irring precession of radio pulsars, Journal of Astrophysics and Astronomy
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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16, pp. – , . Cited on page .
355 G. H
, Zeitschri für Mathematik und Physik 15, p. , , F. T
,
Comptes Rendus 75, p. , , and Comptes Rendus 110, p. , . Cited on page .
356 B. M
, Gravitoelectromagnetism, http://www.arxiv.org/abs/gr-qc/
. See
also its extensive reference list on gravitomagnetism. Cited on page .
357 D. B
& P. K
, On relativistic gravitation, American Journal of Physics 53,
pp. – , , and P. K
& D. B
, e gravitational Poynting vector and
energy transfer, American Journal of Physics 55, pp. – , . Cited on pages
and .
358 is is told in J A. W . Cited on page .
, A Journey into Gravity and Spacetime, W.H. Freeman,
359 See, for example, K.T. M D
, Answer to question . Why c for gravitational
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, C.M. C & K.S. T
, Laboratory experiments to test relativistic gravity,
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360 A. T
& M.L. R
, Gravito-electromagnetism versus electromagnetism,
European Journal of Physics 25, pp. – , . Cited on page .
361 e original claim is by S.M. K
, e post-Newtonian treatment of the VLBI ex-
periment on September , , Physics Letters A 312, pp. – , , or http://www.
arxiv.org/abs/gr-qc/
. An argument against the claim was published, among others,
by S
S
, On the speed of gravity and the v c corrections to the Shapiro time
delay, http://www.arxiv.org/abs/astro-ph/
. Cited on pages and .
362 e quadrupole formula is explained in the text by Goenner. See Ref. . Cited on page .
363 For an introduction to gravitational waves, see B.F. S
, Gravitational waves on the
back of an envelope, American Journal of Physics 52, pp. – , . Cited on page .
364 e beautiful summary by D
K
, e gem of general relativity, Physics
Today 46, pp. – , April , appeared half a year before the authors of the cited work,
Joseph Taylor and Russel Hulse, received the Nobel Prize for the discovery of millisecond
pulsars. A more detailed review article is J.H. T
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ity, Philosophical Transactions of the Royal Society, London A 341, pp. – , . e
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det Gravitationswellen, Spektrum der Wissenscha , pp. – , December . Cited on page
.
365 W.B. B ity 14, pp.
& M.S. P , e gravitational wave rocket, Classical and Quantum Grav-
– , , or http://www.arxiv.org/abs/gr-qc/
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366 L. L
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eld, American Journal of Physics 65, pp. – , . Cited on page .
367 A. E
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der Physik 35, p. , . Cited on page .
368 I.I. S
& al., Fourth test of general relativity, Physical Review Letters 13, pp. –
, . Cited on page .
369 I.I. S
& al., Fourth test of general relativity: preliminary results, Physical Review
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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370 J.H. T
, Pulsar timing and relativistic gravity, Proceedings of the Royal Society, London
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371 W. S
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Monthly Notes of the Royal Astrononmical Society 77, pp. – , p. E, . For a dis-
cussion of de Sitter precession and irring–Lense precession, see also B.R. H
,
Gyroscope precession in general relativity, American Journal of Physics 69, pp. – ,
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372 B. B
,I. C
& P.L. B
, New test of general relativity: measurement
of de Sitter geodetic precession rate for lunar perigee, Physical Review Letters 58, pp. –
, . Later it was con rmed by I.I. S
& al., Measurement of the de Sitter
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, . Cited on pages and .
373 W
R
on page .
, Essential Relativity, Springer, revised second edition, . Cited
374 is is told (without the riddle solution) on p. , in W
P , Relativitätsthe-
orie, Springer Verlag, Berlin, , the edited reprint of a famous text originally published in
. e reference is H. V
, Notiz über das mittlere Krümmungsmaß einer n-fach
ausgedehnten Riemannschen Mannigfalktigkeit, Göttinger Nachrichten, mathematische–
physikalische Klasse p. , . Cited on page .
375 M. S
, L.M. N
& N.A. C
, A curvature based derivation of the
Schwarzschild metric, American Journal of Physics 65, pp. – , . Cited on pages
and .
376 M Verlag,
H. S
, Relativity in Astronomy, Celestial Mechanics and Geodesy, Springer
. Cited on page .
377 R H
P. F
,F
B. M
,W
G. W
&B
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378 C.G. T
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.
, Symmetries of the Einstein equations, Physical Review
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379 H. N
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org//abs/gr-qc/
. Cited on page .
380 Y. W & M. T
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background and galaxy clustering, Physical Review Letters 92, p.
, , or http://
www.arxiv.org/astro-ph/
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381 Arguments for the emptiness of general covariance are given by J D. N
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eral covariance and the foundations of general relativity, Reports on Progress in Physics 56,
pp. – , . e opposite point, including the discussion of ‘absolute elements’, is
made in the book by J.L. A
, Principles of Relativity Physics, chapter , Academic
Press, . Cited on page .
382 For a good introduction to mathematical physics, see the famous three-women text in
two volumes by Y
C
-B
,C
D W -M
&M -
D
-B
, Analysis, Manifolds, and Physics, North-Holland,
and
. e rst edition of this classic appeared in . Cited on page .
383 See for example R.A. K & al., New constraints on ΩM, ΩΛ, and w from an independent set of eleven high-redshi supernovae observed with HST, Astrophysical Journal 598,
pp. – , . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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384 R.P. F Wesley,
,R.B. L
& M. S , e Feynman Lectures on Physics, Addison
, volume II, p. – . Cited on page .
385 A recent overview on the experimental tests of the universality of free fall is that by R.J.
H
, e equivalence principle, Contemporary Physics 4, pp. – , . Cited on
page .
386 See for example H.L. B , Black holes, geometric ows, and the Penrose inequality in general relativity, Notices of the AMS 49, pp. – , . Cited on page .
387 See for example the paper by K. D
, Gravity, geometry and equivalence, preprint to
be found at http://www.arxiv.org/abs/gr-qc/
, and L. L
& E. L
,e
Classical eory of Fields, Pergamon, th edition, , p. . Cited on page .
388 E
K
, Kontinuumstheorie der Versetzungen und Eigenspannungen,
Springer, . Kröner shows how to use the Ricci formalism in the solid state. Cited
on page .
389 Black Cited on page .
390 e equivalence of the various de nitions of the Riemann tensor is explained in ...Cited on page .
391 K. T
, Can the Pioneer anomaly have a gravitational origin?, http://www.arxiv.org/
abs/gr-qc/
. Cited on page .
392 is famous quote is the rst sentence of the nal chapter, the ‘Beschluß’, of I K , Kritik der praktischen Vernun , . Cited on page .
393 A
, Opinions, III, I, . See J -P D
sais, Gallimard, , p. . Cited on page .
, Les écoles présocratiques, Folio Es-
394 Galaxy mass measurements are described by ... Cited on page .
395 A beautiful introduction to modern astronomy is P
M
, I mostri del cielo,
Mondadori Editore, . Cited on page .
396 See for example A.N. C , ed., Allen’s Astrophysical Quantities, AIP Press and Springer Verlag, . Cited on page .
397 P. J
, Gravitational microlensing, Naturwissenscha en 86, pp. – ,
page .
. Cited on
398 D. F , An upper limit to the masses of stars, Nature 434, pp. – , page .
. Cited on
399 G. B , e discovery of brown dwarfs, Scienti c American 282, pp. – , April . Cited on page .
400 P.M. W
& C. T
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netar candidates, http://www.arxiv.org/abs/astro-ph/
. Cited on page .
401 B.M. G
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, M.S. O , M. H
, J.M.
D
& A.J. G , A stellar wind bubble coincident with the anomalous X-ray pulsar
E . - : are magnetars formed from massive progenitors?, e Astrophysical Journal
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page .
402 An opposite idea is defended by ... Cited on page .
403 C. W , Scientia 38, p. , , and K. L
, e motions and the distances of
the spiral nebulae, Monthly Notices of the Royal Astronomical Society 85, pp. – , .
See also G. S
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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404 G. G p. ,
, e origin of the elements and the separation of galaxies, Physical Review 74, . Cited on page .
405 A.G. D
& I.D. N
, Dokl. Akad. Nauk. SSSR 154, p. , . It
appeared translated into English a few months later. e story of the prediction was told by
Penzias in his Nobel lecture. Cited on page .
406 A
A. P
&R
W. W
, A measurement of excess antenna temper-
ature at Mcs, Astrophysical Journal 142, pp. – , . Cited on page .
407 M
, Somnium Scipionis, XIV, . See J -P D
cratiques, Folio Essais, Gallimard, , p. . Cited on page .
, Les écoles préso-
408 On the remote history of the universe, see the excellent texts by G. B
, e Early
Universe – Facts & Fiction, Springer Verlag, rd edition, , or B
P
, Creation –
e Story of the Origin and the Evolution of the Universe, Plenum Press, . For an excellent
popular text, see M. L
, Our Evolving Universe, Cambridge University Press, .
Cited on page .
409 e rst oxygen seems to have appeared in the atmosphere, produced by microorganisms,
. thousand million years ago. See A. B
& al., Dating the rise of atmospheric
oxygen, Nature 427, pp. – , . Cited on page .
410 G
W
, Snowball Earth – e Story of the Great Global Catastrophe at
Spawned Life as We Know It, Crown Publishing, . Cited on pages and .
411 K. K , Spuren einer Sternexplosion, Physik in unserer Zeit 36, p. , . e rst step of
this connection is found in K. K , G. K
, T. F
, E.A. D ,
G. R
& A. W
, Fe anomaly in a deep-sea manganese crust and implica-
tions for a nearby supernova source, Physics Review Letters 93, p.
, , the second
step in N.D. M
& H. S
, Low cloud properties in uenced by cosmic rays,
Physics Review Letters 85, pp. – , , and the third step in P.B. M
,
Plio–Pleistocene African climate, Science 270, pp. – , . Cited on page .
412 A. F
, Über die Krümmung des Raumes, Zeitschri für Physik 10, pp. – ,
, and A. F
, Über die Möglichkeit einer Welt mit konstanter negativer Krüm-
mung des Raumes, Zeitschri für Physik 21, pp. – , . (In the Latin transliteration,
the author aquired a second ‘n’ in his second paper.) Cited on page .
413 H. K on pages
, Darkness at night, European Journal of Physics 18, pp. – , and .
. Cited
414 See for example P.D. P , Brightness at night, American Journal of Physics 66, pp. – , . Cited on pages and .
415 P W
, Olbers’ paradox and the spectral intensity of extra-galactic background
light, Astrophysical Journal 367, p. , . Cited on page .
416 S
W
, Gravitation and Cosmology, John Wiley, . An excellent book writ-
ten with a strong personal touch and stressing most of all the relation with experimental
data. It does not develop a strong feeling for space-time curvature, and does not address the
basic problems of space and time in general relativity. Excellent for learning how to actually
calculate things, but less for the aims of our mountain ascent. Cited on pages and .
417 Supernova searches are being performed by ... Cited on page .
418 e experiments are discussed in detail in the excellent review by D. G
& N.
S
, Das Rätsel der kosmischen Vakuumenergiedichte und die beschleunigte Ex-
pansion des Universums, Physikalische Blätter 556, pp. – , . See also N. S
-
, e mystery of the cosmic vacuum energy density and the accelerated expansion of
the universe, European Journal of Physics 20, pp. – , . Cited on pages and .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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419 A. H
& E. S
, Einstein’s mistake and the cosmological contant, American
Journal of Physics 68, pp. – , . Cited on page .
420 e author of the bible explains rain in this way, as can be deduced from its very rst page, Genesis : - . Cited on page .
421 Up to his death, Fred Hoyle defended his belief that the universe is static, e.g. in G. B -
, F. H
& J.V. N
, A di erent approach to cosmology, Physics Today
52, pp. – , . is team has also written a book with the same title, published in
by Cambridge University Press. Cited on pages and .
422 S
W. H
& G.F.R. E , e Large Scale Structure of Space-Time, Cam-
bridge University Press, Cambridge, . Among other things, this reference text discusses
the singularities of space-time, and their necessity in the history of the universe. Cited on
pages , , and .
423 A
, Confessions, , writes: ‘My answer to those who ask ‘What was god doing
before he made Heaven and Earth?’ is not ‘He was preparing Hell for people who pry into
mysteries’. is frivolous retort has been made before now, so we are told, in order to evade
the point of the question. But it is one thing to make fun of the questioner and another to
nd the answer. So I shall refrain from giving this reply. [...] But if before Heaven and Earth
there was no time, why is it demanded what you [god] did then? For there was no “then”
when there was no time.’ (Book XI, chapter and ). Cited on page .
424 S
H
, A Brief History of Time – From the Big Bang to Black Holes, .
Reading this bestseller is almost a must for any physicist, as it is a frequent topic at dinner
parties. Cited on page .
425 Star details are explained in many texts. See for example ... Cited on page .
426 J. P , R. K
, S. R
& T. S
, e light curve and the time delay of
QSO + , Astronomy and Astrophysics 305, p. , . Cited on page .
427 F. Z
, Nebulae as gravitational lenses, Physical Review Letters 51, p. , and F.
Z
, On the probability to detect nebulae which act as gravitational lenses, p. ,
. e negative view by Einstein is found in A. E
, Lens-like action of a star
by the deviatioin of light in the gravitational eld, Science 84, pp. – , . A review
on gravitational lensing can even be found online, in the paper by J. W
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itational lensing in astronomy, Living Reviews in Relativity 1-12, pp. – , , to be found
on the http://www.livingreviews.org/Articles/Volume / - wamb/ website.
ere is also the book by P. S
, J. E
& E.E. F , Gravitational
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428 M. L
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, . See also B.F. R
, e topology of the universe, http://www.arxiv.org/
abs/astro-ph/
preprint. Cited on page .
429 anks to Steve Carlip for clarifying this point. Cited on page .
430 G.F.R. E & T. R
, Lost horizons, American Journal of Physics 61, pp. –
, . Cited on page .
431 A. G , Die Geburt des Kosmos aus dem Nichts – Die eorie des in ationären Universums, Droemer Knaur, . Cited on page .
432 Entropy values for the universe have been discussed by I P
, Is Future
Given?, World Scienti c, . is was his last book. For a di erent approach, see G.A.
MM
& S. C
, Holography and the large number hypothesis, http://
www.arxiv.org/abs/gr-qc/
. is paper also repeats the o en heard statement that the
universe has an entropy that is much smaller than the theoretical maximum. e maximum
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 916 ny
is ‘generally’ estimated to be k, whereas the actual value is ‘estimated’ to be k. However, other authors give k.
In , Roger Penrose also made statements about the entropy of the universe. Cited on page .
433 C.L. B
, M.S. T
& M. W , e cosmic rosetta stone, Physics Today 50,
pp. – , November . e cosmic background radiation di ers from black hole radi-
ation by less than . %. Cited on page .
434 e lack of expansion in the solar system is shown in ... Cited on page .
435 A pretty article explaining how one can make experiments to nd out how the human body
senses rotation even when blindfolded and earphoned is described by M.-L. M
-
& H. M
, e e ect of centrifugal force on the perception of rota-
tion about a vertical axis, Naturwissenscha en 84, pp. – , . Cited on page .
436 e independence of inertia has been tested ... Cited on page .
437 e present status is given in the conference proceedings by J
B
&
H
P
, eds., Mach’s Principle: From Newton’s Bucket to Quantum Gravity,
Birkhäuser, . Various formulations of Mach’s principle – in fact, di erent ones – are
compared on page .
In a related development, in , Dennis Sciama published a paper in which he argues
that inertia of a particle is due to the gravitational attraction of all other matter in the uni-
verse. e paper is widely quoted, but makes no new statements on the issue. See D.W.
S
, On the origin of inertia, Monthly Notices of the Royal Astronomical Society 113,
pp. – , . Cited on page .
438 Information on the rotation of the universe is given in A. K
, G. H
& A.J.
B
, Limits to global rotation and shear from the COBE DMR four-year sky maps,
Physical Review D 55, pp. – , . Earlier information is found in J.D. B
,
R. J
& D.H. S
, Universal rotation: how large can it be?, Monthly No-
tices of the Royal Astronomical Society 213, pp. – , . See also J.D. B
, R.
J
& D.H. S
, Structure of the cosmic microwave background, Nature
309, pp. – , , or E.F. B , P.G. F
& J. S , How anisotropic is the
universe?, Physical Review Letters 77, pp. – , . Cited on page .
439 e issue has been discussed within linearized gravity by R
T
, in his text-
book Relativity, ermodynamics, and Cosmology, Clarendon Press, , on pp. – .
e exact problem has been solved by A. P , Null electromagnetic elds in general re-
lativity theory, Physical Review 118, pp. – , , and by W.B. B
, e gravita-
tional eld of light, Commun. Math. Phys. 13, pp. – , . See also N.V. M
& K.K. K
, e gravitational eld of a spinning pencil of light, Journal of Math-
ematical Physics 30, pp. – , , and P.C. A
& R.U. S , On the
gravitational eld of a spinning particle, General Relativity and Gravitation 2, pp. – ,
. Cited on page .
440 See the delightful popular account by I N
, Black Holes and the Universe, Cam-
bridge University Press, . e consequences of light decay were studied by M. B -
, Die Ausdehnung des Weltalls, Physikalische Zeitschri der Sowjetunion 3, pp. – ,
. Cited on pages and .
441 C.L. C
, K.M. M
, J.T. S
, E. P
, R. V
, F.
B
, A.G. B
, J. C
& C.P. M
, Astronomical constraints on the
cosmic evolution of the ne structure constant and possible quantum dimensions, Physical
Review Letters 85, pp. – , December . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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442 e observations of black holes at the centre of galaxies and elsewhere are summarised by
R. B
& N. G
, Revisiting the black hole, Physics Today 52, pp. – ,
June . Cited on pages and .
443 An excellent and entertaining book on black holes, without any formulae, but nevertheless
accurate and detailed, is the paperback by I N
, Black Holes and the Universe,
Cambridge University Press, . See also E
F. T
&J
A. W
,
Exploring Black Holes: Introduction to General Relativity, Addison Wesley Longman .
For a historical introduction, see the paper by R. R
, e physics of gravitationally
collapsed objects, pp. – , in Neutron Stars, Black Holes and Binary X-Ray Sources, Pro-
ceedings of the Annual Meeting, San Francisco, Calif., February , , Reidel Publishing,
. Cited on page .
444 J. M
, On the means of discovering the distance, magnitude, etc of the xed stars,
Philosophical Transactions of the Royal Society London 74, p. , , reprinted in S. D -
, Black Holes – Selected Reprints, American Association of Physics Teachers, .
Cited on page .
445 e beautiful paper is R. O
& H. S
, On continued gravitational con-
traction, Physical Review 56, pp. – , . Cited on page .
446 R.P. K , Gravitational eld of a spinning mass as an example of algebraically special metrics, Physical Review Letters 11, pp. – , . Cited on page .
447 E.T. N
, E. C
, R. C
, A. E
, A. P
& R. T -
, Metric of a rotating, charged mass, Journal of Mathematical Physics 6, pp. – ,
. Cited on page .
448 For a summary, see P.O. M
, Black hole uniqueness theorems, pp. – , in M.A.H.
MC
, editor, General Relativity and Gravitation, Cambridge University Press,
, or the update at http://www.arxiv.org/abs/hep-th/
. See also D.C. R
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Four decades of black hole uniqueness theorems,, available at http://www.mth.kcl.ac.uk/
sta /dc_robinson/blackholes.pdf Cited on page .
449 H.P. K
& A.K.M. M
- -A , Spherically symmetric static SU( )
Einstein-Yang-Mills elds, Journal of Mathematical Physics 31, pp. – , . Cited on
page .
450 For information about the tendency of gravitational radiation to produce spherical shapes, see for example ... Cited on page .
451 R. P
& R.M. F
, Extraction of rotational energy from a black hole, Nature
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452 e mass–energy relation for a rotating black hole is due to D. C
, Re-
versible and irreversible transformations in black hole physics, Physical Review Letters 25,
pp. – , . For a general, charged and rotating black hole it is due to D. C -
& R. R
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453 J.D. B on page .
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454 e paradox is discussed in M.A. A
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P , Relative alternatives, Scienti c American 266, p. , August . See also M.A. A -
& E. S
, e wall of death, American Journal of Physics 61,
pp. – , , and M.A. A
& J.P. L
, On traveling round without
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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feeling it and uncurving curves, American Journal of Physics 54, pp. – , on page .
. Cited
455 For information about black holes in the early universe, see ... Cited on page .
456 For information about black holes formation via star collapse, see ... Cited on page .
457 F
L , APS meeting press conference: Binary star U - ,
light years from Earth, Physics News Update, April , . Cited on page .
458 e rst direct evidence for matter falling into a black hole was publicised in early . ... Cited on page .
459 For a readable summary of the Penrose–Hawking singularity theorems, see ... Details can be found in Ref. . Cited on page .
460 For an overview of cosmic censorship, see T.P. S , Gravitational collapse, black holes
and naked singularities, http://www.arxiv.org/abs/gr-qc/
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. e ori-
ginal idea is due to R. P
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461 G.J. S
, On the physical units of nature, Philosophical Magazine 11, pp. – , .
Cited on page .
462 e geometrodynamic clock is discussed in D.E. B
& R.P. G
, Limitations of
the geometrodynamic clock, General Relativity and Gravitation 24, pp. – , . e
clock itself was introduced by R.F. M
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463 R. G .
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464 A. M
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465 R
U. S , Die Hohlwelttheorie, Der mathematisch-naturwissenscha liche Unter-
richt 368, pp. – , . R
U. S , Universal conventionalism and space-time.,
General Relativity and Gravitation 1, pp. – , . See also R
U. S , Die
Hohlwelttheorie, in A
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466 T. D
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. It is the latest in a series of his papers on the topic; the rst was T. D
,
Was Einstein % right?, http://www.arxiv.org/abs/gr-qc/
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and .
467 H. D
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Physikalische Blätter 55, pp. – ,
& G. S
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468 See S. B
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ergy, Physical Review Letters 83, pp. – , . See also C.M. W , Gravitational
radiation and the validity of general relativity, Physics Today 52, p. , October . Cited
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469 e inverse square dependence has been checked down to µm, as reported by E. A -
, B. H
& C.D. H , Testing the gravitational inverse-square law, Physics
World 18, pp. – , . Cited on page .
470 For theories competing with general relativity, see for example C.M. W , e confrontation between general relativity and experiment, Living Reviews of Relativity ,
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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http://www.livingreviews.org/lrr- - . For example, the absence of the Nordtvedt e ect,
a hypothetical -day oscillation in the Earth–Moon distance, which was looked for by
laser ranging experiments without any result, ‘killed’ several competing theories. is ef-
fect, predicted by Kenneth Nordtvedt, would only appear if the gravitational energy in the
Earth–Moon system would fall in a di erent way than the Earth and the Moon themselves.
For a summary of the measurements, see J. M
, M. S
, M. S
& H.
R
, Astrophysical Journal Letters 382, p. L , . Cited on page .
471 Almost everything of importance in general relativity is published in the Journal Classical and Quantum Gravity. Cited on page .
472 Collisions and many body problems ... Cited on page .
473 In ation and early universe ... Cited on page .
474 e study of chaos in Einstein’s eld equations is just beginning. See e.g. L. B
,
F. L
& M. C
, Chaos in Robertson-Walker cosmology, http://www.
arxiv.org/abs/gr-qc/
. Cited on page .
475 e ESA satellite called ‘Planck’ will measure the polarization of the cosmic microwave background. Cited on page .
476 A good introduction to the topic of gamma-ray bursts is S. K , J. G
& D.
H
, Kosmische Gammastrahlenausbrüche – Beobachtungen und Modelle, Teil I
und II, Sterne und Weltraum March and April . Cited on page .
477 e eld solution database is built around the work of A. Karlhede, which allows one to distinguish between solutions with a limited amount of mathematical computation. Cited on page .
478 Torsion is presented in ... Cited on page .
479 G.E. P
& M. J , Generalising Raychaudhuri’s equation, in Di erential Geometry
and Its Applications, Proc. Conf., Opava (Czech Republic), August - , , Silesian Uni-
versity, Opava, , pp. – . Cited on page .
480 Wormholes and nontrivial topologies ... A basic approach is the one by T. D
&
M. H
, Charge and the topology of spacetime, Classical and Quantum Gravity 16,
pp. – , , or http://www.arxive.org/abs/gr-qc/
and M. H
, Spin
half in classical general relativity, Classical and Quantum Gravity 17, pp. – , ,
or http://www.arxive.org/abs/gr-qc/
. Cited on page .
481 An important formulation of relativity is A. A quantum gravity, Physical Review Letters 57, pp.
, New variables for classical and – , . Cited on page .
482 A well written text on the connections between the big bang and particle physics is by
I.L. R
, Big Bang – Big Bounce, How Particles and Fields Drive Cosmic Evolution,
Springer, . For another connection, see M. N
& A.A. W
, Observations
and implications of the ultrahigh energy cosmic rays, Reviews of Modern Physics 72, pp. –
, . Cited on page .
483 Teaching will bene t in particular from new formulations, from concentration on principles and their consequences, as has happened in special relativity, from simpler descriptions at the weak eld level, and from future research in the theory of general relativity. e newer textbooks cited above are all steps in these directions. Cited on page .
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
C
IV
CLASSICAL ELECTRODYNAMICS
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W is light? e study of relativity le us completely in the dark, even though e had embarked in it precisely to nd an answer to that question. True, e have learned how the motion of light compares with that of objects. We also learned that light is a moving entity that cannot be stopped; but we haven’t learned anything about its nature. e answer to this long-standing question emerges only from the study of those types of motion that are not related to gravitation, such as the ways magicians levitate objects.
.
,
Page 150 Challenge 917 e
Ref. 484
Revisiting the list of motors found in this world, we remark that gravitation hardly describes any of them. Neither the motion of sea waves, re and earthquakes, nor that of a gentle breeze are due to gravity. e same applies to the motion of muscles. Have you ever listened to your own heart beat with a stethoscope? Without having done so, you cannot claim to have experienced the mystery of motion. Your heart has about million beats in your lifetime. en it stops.
It was one of the most astonishing discoveries of science that heart beats, sea waves and most other cases of everyday motion, as well as the nature of light itself, are connected to observations made thousands of years ago using two strange stones. ese stones show that all examples of motion, which are called mechanical in everyday life, are, without exception, of electrical origin.
In particular, the solidity, the so ness and the impenetrability of matter are due to internal electricity; also the emission of light is an electrical process. As these aspects are part of everyday life, we will leave aside all complications due to gravity and curved space-time. e most productive way to study electrical motion is to start, as in the case of gravity, with those types of motion which are generated without any contact between the bodies involved.
A
,
e story of electricity starts with trees. Trees have a special relation to electricity. When a tree is cut, a viscous resin appears. With time it solidi es and, a er millions of years, it forms amber. When amber is rubbed with a cat fur, it acquires the ability to attract small objects, such as saw dust or pieces of paper. is was already known to ales of
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•.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 221 Objects surrounded by fields: amber, lodestone and mobile phone
Challenge 918 ny
Miletus, one of the original seven sages, in the sixth century . e same observation
can be made with many other polymer combinations, for example with combs and hair,
with soles of the shoe on carpets, and with a TV screen and dust. Children are always
surprised by the e ect, shown in Figure , that a comb rubbed on wool has on running
tap water. Another interesting e ect can be observed when a rubbed comb is put near a
burning candle. (Can you imagine what happens?)
Another part of the story of electricity involves an iron mineral
found in certain caves around the world, e.g. in a region (still)
water
called Magnesia in the Greek province of essalia, and in some
pipe
regions in central Asia. When two stones of this mineral are put rubbed near each other, they attract or repel each other, depending on comb
their relative orientation. In addition, these stones attract objects
made of cobalt, nickel or iron.
Today we also nd various small objects in nature with more
sophisticated properties, as shown in Figure . Some objects en-
able you to switch on a television, others unlock car doors, still
others allow you to talk with far away friends. All these observations show that in nature there are situations FIGURE 222 How to
where bodies exert in uence on others at a distance. e space amaze kids
surrounding a body exerting such an in uence is said to contain a eld. A (physical) eld
is thus an entity that manifests itself by accelerating other bodies in its region of space. A
eld is some ‘stu ’ taking up space. Experiments show that elds have no mass. e eld
surrounding the mineral found in Magnesia is called a magnetic eld and the stones are
called magnets.* e eld around amber – called ἤλεκτρον in Greek, from a root meaning
‘brilliant, shining’ – is called an electric eld. e name is due to a proposal by the famous
English part-time physicist William Gilbert ( – ) who was physician to Queen
Elizabeth I. Objects surrounded by a permanent electric eld are called electrets. ey
are much less common than magnets; among others, they are used in certain loudspeaker
systems.**
* A pretty book about the history of magnetism and the excitement it generates is J
D. L
,
Driving Force – the Natural Magic of Magnets, Harvard University Press, 1996.
** e Kirlian e ect, which allows one to make such intriguingly beautiful photographs, is due to a time-
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TA B L E 41 Searches for magnetic monopoles, i.e., for magnetic charges
S
M
Smallest magnetic charge suggested by quantum theory Search in minerals Search in meteorites Search in cosmic rays Search with particle accelerators
=
h e
=
eZ α
=
none Ref. 485
none Ref. 485
none Ref. 485
none Ref. 485
. pWb
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 223 Lightning: a picture taken with a moving camera, showing its multiple strokes (© Steven Horsburgh)
e eld around a mobile phone is called a radio eld or, as we will see later, an electromagnetic eld. In contrast to the previous elds, it oscillates over time. We will nd out later that many other objects are surrounded by such elds, though these are o en very weak. Objects that emit oscillating elds, such as mobile phones, are called radio transmitters or radio emitters.
Fields in uence bodies over a distance, without any material support. For a long time, this was rarely found in everyday life, as most countries have laws to restrict machines that use and produce such elds. e laws require that for any device that moves, produces sound, or creates moving pictures, the elds need to remain inside them. For this reason a magician moving an object on a table via a hidden magnet still surprises and entertains his audience. To feel the fascination of elds more strongly, a deeper look into a few experimental results is worthwhile.
varying electric eld.
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•.
TA B L E 42 Some observed magnetic fields
O
M
Lowest measured magnetic eld (e.g., elds of the Schumann reson- fT
ances)
Magnetic eld produced by brain currents
. pT to pT
Intergalactic magnetic elds
pT to pT
Magnetic eld in the human chest, due to heart currents
pT
Magnetic eld of our galaxy
. nT
Magnetic eld due to solar wind
0.2 to nT
Magnetic eld directly below high voltage power line
0.1 to µT
Magnetic eld of Earth
20 to µT
Magnetic eld inside home with electricity
0.1 to µT
Magnetic eld near mobile phone
µT
Magnetic eld that in uences visual image quality in the dark
µT
Magnetic eld near iron magnet
mT
Solar spots
T
Magnetic elds near high technology permanent magnet
max . T
Magnetic elds that produces sense of coldness in humans
T or more
Magnetic elds in particle accelerator
T
Maximum static magnetic eld produced with superconducting coils T
Highest static magnetic elds produced in laboratory, using hybrid T magnets
Highest pulsed magnetic elds produced without coil destruction
T
Pulsed magnetic elds produced, lasting about µs, using imploding
T
coils
Field of white dwarf
T
Fields in petawatt laser pulses
kT
Field of neutron star
from T to T
Quantum critical magnetic eld
. GT
Highest eld ever measured, on magnetar and so gamma repeater 0.8 to ë T SGR-1806-20
Field near nucleus
TT
Maximum (Planck) magnetic eld
.ë T
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H
?
Everybody has seen a lightning ash or has observed the e ect it can have on striking a tree. Obviously lightning is a moving phenomenon. Photographs such as that of Figure show that the tip of a lightning ash advance with an average speed of around
km s. But what is moving? To nd out, we have to nd a way of making lightning for ourselves.
In , the car company General Motors accidentally rediscovered an old and simple
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,
nylon ropes
water pipe
nylon ropes
metal cylinders
bang! metal wires
metal cans
F I G U R E 224 A simple Kelvin generator
pendulum with metal ball
on the roof
in the hall
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in the ground
F I G U R E 225 Franklin’s personal lightning rod
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 486 Ref. 487 Challenge 919 n
method of achieving this. eir engineers had inadvertently built a spark generating mechanism into their cars; when lling the petrol tank, sparks were generated, which sometimes lead to the explosion of the fuel. ey had to recall million vehicles of its Opel brand.
What had the engineers done wrong? ey had unwittingly copied the conditions for a electrical device which anyone can build at home and which was originally invented by William omson.* Repeating his experiment today, we would take two water taps, four empty bean or co ee cans, of which two have been opened at both sides, some nylon rope and some metal wire.
Putting this all together as shown in Figure , and letting the water ow, we nd a strange e ect: large sparks periodically jump between the two copper wires at the point where they are nearest to each other, giving out loud bangs. Can you guess what condition for the ow has to be realized for this to work? And what did Opel do to repair the cars they recalled?
If we stop the water owing just before the next spark is due, we nd that both buckets are able to attract sawdust and pieces of paper. e generator thus does the same that rubbing amber does, just with more bang for the buck(et). Both buckets are surrounded by electric elds. e elds increase with time, until the spark jumps. Just a er the spark,
* William omson (1824–1907), important Irish Unionist physicist and professor at Glasgow University. He worked on the determination of the age of the Earth, showing that it was much older than 6000 years, as several sects believed. He strongly in uenced the development of the theory of magnetism and electricity, the description of the aether and thermodynamics. He propagated the use of the term ‘energy’ as it is used today, instead of the confusing older terms. He was one of the last scientists to propagate mechanical analogies for the explanation of phenomena, and thus strongly opposed Maxwell’s description of electromagnetism. It was mainly for this reason that he failed to receive a Nobel Prize. He was also one of the minds behind the laying of the rst transatlantic telegraphic cable. Victorian to his bones, when he was knighted, he chose the name of a small brook near his home as his new name; thus he became Lord Kelvin of Largs. erefore the unit of temperature obtained its name from a small Scottish river.
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•.
Ref. 488 Challenge 920 n
the buckets are (almost) without electric eld. Obviously, the ow of water somehow builds up an entity on each bucket; today we call this electric charge. Charge can ow in metals and, when the elds are high enough, through air. We also nd that the two buckets are surrounded by two di erent types of electric elds: bodies that are attracted by one bucket are repelled by the other. All other experiments con rm that there are two types of charges. e US politician and part-time physicist Benjamin Franklin ( – ) called the electricity created on a glass rod rubbed with a dry cloth positive, and that on a piece of amber negative. (Previously, the two types of charges were called ‘vitreous’ and ‘resinous’.) Bodies with charges of the same sign repel each other, bodies with opposite charges attract each other; charges of opposite sign owing together cancel each other out.*
In summary, electric elds start at bodies, provided they are charged. Charging can be achieved by rubbing and similar processes. Charge can ow: it is then called an electric current. e worst conductors of current are polymers; they are called insulators or dielectrics. A charge put on an insulator remains at the place where it was put. In contrast, metals are good conductors; a charge placed on a conductor spreads all over its surface.
e best conductors are silver and copper. is is the reason that at present, a er a hundred years of use of electricity, the highest concentration of copper in the world is below the surface of Manhattan.
Of course, one has to check whether natural lightning is actually electrical in origin. In , experiments performed in France, following a suggestion by Benjamin Franklin, published in London in , showed that one can indeed draw electricity from a thunderstorm via a long rod.** ese French experiments made Franklin famous worldwide; they were also the start of the use of lightning rods all over the world. Later, Franklin had a lightning rod built through his own house, but of a somewhat unusual type, as shown in Figure . Can you guess what it did in his hall during bad weather, all parts being made of metal? (Do not repeat this experiment; the device can kill.)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
E
If all experiments with charge can be explained by calling the two charges positive and negative, the implication is that some bodies have more, and some less charge than an uncharged, neutral body. Electricity thus only ows when two di erently charged bodies are brought into contact. Now, if charge can ow and accumulate, we must be able to somehow measure its amount. Obviously, the amount of charge on a body, usually abbreviated q, is de ned via the in uence the body, say a piece of sawdust, feels when subjected to a
eld. Charge is thus de ned by comparing it to a standard reference charge. For a charged body of mass m accelerated in a eld, its charge q is determined by the relation
q qref
=
ma mref aref
,
(390)
* In fact, there are many other ways to produces sparks or even arcs, i.e. sustained sparks; there is even a complete subculture of people who do this as a hobby at home. ose who have a larger budget do it professionally, in particle accelerators. See the http://www.kronjaeger.com/hv/ website. ** e details of how lightning is generated and how it propagates are still a topic of research. An introduction is given on page 597.
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TA B L E 43 Properties of classical electric charge
E
P
M
D
Can be distinguished Can be ordered Can be compared Can change gradually Can be added Do not change Can be separated
distinguishability sequence measurability continuity accumulability conservation separability
element of set order metricity completeness additivity invariance positive or negative
Page 646 Page 1195 Page 1205 Page 1214 Page 69
q = const
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i.e., by comparing it with the acceleration and mass of the reference charge. is de nition re ects the observation that mass alone is not su cient for a complete characterization of a body. For a full description of motion we need to know its electric charge; charge is therefore the second intrinsic property of bodies that we discover in our walk.
Nowadays the unit of charge, the coulomb, is de ned through a standard ow through metal wires, as explained in Appendix B. is is possible because all experiments show that charge is conserved, that it ows, that it ows continuously and that it can accumulate. Charge thus behaves like a uid substance. erefore we are forced to use for its description a scalar quantity q, which can take positive, vanishing, or negative values.
In everyday life these properties of electric charge, listed also in Table , describe observations with su cient accuracy. However, as in the case of all previously encountered classical concepts, these experimental results for electrical charge will turn out to be only approximate. More precise experiments will require a revision of several properties. However, no counter-example to charge conservation has as yet been observed.
A charged object brought near a neutral one polarizes it. Electrical polarization is the separation of the positive and negative charges in a body. For this reason, even neutral objects, such as hair, can be attracted to a charged body, such as a comb. Generally, both insulators and conductors can be polarized; this occurs for whole stars down to single molecules.
Attraction is a form of acceleration. Experiments show that the entity that accelerates charged bodies, the electric eld, behaves like a small arrow xed at each point x in space; its length and direction do not depend on the observer. In short, the electric eld E(x) is a vector eld. Experiments show that it is best de ned by the relation
qE(x) = ma(x)
(391)
Challenge 922 e
taken at every point in space x. e de nition of the electric eld is thus based on how it moves charges.* e eld is measured in multiples of the unit N C or V m.
To describe the motion due to electricity completely, we need a relation explaining how charges produce electric elds. is relation was established with precision (but not for
Challenge 921 ny * Does the de nition of electric eld given here assume a charge speed that is much less than that of light?
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•.
TA B L E 44 Values of electrical charge observed in nature
O
Smallest measured non-vanishing charge Charge per bit in computer memory Charge in small capacitor Charge ow in average lightning stroke Charge stored in a fully-charge car battery Charge of planet Earth Charge separated by modern power station in one year Total charge of positive (or negative) sign observed in universe Total charge observed in universe
C
.ë − C −C −C C to C . MC MC ëC
C C
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TA B L E 45 Some observed electric fields
O
E
Field m away from an electron in vacuum Field values sensed by sharks
Challenge 923 n
down to . µV m
Cosmic noise Field of a W FM radio transmitter at km distance Field inside conductors, such as copper wire Field just beneath a high power line Field of a GSM antenna at m Field inside a typical home Field of a W bulb at m distance Ground eld in Earth’s atmosphere Field inside thunder clouds Maximum electric eld in air before sparks appear Electric elds in biological membranes Electric elds inside capacitors Electric elds in petawatt laser pulses Electric elds in U + ions, at nucleus Maximum practical electric eld in vacuum, limited by electron pair production Maximum possible electric eld in nature (corrected Planck electric eld)
µV m . mV m . Vm 0.1 to V m . Vm 1 to V m
Vm to V m up to over kV m 1 to MV m MV m up to GV m TV m EV m . EV m
. ë Vm
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,
the rst time) by Charles-Augustin de Coulomb on his private estate, during the French Revolution.* He found that around any small-sized or any spherical charge Q at rest there is an electric eld. At a position r, the electric eld E is given by
E(r) =
πε
Q r
r r
where
πε = . GV m C .
(392)
Challenge 924 n
Later we will extend the relation for a charge in motion. e bizarre proportionality con-
stant, built around the so-called permittivity of free space ε , is due to the historical way
the unit of charge was de ned rst.** e essential point of the formula is the decrease of
the eld with the square of the distance; can you imagine the origin of this dependence?
e two previous equations allow one to write the interaction between two charged
bodies as
dp dt
=
πε
qq r
r r
=
−
dp dt
,
(393)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 925 n
where dp is the momentum change, and r is the vector connecting the two centres of mass. is famous expression for electrostatic attraction and repulsion, also due to Coulomb,
is valid only for charged bodies that are of small size or spherical, and most of all, that are at rest.
Electric elds have two main properties: they contain energy and they can polarize bodies. e energy content is due to the electrostatic interaction between charges. e strength of the interaction is considerable. For example, it is the basis for the force of our muscles. Muscular force is a macroscopic e ect of equation . Another example is the material strength of steel or diamond. As we will discover, all atoms are held together by electrostatic attraction. To convince yourself of the strength of electrostatic attraction, answer the following: What is the force between two boxes with a gram of protons each, located on the two poles of the Earth? Try to guess the result before you calculate the astonishing value.
Coulomb’s relation for the eld around a charge can be rephrased in a way that helps to generalize it to non-spherical bodies. Take a closed surface, i.e., a surface than encloses a certain volume. en the integral of the electric eld over this surface, the electric ux, is the enclosed charge Q divided by ε :
∫E closedsurface
dA =
Q ε
.
(394)
Challenge 926 n is mathematical relation, called Gauss’s ‘law’, follows from the result of Coulomb. (In the simpli ed form given here, it is valid only for static situations.) Since inside conductors the electrical eld is zero, Gauss’s ‘law’ implies, for example, that if a charge q is sur-
Ref. 489
* Charles-Augustin de Coulomb (b. 1736 Angoulême, d. 1806 Paris), French engineer and physicist. His careful experiments on electric charges provided a rm basis for the study of electricity. ** Other de nitions of this and other proportionality constants to be encountered later are possible, leading to unit systems di erent from the SI system used here. e SI system is presented in detail in Appendix B. Among the older competitors, the Gaussian unit system o en used in theoretical calculations, the Heaviside–Lorentz unit system, the electrostatic unit system and the electromagnetic unit system are the most important ones.
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Challenge 927 e Challenge 929 n
rounded by a uncharged metal sphere, the outer surface of the metal sphere shows the same charge q.
Owing to the strength of electromagnetic interactions, separating charges is not an easy task. is is the reason that electrical e ects have only been commonly used for about a hundred years. We had to wait for practical and e cient devices to be invented for separating charges and putting them into motion. Of course this requires energy. Batteries, as used in mobile phones, use chemical energy to do the trick.* ermoelectric elements, as used in some watches, use the temperature di erence between the wrist and the air to separate charges; solar cells use light, and dynamos or Kelvin generators use kinetic energy.
Do uncharged bodies attract one other? In rst approximation they do not. But when the question is investigated more precisely, one nds that they can attract one other. Can you nd the conditions for this to happen? In fact, the conditions are quite important, as our own bodies, which are made of neutral molecules, are held together in this way.
What then is electricity? e answer is simple: electricity is nothing in particular. It is the name for a eld of inquiry, but not the name for any speci c observation or e ect. Electricity is neither electric current, nor electric charge, nor electric eld. Electricity is not a speci c term; it applies to all of these phenomena. In fact the vocabulary issue hides a deeper question that remains unanswered at the beginning of the twenty- rst century: what is the nature of electric charge? In order to reach this issue, we start with the following question.
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C
?
Ref. 490
If electric charge really is something owing through metals, we should be able to observe the e ects shown in Figure . Maxwell has predicted most of these e ects: electric charge should fall, have inertia and be separable from matter. Indeed, each of these e ects has been observed.** For example, when a long metal rod is kept vertically, we can measure an electrical potential di erence, a voltage, between the top and the bottom. In other words, we can measure the weight of electricity in this way. Similarly, we can measure the potential di erence between the ends of an accelerated rod. Alternatively, we can measure the potential di erence between the centre and the rim of a rotating metal disc.
e last experiment was, in fact, the way in which the ratio q m for currents in metals was rst measured with precision. e result is
q m = . ë C kg
(395)
Ref. 491 Ref. 492
for all metals, with small variations in the second digit. In short, electrical current has mass. erefore, whenever we switch on an electrical current, we get a recoil. is simple e ect can easily be measured and con rms the mass to charge ratio just given. Also, the emission of current into air or into vacuum is observed; in fact, every television tube uses this principle to generate the beam producing the picture. It works best for metal objects with sharp, pointed tips. e rays created this way – we could say that they are
Challenge 928 n * Incidentally, are batteries sources of charges? ** Maxwell also performed experiments to detect these e ects (apart from the last one, which he did not predict), but his apparatuses where not sensitive enough.
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,
If electric charge in metals moves like a fluid, it should:
fall under gravity
be subject to centrifugation
a
resist acceleration
spray when pumped
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lead to recoil just after
a
switching on a currrent
q
prevent free charges from falling through
a thin hollow tube
F I G U R E 226 Consequences of the flow of electricity
Challenge 930 e Ref. 493
Challenge 931 n
‘free’ electricity – are called cathode rays. Within a few per cent, they show the same mass to charge ratio as expression ( ). is correspondence thus shows that charges move almost as freely in metals as in air; this is the reason metals are such good conductors.
If electric charge falls inside vertical metal rods, we can make the astonishing deduction that cathode rays – as we will see later, they consist of free electrons* – should not be able to fall through a vertical metal tube. is is due to exact compensation of the acceleration by the electrical eld generated by the displaced electricity in the tube and the acceleration of gravity. us electrons should not be able to fall through a long thin cylinder. is would not be the case if electricity in metals did not behave like a uid. e experiment has indeed been performed, and a reduction of the acceleration of free fall for electrons of % has been observed. Can you imagine why the ideal value of % is not achieved?
If electric current behaves like a liquid, one should be able to measure its speed. e rst to do so, in , was Charles Wheatstone. In a famous experiment, he wire of a
* e name ‘electron’ is due to George Stoney. Electrons are the smallest and lightest charges moving in metals; they are, usually – but not always – the ‘atoms’ of electricity – for example in metals. eir charge is small, . aC, so that ows of charge typical of everyday life consist of large numbers of electrons; as a result, electrical charge behaves like a continuous uid. e particle itself was discovered and presented in 1897 by the Prussian physicist Johann Emil Wiechert (1861–1928) and, independently, three months later, by the British physicist Joseph John omson (1856–1940).
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TA B L E 46 Some observed electric current values
O
C
Smallest regularly measured currents Human nerve signals Lethal current for humans
Current drawn by a train engine Current in a lightning bolt Highest current produced by humans Current inside the Earth
fA µA
as low as mA, typically mA A
10 to kA MA
around MA
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Ref. 494 Challenge 932 e
quarter of a mile length, to produce three sparks: one at the start, one at the middle, and one at the end. He then mounted a rapidly moving mirror on a mechanical watch; by noting who much the three spark images were shi ed against each other on a screen, he determined the speed to be Mm s, though with a large error. Latter, more precise measurements showed that the speed is always below Mm s, and that it depends on the metal and the type of insulation of the wire. e high value of the speed convinced many people to use electricity for transmitting messages. A modern version of the experiment, for computer fans, uses the ‘ping’ command. e ‘ping’ command measures the time for a computer signal to reach another computer and return back. If the cable length between two computers is known, the signal speed can be deduced. Just try.
F
Why is electricity dangerous to humans? e main reason is that the human body is controlled by ‘electric wires’ itself. As a result, outside electricity interferes with the internal signals. is has been known since . In that year the Italian medical doctor Luigi Galvani ( – ) discovered that electrical current makes the muscles of a dead animal contract. e famous rst experiment used frog legs: when electricity was applied to them, they twitched violently. Subsequent investigations con rmed that all nerves make use of electrical signals. Nerves are the ‘control wires’ of animals. However, nerves are not made of metal: metals are not su ciently exible. As a result, nerves do not conduct electricity using electrons but by using ions. e ner details were clari ed only in the twentieth century. Nerve signals propagate using the motion of sodium and potassium ions in the cell membrane of the nerve. e resulting signal speed is between . m s and
m s, depending on the type of nerve. is speed is su cient for the survival of most species – it signals the body to run away in case of danger.
Being electrically controlled, all mammals can sense strong electric elds. Humans can sense elds down to around kV m, when hair stands on end. In contrast, several animals can sense weak electric and magnetic elds. Sharks, for example, can detect
elds down to µV m using special sensors, the Ampullae of Lorenzini, which are found around their mouth. Sharks use them to detect the eld created by prey moving in water; this allows them to catch their prey even in the dark. Several freshwater sh are also able to detect electric elds. e salamander and the platypus, the famous duck-billed
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Challenge 933 ny Page 564
mammal, can also sense electric elds. Like sharks, they use them to detect prey in water which is too muddy to see through. Certain sh, the so-called weakly-electric sh, even generate a weak eld in order to achieve better prey detection.*
No land animal has special sensors for electric elds, because any electric eld in air is strongly damped when it encounters a water- lled animal body. Indeed, the usual atmosphere has an electric eld of around V m; inside the human body this eld is damped to the µV m range, which is much less than an animal’s internal electric elds. In other words, humans do not have sensors for low electric elds because they are land animals. (Do humans have the ability to sense electric elds in water? Nobody seems to know.) However, there a few exceptions. You might know that some older people can sense approaching thunderstorms in their joints. is is due the coincidence between the electromagnetic eld frequency emitted by thunderclouds – around kHz – and the resonant frequency of nerve cell membranes.
e water content of the human body also means that the electric elds in air that are found in nature are rarely dangerous to humans. Whenever humans do sense electric
elds, such as when high voltage makes their hair stand on end, the situation is potentially dangerous.
e high impedance of air also means that, in the case of time-varying electromagnetic elds, humans are much more prone to be a ected by the magnetic component than by the electric component.
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Ref. 495
e study of magnetism progressed across the world independently of the study of electricity. Towards the end of the th century, the compass came into use in Europe. At that time, there were heated debates on whether it pointed to the north or the south. In , the French military engineer Pierre de Maricourt ( – ) published his study of magnetic materials. He found that every magnet has two points of highest magnetization, and he called them poles. He found that even a er a magnet is cut, the resulting pieces always retain two poles: one points to the north and the other to the south when the stone is le free to rotate. Magnets are dipoles. Atoms are either dipoles or unmagnetic. ere are no magnetic monopoles. Despite the promise of eternal fame, no magnetic monopole has ever been found, as shown in Table . Like poles repel, and unlike poles attract.
Magnets have a second property: magnets transform unmagnetic materials into magnetic ones. ere is thus also a magnetic polarization, similar to the electric polarization. Unlike the electric case, some magnetic materials retain the induced magnetic polarization: they become magnetized. is happens when the atoms in the material get aligned by the external magnet.
* It took until the year 2000 for technology to make use of the same e ect. Nowadays, airbag sensors in cars o en use electric elds to sense whether the person sitting in the seat is a child or an adult, thus changing the way that the bag behaves in an accident.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 227 The magentotactic bacterium Magnetobacterium bavaricum with its magnetosomes (© Marianne Hanzlik)
C
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Ref. 496 Challenge 934 ny
Any fool can ask more questions than seven sages can answer.
“ Antiquity ” It is known that honey bees, sharks, pigeons, salmon, trout, sea turtles and certain bac-
teria can feel magnetic elds. One speaks of the ability for magnetoreception. All these life forms use this ability for navigation. e most common detection method is the use of small magnetic particles inside a cell; the cell then senses how these small built-in magnets move in a magnetic eld. e magnets are tiny, typically around nm in size. ese small magnets are used to navigate along the magnetic eld of the Earth. For higher animals, the variations of the magnetic eld of the Earth, to µT, produce a landscape that is similar to the visible landscape for humans. ey can remember it and use it for navigation.
Can humans feel magnetic elds? Magnetic material seems to be present in the human brain, but whether humans can feel magnetic elds is still an open issue. Maybe you can devise a way to check this?
Are magnetism and electricity related? François Arago* found out that they were. He observed that a ship that had survived a bad thunderstorm and had been struck by lightning, needed a new compass. us lightning has the ability to demagnetize compasses. Arago knew, like Franklin, that lightning is an electrical phenomena. In other words, electricity and magnetism must be related. More precisely, magnetism must be related to the motion of electricity.
* Dominique-François Arago (1786–1853) French physicist.
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Oersted's motor
currentcarrying metal wire
Modern motor NS
NS magnet
current-carrying metal wire
F I G U R E 228 An old and a newer version of an electric motor
battery wire
compass needle
F I G U R E 229 An electrical current always produces a magnetic field
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?
Communism is the power of the local councils plus electrici cation of the whole country.
“ Lenin.* ” e reason for Lenin’s famous statement were two discoveries. One was made in
by the Danish physicist Hans Christian Oersted ( – ) and the other in by the English physicist Michael Faraday.** e consequences of these experiments changed the world completely in less than one century.
On the st of July of , Oersted published a lea et, in Latin, which took Europe by storm. Oersted had found (during a lecture demonstration to his students) that when a current is sent through a wire, a nearby magnet is put into motion. In other words, he found that the ow of electricity can move bodies.
Further experiments show that two wires in which charges ow attract or repel each other, depending on whether the currents are parallel or antiparallel. ese and other experiments show that wires in which electricity ows behave like magnets.*** In other words, Oersted had found the de nite proof that electricity could be turned into magnetism.
Shortly a erwards, Ampère**** found that coils increase these e ects dramatically.
* Lenin (b. 1870 Simbirsk, d. 1924 Gorki), founder of the Union of Soviet Socialist Republics, in 1920 stated this as the centre of his development plan for the country. In Russian, the local councils of that time were called soviets. ** Michael Faraday (b. 1791 Newington Butts, d. 1867 London) was born to a simple family, without schooling, and of deep and naive religious ideas. As a boy he became assistant to the most famous chemist of his time, Humphry Davy (1778–1829). He had no mathematical training, but late in his life he became member of the Royal Society. A modest man, he refused all other honours in his life. He worked on chemical topics, the atomic structure of matter and, most of all, developed the idea of (magnetic) elds and eld lines through all his experimental discoveries, such as e ect. Fields were later described mathematically by Maxwell, who at that time was the only person in Europe to take over Faraday’s eld concept. *** In fact, if one imagines tiny currents moving in circles inside magnets, one gets a unique description for all magnetic elds observed in nature. **** André-Marie Ampère (b. 1775 Lyon, d. 1836 Marseille), French physicist and mathematician. Autodidact, he read the famous Encyclopédie as a child; in a life full of personal tragedies, he wandered from maths to chemistry and physics, worked as a school teacher, and published nothing of importance until 1820. en
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Ref. 497
Ref. 498 Page 760
Coils behave like small magnets. In particular, coils, like magnetic elds, always have
two poles, usually called the north and the south pole. Opposite poles attract, like poles
repel each other. As is well known, the Earth is itself a large magnet, with its magnetic
north pole near the geographic south pole, and vice versa.
Moving electric charge produces magnetic elds. is result explains why magnetic
elds always have two poles. e lack of magnetic monopoles thus becomes clear. But
one topic is strange. If magnetic elds are due to the motion of charges, this must be also
the case for a normal magnet. Can this be shown?
In , two men in the Netherlands found a simple way to prove that
even in a magnet, something is moving. ey suspended a metal rod
from the ceiling by a thin thread and then put a coil around the rod, as
shown in Figure . ey predicted that the tiny currents inside the
rod would become aligned by the magnetic eld of the coil. As a res-
ult, they expected that a current passing through the coil would make
the rod turn around its axis. Indeed, when they sent a strong current
through the coil, the rod rotated. (As a result of the current, the rod was
magnetized.) Today, this e ect is called the Einstein–de Haas e ect a er
the two physicists who imagined, measured and explained it.* e e ect
thus shows that even in the case of a permanent magnet, the magnetic eld is due to the internal motion of charges. e size of the e ect also
shows that the moving particles are electrons. (Twelve years later it became clear that the angular momentum of the electrons responsible for
F I G U R E 230 Current makes a metal rods rotate
the e ect is a mixture of orbital and spin angular momentum; in fact, the electron spin
plays a central role in the e ect.)
Since magnetism is due to the alignment of microscopic rotational motions, an even
more surprising e ect can be predicted. Simply rotating a ferromagnetic material**
should magnetize it, because the tiny rotating currents would then be aligned along the
axis of rotation. is e ect has indeed been observed; it is called the Barnett e ect a er its
discoverer. Like the Einstein–de Haas e ect, the Barnett e ect can also be used to determ-
ine the gyromagnetic ratio of the electron; thus it also proves that the spins of electrons
(usually) play a larger role in magnetism than their orbital angular momentum.
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Experiments show that the magnetic eld always has a given direction in space, and a magnitude common to all (resting) observers, whatever their orientation. We are tempted
the discovery of Oersted reached all over Europe: electrical current can deviate magnetic needles. Ampère worked for years on the problem, and in 1826 published the summary of his ndings, which lead Maxwell to call him the ‘Newton of electricity’. Ampère named and developed many areas of electrodynamics. In 1832, he and his technician also built the rst dynamo, or rotative current generator. Of course, the unit of electrical current is named a er him. * Wander Johannes de Haas (1878–1960), Dutch physicist. De Haas is best known for two additional magneto-electric e ects named a er him, the Shubnikov–de Haas e ect (the strong increase of the magnetic resistance of bismuth at low temperatures and high magnetic elds) and the de Haas–Van Alphen e ect (the diamagnetic susceptibility of bismuth at low temperatures is a periodic function of the magnetic
eld). ** A ferromagnetic material is a special kind of paramagnetic material that has a permanent magnetization.
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magnet
magnet
diamagnetic material
paramagnetic material
F I G U R E 231 The two basic types of magnetic material behaviour (tested in an inhomogeneous field): diamagnetism and paramagnetism
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Challenge 935 e
to describe the magnetic eld by a vector. However, this would be wrong, since a magnetic eld does not behave like an arrow when placed before a mirror. Imagine that a system
produces a magnetic eld directed to the right. You can take any system, a coil, a machine, etc. Now build or imagine a second system that is the exact mirror version of the rst: a mirror coil, a mirror machine, etc. e magnetic system produced by the mirror system does not point to the le , as maybe you expected: it still points to the right. (Check by yourself.) In simple words, magnetic elds do not behave like arrows.
In other words, it is not completely correct to describe a magnetic eld by a vector B = (Bx , By , Bz), as vectors behave like arrows. One also speaks of a pseudovector; angular momentum and torque are also examples of such quantities. e precise way is to describe the magnetic eld by the quantity*
−Bz By
B = Bz
−Bx ,
−By Bx
(396)
called an antisymmetric tensor. In summary, magnetic elds are de ned by the acceleration they impart on moving charges. is acceleration turns out to follow
a
=
e m
vB
=
e m
v
B
(397)
a relation which is o en called Lorentz acceleration, a er the important Dutch physicist Hendrik A. Lorentz (b. Arnhem, d. Haarlem) who rst stated it clearly.** ( e relation is also called the Laplace acceleration.) e unit of the magnetic eld is called tesla and is abbreviated T. One has T = N s C m = V s m = V s A m.
e Lorentz acceleration is the e ect at the root of any electric motor. An electric motor is a device that uses a magnetic eld as e ciently as possible to accelerate charges owing
Challenge 936 ny
* e quantity B was not called the ‘magnetic eld’ until recently. We follow here the modern, logical de nition, which supersedes the traditional one, where B was called the ‘magnetic ux density’ or ‘magnetic induction’ and another quantity, H, was called – incorrectly, but for over a century – the magnetic eld. is quantity H will not appear in this walk, but it is important for the description of magnetism in materials. ** Does the de nition of magnetic eld given here assume a charge speed much lower than that of light?
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
in a wire. rough the motion of the charges, the wire is then also moved. Electricity is thus transformed into magnetism and then into motion. e rst e cient motor was built back in .
As in the electric case, we need to know how the strength of a magnetic eld is determined. Experiments such as Oersted’s show that the magnetic eld is due to moving charges, and that a charge moving with velocity v produces a eld B given by
B(r) =
µ π
q
v r
r
where
µ= π
− NA .
(398)
Challenge 937 e Challenge 938 n
is is called Ampère’s ‘law’. Again, the strange factor µ π is due to the historical way in which the electrical units were de ned. e constant µ is called the permeability of the vacuum and is de ned by the fraction of newtons per ampere squared given in the formula. It is easy to see that the magnetic eld has an intensity given by vE c , where E is the electric eld measured by an observer moving with the charge. is is the rst hint that magnetism is a relativistic e ect.
We note that equation ( ) is valid only for small velocities and accelerations. Can you nd the general one?
In , Michael Faraday discovered an additional piece of the puzzle, one that even the great Ampère had overlooked. He found that a moving magnet could cause a current
ow in an electrical circuit. Magnetism can thus be turned into electricity. is important discovery allowed the production of electrical current ow by generators, so-called dynamos, using water power, wind power or steam power. In fact, the rst dynamo was built in by Ampère and his technician. Dynamos started the use of electricity throughout the world. Behind every electrical plug there is a dynamo somewhere.
Additional experiments show that magnetic elds also lead to electric elds when one changes to a moving viewpoint. You might check this on any of the examples of Figures to . Magnetism indeed is relativistic electricity. Electric and magnetic elds are partly transformed into each other when switching from one inertial reference frame to the other. Magnetic and electrical elds thus behave like space and time, which are also mixed up when changing from one inertial frame to the other. e theory of special relativity thus tells us that there must be a single concept, an electromagnetic eld, describing them both. Investigating the details, one nds that the electromagnetic eld F surrounding charged bodies has to be described by an antisymmetric -tensor
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−Ex c −Ey c −Ez c
Ex c Ey c Ez c
Fµν =
Ex c Ey c
Bz
−Bz
By −Bx
or Fµν =
−Ex c −Ey c
Bz
−Bz By −Bx
.
Ez c −By
Bx
−Ez c −By Bx
(399)
Obviously, the electromagnetic eld F, and thus every component of these matrices, de-
pends on space and time. e matrices show that electricity and magnetism are two faces
of the same e ect.* In addition, since electric elds appear only in the topmost row and
* Actually, the expression for the eld contains everywhere the expression light c. We will explain the reason for this substitution shortly.
µoε instead of the speed of
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Challenge 939 n
le most column, the expressions show that in everyday life, for small speeds, electricity and magnetism can be separated. (Why?)
e total in uence of electric and magnetic elds on xed or moving charges is then given by the following expression for the relativistic force-acceleration relation K = mb:
mb = qFu or
m
du µ dτ
=
qFµ ν uν
or
γc
Ex c Ey c Ez c
γc
m
d dτ
γvx γvy
=q
Ex c Ey c −Bz
Bz
−By Bx
γvx γvy
or
γvz
Ez c By −Bx
γvz
W = qEv and dp dt = q(E + v B) ,
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Ref. 499, Ref. 500 Challenge 940 ny Challenge 941 n
which show how the work W and the three-force d p dt depend on the electric and magnetic elds. All four expressions describe the same content; the simplicity of the rst one is the reason for the involved matrices ( ) of the electromagnetic eld. In fact, the extended Lorentz relation ( ) is the de nition of the electromagnetic eld, since the eld is de ned as that ‘stu ’ which accelerates charges. In particular, all devices that put charges into motion, such as batteries and dynamos, as well as all devices that are put into motion by owing charges, such as electric motors and muscles, are described by this relation.
at is why this relation is usually studied, in simple form, already in school. e Lorentz relation describes all cases in which the motion of objects can be seen by the naked eye or felt by our senses, such as the movement of an electrical motor in a high speed train, in a li and in a dental drills, the motion of the picture generating electron beam in a television tube, or the travelling of an electrical signal in a cable and in the nerves of the body.
In equation ( ) it is understood that one sums over indices that appear twice. e electromagnetic eld tensor F is an antisymmetric -tensor. (Can you write down the relation between Fµν, Fµν and Fµ ν?) Like any such tensor, it has two invariants, i.e., two deduced properties that are the same for every observer: the expression B − E c =
tr F and the product EB = −c tr F F. (Can you con rm this, using the de nition of trace as the sum of the diagonal elements?)
e rst invariant expression turns out to be the Lagrangian of the electromagnetic eld. It is a scalar and implies that if E is larger, smaller, or equal to cB for one observer, it also is for all other observers. e second invariant, a pseudoscalar, describes whether the angle between the electric and the magnetic eld is acute or obtuse for all observers.*
* ere is in fact a third Lorentz invariant, much less known. It is speci c to the electromagnetic eld and is a combination of the eld and its vector potential:
Ref. 501
κ = Aµ AµFρνFνρ − AρFρνFνµ Aµ
= (AE) + (AB) − A
E
−A
B
+
φ c
(AE
B)
−
(
φ c
)
(E
+B ).
(401)
is expression is Lorentz (but not gauge) invariant; knowing it can help clarify unclear issues, such as the lack of existence of waves in which the electric and magnetic elds are parallel. Indeed, for plane mono-
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e application of electromagnetic e ects to daily life has opened up a whole new world that did not exist before. Electrical light, electric motors, radio, telephone, X-rays, television and computers have changed human life completely in less than one century. For example, the installation of electric lighting in city streets has almost eliminated the previously so common night assaults. ese and all other electrical devices exploit the fact that charges can ow in metals and, in particular, that electromagnetic energy can be transformed
— into mechanical energy – as used in loudspeakers, motors, piezo crystals; — into light – as in lamps and lasers; — into heat – as in ovens and tea pots; — into chemical e ects – as in hydrolysis, battery charging and electroplating; — into coldness – as in refrigerators and Peltier elements; — into radiation signals – as in radio and television; — into stored information – as in magnetic records and in computers.
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Ref. 502 Challenge 943 e
e only mathematical operation I performed in my life was to turn the handle of a calculator.
“ Michael Faraday ” All electric motors are based on the result that electric currents interact with magnetic
elds. e simplest example is the attraction of two wires carrying parallel currents. is
observation alone, made in by Ampère, is su cient to make motion larger than a
certain maximal speed impossible.
e argument is beautifully simple. We
change the original experiment and imagine
two long, electrically charged rods of mass m,
v
moving in the same direction with velocity v d
charged rods
and separation d. An observer moving with
the rods would see an electrostatic repulsion
v
between the rods given by
F I G U R E 232 The relativistic aspect of magnetism
mae = − πε
λ d
(402)
Challenge 944 e
where λ is the charge per length of the rods. A second, resting observer sees two e ects:
the electrostatic repulsion and the attraction discovered by Ampère. e second observer
therefore observes
maem = − πε
λ d
+
µ π
λv d
.
(403)
is expression must be consistent with the observation of the rst observer. is is the
Challenge 942 n
chromatic waves all three invariants vanish in the Lorentz gauge. Also the quantities ∂µ J µ, Jµ Aµ and ∂µ Aµ are Lorentz invariants. (Why?) e latter, the frame independence of the divergence of the four-potential, re ects the invariance of gauge choice. e gauge in which the expression is set to zero is called the Lorentz gauge.
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case only if both observers nd repulsions. It is easy to check that the second observer sees a repulsion, as does the rst one, only if
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
v <ε µ .
(404)
Challenge 945 ny
is maximum speed, with a value of . GM s, is thus valid for any object carrying charges. But all everyday objects contain charges: there is thus a maximum speed for matter.
Are you able to extend the argument for a maximum speed to neutral particles as well? We will nd out more on this limit velocity, which we know already, in a minute.
Another argument for magnetism as a relativistic e ect is the following. In a wire with electrical current, the charge is zero for an observer at rest with respect to the wire. e reason is that the charges enter and exit the wire at the same time for that observer. Now imagine an observer who ies along the wire. e entrance and exit events do not occur simultaneously any more; the wire is charged for a moving observer. ( e charge depends on the direction of the observer’s motion.) In other words, if the observer himself were charged, he would experience a force. Moving charges experience forces from currentcarrying wires. is is exactly why magnetic elds were introduced: they only produce forces on moving charges. In short, current carrying wires are surrounded by magnetic
elds. In summary, electric e ects are due to ow of electric charges and to electric elds;
magnetism is due to moving electric charges. It is not due to magnetic charges.* e strength of magnetism, used in any running electric motor, proves relativity right: there is a maximum speed in nature. Both electric and magnetic elds carry energy and momentum. ey are two faces of the same coin. However, our description of electromagnetism is not complete yet: we need the nal description of the way charges produce an electromagnetic eld.
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Et facta mirari et intellectua assequi.
“ ” Augustine of Hippo
Before we study the motion of an electromagnetic eld in detail, let’s have some fun with electricity.
**
Nowadays, having fun with sparks is straightforward. Tesla coils, named a er Nikola Tesla ** are the simplest devices that allow to produce long sparks at home. Attention: this is dangerous; that is the reason that such devices cannot be bought anywhere. e basic
Page 590 Challenge 946 ny
* ‘Electrons move in metal with a speed of about µm s; thus if I walk with the same speed along a cable carrying a constant current, I should not be able to sense any magnetic eld.’ What is wrong with this argument? ** Никола Тесла (1856 Smiljan–1943 New York City), Serbian engineer and inventor. He invented and promoted the polyphase alternating current system, the alternating current electric motor, wireless communic-
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TA B L E 47 Voltage values observed in nature
O
Smallest measured voltage Human nerves Voltaic cell (‘battery’) Mains in households Electric eel Sparks when rubbing a polymer pullover Electric fence Colour television tube X-ray tube Electron microscopes Stun gun Lightning stroke Record accelerator voltage Planck voltage, highest value possible in nature
V
c. f V mV
.V V or V
100 to V kV 0.7 to kV
kV 30 to kV
. kV to MV 65 to kV 10 to MV TV .ë V
capacitive head (c.10-20 pF to earth)
230 V 50 Hz
10-100nF
c.10 kV 50 Hz
spark gap for switching
c.1000 turns
large sparks
c.10 turns
resonance frequencies 100 - 500 kHz
ground
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F I G U R E 233 The schematics, the realization and the operation of a Tesla coil, including spark and corona discharges (© Robert Billon)
ation, uorescent lighting and many other applications of electricity. He is also one of the inventors of radio. e SI unit of the magnetic eld is named a er him. A amboyant character, his ideas were sometimes
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diagram and an example is shown in Figure 233. Tesla coils look like large metal mushrooms (to avoid unwanted discharges) and plans for their construction can be found on numerous websites or from numerous enthusiast’s clubs, such as http://www.stefan-kluge. de.
**
If even knocking on a wooden door is an electric e ect, we should be able to detect elds Challenge 947 ny when doing so. Can you devise an experiment to check this?
Challenge 948 n
**
Birds come to no harm when they sit on unprotected electricity lines. Nevertheless, one almost never observes any birds on tall, high voltage lines of kV or more, which transport power across longer distances. Why?
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**
How can you distinguish a magnet from an unmagnetized metal bar of the same size and Challenge 949 n material, using no external means?
Challenge 950 n
**
How do you wire up a light bulb to the mains and three switches so that the light can be switched on at any of the switches and o at any other switch? And for four switches? Nobody will take a physicist seriously who is able to write Maxwell’s equations but cannot solve this little problem.
**
e rst appliances built to generate electric currents were large rubbing machines. en, in 1799 the Italian scientist Alessandro Volta (1745–1827) invented a new device to generate electricity and called it a pile; today it is called a (voltaic) cell or, less correctly, a battery. Voltaic cells are based on chemical processes; they provide much more current and are smaller and easier to handle than electrostatic machines. e invention of the battery changed the investigation of electricity so profoundly that Volta became world famous. At last, a simple and reliable source of electricity was available for use in experiments; unlike rubbing machines, piles are compact, work in all weather conditions and make no noise.
An apple or a potato with a piece of copper and one of zinc inserted is one of the simplest possible voltaic cells. It provides a voltage of about V and can be used to run digital clocks or to produce clicks in headphones. Volta was also the discoverer of the charge law q = CU of capacitors (C being the capacity, and U the voltage) and the inventor of the high sensitivity capacitor electroscope. A modest man, nevertheless, the unit of electrical potential, or ‘tension’, as Volta used to call it, was deduced from his name. A ‘battery’ is a large number of voltaic cells; the term was taken from an earlier, almost purely military use.* A battery in a mobile phone is just an elaborated replacement for a
unrealistic; for example he imagined that Tesla coils could be used for wireless power transmission. * A pile made of sets of a zinc plate, a sheet of blotting paper soaked with salt water and a copper coin is Challenge 951 ny easily constructed at home.
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•.
number of apples or potatoes.
Challenge 952 n
**
A PC or a telephone can communicate without wires, by using radio waves. Why are these and other electrical appliances not able to obtain their power via radio waves, thus eliminating power cables?
**
Objects that are not right–le symmetric are called chiral, from the Greek word for ‘hand’. Challenge 953 n Can you make a mirror that does not exchange le and right? In two di erent ways?
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**
A Scotch tape roll is a dangerous device. Pulling the roll quickly leads to light emission (through triboluminescence) and to small sparks. It is suspected that several explosions in mines were triggered when such a spark ignited a combustible gas mixture.
Challenge 954 n
**
Take an envelope, wet it and seal it. A er letting it dry for a day or more, open it in the dark. At the place where the two sides of paper are being separated from each other, the envelope glows with a blue colour. Why? Is it possible to speed up the test using a hair dryer?
**
Electromagnetism is full of surprises and o ers many e ects that can be reproduced at home. e internet is full of descriptions of how to construct Tesla coils to produce sparks, coil guns or rail guns to shoot objects, electrostatic machines to make your hair stand on end, glass spheres with touch-sensitive discharges and much more. If you like experiments, just search for these terms.
**
Challenge 955 e
A high voltage can lead to current ow through air, because air becomes conductive in high electric elds. In such discharges, air molecules are put in motion. As a result, one can make objects that are attached to a pulsed high tension source li up in the air, if one optimizes this air motion so that it points downwards everywhere. e high tension is thus e ectively used to accelerate ionized air in one direction and, as a result, an object will move in the opposite direction, using the same principle as a rocket. An example is shown in Figure 234, using the power supply of a PC monitor. (Watch out: danger!) Numerous websites explain how to build these so-called li ers at home; in Figure 234, the bottle and the candle are used as high voltage insulator to keep one of the two thin high voltage wires (not visible in the photograph) high enough in the air, in order to avoid discharges to the environment or to interfere with the li er’s motion. Unfortunately, the majority of websites – not all – give incorrect or confused explanations of the phenomenon.
ese websites thus provide a good challenge for one to learn to distinguish fact from speculation.
**
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,
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 234 Lifting a light object – covered with aluminium foil – using high a tension discharge (© Jean-Louis Naudin at http://www.jlnlabs.org)
e electric e ects produced by friction and by liquid ow are usually small. However, in the 1990s, a number oil tankers disappeared suddenly. e sailors had washed out the oil tanks by hosing sea water onto the tank walls. e spraying led to charging of the tank; a discharge then led to the oil fumes in the tank igniting. is led to an explosion and subsequently the tankers sank. Similar accidents also happen regularly when chemicals are moved from one tank to another.
Challenge 956 n
**
Rubbing a plastic spoon with a piece of wool charges it. Such a charged spoon can be used to extract pepper from a salt–pepper mixture by holding the spoon over the mixture. Why?
Ref. 503
**
When charges move, they produce a magnetic eld. In particular, when ions inside the Earth move due to heat convection, they produce the Earth’s magnetic eld. When the ions high up in the stratosphere are moved by solar wind, a geomagnetic storm appears; its eld strength can be as high as that of the Earth itself. In 2003, an additional mechanism was discovered. When the tides move the water of the oceans, the ions in the salt water produce a tiny magnetic eld; it can be measured by highly sensitive magnetometers in satellites orbiting the Earth. A er two years of measurements from a small satellite it was possible to make a beautiful lm of the oceanic ows. Figure 235 gives an impression.
**
e names electrode, electrolyte, ion, anode and cathode were suggested by William Whewell (1794–1866) on demand of Michael Faraday; Faraday had no formal education and asked his friend Whewell to form two Greek words for him. For anode and cathode, Whewell took words that literally mean ‘upward street’ and ‘downward street’. Faraday
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•.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 235 The magnetic field due to the tides (© Stefan Maus)
then popularized these terms, like the other words mentioned above.
**
e shortest light pulse produced so far had a length of Challenge 957 n of green light would that correspond?
as. To how many wavelengths
**
Why do we o en see shadows of houses and shadows of trees, but never shadows of the Challenge 958 n electrical cables hanging over streets?
**
How would you measure the speed of the tip of a lightning bolt? What range of values do Challenge 959 n you expect?
Ref. 504
**
One of the simplest possible electric motors was discovered by Faraday in 1831. A magnet suspended in mercury will start to turn around its axis if a current ows through it. (See Figure 236.) In addition, when the magnet is forced to turn, the device (o en also called Barlow’s wheel) also works as a current generator; people have even tried to generate
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,
battery
suspending wire
N
S mercury F I G U R E 236 A unipolar motor
F I G U R E 237 The simplest motor (© Stefan Kluge)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 960 n domestic current with such a system! Can you explain how it works? e modern version of this motor makes use of a battery, a wire, a conductive
samarium–cobalt magnet and a screw. e result is shown in Figure 237.
Ref. 505
**
e magnetic eld of the Earth is much higher than that of other planets because of the Earth’s Moon. e eld has a dipole strength of . ë A m . It shields us from lethal solar winds and cosmic radiation particles. Today, a lack of magnetic eld would lead to high radiation on sunny days; but in the past, its lack would have prevented the evolution of the human species. We owe our existence to the magnetic eld.
Challenge 961 n
**
e ionosphere around the Earth has a resonant frequency of Hz; for this reason any apparatus measuring low frequencies always gets a strong signal at this value. Can you give an explanation of the frequency?
Challenge 962 ny
** e Sun is visible to the naked eye only up to a distance of 50 light years. Is this true?
**
Page 590 Challenge 963 e
At home, electricity is mostly used as alternating current. In other words, no electrons actually ow through cables; as the dri speed of electrons in copper wires is of the order of µm s, electrons just move back and forward by nm. Nothing ows in or out of the cables! Why do the electricity companies require a real ow of money in return, instead of being satis ed with a back and forth motion of money?
Challenge 964 n
**
Comparing electricity with water is a good way of understanding electronics. Figure 238 shows a few examples that even a teenager can use. Can you ll in the correspondence for the coil, and thus for a transformer?
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current, voltage wire
resistor
electrical component
capacitor
battery
diode
transistor
•.
hydraulic component
mass flow, pressure tube
restriction
flexible & elastic closure
pump
one-way valve
activated valve
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inductor
challenge
F I G U R E 238 The correspondence of electronics and water flow
** Do electrons and protons have the same charge? Experiments show that the values are Challenge 965 ny equal to within at least twenty digits. How would you check this?
** Challenge 966 ny Charge is also velocity-independent. How would you check this?
** Magnets can be used, even by school children, to climb steel walls. Have a look at the http://www.physicslessons.com/TPNN.htm website.
** Extremely high magnetic elds have strange e ects. At elds of T, vacuum becomes e ectively birefringent, photons can split and coalesce, and atoms get squeezed. Hydrogen atoms, for example, are estimated to get two hundred times narrower in one direction. Fortunately, these conditions exist only in speci c neutron stars, called magnetars.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
electric field E
wire with current I
current I
object with charge ρ
speed v N
S
magnetic field B
F I G U R E 239 The first of Maxwell’s equations
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**
Ref. 506 Challenge 967 n
A good way to make money is to produce electricity and sell it. In 1964, a completely new method was invented by Fletcher Osterle. e method was presented to a larger public in a beautiful experiment in 2003. One can take a plate of glass, add a conducting layers on each side, and then etch a few hundred thousand tiny channels through the plate, each around µm in diameter. When water is made to ow through the channels, a current is generated. e contacts at the two conducting plates can be used like battery contacts.
is simple device uses the e ect that glass, like most insulators, is covered with a charged layer when it is immersed in a liquid. Can you imagine why a current is generated? Unfortunately, the e ciency of electricity generation is only about 1%, making the method much less interesting than a simple blade wheel powering a dynamo.
T
Page 1201
In the years between and , taking in the details of all the experiments known to him, James Clerk Maxwell produced a description of electromagnetism that forms one of the pillars of physics.* Maxwell took all the experimental results and extracted their common basic principles, as shown in Figures and . Twenty years later, Heaviside and Hertz extracted the main points of Maxwell ideas, calling their summary Maxwell’s theory of the electromagnetic eld. It consists of two equations (four in the non-relativistic case).
e rst equation is the precise statement that electromagnetic elds originate at
* James Clerk Maxwell (b. 1831 Edinburgh, d. 1879 Cambridge), Scottish physicist. He founded electromagnetism by theoretically unifying electricity and magnetism, as described in this chapter. His work on thermodynamics forms the second pillar of his activity. In addition, he studied the theory of colours and developed the now standard horseshoe colour diagram; he was one of the rst people to make a colour photograph. He is regarded by many as the greatest physicist ever. Both ‘Clerk’ and ‘Maxwell’ were his family names.
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•.
no magnetic charges
I1(t)
I2(t)
F I G U R E 240 The second of Maxwell’s equations
charges, and nowhere else. e corresponding equation is variously written*
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dF = j
µ ε
or
dνFµν = jµ
µ ε
or
Ex c Ey c Ez c
(∂t c, −∂x , −∂y, −∂z)
−Ex c −Ey c
Bz
−Bz By −Bx
=
−Ez c −By Bx
∇E
=
ρ ε
and
∇
B− c
∂E ∂t
=
µ
j
.
(405)
µ ε
(ρ, jx
c, jy c, jz c) or
Challenge 968 ny Challenge 969 e
Each of these four equivalent ways to write the equation makes a simple statement: electrical charge carries the electromagnetic eld. is statement, including its equations, are equivalent to the three basic observations of Figure . It describes Coulomb’s relation, Ampère’s relation, and the way changing electrical elds induce magnetic e ects, as you may want to check for yourself.
e second half of equation ( ) contains the right hand rule for magnetic elds around wires, through the vector product. e equation also states that changing electric
elds induce magnetic elds. e e ect is essential for the primary side of transformers. e factor c implies that the e ect is small; that is why coils with many windings or strong electric currents are needed to nd it. Due to the vector product, all induced magnetic eld lines are closed lines.
e second result by Maxwell is the precise description of how changing electric elds create magnetic elds, and vice versa. In particular, an electric eld can have vortices only when there is a changing magnetic eld. In addition it expresses the observation that in nature there are no magnetic charges, i.e. that magnetic elds have no sources. All these
* Maxwell generalized this equation to cases where the charges are not surrounded by vacuum, but located inside matter. We will not explore these situations in our walk because, as we will see during our mountain ascent, the apparently special case of vacuum in fact describes all of nature.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
results are described by the relation variously written
d F = with Fρσ = ερσ µνFµν or
εµνρ ∂µFνρ = ∂µFνρ + ∂νFρµ + ∂ρFµν = or
γ c ∂t γ∂x γ∂y γ∂z
Bx
By
Bz
−Bx −By
Ez c
−Ez c Ey c −Ex c
=
−Bz −Ey c Ex c
∇B =
and
∇
E
=
−
∂B ∂t
.
or
(406)
Challenge 970 ny
e relation expresses the lack of sources for the dual eld tensor, usually written F: there are no magnetic charges, i.e. no magnetic monopoles in nature. In practice, this equation is always needed together with the previous one. Can you see why?
Since there are no magnetic charges, magnetic eld lines are always closed; they never start or end. For example, eld lines continue inside magnets. is is o en expressed mathematically by stating that the magnetic ux through a closed surface S – such as a
sphere or a cube – always vanishes: ∫S BdA = .
e second half of equation ( ), also shown in Figure , expresses that changes in magnetic elds produce electric elds: this e ect is used in the secondary side of transformers and in dynamos. e cross product in the expression implies that an electric eld generated in this way – also called an electromotive eld – has no start and endpoints. e electromotive eld lines thus run in circles: in most practical cases they run along electric circuits.
Together with Lorentz’ evolution equation ( ), which describes how charges move given the motion of the elds, Maxwell’s evolution equations ( ) and ( ) describe all electromagnetic phenomena occurring on everyday scales, from mobile phones, car batteries, to personal computers, lasers, lightning, holograms and rainbows. We now have a system as organized as the expression a = GM r or as Einstein’s eld equations for gravitation.
We will not study many applications of the eld equations but will continue directly towards our aim to understand the connection to everyday motion and to the motion of light. In fact, the electromagnetic eld has an important property that we mentioned right at the start: the eld itself itself can move.
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Challenge 971 ny
A simple experiment clari es the properties of electromagnetic elds de ned above. When two charged particles collide, their total momentum is not conserved.
Imagine two particles of identical mass and identical charge just a er a collision, when they are moving away from one another. Imagine also that the two masses are large, so that the acceleration due to their electrical repulsion is small. For an observer at the centre of gravity of the two, each particle feels an acceleration from the electric eld of the other.
e electric eld E is given by the so-called Heaviside formula
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m, q
m, q
v
v
0 distance r F I G U R E 241 Charged particles after a collision
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
E=
q(
−v c πe r
)
.
(407)
Ref. 507 Challenge 972 n
Ref. 508
In other words, the total system has a vanishing total momentum. Take a second observer, moving with respect to the rst with velocity v, so that the rst
charge will be at rest. Expression ( ) leads to two di erent values for the electric elds, one at the position of each particle. In other words, the system of the two particles is not in inertial motion, as we would expect; the total momentum is not conserved. Where did it go?
is at rst surprising e ect has been put in the form of a theorem by Van Dam and Wigner. ey showed that for a system of particles interacting at a distance the total particle energy–momentum cannot remain constant in all inertial frames.
e total momentum of the system is conserved only because the electromagnetic eld itself also carries momentum. If electromagnetic elds have momentum, they are able to strike objects and to be struck by them. As we will show below, light is also an electromagnetic eld. us we should be able to move objects by shining light on to them. We should even be able to suspend particles in mid air by shining light on to them from below. Both predictions are correct, and some experiments will be presented shortly.
We conclude that any sort of eld leading to particle interactions must carry energy and momentum, as the argument applies to all such cases. In particular, it applies to nuclear interactions. Indeed, in the second part of our mountain ascent we will even nd an additional result: all elds are themselves composed of particles. e energy and momentum of elds then become an obvious state of a airs.
T
–
e study of moving elds is called eld theory and electrodynamics is the prime example. ( e other classical example is uid dynamics; moving electromagnetic elds and moving uids are very similar mathematically.) Field theory is a beautiful topic; eld lines, equipotential lines and vortex lines are some of the concepts introduced in this domain.
ey fascinate many.* However, in this mountain ascent we keep the discussion focused
Challenge 973 n Ref. 48949
* What is the relation, for static elds, between eld lines and (equi-) potential surfaces? Can a eld line cross a potential surface twice? For more details on topics such as these, see the free textbook by B T , Electromagnetic Field eory, on his http://www.plasma.uu.se/CED/Book website. And of course, in English, have a look at the texts by Schwinger and by Jackson.
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current
current
magnets
N
vector potential
S
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F I G U R E 242 Vector potentials for selected situations
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 509
on motion.
We have seen that elds force us to extend our concept of motion. Motion is not only
the change in state of objects and of space-time, but also the change in state of elds. We
therefore need, also for elds, a complete and precise description of their state. e obser-
vations with amber and magnets have shown us that elds possess energy and momentum.
ey can impart it to particles. e experiments with motors have shown that objects can
add energy and momentum to elds. We therefore have to de ne a state function which
allows us to de ne energy and momentum for electric and magnetic elds. Since electric
and magnetic elds transport energy, their motion follows the speed limit in nature.
Maxwell de ned the state function in two standard steps. e rst step is the de nition
of the (magnetic) vector potential, which describes the momentum per charge that the
eld provides:
A
=
p q
.
(408)
When a charged particle moves through a magnetic potential A(x), its momentum
changes by q∆A; it changes by the di erence between the potential values at the start
and end points, multiplied by its charge. Owing to this de nition, the vector potential
has the property that
B = ∇ A = curl A
(409)
Challenge 974 ny Ref. 510
Challenge 975 d
i.e. that the magnetic eld is the curl of the magnetic potential. e curl is called the
rotation, abbreviated rot in most languages. e curl (or rotation) of a eld describes, for
each point of space, the direction of the local, imagined axis of rotation, as well as (twice)
the rotation speed around that axis. For example, the curl for the velocities of a rotating
solid body is everywhere ω, or twice the angular velocity.
e vector potential for a long straight current-carrying wire is parallel to the wire; it
has the magnitude
A(r)
=
−
µ
I π
ln
r r
,
(410)
which depends on the radial distance r from the wire and an integration constant r . is expression for the vector potential, pictured in Figure , shows how the moving current
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•.
produces a linear momentum in the (electro-) magnetic eld around it. In the case of a solenoid, the vector potential ‘circulates’ around the solenoid. e magnitude obeys
A(r)
=
−
Φ π
r
,
(411)
Challenge 976 ny
where Φ is the magnetic ux inside the solenoid. We see that, in general, the vector potential is dragged along by moving charges. e dragging e ect decreases for larger distances.
is ts well with the image of the vector potential as the momentum of the electromagnetic eld.
is behaviour of the vector potential around charges is reminiscent of the way honey is dragged along by a spoon moving in it. In both cases, the dragging e ect decreases with distance. However, the vector potential, unlike the honey, does not produce any friction that slows down charge motion. e vector potential thus behaves like a frictionless liquid.
Inside the solenoid, the magnetic eld is constant and uniform. For such a eld B we nd the vector potential
A(r) = − B r .
(412)
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In this case, the magnetic potential thus increases with increasing distance from the origin.* In the centre of the solenoid, the potential vanishes. e analogy of the dragged honey gives exactly the same behaviour.
However, there is a catch. e magnetic potential is not de ned uniquely. If A(x) is a vector potential, then the di erent vector potential
A′(x) = A(x) + grad Λ ,
(413)
Challenge 977 ny Ref. 509
where Λ(t, x) is some scalar function, is also a vector potential for the same situation. ( e magnetic eld B stays the same, though.) Worse, can you con rm that the corresponding (absolute) momentum values also change? is unavoidable ambiguity, called gauge invariance, is a central property of the electromagnetic eld. We will explore it in more detail below.
Not only the momentum, but also the energy of the electromagnetic eld is de ned ambiguously. Indeed, the second step in the speci cation of a state for the electromagnetic
eld is the de nition of the electric potential as the energy U per charge:
φ
=
U q
(414)
In other words, the potential φ(x) at a point x is the energy needed to move a unit charge to the point x starting from a point where the potential vanishes. e potential energy is thus given by qφ. From this de nition, the electric eld E is simply the change of the
* is is only possible as long as the eld is constant; since all elds drop again at large distances – because the energy of a eld is always nite – also the vector potential drops at large distances.
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potential with position corrected by the time dependence of momentum, i.e.
E
=
−∇φ
−
∂ ∂t
A
,
(415)
Obviously, there is a freedom in the choice of the de nition of the potential. If φ(x) is a
possible potential, then
φ′(x)
=
φ(x)
−
∂ ∂t
Λ
(416)
Ref. 509 Challenge 978 ny
is also a potential function for the same situation. is freedom is the generalization of the freedom to de ne energy up to a constant. Nevertheless, the electric eld E remains the same for all potentials.
To be convinced that the potentials really are the energy and momentum of the electromagnetic eld, we note that for a moving charge we have
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
d dt
(
mv
+
qφ)
=
∂ ∂t
q(φ
−
vA)
d dt
(mv
+
qA)
=
−∇q(φ
−
vA)
,
(417)
which show that the changes of generalized energy and momentum of a particle (on the le -hand side) are due to the change of the energy and momentum of the electromagnetic
eld (on the right-hand side).* In relativistic -vector notation, the energy and the momentum of the eld appear
together in one quantity. e state function of the electromagnetic eld becomes
Aµ = (φ c, A) .
(418)
It is easy to see that the description of the eld is complete, since we have
F = d A or Fµν = ∂µ Aν − ∂ν Aµ ,
(419)
which means that the electromagnetic eld F is completely speci ed by the -potential A. But as just said, the -potential itself is not uniquely de ned. Indeed, any other gauge
eld A′ is related to A by the gauge transformation
A′ µ = Aµ + ∂µ Λ
(420)
where Λ = Λ(t, x) is any arbitrarily chosen scalar eld. e new eld A′ leads to the same electromagnetic eld, and to the same accelerations and evolutions. e gauge -
eld A is thus an overdescription of the physical situation as several di erent gauge elds correspond to the same physical situation. erefore we have to check that all measurement results are independent of gauge transformations, i.e. that all observables are gauge
* is connection also shows why the expression Pµ − qAµ appears so regularly in formulae; indeed, it plays a central role in the quantum theory of a particle in the electromagnetic eld.
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•.
Challenge 979 e
invariant quantities. Such gauge invariant quantities are, as we just saw, the elds F and
F, and in general all classical quantities. We add that many theoretical physicists use the
term ‘electromagnetic eld’ loosely for both the quantities F µν and Aµ. ere is a simple image, due to Maxwell, to help overcoming the conceptual di culties
of the vector potential. It turns out that the closed line integral over Aµ is gauge invariant,
because
∫ ∫ ∫ Aµdx µ = (Aµ + ∂µ Λ)dx µ = A′µdx µ .
(421)
In other words, if we picture the vector potential as a quantity allowing one to associate a number to a tiny ring at each point in space, we get a good, gauge invariant picture of the vector potential.*
Now that we have de ned a state function that describes the energy and momentum of the electromagnetic eld, let us look at what happens in more detail when electromagnetic elds move.
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E
e description so far allows us to write the total energy Energy of the electromagnetic
eld as
∫ Energy = π
εE
+
B µ
dV .
(422)
Energy is thus quadratic in the elds. For the total linear momentum one obtains
∫ P = ε E B dV . π
e expression is also called the Poynting vector.** For the total angular momentum one has
(423)
∫ L =
ε π
E A dV ,
where A is the magnetic vector potential.
(424)
TL
Challenge 980 ny
e motion of charged particles and the motion of the electromagnetic eld can also
be described using a Lagrangian instead of using the three equations given above. It is
not hard to see that the action SCED for a particle in classical electrodynamics can be symbolically de ned by***
Ref. 511 * In the second part of the text, on quantum mechanics, we will see that the exponent of this expression, namely exp(iq ∫ Aµdx µ) ħ, usually called the phase factor, can indeed be directly observed in experiments. ** John Henry Poynting (1852–1914) introduced the concept in 1884.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
∫ ∫ ∫ SCED = −mc dτ − µ F F − j A ,
(425)
which in index notation becomes
∫ SCED = −mc −
∫ η
µ
ν
dx
µ n
(s)
ds
dxnν (s) ds
ds −
M
µ FµνFµν + jµ Aµ d x .
(426)
What is new is the measure of the change produced by the electromagnetic eld. Its internal change is given by the term F F, and the change due to interaction with matter is given by the term jA.
e least action principle, as usual, states that the change in a system is always as small
as possible. e action SCED leads to the evolution equations by requiring that the action be stationary under variations δ and δ′ of the positions and of the elds which vanish at in nity. In other terms, the principle of least action requires that
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δS =
when
xµ = xµ + δµ
and
Aµ
=
Aµ
+
δ
′ µ
,
provided δxµ(θ)
for θ
and δAµ(xν)
for xν
.
(427)
Page 180 In the same way as in the case of mechanics, using the variational method for the two Challenge 981 ny variables A and x, we recover the evolution equations for particle and elds
bµ
=
q m
Fνµ
uν
,
∂µFµν = jν
µ ε
, and εµνρσ ∂νFρσ = ,
(428)
Challenge 982 ny Challenge 983 ny
which we know already. Obviously, they are equivalent to the variational principle based on SCED. Both descriptions have to be completed by specifying initial conditions for the particles and the elds, as well as boundary conditions for the latter. We need the rst and zeroth derivatives of the position of the particles, and the zeroth derivative for the electromagnetic eld.
Are you able to specify the Lagrangian of the pure electrodynamic eld using the elds E and B instead of F and F?
e form of the Lagrangian implies that electromagnetism is time reversible. is means that every example of motion due to electric or magnetic causes can also take place backwards. is is easily deduced from the properties of the Lagrangian. On the other hand, everyday life shows many electric and magnetic e ects which are not time invariant, such as the breaking of bodies or the burning of electric light bulbs. Can you explain how this ts together?
In summary, with the Lagrangian ( ) all of classical electrodynamics can be described and understood. For the rest of this chapter, we look at some speci c topics from this vast eld.
*** e product described by the symbol , ‘wedge’ or ‘hat’, has a precise mathematical meaning, de ned for this case in equation (426). Its background, the concept of (mathematical) form, carries us too far from Ref. 512 our walk.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•.
S
–
–
Page 424
We know from classical mechanics that we get the de nition of energy and momentum tensor by using Noether’s theorem, if we determine the conserved quantity from the Lorentz symmetry of the Lagrangian. For example, we found that relativistic particles have an energy–momentum vector. At the point at which the particle is located, it describes the energy and momentum.
Since the electromagnetic eld is not a localized entity, like a point particle, but an extended entity, we need to know the ow of energy and momentum at every point in space, separately for each direction. is makes a description with a tensor necessary. e result is the energy–momentum tensor of the electromagnetic eld
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Tµν = =
energy density
energy ow or momentum density
energy ow or momentum density
momentum ow density
u S c = cp cp T
=
ε cë EB
ε cE B −ε Ei Ej − BiBj µ
δi j(ε E + B µ )
(429)
e energy–momentum tensor shows again that electrodynamics is both Lorentz and gauge invariant.
Both the Lagrangian and the energy–momentum tensor show that electrodynamics is symmetric under motion inversion. If all charges change direction of motion – a situation o en incorrectly called ‘time inversion’ – they move backwards along the exact paths they took when moving forward.
We also note that charges and mass destroy a symmetry of the vacuum that we mentioned in special relativity: only the vacuum is invariant under conformal symmetries. In particular, only the vacuum is invariant under the spatial inversion r r.
To sum up, electrodynamic motion, like all other examples of motion that we have encountered so far, is deterministic, reversible and conserved. is is no big surprise. Nevertheless, two symmetries of electromagnetism deserve special mention.
W
?
Challenge 984 n
We will study the strange properties of mirrors several times during our walk. We start
with the simplest one rst. Everybody can observe, by painting each of their hands in a
di erent colour, that a mirror does not exchange right and le , as little as it exchanges up
and down; however, a mirror does exchange right and le handedness. In fact, it does so
by exchanging front and back.
Electrodynamics give a second answer: a mirror is a device that switches magnetic
north and south poles. Can you con rm this with a diagram?
But is it always possible to distinguish le from right? is seems easy: this text is
quite di erent from a
version, as are many other objects in our surroundings.
mirrored
But take a simple landscape. Are you able to say which of the two pictures of Figure
is the original?
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F I G U R E 243 Which one is the original landscape? (NOAA)
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Astonishingly, it is actually impossible to distinguish an original picture of nature from its mirror image if it does not contain any human traces. In other words, everyday nature is somehow le –right symmetric. is observation is so common that all candidate excepPage 939 tions, from the jaw movement of ruminating cows to the helical growth of plants, such as hops, or the spiral direction of snail shells, have been extensively studied.* Can you name Challenge 985 n a few more?
e le –right symmetry of nature appears because everyday nature is described by gravitation and, as we will see, by electromagnetism. Both interactions share an important property: substituting all coordinates in their equations by the negative of their values leaves the equations unchanged. is means that for any solution of these equations, i.e. for any naturally occurring system, a mirror image is a possibility that can also occur naturally. Everyday nature thus cannot distinguish between right and le . Indeed, there are right and le handers, people with their heart on the le and others with their heart on the right side, etc.
To explore further this strange aspect of nature, try the following experiment: imagine you are exchanging radio messages with a Martian; are you able to explain to him what right and le are, so that when you meet, you are sure you are talking about the same Challenge 986 n thing?
Actually, the mirror symmetry of everyday nature – also called its parity invariance Ref. 514 – is so pervasive that most animals cannot distinguish le from right in a deeper sense.
Most animals react to mirror stimuli with mirror responses. It is hard to teach them different ways to react, and it is possible almost only for mammals. e many experiments performed in this area gave the result that animals have symmetrical nervous systems,
Ref. 513
* e most famous is the position of the heart. e mechanisms leading to this disposition are still being investigated. Recent research suggests that the oriented motion of the cilia on embryos, probably in the region called the node, determines the right–le asymmetry. e deep origin of this asymmetry is not yet elucidated, however.
Most human bodies have more muscles on the right side for right-handers, such as Albert Einstein and Pablo Picasso, and correspondingly on the le side for le -handers, such as Charlie Chaplin and Peter Ustinov. is asymmetry re ects an asymmetry of the human brain, called lateralization, which is essential to human nature.
Another asymmetry of the human body is the hair whirl on the back of the head; the majority of humans have only one, and in 80 % of the cases it is le turning. But many people have more than one.
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•.
Challenge 987 n Challenge 988 n
and possibly only humans show lateralization, i.e. a preferred hand and di erent uses for the le and the right parts of the brain.
To sum up this digression, classical electrodynamics is le –right symmetric, or parity invariant. Can you show this using its Lagrangian?
A concave mirror shows an inverted image; so does a plane mirror if it is partly folded along the horizontal. What happens if this mirror is rotated around the line of sight?
Why do metals provide good mirrors? Metals are strong absorbers of light. Any strong absorber has a metallic shine. is is true for metals, if they are thick enough, but also for dye or ink crystals. Any material that strongly absorbs a light wavelength also re ects it e ciently. e cause of the strong absorption of a metal is the electrons inside it; they can move almost freely and thus absorb most visible light frequencies.
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Page 942
Obviously, the standard answer is that electric elds have sources, and magnetic elds do not; moreover, magnetic elds are small relativistic e ects of importance only when charge velocities are high or when electrical elds cancel out.
For situations involving matter, this clear distinction is correct. Up to the present day, no particle with a magnetic charge, called a magnetic monopole, has ever been found, even though its existence is possible in several uni ed models of nature. If found, the action ( ) would have to be modi ed by the addition of a fourth term, namely the magnetic current density. However, no such particle has yet been detected, despite intensive search e orts.
In empty space, when matter is not around, it is possible to take a completely di erent view. In empty space the electric and the magnetic elds can be seen as two faces of the same quantity, since a transformation such as
E cB B −E c
(430)
Challenge 989 n
called (electromagnetic) duality transformation, transforms each vacuum Maxwell equation into the other. e minus sign is necessary for this. (In fact, there are even more such transformations; can you spot them?) Alternatively, the duality transformation transforms F into F. In other words, in empty space we cannot distinguish electric from magnetic elds.
Matter would be symmetric under duality only if magnetic charges, also called magnetic monopoles, could exist. In that case the transformation ( ) could be extended to
cρe ρm , ρm −cρe .
(431)
It was one of the great discoveries of theoretical physics that even though classical electrodynamics with matter is not symmetric under duality, nature is. In , Claus Montonen and David Olive showed that quantum theory allows duality transformations even with the inclusion of matter. It has been known since the s that quantum theory allows magnetic monopoles. We will discover the important rami cations of this result in the
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Challenge 990 ny
third part of the text. is duality turns out to be one of the essential stepping stones
that leads to a uni ed description of motion. (A somewhat di cult question: extending
this duality to quantum theory, can you deduce what transformation is found for the ne
structure constant, and why it is so interesting?)
Duality, by the way, is a symmetry that works only in Minkowski space-time, i.e. in space-times of + dimensions. Mathematically, duality is closely related to the existence of quaternions, to the possibility of interpreting Lorentz boosts as rotations in + dimensions, and last, but not least, to the possibility of de ning other smooth mathemat-
ical structures than the standard one on the space R . ese mathematical connections are mysterious for the time being; they somehow point to the special role that four spacetime dimensions play in nature. More details will become apparent in the third part of
our mountain ascent.
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E
C
?
Ref. 515 Ref. 516
Ref. 516
Any interaction such as Coulomb’s rule ( ), which acts, for one given observer, between two particles independently of -velocity, must depend on -velocity for other inertial observers.* It turns out that such an interaction cannot be independent of the -velocity either. Such an interaction, even though it would indeed be -velocity dependent, would change the rest mass, since the -acceleration would not be -orthogonal to the -velocity.
e next simplest case is the one in which the acceleration is proportional to the velocity. Together with the request that the interaction leaves the rest mass constant, we then recover electrodynamics.
In fact, the requirements of gauge symmetry and of relativity symmetry also make it impossible to modify electrodynamics. In short, it does not seem possible to have a behaviour di erent from r for a classical interaction.
An inverse square dependence implies a vanishing mass of light and light particles, the photons. Is the mass really zero? e issue has been extensively studied. A massive photon would lead to a wavelength dependence of the speed of light in vacuum, to deviations from the inverse square ‘law’, to deviations from Ampère’s ‘law’, to the existence of longitudinal electromagnetic waves and more. No evidence for these e ects has ever been found. A summary of these studies shows that the photon mass is below − kg, or maybe − kg. Some arguments are not universally accepted, thus the limit varies somewhat from researcher to researcher.
A small non-vanishing mass for the photon would change electrodynamics somewhat. e inclusion of a tiny mass poses no special problems, and the corresponding Lagrangian, the so-called Proca Lagrangian, has already been studied, just in case.
Strictly speaking, the photon mass cannot be said to vanish. In particular, a photon with a Compton wavelength of the radius of the visible universe cannot be distinguished from one with zero mass through any experiment. is gives a mass of − kg for the photon. One notes that the experimental limits are still much larger. Photons with such a small mass value would not invalidate electrodynamics as we know it.
* is can be deduced from special relativity from the reasoning of page 536 or from the formula in the footnote of page 324.
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•.
Interestingly, a non-zero mass of the photon implies the lack of magnetic monopoles, as the symmetry between electric and magnetic elds is broken. It is therefore important on the one hand to try to improve the experimental mass limit, and on the other hand to explore whether the limit due to the universe’s size has any implications for this issue.
is question is still open.
T
Electrodynamics faces an experimental and theoretical issue that physicist o en avoid. e process of thought is electric in nature. Physics faces two challenges in this domain.
First, physicists must nd ways of modelling the thought process. Second, measurement technology must be extended to allow one to measure the currents in the brain.
Even though important research has been carried out in these domains, researchers are still far from a full understanding. Research using computer tomography has shown, for example, that the distinction between the conscious and the unconscious can be measured and that it has a biological basis. Psychological concepts such as repression can be observed in actual brain scans. Modellers of the brain mechanisms must thus learn to have the courage to take some of the concepts of psychology as descriptions for actual physical processes. is approach requires one to translate psychology into physical models, an approach that is still in its infancy.
Similarly, research into magnetoencephalography devices is making steady progress. e magnetic elds produced by brain currents are as low as fT, which require sensors at liquid helium temperature and a good shielding of background noise. Also the spatial resolution of these systems needs to be improved.
e whole programme would be considered complete as soon as, in a distant future, it was possible to use sensitive measuring apparatus to detect what is going on inside the brain and to deduce or ‘read’ the thoughts of a person from these measurements. In fact, this challenge might be the most complex of all challenges that science is facing. Clearly, the experiment will require involved and expensive machinery, so that there is no danger for a misuse of the technique. It could also be that the spatial resolution required is beyond the abilities of technology. However, the understanding and modelling of the brain will be a useful technology in other aspects of daily life as well.*
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.
?
Ref. 517
e nature of light has fascinated explorers of nature since at least the time of the ancient Greeks. In , Maxwell summarized all data collected in the years before him by deducing a basic consequence of the equations of electrodynamics. He found that in the case of empty space, the equations of the electrodynamic eld could be written as
A=
or, equivalently
ε
µ
∂ φ + ∂ Ax ∂t ∂x
+ ∂ Ay ∂y
+ ∂ Az ∂z
=
.
(432)
* is vision, formulated here in 2005, is so far from realization that it is unclear whether it will come true in the twenty- rst or in any subsequent century.
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Figure to be added in future
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 244 A plane, monochromatic and linearly polarized electromagnetic wave, with the fields as described by the field equations of electrodynamics
Challenge 991 e is is called a wave equation, because it admits solutions of the type
A(t, x) = A sin(ωt − kx + δ) = A sin( π f t − πx λ + δ) ,
(433)
which are commonly called plane waves. Such a wave satis es equation ( ) for any value of amplitude A , of phase δ, and of angular frequency ω, provided the wave vector k satis es the relation
ω(k) = ε µ k or ω(k) = ε µ k .
(434)
Page 562
e relation ω(k) between the angular frequency and the wave vector, the so-called dispersion relation, is the main property of any type of wave, be it a sound wave, a water wave, an electromagnetic wave, or any other kind. Relation ( ) speci cally characterizes electromagnetic waves in empty space, and distinguishes them from all other types of waves.*
Equation ( ) for the electromagnetic eld is linear in the eld; this means that the sum of two situations allowed by it is itself an allowed situation. Mathematically speaking, any superposition of two solutions is also a solution. For example, this means that two waves can cross each other without disturbing each other, and that waves can travel across static electromagnetic elds. Linearity also means that any electromagnetic wave can be described as a superposition of pure sine waves, each of which is described by expression ( ). e simplest possible electromagnetic wave, a harmonic plane wave with linear polarization, is illustrated in Figure .
A er Maxwell predicted the existence of electromagnetic waves, in the years between and Heinrich Hertz** discovered and studied them. He fabricated a very simple
Page 205
* For completeness, we remember that a wave in physics is any propagating imbalance. ** Heinrich Rudolf Hertz (b. 1857 Hamburg, d. 1894 Bonn), important Hamburger theoretical and experimental physicist. e unit of frequency is named a er him. Despite his early death, Hertz was a central
gure in the development of electromagnetism, in the explanation of Maxwell’s theory and in the unfolding of radio communication technology. More about him on page 152.
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F I G U R E 245 The first transmitter (left) and receiver (right) of electromagnetic (micro-) waves
Page 78
transmitter and receiver for GHz waves. Waves around this frequency are used in the
last generation of mobile phones. ese waves are now called radio waves, since physicists
tend to call all moving force elds radiation, recycling somewhat incorrectly a Greek term
that originally meant ‘light emission.’
Hertz also measured the speed of these waves. In fact, you can also
measure the speed at home, with a chocolate bar and a kitchen mi-
crowave oven. A microwave oven emits radio waves at . GHz – not
far from Hertz’s value. Inside the oven, these waves form standing
waves. Just put the chocolate bar (or a piece of cheese) in the oven and
switch the power o as soon as melting begins. You will notice that the
bar melts at regularly spaced spots. ese spots are half a wavelength
apart. From the measured wavelength value and the frequency, the
speed of light and radio waves simply follows as the product of the
two.
Heinrich Hertz
If you are not convinced, you can measure the speed directly, by
telephoning a friend on another continent, if you can make sure of using a satellite line
(choose a low cost provider). ere is about half a second additional delay between the
end of a sentence and the answer of the friend, compared with normal conversation. In
this half second, the signal goes up to the geostationary satellite, down again and returns
the same way. is half second gives a speed of c ë
km . s ë km s,
which is close to the precise value. Radio amateurs who re ect their signals from the
Moon can perform the same experiment and achieve higher precision.
But Maxwell did more. He strengthened earlier predictions that light itself is a solu-
tion of equation ( ) and therefore an electromagnetic wave, albeit with a much higher
frequency. Let us see how we can check this.
It is easy to con rm the wave properties of light; indeed they were known already
long before Maxwell. In fact, the rst to suggest that light is a wave was, around the year
, the important physicist Christiaan Huygens. You can con rm that light is a wave
with your own ngers. Simply place your hand one or two centimetres in front of your
eye, look towards the sky through the gap between the middle and the index nger and
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F I G U R E 246 The primary and secondary rainbow, and the supernumerary bows below the primary bow (© Antonio Martos and Wolfgang Hinz)
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Challenge 993 n Ref. 518
Page 574
Ref. 519 Challenge 994 ny
let the two ngers almost touch. You will see a number of dark lines crossing the gap. ese lines are the interference pattern formed by the light behind the slit created by the ngers. Interference is the name given to the amplitude patterns that appear when several
waves superpose.* e interference patterns depend on the spacing between the ngers. is experiment therefore allows you to estimate the wavelength of light, and thus, if you
know its speed, its frequency. Can you do this? Historically, another e ect was central in convincing everybody that light was a wave:
supernumerary rainbows, the additional bows below the main or primary rainbow. If we look carefully at a rainbow, below the main red–yellow–green–blue–violet bow, we observe weaker, additional green, blue and violet bows. Depending on the intensity of the rainbow, several of these supernumerary rainbows can be observed. ey are due to an interference e ect, as omas Young showed around .** Indeed, the repetition distance of the supernumerary bows depends on the radius of the average water droplets that form them. (Details about the normal rainbows are given below.) Supernumerary rainbows were central in convincing people that light is a wave. It seems that in those times scientists either did not trust their own ngers, or did not have any.
ere are many other ways in which the wave character of light can be made apparent. Maybe the most beautiful is an experiment carried out by a team of Dutch physicists in
. ey simply measured the light transmitted through a slit in a metal plate. It turns out that the transmitted intensity depends on the width of the slit. eir surprising result is shown in Figure . Can you explain the origin of the unexpected intensity steps in the curve?
Challenge 992 n
* Where does the energy go in an interference pattern? ** omas Young (1773 Milverton–1829), read the bible at two, spoke Latin at four; a doctor of medicine, he became a professor of physics. He introduced the concept of interference into optics, explaining Newtonian rings and supernumerary rainbows; he was the rst person to determine light’s wavelength, a concept that he also introduced, and its dependence on colour. He was the rst to deduce the three-colour vision explanation of the eye and, a er reading of the discovery of polarization, explained light as a transverse wave. In short, Young discovered most of what people learn at secondary school about light. He was a universal talent: he also worked on the deciphering of hieroglyphs, on ship building and on engineering problems. Young collaborated with Fraunhofer and Fresnel. In Britain his ideas on light were not accepted, since Newton and his followers crushed all opposing views. Towards the end of his life, his results were nally made known to the physics community by Fresnel and Helmholtz.
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Challenge 995 n
Page 564 Ref. 520 Ref. 521
Ref. 522 Challenge 996 n
Ref. 523
Numerous other experiments on the creation, detection and measurement of elec-
transmitted light power
tromagnetic waves were performed in the
nineteenth and twentieth centuries. For
example, in , William Herschel dis-
covered infrared light using a prism and a
thermometer. (Can you guess how?) In ,
Johann Wilhelm Ritter ( – ) a col-
ourful gure of natural Romanticism, dis-
(preliminary figure)
covered ultraviolet light using silver chlor-
ide, AgCl, and again a prism. e result
of all these experiments is that electromag-
netic waves can be primarily distinguished
slit width
by their wavelength or frequency. e main F I G U R E 247 The light power transmitted
categories are listed in Table . For visible through a slit as function of its width
light, the wavelength lies between . µm
(pure violet) and . µm (pure red).
At the end of the twentieth century the nal con rmation of the wave character of
light became possible. Using quite sophisticated experiments researchers, measured the
oscillation frequency of light directly. e value, between and THz, is so high
that its detection was impossible for a long time. But with these modern experiments the
dispersion relation ( ) of light has nally been con rmed in all its details.
We are le with one additional question about light. If light oscillates, in which direc-
tion does this occur? e answer is hidden in the parameter A in expression ( ). Elec-
tromagnetic waves oscillate in directions perpendicular to their motion. erefore, even
for identical frequency and phase, waves can still di er: they can have di erent polariza-
tion directions. For example, the polarization of radio transmitters determines whether
radio antennas of receivers have to be kept horizontal or vertical. Also for light, polar-
ization is easily achieved, e.g. by shining it through a stretched plastic lm. When the
polarization of light was discovered in by the French physicist Louis Malus ( –
), it de nitively established the wave nature of light. Malus discovered it when he
looked at the strange double images produced by feldspar, a transparent crystal found in
many minerals. Feldspar (KAlSi O ) splits light beams into two – it is birefringent – and
polarizes them di erently. at is the reason that feldspar is part of every crystal collec-
tion. Calcite (CaCO ) shows the same e ect. If you ever get hold of a piece of feldspar or
transparent calcite, do look through it at some written text.
By the way, the human eye is unable to detect polarization, in contrast to the eyes
of many insects, spiders and certain birds. Honey bees use polarization to deduce the
position of the Sun, even when it is hidden behind clouds. Some beetles of the genus
Scarabeus use the polarization of the Moon light for navigation, and many insects use
polarization to distinguish water surfaces from mirages. Can you nd out how? Despite
the human inability to detect polarization, both the cornea and the lens of the human eye
are birefringent.
Note that all possible polarizations of light form a continuous set. However, a general
plane wave can be seen as the superposition of two orthogonal, linearly polarized waves
with di erent amplitudes and di erent phases. Most books show pictures of plane, linear-
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Challenge 997 ny Challenge 998 n
ized electrodynamic waves. Essentially, electric elds look like water waves generalized to three dimensions, the same for magnetic elds, and the two are perpendicular to each other. Can you con rm this?
Interestingly, a generally polarized plane wave can also be seen as the superposition of right and le circularly polarized waves. However, no illustrations of circularly polarized waves are found in any textbook. Can you explain why?
So far it is clear that light is a wave. To con rm that light waves are indeed electromagnetic is more di cult. e rst argument was given by Bernhard Riemann in ;* he deduced that any electromagnetic wave must propagate with a speed c given by
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c= ε µ .
(435)
Challenge 999 ny Challenge 1000 e
Page 553
Already ten years before him, in , Kircho had noted that the measured values on both sides agreed within measurement errors. A few years later, Maxwell gave a beautiful con rmation by deducing the expression from equation ( ). You should be able to repeat the feat. Note that the right-hand side contains electric and magnetic quantities, and the le -hand side is an optical entity. Riemann’s expression thus uni es electromagnetism and optics.
Of course, people were not yet completely convinced. ey looked for more ways to show that light is electromagnetic in nature. Now, since the evolution equations of the electrodynamic eld are linear, additional electric or magnetic elds alone do not in uence the motion of light. On the other hand, we know that electromagnetic waves are emitted only by accelerated charges, and that all light is emitted from matter. It thus follows that matter is full of electromagnetic elds and accelerated electric charges. is in turn implies that the in uence of matter on light can be understood from its internal electromagnetic elds and, in particular, that subjecting matter to an external electromagnetic
eld should change the light it emits, the way matter interacts with light, or generally, the material properties as a whole.
Searching for e ects of electricity and magnetism on matter has been a main e ort of physicists for over a hundred years. For example, electric elds in uence the light transmission of oil, an e ect discovered by John Kerr in .** e discovery that certain gases change colour when subject to a eld yielded several Nobel Prizes for physics. With time, many more in uences on light-related properties by matter subjected to elds were found. An extensive list is given in the table on page . It turns out that apart from a few exceptions the e ects can all be described by the electromagnetic Lagrangian ( ), or equivalently, by Maxwell’s equations ( ). In summary, classical electrodynamics indeed uni es the description of electricity, magnetism and optics; all phenomena in these
elds, from the rainbow to radio and from lightning to electric motors, are found to be di erent aspects of the evolution of the electromagnetic eld.
* Bernhard Riemann (b. 1826 Breselenz, d. 1866 Selasca), important German mathematician. He studied curved space, providing several of the mathematical and conceptual foundations of general relativity, but then died at an early age. ** John Kerr (1824–1907), Scottish physicist, friend and collaborator of William omson.
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•.
TA B L E 48 The electromagnetic spectrum
F- W -N
M
A
U
ë − Hz < Hz
m
lower frequency limit
Mm quasistatic elds
see the section on cosmology
intergalactic,
power transmission,
galactic, stellar and accelerating and
planetary elds, de ecting cosmic
brain, electrical sh radiation
Hz– kHz
– kHz
– kHz .– MHz
– MHz
– MHz
Mm– km
radio waves
ELW
go round the
globe, penetrate
into water,
penetrate metal
electronic devices
nerve cells, electromechanical devices
km–
LW
. km
m– MW m
m– m SW
m– m VHF
follow Earth’s emitted by curvature, felt by thunderstorms nerves (‘bad weather nerves’)
re ected by night sky
circle world if re ected by the ionosphere, destroy hot air balloons
emitted by stars
allow battery operated transmitters
emitted by Jupiter
m– . m UHF
idem, line of sight propagation
power transmission, communication through metal walls, communication with submarines http:// www.vlf.it radio communications, telegraphy, inductive heating radio
radio transmissions, radio amateurs, spying
remote controls, closed networks, tv, radio amateurs, radio navigation, military, police, taxi radio, walkie-talkies, tv, mobile phones, internet via cable, satellite communication, bicycle speedometers
.– GHz
– GHz
microwaves
cm– cm SHF
idem, absorbed by water
mm– EHF mm
idem, absorbed by water
night sky, emitted radio astronomy, by hydrogen atoms used for cooking
( . GHz), telecommunications, radar
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F- W -N
M
A
U
.– THz – THz
– THz
– THz
– THz
– THz – THz
– THz
– THz
infrared go through clouds
– µm IRC or far infrared
µm– . µm
IRB or medium infrared
–
IRA or penetrates for
nm near several cm into
infrared human skin
emitted by every satellite
warm object
photography of Earth,
astronomy
sunlight, living beings
seeing through clothes, envelopes
and teeth
sunlight
used for optical bre communications for
telephone and cable TV
sunlight, radiation healing of wounds,
from hot bodies
rheumatism, sport physiotherapy, hidden illumination
–
light not absorbed by heat (‘hot light’), de nition of
nm
air, detected by lasers & chemical straightness,
the eye (up to reactions
enhancing
nm at
e.g. phosphor
photosynthesis in
su cient power) oxidation, re ies agriculture,
(‘cold light’)
photodynamic
therapy,
hyperbilirubinaemia
treatment
– nm red
penetrate esh blood
alarm signal, used for breast imaging
nm pure red
rainbow
colour reference for printing, painting, illumination and displays
– nm orange
various fruit
attracts birds and insects
nm standard orange
– nm yellow
majority of owers idem; best background for reading black text
nm standard yellow
–
green maximum eye algae and plants highest brightness
nm
sensitivity
per light energy for
the human eye
. nm pure green
rainbow
colour reference
– nm blue
sky, gems, water
nm standard cyan
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•.
F- W -N
M
A
U
. nm pure blue
–
–
indigo,
THz
nm violet
rainbow owers, gems
colour reference
– THz
.– . PHz .– . PHz
– PHz
ultraviolet
– nm UVA
penetrate mm emitted by Sun and seen by certain birds,
into skin, darken stars
integrated circuit
it, produce
fabrication
vitamin D,
suppress immune
system, cause
skin cancer,
destroy eye lens
– nm UVB
idem, destroy idem DNA, cause skin cancer
idem
– nm UVC
form oxygen idem radicals from air, kill bacteria, penetrate µm into skin
disinfection, water puri cation, waste disposal, integrated circuit fabrication
– nm EUV
sky maps, silicon lithography
X-rays penetrate materials
– PHz
– . nm
PHz < . nm or keV
so X-rays
hard X-rays
idem idem
emitted by stars, imaging human plasmas and black tissue holes
synchrotron
idem
radiation
emitted when fast crystallography, electrons hit matter structure
determination
EHz < pm or
keV
ë Hz − m
γ-rays idem Planck limit
radioactivity, cosmic rays
chemical analysis, disinfection, astronomy
see part three of this text
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T e well-known expression
c= εµ
(436)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
for the speed of light is so strange that one should be astonished when one sees it. Something essential is missing.
Indeed, the speed is independent of the proper motion of the observer measuring the electromagnetic eld. In other words, the speed of light is independent of the speed of the lamp and independent of the speed of the observer. All this is contained in expression ( ). Incredibly, for ve decades, nobody explored this strange result. In this way, the theory of relativity remained undiscovered from to . As in so many other cases, the progress of physics was much slower than necessary.
e constancy of the speed of light is the essential point that distinguishes special relativity from Galilean physics. In this sense, any electromagnetic device, making use of expression ( ), is a working proof of special relativity.
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H
?
Challenge 1001 n
At the end of the nineteenth century, the teenager Albert Einstein read a book discussing the speed of light. e book asked what would happen if an observer moved at the same speed as light.* Einstein thought much about the issue, and in particular, asked himself what kind of electromagnetic eld he would observe in that case. Einstein later explained that this Gedanken experiment convinced him already at that young age that nothing could travel at the speed of light, since the eld observed would have a property not found in nature. Can you nd out which one he meant?
Riding on a light beam situation would have strange consequences.
— You would have no mirror image, like a vampire. — Light would not be oscillating, but would be a static eld. — Nothing would move, like in the tale of sleeping beauty.
But also at speeds near the velocity of light observations would be interesting. You would:
— see a lot of light coming towards you and almost no light from the sides or from behind; the sky would be blue/white in the front and red/black behind;
— observe that everything around happens very very slowly; — experience the smallest dust particle as a deadly bullet.
Challenge 1002 n Can you think of more strange consequences? It is rather reassuring that our planet moves rather slowly through its environment.
D
?
Ref. 524
Usually light moves in straight lines. Indeed, we even use light to de ne ‘straightness.’ However, there are a number of exceptions that every expert on motion should know.
In sugar syrup, light beams curve, as shown in Figure . In fact, light beams bend at any material interface. is e ect, called refraction, also changes the appearance of the shape of our feet when we are in the bath tub and makes aquaria seem less deep than they actually are. Refraction is a consequence of the change of light speed from material
* is was the book series in twenty volumes by Aaron Bernstein, Naturwissenscha liche Volksbücher, Duncker, 1873-1874. e young Einstein read them, between 1892 and 1894, with ‘breathless attention’, as he wrote later on.
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air
light beam
sugar and water
F I G U R E 248 Sugar water bends light
•.
object and light departing from it
do
f
focus
real image with optional screen di
f
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object and light departing from it
focus
virtual image
di
do
F I G U R E 249 The real image produced by a converging lens and the virtual image produced by a diverging lens
Challenge 1003 n Challenge 1004 n
to material. Are you able to explain refraction, and thus explain the syrup e ect? Refraction is chie y used in the design of lenses. Using glass instead of water, one can
produce curved surfaces, so that light can be focused. Focusing devices can be used to produce images. e two main types of lenses, with their focal points and the images they produce, are shown in Figure . When an object is put between a converging lens and its focus, works as a magnifying glass. It also produces a real image, i.e., an image that can be projected onto a screen. In all other cases lenses produce so-called virtual images: such images can be seen with the eye but not be projected onto a screen. Figure also allows one to deduce the thin lens formula that connects the lengths do, do and f . What is it?
Even though glasses and lenses have been known since antiquity, the Middle Ages had to pass by before two lenses were combined to make more elaborate optical instruments.
e telescope was invented in, or just before, by the German–Dutch lens grinder
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object
final, enlarged virtual image
common focus of both lenses
intermediate, real image
objective lens
ocular
lens
to human eye
F I G U R E 250 Refraction as the basis of the telescope – shown here in the original Dutch design
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Challenge 1005 n
Challenge 1006 ny Page 574 Ref. 525
Johannes Lipperhey (c. – ), who made a fortune by selling it to the Dutch military. When Galileo heard about the discovery, he quickly took it over and improved it. In , Galileo performed the rst astronomical observations; they made him worldfamous. e Dutch telescope design has a short tube yielding a bright and upright image. It is still used today in opera glasses. Many other ways of building telescopes have been developed over the years. *
Another way to combine two lenses leads to the microscope. Can you explain to a nonphysicist how a microscope works?** Werner Heisenberg almost failed his Ph.D. exam because he could not. e problem is not di cult, though. Indeed, the inventor of the microscope was an autodidact of the seventeenth century: the Dutch technician Antoni van Leeuwenhoek ( – ) made a living by selling over ve hundred of his microscopes to his contemporaries.
No ray tracing diagram, be it that of a simple lens, of a telescope or a microscope, is really complete if the eye, with its lens and retina, is missing. Can you add it and convince yourself that these devices really work?
Refraction is o en colour-dependent. For that reason, microscopes or photographic cameras have several lenses, made of di erent materials. ey compensate the colour effects, which otherwise yield coloured image borders. e colour dependence of refraction in water droplets is also the basis of the rainbow, as shown below, and refraction in ice crystals in the atmosphere is at the basis of the halos and the many other light patterns o en seen around the Sun and the Moon.
Also the lens in the human eye has colour-dependent di raction. is is e ect is not corrected in the eye, but in the brain. e dispersion of the eye lens can be noticed if this correction is made impossible, for example when red or blue letters are printed on a black
* A fascinating overview about what people have achieved in this domain up to now is given by P
M , Unusual Telescopes, Cambridge University Press, 1991. Images can also be made with mirrors.
Since mirrors are cheaper and more easy to fabricate with high precision, most large telescopes have a mirror
instead of the rst lens.
By the way, telescopes also exist in nature. Many spiders have two types of eyes. e large ones, made to
see far away, have two lenses arranged in the same way as in the telescope.
** If not, read the beautiful text by E
M. S
&H
S. S
, Light and Electron
Microscopy, Cambridge University Press, 1993.
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•.
Eye lens dispersion
F I G U R E 251 Watching this graphic at higher magnification shows the dispersion of the lens in the human eye: the letters float at different depths
preliminary drawing
F I G U R E 252 In certain materials, light beams can spiral around each other
α b M
F I G U R E 253 Masses bend light
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Challenge 1007 n
Ref. 526 Challenge 1008 n Challenge 1009 n
Ref. 527 Challenge 1010 ny
background, as shown in Figure . One gets the impression that the red letters oat in front of the blue letters. Can you show how dispersion leads to the oating e ect?
A second important observation is that light goes around corners, and the more so the sharper they are. is e ect is called di raction. In fact, light goes around corners in the same way that sound does. Di raction is due to the wave nature of light (and sound). You probably remember the topic from secondary school.
Because of di raction, it is impossible to produce strictly parallel light beams. For example, every laser beam diverges by a certain minimum amount, called the di raction limit. Maybe you know that the world’s most expensive Cat’s-eyes are on the Moon, where they have been deposited by the Lunakhod and the Apollo missions. Can you determine how wide a laser beam with minimum divergence has become when it arrives at the Moon and returns back to Earth, assuming that it was m wide when it le Earth? How wide would it be when it came back if it had been mm wide at the start?
Di raction implies that there are no perfectly sharp images: there exists a limit on resolution. is is true for every optical instrument, including the eye. e resolution of the eye is between one and two minutes of arc, i.e. between . and . mrad. e limit is due to the nite size of the pupil. erefore, for example, there is a maximum distance at which humans can distinguish the two headlights of a car. Can you estimate it?
Resolution limits also make it impossible to see the Great Wall in northern China from the Moon, contrary to what is o en claimed. In the few parts that are not yet in ruins, the wall is about metres wide, and even if it casts a wide shadow during the morning or the evening, the angle it subtends is way below a second of arc, so that it is completely invisible to the human eye. In fact, three di erent cosmonauts who travelled to the Moon performed careful searches and con rmed that the claim is absurd. e story is one of the most tenacious urban legends. (Is it possible to see the Wall from the space shuttle?) e largest human-made objects are the polders of reclaimed land in the Netherlands; they are visible from outer space. So are most large cities as well as the highways in Belgium at night; their bright illumination makes them stand out clearly from the dark side of the
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F I G U R E 254 Reflection at air interfaces is the basis of the Fata Morgana
Ref. 528 Page 475
Earth. Di raction also means that behind a small disc illuminated along its axis, the centre
of the shadow shows, against all expectations, a bright spot. is ‘hole’ in the shadow was predicted in by Denis Poisson ( – ) in order to show to what absurd consequences the wave theory of light would lead. He had just read the mathematical description of di raction developed by Augustin Fresnel* on the basis of the wave description of light. But shortly a erwards, François Arago ( – ) actually observed Poisson’s point, converting Poisson, making Fresnel famous and starting the general acceptance of the wave properties of light.
Additional electromagnetic elds usually do not in uence light directly, since light has no charge and since Maxwell’s equations are linear. But in some materials the equations are non-linear, and the story changes. For example, in certain photorefractive materials, two nearby light beams can even twist around each other, as was shown by Segev and coworkers in .
A nal way to bend light is gravity, as discussed already in the chapters on universal gravity and on general relativity. e e ect of gravity between two light beams was also discussed there.
In summary, light travels straight only if it travels far from other matter. In everyday life, ‘far’ simply means more than a few millimetres, because all gravitational and electromagnetic e ects are negligible at these distances, mainly due to lights’ truly supersonic speed.
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T
If one builds a large lens or a curved mirror, one can collect the light of the Sun and focus it on a single spot. Everybody has used a converging lens as a child to burn black spots on newspapers in this way. In Spain, wealthier researchers have even built a curved mirror as large as a house, in order to study solar energy use and material behaviour at high temperature. Essentially, the mirror provides a cheap way to re an oven. Indeed, ‘focus’ is the Latin word for ‘oven’.
Kids nd out quite rapidly that large lenses allow them to burn things more easily than small ones. It is obvious that the Spanish site is the record holder in this game. However, building a larger mirror does not make sense. Whatever its size may be, such a set-up
* Augustin Jean Fresnel (1788–1827), engineer and part time physicist; he published in 1818 his great paper on wave theory for which he got the prize of the French Academy of Sciences in 1819. To improve his nances, he worked in the commission responsible for lighthouses, for which he developed the well-known Fresnel lens. He died prematurely, partly of exhaustion due to overwork.
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Challenge 1011 ny
cannot reach a higher temperature than that of the original light source. e surface temperature of the Sun is about K; indeed, the highest temperature reached so far is about K. Are you able to show that this limitation follows from the second law of thermodynamics?
In short, nature provides a limit to the concentration of light energy. In fact, we will encounter additional limits in the course of our exploration.
C
?
Challenge 1012 e
If a little glass bead is put on top of a powerful laser, the bead remains suspended in mid-air, as shown in Figure .* is means that light has momentum. erefore, contrary to what we said in the begin- F I G U R E 255 The last mirror of the solar furnace ning of our mountain ascent, images can be at Odeillo, in the French Pyrenees (© Gerhard touched! In fact, the ease with which objects Weinrebe) can be pushed even has a special name. For stars, it is called the albedo, and for general objects it is called the re ectivity, abbreviated as r.
Like each type of electromagnetic eld, and like every kind of wave, light carries energy; the energy ow T per surface and time preliminary figure is
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Challenge 1013 e
T = µ E B giving an average T = µ EmaxBmax . (437)
Obviously, light also has a momentum P. It is related to the energy
E by
P
=
E c
.
(438)
As a result, the pressure p exerted by light on a body is given by
light
F I G U R E 256 Levitating a small glass bead with a laser
p=
T c
(
+ r)
(439)
Challenge 1014 n
where for black bodies we have that a re ectivity r = and for mirrors r = ; other bodies have values in between. What is your guess for the amount of pressure due to sunlight on a black surface of one square metre? Is this the reason that we feel more pressure during the day than during the night?
In fact, rather delicate equipment is needed to detect the momentum of light, in other words, its radiation pressure. Already around , Johannes Kepler had suggested in De cometis that the tails of comets exist only because the light of the Sun hits the small dust
* e heaviest object that has been levitated with a laser had a mass of g; the laser used was enormous, and the method also made use of a few additional e ects, such as shock waves, to keep the object in the air.
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Challenge 1015 e
Challenge 1016 n Ref. 529 Ref. 530 Ref. 531 Ref. 532
Ref. 532
particles that detach from it. For this reason, the tail always points away from the Sun, as
you might want to check at the next opportunity. Today, we know that Kepler was right;
but proving the hypothesis is not easy.
In , William Crookes * invented the light
mill radiometer. He had the intention of demon-
strating the radiation pressure of light. e light
mill consists of four thin plates, black on one
side and shiny on the other, that are mounted
on a vertical axis, as shown in Figure . How-
ever, when Crookes nished building it – it was
light
similar to those sold in shops today – he found,
like everybody else, that it turned in the wrong
direction, namely with the shiny side towards
the light! (Why is it wrong?) You can check it by
yourself by shining a laser pointer on to it. e
behaviour has been a puzzle for quite some time.
Explaining it involves the tiny amount of gas le
over in the glass bulb and takes us too far from
the topic of our mountain ascent. It was only in
, with the advent of much better pumps, that the Russian physicist Peter/Pyotr Lebedev man-
F I G U R E 257 A commercial light mill turns against the light
aged to create a su ciently good vacuum to al-
low him to measure the light pressure with such an improved, true radiometer. Lebedev
also con rmed the predicted value of the light pressure and proved the correctness of
Kepler’s hypothesis. Today it is even possible to build tiny propellers that start to turn
when light shines on to them, in exactly the same way that the wind turns windmills.
But light cannot only touch and be touched, it can also grab. In the s, Arthur
Ashkin and his research group developed actual optical tweezers that allow one to grab,
suspend and move small transparent spheres of to µm diameter using laser beams. It
is possible to do this through a microscope, so that one can also observe at the same time
what is happening. is technique is now routinely used in biological research around
the world, and has been used, for example, to measure the force of single muscle bres,
by chemically attaching their ends to glass or Te on spheres and then pulling them apart
with such optical tweezers.
But that is not all. In the last decade of the twentieth century, several groups even
managed to rotate objects, thus realizing actual optical spanners. ey are able to rotate
particles at will in one direction or the other, by changing the optical properties of the
laser beam used to trap the particle.
In fact, it does not take much to deduce that if light has linear momentum, circularly
polarized light also has angular momentum. In fact, for such a wave the angular mo-
mentum L is given by
L=
Energy ω
.
(440)
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* William Crookes (b. 1832 London, d. 1919 London), English chemist and physicist, president of the Royal Society, discoverer of thallium.
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•.
Challenge 1017 e Equivalently, the angular momentum of a wave is λ π times its linear momentum. For
Ref. 533 light, this result was already con rmed in the early twentieth century: a light beam can
Challenge 1018 ny put certain materials (which ones?) into rotation, as shown in Figure . Of course, the
whole thing works even better with a laser beam. In the s, a beautiful demonstration
was performed with microwaves. A circularly polarized microwave beam from a maser
– the microwave equivalent of a laser – can put a metal piece absorbing it into rotation.
Indeed, for a beam with cylindrical symmetry, depending on the sense of rotation, the
angular momentum is either parallel or antiparallel to the direction of propagation. All
these experiments con rm that light also carries angular momentum, an e ect which will
play an important role in the second part of our mountain ascent.
We note that not for all waves angular momentum is
energy per angular frequency. is is only the case for
waves made of what in quantum theory will be called spin particles. For example, for gravity waves the angular mo-
suspension wire
mentum is twice this value, and they are therefore expec-
ted to be made of spin particles.
In summary, light can touch and be touched. Obviously,
if light can rotate bodies, it can also be rotated. Could you
Challenge 1019 n imagine how this can be achieved?
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Challenge 1020 ny Challenge 1021 ny
W,
circularly
From the tiny e ects of equation ( ) for light pressure we deduce that light is not an e cient tool for hitting ob-
polarized light beam
jects. On the other hand, light is able to heat up objects, F I G U RE 258 Light can rotate
as we can feel when the skin is touched by a laser beam of objects
about mW or more. For the same reason even cheap
laser pointers are dangerous to the eye.
In the s, and again in , a group of people who had read too many science
ction novels managed to persuade the military – who also indulge in this habit – that
lasers could be used to shoot down missiles, and that a lot of tax money should be spent
on developing such lasers. Using the de nition of the Poynting vector and a hitting time
of about . s, are you able to estimate the weight and size of the battery necessary for
such a device to work? What would happen in cloudy or rainy weather?
Other people tried to persuade NASA to study the possibility of propelling a rocket
using emitted light instead of ejected gas. Are you able to estimate whether this is feasible?
W
?
Challenge 1022 n
We saw that radio waves of certain frequencies are visible. Within that range, di erent frequencies correspond to di erent colours. (Are you able to convince a friend about this?) But the story does not nish here. Numerous colours can be produced either by a single wavelength, i.e. by monochromatic light, or by a mixture of several di erent colours. For example, standard yellow can be, if it is pure, an electromagnetic beam of nm wavelength or it can be a mixture of standard green of . nm and standard red of
nm. e eye cannot distinguish between the two cases; only spectrometers can. In everyday life, all colours turn out to be mixed, with the exceptions of those of yellow
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colour-dependent refraction in glass
white
glass
colour-dependent refraction in the eye (watch pattern at 1 cm distance)
red
green violet
internal reflection and colour-dependent refraction in the primary rainbow
white (Sun) water droplet
40.5°
42.4°
violet green red
internal reflection and colour-dependent refraction in the secondary rainbow
white (Sun) water droplet
50.3°
53.6°
F I G U R E 260 Proving that white light is a mixture of colours
red green
violet
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Challenge 1023 e Challenge 1024 n
street lamps, of laser beams and of the rainbow. You can check this for yourself, using an umbrella or a compact disc: they decompose light mixtures, but not pure colours.
In particular, white light is a mixture of a continuous range of colours with a given intensity per wavelength. To check that white light is a mixture of colours, simply hold the le -hand side of Figure so near to your eye that you cannot focus the stripes any more. e unsharp borders of the white stripes have a pink or a green shade. ese colours are due to the imperfections of the lens in the human eye, its so-called chromatic aberrations. Aberrations have the consequence that not all light frequencies follow the same path through the lens of the eye, and therefore they hit the retina at di erent spots. is is the F I G URE 259 Umbrellas same e ect that occurs in prisms or in water drops showing a decompose white light rainbow. By the way, the shape of the rainbow tells something about the shape of the water droplets. Can you deduce the connection?
e right side of Figure explains how rainbows form. e main idea is that internal re ection inside the water droplets in the sky is responsible for throwing back the light coming from the Sun, whereas the wavelength-dependent refraction at the air–water sur-
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Ref. 534 Challenge 1025 e
Page 561 Challenge 1027 e
Ref. 536
face is responsible for the di erent paths of each colour. e rst two persons to verify this
explanation were eodoricus Teutonicus de Vriberg (c. to c. ), in the years from
to and, at the same time, the Persian mathematician Kamal al-Din al-Farisi. To
check the explanation, they did something smart and simple; anybody can repeat this at
home. ey built an enlarged water droplet by lling a thin spherical (or cylindrical) glass
container with water; then they shone a beam of white light through it. eodoricus and
al-Farisi found exactly what is shown in Figure . With this experiment, each of them
was able to reproduce the angle of the main or primary rainbow, its colour sequence, as
well as the existence of a secondary rainbow, its observed angle and its inverted colour
sequence.* All these bows are visible in Figure . eodoricus’s beautiful experiment is
sometimes called the most important contribution of natural science in the Middle Ages.
Even pure air splits white light. is is the reason that
the sky and far away mountains look blue or that the Sun
looks red at sunset and at sunrise. ( e sky looks black even
during the day from the Moon.) You can repeat this e ect
by looking through water at a black surface or at a lamp.
Adding a few drops of milk to the water makes the lamp yel-
low and then red, and makes the black surface blue (like the
sky seen from the Earth as compared to the sky seen from
the Moon). More milk increases the e ect. For the same
reason, sunsets are especially red a er volcanic eruptions.
Incidentally, at sunset the atmosphere itself also acts as a
prism; that means that the Sun is split into di erent images,
one for each colour, which are slightly shi ed with respect
to each other, a bit like a giant rainbow in which not only the
rim, but the whole disc is coloured. e total shi is about
/ th of the diameter. If the weather is favourable and if the air is clear and quiet up to and beyond the horizon, for a few seconds it is possible to see, a er the red, orange and yellow images of the setting Sun, the rim of the green–blue image.
F I G U R E 261 Milk and water simulate the evening sky (© Antonio Martos)
is is the famous ‘rayon vert’ described by Jules Verne in his novel of the same title. It is
o en seen on islands, for example in Hawaii.**
To clarify the di erence between colours in physics and colour in human perception
and language, a famous linguistic discovery deserves to be mentioned: colours in human
language have a natural order. Colours are ordered by all peoples of the world, whether
they come from the sea, the desert or the mountains, in the following order: st black and
white, nd red, rd green and yellow, th blue, th brown; th come mauve, pink, orange,
grey and sometimes a twel h term that di ers from language to language. (Colours that
refer to objects, such as aubergine or sepia, or colours that are not generally applicable,
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Challenge 1026 ny Ref. 535
* Can you guess where the tertiary and quaternary rainbows are to be seen? ere are rare reported sightings of them. e hunt to observe the h-order rainbow is still open. (In the laboratory, bows around droplets up to the 13th order have been observed.) For more details, see the beautiful website at http://www.sundog.clara. co.uk/atoptics/phenom.htm. ere are several formulae for the angles of the various orders of rainbows; they follow from straightforward geometric considerations, but are too involved to be given here. ** For this and many other topics on colours in nature, such as, for example, the halos around the Moon and the Sun or the colour of shadows,, see the beautiful book by Marcel Minnaert mentioned on page 74.
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v ph
v gr v So v fr
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F I G U R E 262 The definition of important velocities in wave phenomena
Ref. 537
such as blond, are excluded in this discussion.) e precise discovery is the following: if a particular language has a word for any of these colours, then it also has a word for all the preceding ones. e result also implies that people use these basic colour classes even if their language does not have a word for each of them. ese strong statements have been con rmed for over languages.
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?–A
Physics talks about motion. Talking is the exchange of sound; and sound is an example of
a signal. A (physical) signal is the transport of information using the transport of energy.
ere are no signals without a motion of energy. Indeed, there is no way to store inform-
ation without storing energy. To any signal we can thus ascribe a propagation speed. e
highest possible signal speed is also the maximal velocity of the general in uences, or, to
use sloppy language, the maximal velocity with which e ects spread causes.
If the signal is carried by matter, such as by the written text in a letter, the signal velocity
is then the velocity of the material carrier, and experiments show that it is limited by the
speed of light.
For a wave carrier, such as water waves, sound, light or radio waves, the situation is less
evident. What is the speed of a wave? e rst answer that comes to mind is the speed
with which wave crests of a sine wave move. is already introduced phase velocity is
given by the ratio between the frequency and the wavelength of a monochromatic wave,
i.e. by
vph
=
ω k
.
(441)
Challenge 1028 n Ref. 538
For example, the phase velocity determines interference phenomena. Light in a vacuum has the same phase velocity vph = c for all frequencies. Are you able to imagine an experiment to test this to high precision?
On the other hand, there are cases where the phase velocity is greater than c, most notably when light travels through an absorbing substance, and when at the same time the frequency is near to an absorption maximum. In these cases, experiments show that the phase velocity is not the signal velocity. For such situations, a better approximation
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to the signal speed is the group velocity, i.e. the velocity at which a group maximum will
travel. is velocity is given by
vgr
=
dω dk
k
,
(442)
where k is the central wavelength of the wave packet. We observe that ω = c(k)k = πvph λ implies the relation
vgr =
dω dk
k
=
vph
−
λ
dvph dλ
.
(443)
Challenge 1029 ny Ref. 539 Ref. 538
is means that the sign of the last term determines whether the group velocity is larger or smaller than the phase velocity. For a travelling group, as shown by the dashed line in Figure , this means that new maxima appear either at the end or at the front of the group. Experiments show that this is only the case for light passing through matter; for light in vacuum, the group velocity has the same value vgr = c for all values of the wave vector k.
You should be warned that many publications are still propagating the incorrect statement that the group velocity in a material is never greater than c, the speed of light in vacuum. Actually, the group velocity in a material can be zero, in nite or even negative; this happens when the light pulse is very narrow, i.e. when it includes a wide range of frequencies, or again when the frequency is near an absorption transition. In many (but not all) cases the group is found to widen substantially or even to split, making it di cult to de ne precisely the group maximum and thus its velocity. Many experiments have con-
rmed these predictions. For example, the group velocity in certain materials has been measured to be ten times that of light. e refractive index then is smaller than . However, in all these cases the group velocity is not the same as the signal speed.*
What then is the best velocity describing signal propagation? e German physicist Arnold Sommerfeld** almost solved the main problem in the beginning of the twentieth century. He de ned the signal velocity as the velocity vSo of the front slope of the pulse, as shown in Figure . e de nition cannot be summarized in a formula, but it does have the property that it describes signal propagation for almost all experiments, in particular those in which the group and phase velocity are larger than the speed of light. When studying its properties, it was found that for no material is Sommerfeld’s signal velocity greater than the speed of light in vacuum.
Sometimes it is conceptually easier to describe signal propagation with the help of the energy velocity. As previously mentioned, every signal transports energy. e energy velocity ven is de ned as the ratio between the power ow density P, i.e. the Poynting
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* In quantum mechanics, Schrödinger proved that the velocity of an electron is given by the group velocity of its wave function. erefore the same discussion reappeared in quantum theory, as we will nd out in the second part of the mountain ascent. ** Arnold Sommerfeld (b. 1868 Königsberg, d. 1951 München) was a central gure in the spread of special and general relativity, of quantum theory, and of their applications. A professor in Munich, an excellent teacher and text book writer, he worked on atomic theory, on the theory of metals and on electrodynamics, and was the rst to understand the importance and the mystery around ‘Sommerfeld’s famous ne structure constant.’
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
vector, and the energy density W, both taken in the direction of propagation. For electro-
magnetic elds – the only ones fast enough to be interesting for eventual superluminal
signals – this ratio is
ven
=
Re(P) W
=
c E
E +c
B B
.
(444)
Ref. 538 Challenge 1030 n
However, as in the case of the front velocity, in the case of the energy velocity we have to specify if we mean the energy transported by the main pulse or by the front of it. In vacuum, neither is ever greater than the speed of light.* (In general, the velocity of energy in matter has a value slightly di erent from Sommerfeld’s signal velocity.)
In recent years, the progress in light detector technology, allowing one to detect even the tiniest energies, has forced scientists to take the fastest of all these energy velocities to describe signal velocity. Using detectors with the highest possible sensitivity we can use as signal the rst point of the wave train whose amplitude is di erent from zero, i.e. the rst tiny amount of energy arriving. is point’s velocity, conceptually similar to Sommerfeld’s signal velocity, is commonly called the front velocity or, to distinguish it even more clearly from Sommerfeld’s case, the forerunner velocity. It is simply given by
vfr
=
lim
ω
ω k
.
(445)
Challenge 1031 n Challenge 1032 n
e forerunner velocity is never greater than the speed of light in a vacuum, even in materials. In fact it is precisely c because, for extremely high frequencies, the ratio ω k is independent of the material, and vacuum properties take over. e forerunner velocity is the true signal velocity or the true velocity of light. Using it, all discussions on light speed become clear and unambiguous.
To end this section, here are two challenges for you. Which of all the velocities of light is measured in experiments determining the velocity of light, e.g. when light is sent to the Moon and re ected back? And now a more di cult one: why is the signal speed of light less inside matter, as all experiments show?
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Ref. 542
In the Soviet physicist Victor Veselago made a strange prediction: the index of refraction could have negative values without invalidating any known law of physics. A negative index means that a beam is refracted to the same side of the vertical, as shown in Figure .
In , John Pendry and his group proposed ways of realizing such materials. In , a rst experimental con rmation for microwave refraction was published, but it
met with strong disbelief. In the debate was in full swing. It was argued that negative refraction indices imply speeds greater than that of light and are only possible for either phase velocity or group velocity, but not for the energy or true signal velocity. e
Ref. 540 Ref. 541
* Signals not only carry energy, they also carry negative entropy (‘information’). e entropy of a transmitter increases during transmission. e receiver decreases in entropy (but less than the increase at the transmitter, of course).
Note that the negative group velocity implies energy transport against the propagation velocity of light. is is possible only in energy loaded materials.
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α
air
water (n>0)
β
α
air
left-handed
material
β
(n<0)
F I G U R E 263 Positive and negative indices of refraction
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 543
Ref. 544 Ref. 545 Ref. 544 Challenge 1033 ny
conceptual problems would arise only because in some physical systems the refraction angle for phase motion and for energy motion di er.
Today, the consensus is the following: a positive index of refraction less than one is impossible, as it implies an energy speed of greater than one. A negative index of refraction, however, is possible if it is smaller than − . Negative values have indeed been frequently observed; the corresponding systems are being extensively explored all over the world.
e materials showing this property are called le -handed. e reason is that the vectors of the electric eld, the magnetic eld and the wave vector form a le -handed triplet, in contrast to vacuum and most usual materials, where the triplet is right-handed. Such materials consistently have negative magnetic permeability and negative dielectric coe cient (permittivity).
Le -handed materials have negative phase velocities, i.e., a phase velocity opposed to the energy velocity, they show reversed Doppler e ects and yield obtuse angles in the Çerenkov e ect (emitting Çerenkov radiation in the backward instead of the forward direction).
But, most intriguing, negative refraction materials are predicted to allow the construction of lenses that are completely at. In addition, in the year , John Pendry gained the attention of the whole physics community world-wide by predicting that lenses made with such materials, in particular for n = − , would be perfect, thus beating the usual diffraction limit. is would happen because such a lens also images the evanescent parts of the waves, by amplifying them accordingly. First experiments seem to con rm the prediction. Discussion on the topic is still in full swing.
Can you explain how negative refraction di ers from di raction?
S
Ref. 546
When one person reads a text over the phone to a neighbour who listens to it and maybe repeats it, we speak of communication. For any third person, the speed of communication is always less than the speed of light. But if the neighbour already knows the text, he can recite it without having heard the readers’ voice. To the third observer such a situation appears to imply motion that is faster than light. Prediction can thus mimic communication and, in particular, it can mimic faster-than-light (superluminal) communication. Such a situation was demonstrated most spectacularly in by Günter Nimtz, who seemingly transported music – all music is predictable for short time scales – through a
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TA B L E 49 Experimental properties of (flat) vacuum and of the ‘aether’
P
E
permeability permittivity wave impedance/resistance conformal invariance spatial dimensionality topology mass and energy content friction on moving bodies motion relative to space-time
µ = . µH m ε = . pF m Z= .Ω applies
R not detectable not detectable not detectable
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
‘faster-than-light’ system. To distinguish between the two situations, we note that in the
case of prediction, no transport of energy takes place, in contrast to the case of commu-
nication. In other words, the de nition of a signal as a transporter of information is not
as useful and clear-cut as the de nition of a signal as a transporter of energy. In the above-
mentioned experiment, no energy was transported faster than light. e same distinction
between prediction on the one hand and signal or energy propagation on the other will
be used later to clarify some famous experiments in quantum mechanics.
“If the rate at which physics papers are being published continues to increase, physics journals will soon be lling library shelves faster
than the speed of light. is does not violate
relativity since no useful information is being
transmitted.
D
?
” David Mermin
Page 1155 Ref. 547
Challenge 1034 n
Gamma rays, light and radio waves are moving electromagnetic waves. All exist in empty space. What is oscillating when light travels? Maxwell himself called the ‘medium’ in which this happens the aether. e properties of the aether measured in experiments are listed in Table . Of course, the values of the permeability and the permittivity of the vacuum are related to the de nition of the units henry and farad. e last item of the table is the most important: despite intensive e orts, nobody has been able to detect any motion of the aether. In other words, even though the aether supposedly oscillates, it does not move. Together with the other data, all these results can be summed up in one sentence: there is no way to distinguish the aether from the vacuum: they are one and the same.
Sometimes one hears that certain experiments or even the theory of relativity show that the aether does not exist. is is not strictly correct. In fact, experiments show something more important. All the data show that the aether is indistinguishable from the vacuum. Of course, if we use the change of curvature as the de nition for motion of the vacuum, the vacuum can move, as we found out in the section on general relativity; but
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grass
dew (not to scale)
head
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Sun
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 264 The path of light for the dew on grass that is responsible for the aureole
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Ref. 548
the aether still remains indistinguishable from it.* Later we will even nd out that the ability of the vacuum to allow the propagation of
light and its ability to allow the propagation of particles are equivalent: both require the same properties. erefore the aether remains indistinguishable from a vacuum in the rest of our walk. In other words, the aether is a super uous concept; we will drop it from our walk from now on. Despite this result, we have not yet nished the study of the vacuum; vacuum will keep us busy for a long time, starting with the intermezzo following this chapter. Moreover, quite a few of the aspects in Table will require some amendments later.
C
H
’
Ref. 549
Ref. 550 Challenge 1035 n
Light re ection and refraction are responsible for many e ects. e originally Indian symbol of holiness, now used throughout most of the world, is the aureole, also called halo or Heiligenschein, a ring of light surrounding the head. You can easily observe it around your own head. You need only to get up early in the morning and look into the wet grass while turning your back to the Sun. You will see an aureole around your shadow. e e ect is due to the morning dew on the grass, which re ects the light back predominantly in the direction of the light source, as shown in Figure . e fun part is that if you do this in a group, you will see the aureole around only your own head.
Retrore ective paint works in the same way: it contains tiny glass spheres that play the role of the dew. A large surface of retrore ective paint, a tra c sign for example, can also show your halo if the light source is su ciently far away. Also the so-called ‘glow’ of the eyes of a cat at night is due to the same e ect; it is visible only if you look at the cat with a light source behind you. By the way, do Cat’s-eyes work like a cat’s eyes?
Ref. 548
* In fact, the term ‘aether’ has been used as an expression for several di erent ideas, depending on the author. First of all it was used for the idea that a vacuum is not empty, but full; secondly, that this fullness can be described by mechanical models, such as gears, little spheres, vortices, etc.; thirdly, it was imagined that a vacuum is similar to matter, being made of the same substrate. Interestingly, some of these issues will reappear in the third part of our mountain ascent.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 265 A limitation of the eye
D
?
Challenge 1036 n Ref. 551
Challenge 1037 ny
Sometimes we see less than there is. Close your le eye, look at the white spot in Figure , bring the page slowly towards your eye, and pay attention to the middle lines. At a distance of about to cm the middle line will seem uninterrupted. Why?
On the other hand, sometimes we see more than there is, as Figures and show. e rst shows that parallel lines can look skewed, and the second show a so-called Hermann lattice, named a er its discoverer.* e Hermann lattice of Figure , discovered by Elke Lingelbach in , is especially striking. Variations of these lattices are now used to understand the mechanisms at the basis of human vision. For example, they can be used to determine how many light sensitive cells in the retina are united to one signal pathway towards the brain. e illusions are angle dependent because this number is also angle dependent. Our eyes also ‘see’ things di erently: the retina sees an inverted image of the world. ere is a simple method to show this, due to Helmholtz.** You need only a needle and a piece of paper, e.g. this page of text. Use the needle to make two holes inside the two letters ‘oo’. en keep the page as close to your eye as possible, look through the two holes towards the wall, keeping the needle vertical, a few centimetres behind the paper. You will see two images of the needle. If you now cover the le hole with your nger, the right needle will disappear, and vice versa. is shows that the image inside the eye, on the retina, is inverted. Are you able to complete the proof?
* Ludimar Herrmann (1838–1914), Swiss physiologist. e lattices are o en falsely called ‘Hering lattices’
a er the man who made Hermann’s discovery famous.
** See H
H
, Handbuch der physiologischen Optik, 1867. is famous classic is
available in English as Handbook of Physiological Optics, Dover, 1962. e Prussian physician, physicist and
science politician born as Hermann Helmholtz (b. 1821 Potsdam, d. 1894 Charlottenburg) was famous for
his works on optics, acoustics, electrodynamics, thermodynamics, epistemology and geometry. He founded
several physics institutions across Germany. He was one of the rst to propagate the idea of conservation
of energy. His other important book, Die Lehre von den Tonemp ndungen, published in 1863, describes the
basis of acoustics and, like the handbook, is still worth reading.
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F I G U R E 266 What is the angle between adjacent horizontal lines?
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 267 The Lingelbach lattice: do you see white, grey, or black dots?
Two other experiments can show the same result. If you push very lightly on the inside of your eye (careful!), you will see a dark spot appear on the outside of your vision eld. And if you stand in a dark room and ask a friend to look at a burning candle, explore his eye: you will see three re ections: two upright ones, re ected from the cornea and from the lens, and a dim third one, upside-down, re ected form the retina.
Another reason that we do not see the complete image of nature is that our eyes have a limited sensitivity. is sensitivity peaks around nm; outside the red and the violet, the eye does not detect radiation. We thus see only part of nature. For example, infrared photographs of nature, such as the one shown in Figure , are interesting because they show us something di erent from what we see usually.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 268 An example of an infrared photograph, slightly mixed with a colour image (© Serge Augustin)
Ref. 552 Challenge 1038 n
Every expert of motion should also know that the sensitivity of the eye does not correspond to the brightest part of sunlight. is myth has been spread around the world by the numerous textbooks that have copied from each other. Depending on whether frequency or wavelength or wavelength logarithm is used, the solar spectrum peaks at
nm, nm or nm. ey human eye’s spectral sensitivity, like the completely different sensitivity of birds or frogs, is due to the chemicals used for detection. In short, the human eye can only be understood by a careful analysis of its particular evolutionary history.
An urban legend says that newborn babies see everything upside down. Can you explain why this idea is wrong?
In summary, we have to be careful when maintaining that seeing means observing. Examples such as these lead to ask whether there are other limitations of our senses which are less evident. And our walk will indeed uncover several of them.
H
?
Ref. 553
e most beautiful pictures so far of a living human retina, such as that of Figure , were made by the group of David Williams and Austin Roorda at the University at Rochester in New York. ey used adaptive optics, a technique that changes the shape of the imaging lens in order to compensate for the shape variations of the lens in the human eye.*
* Nature uses another trick to get maximum resolution: the eye continuously performs small movements, called micronystagmus. e eye continuously oscillates around the direction of vision with around 40 to
Hz. In addition, this motion is also used to allow the cells in the retina to recharge.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 269 A high quality photograph of a live human retina, including a measured (false colour) indication of the sensitivity of each cone cell (© Austin Roorda)
Page 574 Challenge 1039 n
e eyes see colour by averaging the intensity arriving at the red, blue and green sensitive cones. is explains the possibility, mentioned above, of getting the same impression of colour, e.g. yellow, either by a pure yellow laser beam, or by a suitable mixture of red and green light.
But if the light is focused on to one cone only, the eye makes mistakes. If, using this adaptive optics, a red laser beam is focused such that it hits a green cone only, a strange thing happens: even though the light is red, the eye sees a green colour!
Incidentally, Figure is quite puzzling. In the human eye, the blood vessels are located in front of the cones. Why don’t they appear in the picture? And why don’t they disturb us in everyday life? ( e picture does not show the other type of sensitive light cells, the rods, because the subject was in ambient light; rods come to the front of the retina only in the dark, and then produce black and white pictures.
Of all the mammals, only primates can see colours. Bulls for example, don’t; they cannot distinguish red from blue. On the other hand, the best colour seers overall are the birds. ey have cone receptors for red, blue, green, UV and, depending on the bird, for up to three more sets of colours. A number of birds (but not many) also have a better eye resolution than humans. Several birds also have a faster temporal resolution: humans see continuous motion when the images follow with to Hz (depending on the image content); some insects can distinguish images up to Hz.
Challenge 1040 e
H
-
?
Our sense of sight gives us the impression of depth mainly due to three e ects. First, the two eyes see di erent images. Second, the images formed in the eyes are position dependent. ird, our eye needs to focus di erently for di erent distances.
A simple photograph does not capture any of the three e ects. A photograph corresponds to the picture taken by one eye, from one particular spot and at one particular focus. In fact, all photographic cameras are essentially copies of a single static eye.
Any system wanting to produce the perception of depth must include at least one of
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object
holographic plate
virtual object image
developed film
observer
reference beam
laser
reconstruction beam
laser or point source
F I G U R E 270 The recording and the observation of a hologram
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 1041 n
the three e ects just mentioned. In all systems so far, the third and weakest e ect, varying focus with distance, is never used, as it is too weak. Stereo photography and virtual reality systems extensively use the rst e ect by sending two di erent images to the eyes. Also certain post cards and computer screens are covered by thin cylindrical lenses that allow one to send two di erent images to the two eyes, thus generating the same impression of depth.
But obviously the most spectacular e ect is obtained whenever position dependent images can be created. Some virtual reality systems mimic this e ect using a sensor attached to the head, and creating computer–generated images that depend on this position. However, such systems are not able to reproduce actual situations and thus pale when compared with the impression produced by holograms.
Holograms reproduce all that is seen from any point of a region of space. A hologram is thus a stored set of position dependent pictures of an object. It is produced by storing amplitude and phase of the light emitted by an object. To achieve this, the object is illuminated by a coherent light source, such as a laser, and the interference pattern is stored. Illuminating the developed photographic lm by a coherent light source then allows one to see a full three-dimensional image. In particular, due to the reproduction of the situation, the image appears to oat in free space. Holograms were developed in by the Hungarian physicist Dennis Gabor ( – ), who received the Nobel Prize for physics for this work.
Holograms can be made to work in re ection or transmission. e simplest holograms use only one wavelength. Most coloured holograms are rainbow holograms, showing false colours that are unrelated to the original objects. Real colour holograms, made with three di erent lasers, are rare but possible.
Is it possible to make moving holograms? Yes; however, the technical set-ups are still extremely expensive. So far, they exist only in a few laboratories and cost millions of euro. By the way, can you describe how you would distinguish a moving hologram from a real body, if you ever came across one, without touching it?
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 271 Sub-wavelength optical microscopy using stimulated emission depletion (© MPI für biophysikalische Chemie/Stefan Hell)
I
Producing images is an important part of modern society. e quality images depends on the smart use of optics, electronics, computers and materials science. Despite long experience in this domain, there are still new results in the eld. Images, i.e. two or threedimensional reproductions, can be taken by at least four methods:
— Photography uses a light source, lenses and lm or another large area detector. Photography can be used in re ection, in transmission, with phase-dependence and in many other ways.
— Holography uses lasers and large area detectors, as explained above. Holography allows to take three-dimensional images of objects. It is usually used in re ection, but can also be used in transmission.
— Scanning techniques construct images point by point through the motion of the detector, the light source or both.
— Tomography, usually in transmission, uses a source and a line detector that are both rotated around an object. is allows to image cross sections of materials.
In all methods, the race is for images with the highest resolution possible. e techniques of producing images with resolutions less than the wavelength of light have made great progress in recent years.
A recent technique, called stimulated emission depletion microscopy, allows spot sizes of molecular size. e conventional di raction limit for microscopes is
d
n
λ sin α
,
(446)
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where λ is the wavelength, n the index of refraction and α is the angle of observation.
e new technique, a special type of uorescence microscopy developed by Stefan Hell,
modi es this expression to
d
λ
,
n sin α; I Isat
(447)
Ref. 554
so that a properly chosen saturation intensity allows one to reduce the di raction limit to arbitrary low values. So far, light microscopy with a resolution of nm has been performed, as shown in Figure . is and similar techniques should become commonplace in the near future.
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L
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In many countries, there is more money to study assault weapons than to increase the education and wealth of their citizen. Several types of assault weapons using electromagnetic radiation are being researched. Two are particularly advanced.
e rst weapon is a truck with a movable parabolic antenna on its roof, about m in size, that emits a high power (a few kW) microwave ( GHz) beam. e beam, like all microwave beams, is invisible; depending on power and beam shape, it is painful or lethal, up to a distance of a m. is terrible device, with which the operator can make many many victims without even noticing, was ready in . (Who expects that a parabolic antenna is dangerous?) E orts to ban it across the world are slowly gathering momentum.
e second weapon under development is the so-called pulsed impulse kill laser. e idea is to take a laser that emits radiation that is not absorbed by air, steam or similar obstacles. An example is a pulsed deuterium uoride laser that emits at . µm. is laser burns every material it hits; in addition, the evaporation of the plasma produced by the burn produces a strong hit, so that people hit by such a laser are hurt and hit at the same time. Fortunately, it is still di cult to make such a device rugged enough for practical use. But experts expect battle lasers to appear soon.
In short, it is probable that radiation weapons will appear in the coming years.*
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
.
–
One of the most important results of physics: electric charge is discrete has already been mentioned a number of times. Charge does not vary continuously, but changes in xed steps. Not only does nature show a smallest value of entropy and smallest amounts of matter; nature also shows a smallest charge. Electric charge is quantized.
In metals, the quantization of charge is noticeable in the ow of electrons. In electrolytes, i.e. electrically conducting liquids, the quantization of charge appears in the ow of charged atoms, usually called ions. All batteries have electrolytes inside; also water is an electrolyte, though a poorly conducting one. In plasmas, like re or uorescent lamps,
* What a man working on such developments tells his children when he comes home in the evening is not clear.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•.
both ions and electrons move and show the discreteness of charge. Also in radiation –
from the electron beams inside TVs, channel rays formed in special glass tubes, and cos-
mic radiation, up to radioactivity – charges are quantized.
From all known experiments, the same smallest value for charge change has been
found. e result is
∆q e = . − C .
(448)
Page 753
In short, like all ows in nature, the ow of electricity is due to a ow of discrete particles. A smallest charge change has a simple implication: classical electrodynamics is wrong.
A smallest charge implies that no in nitely small test charges exist. But such in nitely small test charges are necessary to de ne electric and magnetic elds. e limit on charge size also implies that there is no correct way of de ning an instantaneous electric current and, as a consequence, that the values of electric and magnetic eld are always somewhat fuzzy. Maxwell’s evolution equations are thus only approximate.
We will study the main e ects of the discreteness of charge in the part on quantum theory. Only a few e ects of the quantization of charge can be treated in classical physics. An instructive example follows.
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H
?
Challenge 1042 n
In vacuum, such as inside a colour television, charged particles accelerated by a tension of kV move with a third of the speed of light. In modern particle accelerators charges move so rapidly that their speed is indistinguishable from that of light for all practical purposes.
Inside a metal, electric signals move with speeds of the order of the speed of light. e precise value depends on the capacity and impedance of the cable and is usually in the range . c to . c. is high speed is due to the ability of metals to easily take in arriving charges and to let others depart. e ability for rapid reaction is due to the high mobility of the charges inside metals, which in turn is due to the small mass and size of these charges, the electrons.
e high signal speed in metals appears to contradict another determination. e dri speed of the electrons in a metal wire obviously obeys
v
=
I Ane
,
(449)
Challenge 1043 n
where I is the current, A the cross-section of the wire, e the charge of a single electron and n the number density of electrons. e electron density in copper is . ë m− . Using a typical current of . A and a typical cross-section of a square millimetre, we get a dri speed of . µm s. In other words, electrons move a thousand times slower than ketchup inside its bottle. Worse, if a room lamp used direct current instead of alternate current, the electrons would take several days to get from the switch to the bulb! Nevertheless, the lamp goes on or o almost immediately a er the switch is activated. Similarly, the electrons from an email transported with direct current would arrive much later than a paper letter sent at the same time; nevertheless, the email arrives quickly. Are you able to explain the apparent contradiction between dri velocity and signal velocity?
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Page 754
Inside liquids, charges move with a di erent speed from that inside metals, and their charge to mass ratio is also di erent. We all know this from direct experience. Our nerves work by using electric signals and take (only) a few milliseconds to respond to a stimulus, even though they are metres long. A similar speed is observed inside semiconductors and inside batteries. In all these systems, moving charge is transported by ions; they are charged atoms. Ions, like atoms, are large and composed entities, in contrast to the tiny electrons.
In other systems, charges move both as electrons and as ions. Examples are neon lamps, re, plasmas and the Sun. Inside atoms, electrons behave even more strangely. One tends to think that they orbit the nucleus (as we will see later) at a rather high speed, as the orbital radius is so small. However, it turns out that in most atoms many electrons do not orbit the nucleus at all. e strange story behind atoms and their structure will be told in the second part of our mountain ascent.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
C Charge discreteness is one of the central results of physics.
** Challenge 1044 n How would you show experimentally that electrical charge comes in smallest chunks?
**
e discreteness of charge implies that one can estimate the size of atoms by observing Challenge 1045 ny galvanic deposition. How?
Page 897 Ref. 555
Challenge 1046 ny
**
Cosmic radiation consists of charged particles hitting the Earth. (We will discuss this in more detail later.) Astrophysicists explain that these particles are accelerated by the magnetic elds around the Galaxy. However, the expression of the Lorentz acceleration shows that magnetic elds can only change the direction of the velocity of a charge, not its magnitude. How can nature get acceleration nevertheless?
**
What would be the potential of the Earth in volt if we could take away all the electrons of Challenge 1047 n a drop of water?
**
When a voltage is applied to a resistor, how long does it take until the end value of the current, given by Ohm’s ‘law’, is reached? e rst to answer this question was Paul Drude.* in the years around 1900. He reasoned that when the current is switched on, the speed v of an electron increases as v = (eE m)t, where E is the electrical eld, e the charge and m the mass of the electron. Drude’s model assumes that the increase of electron speed stops
* Paul Karl Ludwig Drude (1863–1906), German physicist. A result of his electron gas model of metals was the prediction, roughly correct, that the ratio between the thermal conductivity and the electronic conductivity at a given temperature should be the same for all metals. Drude also introduced c as the symbol for the speed of light.
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when the electron hits an atom, loses its energy and begins to be accelerated again. Drude Challenge 1048 ny deduced that the average time τ up to the collision is related to the speci c resistance by
ρ
=
m τe n
,
(450)
with n being the electron number density. Inserting numbers for copper (n = . ë m− and ρ = . ë − Ωm), one gets a time τ = ps. is time is so
short that the switch-on process can usually be neglected.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Classical electromagnetism and light are almost endless topics. Some aspects are too beautiful to be missed.
**
Since light is a wave, something must happen if it is directed to a hole less than its Challenge 1049 n wavelength in diameter. What exactly happens?
**
Electrodynamics shows that light beams always push; they never pull. Can you con rm Challenge 1050 e that ‘tractor beams’ are impossible in nature?
**
It is well known that the glowing material in light bulbs is tungsten wire in an inert gas. is was the result of a series of experiments that began with the grandmother of all
lamps, namely the cucumber. e older generation knows that a pickled cucumber, when attached to the V of the mains, glows with a bright green light. (Be careful; the experiment is dirty and somewhat dangerous.)
Challenge 1051 n Ref. 556
**
If you calculate the Poynting vector for a charged magnet – or simpler, a point charge near a magnet – you get a surprising result: the electromagnetic energy ows in circles around the magnet. How is this possible? Where does this angular momentum come from?
Worse, any atom is an example of such a system – actually of two such systems. Why is this e ect not taken into account in calculations in quantum theory?
**
Ohm’s law, the observation that for almost all materials the current is proportional to the voltage, is due to a school teacher. Georg Simon Ohm* explored the question in great depth; in those days, such measurements were di cult to perform. is has changed now.
* Georg Simon Ohm (b. 1789 Erlangen, d. 1854 München), Bavarian school teacher and physicist. His efforts were recognized only late in his life, and he eventually was promoted to professor at the University in München. Later the unit of electrical resistance, the proportionality factor between voltage and current, was named a er him.
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C1
C2
F I G U R E 272 Capacitors in series
insulators
high tension line
wires
neon lamp F I G U R E 273 Small neon lamps on a high voltage cable
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Recently, even the electrical resistance of single atoms has been measured: in the case of Ref. 557 xenon it turned out to be about Ω. It was also found that lead atoms are ten times Challenge 1052 ny more conductive than gold atoms. Can you imagine why?
Ref. 558 Challenge 1053 n
**
e charges on two capacitors in series are not generally equal, as naive theory states. For perfect, leak-free capacitors the voltage ratio is given by the inverse capacity ratio V V = C C , due to the equality of the electric charges stored. is is easily deduced from Figure 272. However, in practice this is only correct for times between a few and a few dozen minutes. Why?
Challenge 1054 d
**
Does it make sense to write Maxwell’s equations in vacuum? Both electrical and magnetic elds require charges in order to be measured. But in vacuum there are no charges at all.
In fact, only quantum theory solves this apparent contradiction. Are you able to imagine how?
**
Ref. 559 Grass is usually greener on the other side of the fence. Can you give an explanation based Challenge 1055 n on observations for this statement?
**
e maximum force in nature limits the maximum charge that a black hole can carry. Challenge 1056 ny Can you nd the relation?
Challenge 1057 ny
**
On certain high voltage cables leading across the landscape, small neon lamps shine when the current ows, as shown in Figure 273. (You can see them from the train when riding from Paris to the Roissy airport.) How is this possible?
**
Challenge 1058 n
‘Inside a conductor there is no electric eld.’ is statement is o en found. In fact the truth is not that simple. First, a static eld or a static charge on the metal surface of a body does not in uence elds and charges inside it. A closed metal surface thus forms a shield against an electric eld. Can you give an explanation? In fact, a tight metal layer is
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Ref. 560 Challenge 1059 ny
not required to get the e ect; a cage is su cient. One speaks of a Faraday cage. e detailed mechanism allows you to answer the following question: do Faraday cages
for gravity exist? Why? For moving external elds or charges, the issue is more complex. Fields due to accel-
erated charges – radiation elds – decay exponentially through a shield. Fields due to charges moving at constant speed are strongly reduced, but do not disappear. e reduction depends on the thickness and the resistivity of the metal enclosure used. For sheet metal, the eld suppression is very high; it is not necessarily high for metal sprayed plastic. Such a device will not necessarily survive a close lightning stroke.
In practice, there is no danger if lightning hits an aeroplane or a car, as long they are made of metal. ( ere is one lm on the internet of a car hit by lightning; the driver does not even notice.) However, if your car is hit by lightning in dry weather, you should wait a few minutes before getting out of it. Can you imagine why?
Faraday cages also work the other way round. (Slowly) changing electric elds changing that are inside a Faraday cage are not felt outside. For this reason, radios, mobile phones and computers are surrounded by boxes made of metal or metal-sprayed plastics.
e metal keeps the so-called electromagnetic smog to a minimum. ere are thus three reasons to surround electric appliances by a grounded shield:
to protect the appliance from outside elds, to protect people and other machines from electromagnetic smog, and to protect people against the mains voltage accidentally being fed into the box (for example, when the insulation fails). In high precision experiments, these three functions can be realized by three separate cages.
For purely magnetic elds, the situation is more complex. It is quite di cult to shield the inside of a machine from outside magnetic elds. How would you do it? In practice one o en uses layers of so-called mu-metal; can you guess what this material does?
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**
Electric polarizability is the property of matter responsible for the deviation of water owing from a tap caused by a charged comb. It is de ned as the strength of electric dipole induced by an applied electric eld. e de nition simply translates the observation that many objects acquire a charge when an electric eld is applied. Incidentally, how precisely combs get charged when rubbed, a phenomenon called electri cation, is still one of the mysteries of modern science.
Challenge 1060 ny
**
A pure magnetic eld cannot be transformed into a pure electric eld by change of observation frame. e best that can be achieved is a state similar to an equal mixture of magnetic and electric elds. Can you provide an argument elucidating this relation?
**
Ref. 561 Challenge 1061 ny
Researchers are trying to detect tooth decay with the help of electric currents, using the observation that healthy teeth are bad conductors, in contrast to teeth with decay. How would you make use of this e ect in this case? (By the way, it might be that the totally unrelated technique of imaging with terahertz waves could yield similar results.)
**
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 1062 ny
A team of camera men in the middle of the Sahara were using battery-driven electrical equipment to make sound recordings. Whenever the microphone cable was a few tens of metres long, they also heard a Hz power supply noise, even though the next power supply was thousands of kilometres away. An investigation revealed that the high voltage lines in Europe lose a considerable amount of power by irradiation; these Hz waves are re ected by the ionosphere around the Earth and thus can disturb recording in the middle of the desert. Can you estimate whether this observation implies that living directly near a high voltage line is dangerous?
**
Ref. 562 When two laser beams cross at a small angle, they can form light pulses that seem to move Challenge 1063 n faster than light. Does this contradict special relativity?
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It is said that astronomers have telescopes so powerful that they can see whether someChallenge 1064 ny body is lighting a match on the Moon. Can this be true?
**
When solar plasma storms are seen on the Sun, astronomers rst phone the electricity company. ey know that about 24 to 48 hours later, the charged particles ejected by the storms will arrive on Earth, making the magnetic eld on the surface uctuate. Since power grids o en have closed loops of several thousands of kilometres, additional electric currents are induced, which can make transformers in the grid overheat and then switch o . Other transformers then have to take over the additional power, which can lead to their overheating, etc. On several occasions in the past, millions of people have been le without electrical power due to solar storms. Today, the electricity companies avoid the problems by disconnecting the various grid sections, by avoiding large loops, by reducing the supply voltage to avoid saturation of the transformers and by disallowing load transfer from failed circuits to others.
**
Is it really possible to see stars from the bottom of a deep pit or of a well, even during the Challenge 1065 n day, as is o en stated?
**
If the electric eld is described as a sum of components of di erent frequencies, its soRef. 563 called Fourier components, the amplitudes are given by
∫ Eˆ(k, t) = ( π)
E(x, t)e−ikx d x
(451)
and similarly for the magnetic eld. It then turns out that a Lorentz invariant quantity N, describing the energy per circular frequency ω, can be de ned:
∫ N = π
E(k, t) + B(k, t) ck
d k.
(452)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 274 How natural colours (top) change for three types of colour blind: deutan, protan and tritan (© Michael Douma)
Challenge 1066 n Can you guess what N is physically? (Hint: think about quantum theory.)
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Challenge 1067 ny
Faraday discovered how to change magnetism into electricity, knowing that electricity could be transformed into magnetism. ( e issue is subtle. Faraday’s law is not the dual of Ampère’s, as that would imply the use of magnetic monopoles; neither is it the reciprocal, as that would imply the displacement current. But he was looking for a link and he found a way to relate the two observations – in a novel way, as it turned out.) Faraday also discovered how to transform electricity into light and into chemistry. He then tried to change gravitation into electricity. But he was not successful. Why not?
Challenge 1068 n
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At high altitudes above the Earth, gases are completely ionized; no atom is neutral. One speaks of the ionosphere, as space is full of positive ions and free electrons. Even though both charges appear in exactly the same number, a satellite moving through the ionosphere acquires a negative charge. Why? How does the charging stop?
Challenge 1069 n
**
A capacitor of capacity C is charged with a voltage U. e stored electrostatic energy is E = CU . e capacitor is then detached from the power supply and branched on to an empty capacitor of the same capacity. A er a while, the voltage obviously drops to U . However, the stored energy now is C(U ) , which is half the original value. Where did the energy go?
**
Colour blindness was discovered by the great English scientist John Dalton (1766–1844) Challenge 1070 ny – on himself. Can you imagine how he found out? It a ects, in all its forms, one in 20 men.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 275 Cumulonimbus clouds from ground and from space (NASA)
In many languages, a man who is colour blind is called daltonic. Women are almost never Ref. 564 daltonic, as the property is linked to defects on the X chromosome. If you are colour blind,
you can check to which type you belong with the help of Figure 274.
** Perfectly spherical electromagnetic waves are impossible in nature. Can you show this Challenge 1071 n using Maxwell’s equation of electromagnetism, or even without them?
** Light beams, such as those emitted from lasers, are usually thought of as lines. However, light beams can also be tubes. Tubular laser beams, or Bessel beams of high order, are used in modern research to guide plasma channels.
I
?–E
Ref. 566 Page 521
Looking carefully, the atmosphere is full of electrical e ects. e most impressive electrical phenomenon we observe, lightning, is now reasonably well understood. Inside a thunderstorm cloud, especially inside tall cumulonimbus clouds,* charges are separated by collision between the large ‘graupel’ ice crystals falling due to their weight and the small ‘hail’ ice crystallites rising due to thermal upwinds. Since the collision takes part in an electric eld, charges are separated in a way similar to the mechanism in the Kelvin generator. Discharge takes place when the electric eld becomes too high, taking a strange path in uenced by ions created in the air by cosmic rays. It seems that cosmic rays are at least partly
Ref. 565
* Clouds have Latin names. ey were introduced in 1802 by the English explorer Luke Howard (1772– 1864), who found that all clouds could be seen as variations of three types, which he called cirrus, cumulus and stratus. He called the combination of all three, the rain cloud, nimbus (from the Latin ‘big cloud’). Today’s internationally agreed system has been slightly adjusted and distinguishes clouds by the height of their lower edge. e clouds starting above a height of km are the cirrus, the cirrocumulus and the cirrostratus; those starting at heights of between 2 and km are the altocumulus, the altostratus and the nimbostratus; clouds starting below a height of km are the stratocumulus, the stratus and the cumulus. e rain or thunder cloud, which crosses all heights, is today called cumulonimbus.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 568 Challenge 1072 n
responsible for the zigzag shape of lightning.* Lightning ashes have strange properties. First, they appear at elds around kV m (at low altitude) instead of the MV m of normal sparks. Second, lightning emits radio pulses. ird, they emit gamma rays. Russian researchers, from onwards explained all three e ects by a newly discovered discharge mechanism. At length scales of m and more, cosmic rays can trigger the appearance of lightning; the relativistic energy of these rays allows for a discharge mechanism that does not exist for low energy electrons. At relativistic energy, so-called runaway breakdown leads to discharges at much lower elds than usual laboratory sparks. e multiplication of these relativistic electrons also leads to the observed radio and gamma ray emissions.
Incidentally, you have a % chance of survival a er being hit by lightning, especially if you are completely wet, as in that case the current will ow outside the skin. Usually, wet people who are hit loose all their clothes, as the evaporating water tears them o . Rapid resuscitation is essential to help somebody to recover a er a hit.**
As a note, you might know how to measure the distance of a lightning by counting the seconds between the lightning and the thunder and multiplying this by the speed of sound, m s; it is less well known that one can estimate the length of the lightning bolt by measuring the duration of the thunder, and multiplying it by the same factor.
In the s more electrical details about thunderstorms became known. Airline pilots and passengers sometime see weak and coloured light emissions spreading from the top of thunderclouds. ere are various types of such emissions: blue jets and mostly red sprites and elves, which are somehow due to electric elds between the cloud top and the ionosphere. e details are still under investigation, and the mechanisms are not yet clear.***
All these details are part of the electrical circuit around the Earth. is fascinating part of geophysics would lead us too far from the aim of our mountain ascent. But every physicist should know that there is a vertical electric eld of between and V m on a clear day, as discovered already in . (Can you guess why it is not noticeable in everyday life? And why despite its value it cannot be used to extract large amounts of energy?) e
eld is directed from the ionosphere down towards the ground; in fact the Earth is permanently negatively charged, and in clear weather current ows downwards through the clear atmosphere, trying to discharge our planet. e current of about kA is spread over the whole planet; it is possible due to the ions formed by cosmic radiation. ( e resistance between the ground and the ionosphere is about Ω, so the total voltage drop is about
kV.) At the same time, the Earth is constantly being charged by several e ects: there is a dynamo e ect due to the tides of the atmosphere and there are currents induced by the magnetosphere. But the most important e ect is lightning. In other words, contrary
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Ref. 567
* ere is no ball lightning even though there is a Physics Report about it. Ball lightning is one of the favourite myths of modern pseudo-science. Actually, they would exist if we lived in a giant microwave oven. To show this, just stick a toothpick into a candle, light the toothpick, and put it into (somebody else’s) microwave at maximum power. ** If you are ever hit by lightning and survive, go to the hospital! Many people died three days later having failed to do so. A lightning strike o en leads to coagulation e ects in the blood. ese substances block the kidneys, and one can die three days later because of kidney failure. e remedy is to have dialysis treatment. *** For images, have a look at the interesting http://sprite.gi.alaska.edu/html/sprites.htm, http://www. fma-research.com/spriteres.htm and http://paesko.ee.psu.edu/Nature websites.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 569 Ref. 570
Ref. 571 Ref. 572
to what one may think, lightning does not discharge the ground, it actually charges it up!* Of course, lightning does discharge the cloud to ground potential di erence; but by doing so, it actually sends a negative charge down to the Earth as a whole. underclouds are batteries; the energy from the batteries comes from the the thermal upli s mentioned above, which transport charge against the global ambient electrical eld.
Using a few electrical measurement stations that measure the variations of the electrical eld of the Earth it is possible to locate the position of all the lightning that comes down towards the Earth at a given moment. Present research also aims at measuring the activity of the related electrical sprites and elves in this way.
e ions in air play a role in the charging of thunderclouds via the charging of ice crystals and rain drops. In general, all small particles in the air are electrically charged. When aeroplanes and helicopters y, they usually hit more particles of one charge than of the other. As a result, aeroplanes and helicopters are charged up during ight. When a helicopter is used to rescue people from a ra in high seas, the rope pulling the people upwards must rst be earthed by hanging it in the water; if this is not done, the people on the ra could die from an electrical shock when they touch the rope, as has happened a few times in the past.
e charges in the atmosphere have many other e ects. Recent experiments have conrmed what was predicted back in the early twentieth century: lightning emits X-rays. e con rmation is not easy though; it is necessary to put a detector near the lightning ash. To achieve this, the lightning has to be directed into a given region. is is possible using a missile pulling a metal wire, the other end of which is attached to the ground. ese experimental results are now being collated into a new description of lightning which also explains the red-blue sprites above thunderclouds. In particular, the processes also imply that inside clouds, electrons can be accelerated up to energies of a few MeV.
Why are sparks and lightning blue? is turns out to be a material property: the colour comes from the material that happens to be excited by the energy of the discharge, usually air. is excitation is due to the temperature of kK inside the channel of a typical lightning ash. For everyday sparks, the temperature is much lower. Depending on the situation, the colour may arise from the gas between the two electrodes, such as oxygen or nitrogen, or it may due to the material evaporated from the electrodes by the discharge. For an explanation of such colours, as for the explanation of all colours due to materials, we need to wait for the next part of our walk.
But not only electric elds are dangerous. Also time-varying electromagnetic elds can be. In , in beautiful calm weather, a Dutch hot air balloon approached the powerful radio transmitter in Hilversum. A er travelling for a few minutes near to the antenna, the gondola suddenly detached from the balloon, killing all the passengers inside.
An investigation team reconstructed the facts a few weeks later. In modern gas balloons the gondola is suspended by high quality nylon ropes. To avoid damage by lightning and in order to avoid electrostatic charging problems all these nylon ropes contain thin metal wires which form a large equipotential surface around the whole balloon. Unfortunately, in the face of the radio transmitter, these thin metal wires absorbed radio energy from the transmitter, became red hot, and melted the nylon wires. It was the rst time that this had ever been observed.
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Challenge 1073 ny * e Earth is thus charged to about − MC. Can you con rm this?
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D
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Ref. 573 Ref. 574
We learned in the section on general relativity that gravitation has the same e ects as acceleration. is means that a charge kept xed at a certain height is equivalent to a charge accelerated by . m s , which would imply that it radiates electromagnetically, since all accelerated charges radiate. However, the world around us is full of charges at
xed heights, and there is no such radiation. How is this possible? e question has been a pet topic for many years. Generally speaking, the concept of
radiation is not observer invariant: If one observer detects radiation, a second one does not necessarily do so as well. e exact way a radiation eld changes from one observer to the other depends on the type of relative motion and on the eld itself.
A precise solution of the problem shows that for a uniformly accelerated charge, an observer undergoing the same acceleration only detects an electrostatic eld. In contrast, an inertial observer detects a radiation eld. Since gravity is (to a high precision) equivalent to uniform acceleration, we get a simple result: gravity does not make electrical charges radiate for an observer at rest with respect to the charge, as is observed. e results holds true also in the quantum theoretical description.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Ref. 505 Challenge 1074 ny
e classical description of electrodynamics is coherent and complete; nevertheless there
are still many subjects of research. Here are a few of them.
e origin of magnetic eld of the Earth, the other
planets, the Sun and even of the galaxy is a fascinating topic. e way that the convection of uids inside the
ocean crust
planets generates magnetic elds, an intrinsically three-
dimensional problem, the in uence of turbulence, of
mantle
nonlinearities and of chaos makes it a surprisingly com-
plex question.
liquid core
e details of the generation of the magnetic eld of the Earth, usually called the geodynamo, began to
solid core
appear only in the second half of the twentieth century,
when the knowledge of the Earth’s interior reached a
su cient level. e Earth’s interior starts below the
Earth’s crust. e crust is typically to km thick
(under the continents), though it is thicker under high F I G U RE 276 The structure of our
mountains and thinner near volcanoes or under the planet
oceans. As already mentioned, the crust consists of
large segments, the plates, that move with respect to one other. e Earth’s interior is
divided into the mantle – the rst km from the surface – and the core. e core is
made up of a liquid outer core, km thick, and a solid inner core of km radius.
( e temperature of the core is not well known; it is believed to be to kK. Can you
nd a way to determine it? e temperature might have decreased a few hundred kelvin
during the last million years.)
e Earth’s core consists mainly of iron that has been collected from the asteroids that
collided with the Earth during its youth. It seems that the liquid and electrically con-
ducting outer core acts as a dynamo that keeps the magnetic eld going. e magnetic
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 576 Challenge 1075 ny
energy comes from the kinetic energy of the outer core, which rotates with respect to the
Earth’s surface; the uid can act as a dynamo because, apart from rotating, it also convects
from deep inside the Earth to more shallow depths, driven by the temperature gradients
between the hot inner core and the cooler mantle. Huge electric currents ow in com-
plex ways through these liquid layers, maintained by friction, and create the magnetic
eld. Why this eld switches orientation at irregular intervals of between a few tens of
thousands and a few million years, is one of the central questions. e answers are dif-
cult; experiments are not yet possible, years of measurements is a short time when
compared with the last transition – about
years ago – and computer simulations
are extremely involved. Since the eld measurements started, the dipole moment of the
magnetic eld has steadily diminished, presently by % a year, and the quadrupole mo-
ment has steadily increased. Maybe we are heading towards a surprise.* (By the way, the
study of galactic magnetic elds is even more complex, and still in its infancy.)
Another important puzzle about electricity results from the equivalence of mass and
energy. It is known from experiments that the size d of electrons is surely smaller than
− m. is means that the electric eld surrounding it has an energy content E given
by at least
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∫ ∫ Energy = ε
Electric field dV = ε
d
(
πεo
q r
)
πr dr
=
q πεo d
. µJ .
(453)
Ref. 577 Page 605
On the other hand, the mass of an electron, usually given as keV c , corresponds to an energy of only fJ, ten million times less than the value just calculated. In other words, classical electrodynamics has considerable di culty describing electrons. In fact, a consistent description of charged point particles within classical electrodynamics is impossible. is pretty topic receives only a rare – but then o en passionate – interest nowadays, because the puzzle is solved in a di erent way in the upcoming, second part of our mountain ascent.
Even though the golden days of materials science are over, the various electromagnetic properties of matter and their applications in devices do not seem to be completely explored yet. About once a year a new e ect is discovered that merits inclusion in the list of electromagnetic matter properties of Table . Among others, some newer semiconductor technologies will still have an impact on electronics, such as the recent introduction of low cost light detecting integrated circuits built in CMOS (complementary metal oxide silicon) technology.
e building of light sources of high quality has been a challenge for many centuries and remains one for the future. Light sources that are intense, tunable and with large coherence length or sources that emit extreme wavelengths are central to many research pursuits. As one example of many, the rst X-ray lasers have recently been built; however, they are several hundred metres in size and use modi ed particle accelerators. e construction of compact X-ray lasers is still many years o – if it is possible at all.
* In 2005, it has been reported that the inner core of the Earth seems to rotate faster than the Earth’s crust Ref. 575 by up to half a degree per year.
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Ref. 578 Challenge 1076 r
Electrodynamics and general relativity interact in many ways. Only a few cases have been studied up to now. ey are important for black holes and for empty space. For example, it seems that magnetic elds increase the sti ness of empty space. Many such topics will appear in the future.
But maybe the biggest challenge imaginable in classical electrodynamics is to decode the currents inside the brain. Will it be possible to read our thoughts with an apparatus placed outside the head? One could start with a simpler challenge: Would it be possible to distinguish the thought ‘yes’ from the thought ‘no’ by measuring electrical or magnetic
elds around the head? In other words, is mind-reading possible? Maybe the twenty- rst century will come up with a positive answer. If so, the team performing the feat will be instantly famous.
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Ref. 579 Ref. 580 Ref. 581
Ref. 582
We have seen that it is possible to move certain objects without touching them, using a magnetic or electric eld or, of course, using gravity. Is it also possible, without touching an object, to keep it xed, oating in mid-air? Does this type of rest exist?
It turns out that there are several methods of levitating objects. ese are commonly divided into two groups: those that consume energy and those who do not. Among the methods that consume energy is the oating of objects on a jet of air or of water, the oating of objects through sound waves, e.g. on top of a siren, or through a laser beam coming from below, and the oating of conducting material, even of liquids, in strong radiofrequency elds. Levitation of liquids or solids by strong ultrasound waves is presently becoming popular in laboratories. All these methods give stationary levitation. Another group of energy consuming methods sense the way a body is falling and kick it up again in the right way via a feedback loop; these methods are non-stationary and usually use magnetic elds to keep the objects from falling. e magnetic train being built in Shanghai by a German consortium is levitated this way. e whole train, including the passengers, is levitated and then moved forward using electromagnets. It is thus possible, using magnets, to levitate many tens of tonnes of material.
For levitation methods that do not consume energy – all such methods are necessarily stationary – a well-known limitation can be found by studying Coulomb’s ‘law’ of electrostatics: no static arrangement of electric elds can levitate a charged object in free space or in air. e same result is valid for gravitational elds and massive objects;* in other words, we cannot produce a local minimum of potential energy in the middle of a box using electric or gravitational elds. is impossibility is called Earnshaw’s theorem. Speaking mathematically, the solutions of the Laplace equation ∆φ = , the so-called harmonic functions, have minima or maxima only at the border, and never inside the domain of de nition. (You proved this yourself on page .) e theorem can also be proved by noting that given a potential minimum in free space, Gauss’ theorem for a sphere around that minimum requires that a source of the eld be present inside, which is in contradiction with the original assumption.
Page 80 * To the disappointment of many science- ction addicts, this would even be true if a negative mass existed. And even though gravity is not really due to a eld, but to space-time curvature, the result still holds in general relativity.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
We can deduce that it is also impossible to use electric elds to levitate an electrically neutral body in air: the potential energy U of such a body, with volume V and dielectric constant ε, in an environment of dielectric constant ε , is given by
U V
=−
(ε − ε )E
.
(454)
Challenge 1077 ny Challenge 1079 ny
Since the electric eld E never has a maximum in the absence of space charge, and since for all materials ε ε , there cannot be a minimum of potential energy in free space for a neutral body.*
To sum up, using static electric or static gravitational elds it is impossible to keep an object from falling; neither quantum mechanics, which incorporates phenomena such as antimatter, nor general relativity, including phenomena such as black holes, change this basic result.
For static magnetic elds, the argument is analogous to electrical elds: the potential energy U of a magnetizable body of volume V and permeability µ in a medium with permeability µ containing no current is given by
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U V
=−
(µ − µ
)B
(455)
Challenge 1080 ny Ref. 584
Ref. 581 Ref. 585
Ref. 586 Challenge 1081 ny
and due to the inequality ∆B , isolated maxima of a static magnetic eld are not possible, only isolated minima. erefore, it is impossible to levitate paramagnetic (µ µo) or ferromagnetic (µ µ ) materials such as steel, including bar magnets, which are all attracted, and not repelled to magnetic eld maxima.
ere are thus two ways to get magnetic levitation: levitating a diamagnet or using a time dependent eld. Diamagnetic materials (µ < µo) can be levitated by static magnetic
elds because they are attracted to magnetic eld minima; the best-known example is the levitation of superconductors, which are, at least those of type I, perfects diamagnets (µ = ). Strong forces can be generated, and this method is also being tested for the levitation of passenger trains in Japan. In some cases, superconductors can even be suspended in mid-air, below a magnet. Single atoms with a magnetic moment are also diamagnets; they are routinely levitated this way and have also been photographed in this state.
Also single neutrons, which have a magnetic dipole moment, have been kept in magnetic bottles in this way, until they decay. Recently, scientists have levitated pieces of wood, plastic, strawberries, water droplets, liquid helium droplets as large as cm, grasshoppers,
sh and frogs (all alive and without any harm) in this way. ey are, like humans, all made of diamagnetic material. Humans themselves have not yet been levitated, but the feat is being planned and worked on.
Diamagnets levitate if ∇B µ ρ χ, where ρ is the mass density of the object and χ = − µ µ its magnetic susceptibility. Since χ is typically about − and ρ of order
kg m , eld gradients of about T m are needed. In other words, levitation re-
Ref. 583 * It is possible, however, to ‘levitate’ gas bubbles in liquids – ‘trap’ them to prevent them from rising would be a better expression – because in such a case the dielectric constant of the environment is higher than that
Challenge 1078 ny of the gas. Can you nd a liquid–gas combination where bubbles fall instead of rise?
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F I G U R E 277 Trapping a metal sphere using a variable speed drill and a plastic saddle
F I G U R E 278 Floating ‘magic’ nowadays available in toy shops
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Ref. 579 Ref. 579 Ref. 587 Ref. 588 Ref. 589
Page 922
quires elds changes of T over cm, which is nowadays common for high eld laboratory magnets.
Finally, time dependent electrical or magnetic elds, e.g. periodic elds, can lead to levitation in many di erent ways without any consumption of energy. is is one of the methods used in the magnetic bearings of turbomolecular vacuum pumps. Also single charged particles, such as ions and electrons, are now regularly levitated with Paul traps and Penning traps. e mechanical analogy is shown in Figure .
Figure shows a toy that allows you to personally levitate a spinning top in mid-air above a ring magnet, a quite impressive demonstration of levitation for anybody looking at it. It is not hard to build such a device yourself.
Even free electrons can be levitated, letting them oat above the surface of uid helium. In the most recent twist of the science of levitation, in Stephen Haley predicted that the suspension height of small magnetic particles above a superconducting ring should be quantized. However, the prediction has not been veri ed by experiment yet.
For the sake of completeness we mention that nuclear forces cannot be used for levitation in everyday life, as their range is limited to a few femtometres. However, we will see later that the surface matter of the Sun is prevented from falling into the centre by these interactions; we could thus say that it is indeed levitated by nuclear interactions.
M
,
Ref. 590
e levitation used by magicians mostly falls into another class. When David Coppereld, a magician performing for the MTV generation at the end of the twentieth century, ‘ ies’ during his performances, he does so by being suspended on thin shing lines that are rendered invisible by clever lighting arrangements. In fact, if we want to be precise, we should count shing lines, plastic bags, as well as every table and chair as levitation devices. (Journalists would even call them ‘anti-gravity’ devices.) Contrary to our impression, a hanging or lying object is not really in contact with the suspension, if we look at the critical points with a microscope.* More about this in the second part of our walk.
Challenge 1082 ny * e issue is far from simple: which one of the levitation methods described above is used by tables or chairs?
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Ref. 591 Challenge 1083 r
But if this is the case, why don’t we fall through a table or through the oor? We started the study of mechanics by stating that a key property of matter its solidity, i.e. the impossibility of having more than one body at the same place at the same time. But what is the origin of solidity? Again, we will be able to answer the question only in the second part of our adventure, but we can already collect the rst clues at this point.
Solidity is due to electricity. Many experiments show that matter is constituted of charged particles; indeed, matter can be moved and in uenced by electromagnetic elds in many ways. Over the years, material scientists have produced a long list of such e ects, all of which are based on the existence of charged constituents. Can you nd or imagine a new one? For example, can electric charge change the colour of objects?
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TA B L E 50 Selected matter properties related to electromagnetism, showing among other things the role it plays in the constitution of matter; at the same time a short overview of atomic, solid state, fluid and business physics
P
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thermal radiation or heat every object radiation or incandescence
temperature-dependent radiation emitted by any macroscopic amount of matter
Interactions with charges and currents
electri cation
separating metals from spontaneous charging insulators
triboelectricity
glass rubbed on cat fur charging through rubbing
barometer light
mercury slipping along gas discharge due to triboelectricity Ref. 592 glass
insulation
air
no current ow below critical voltage drop
semiconductivity
diamond, silicon or current ows only when material is impure
gallium arsenide
(‘doped’)
conductivity
copper, metals
current ows easily
superconductivity
niobium
current ows inde nitely
ionization
re ames
current ows easily
localization (weak, Anderson)
disordered solids
resistance of disordered solids
resistivity, Joule e ect
graphite
heating due to current ow
thermoelectric e ects: ZnSb, PbTe, PbSe, Peltier e ect, Seebeck e ect, BiSeTe, Bi Te , etc.
omson e ect
cooling due to current ow, current ow due to temperature di erence, or due to temperature gradients
acoustoelectric e ect
CdS
sound generation by currents, and vice versa
magnetoresistance
iron, metal multilayers resistance changes with applied magnetic eld Ref. 593
recombination
re alarms
charge carriers combine to make neutral atoms or molecules
annihilation
positron tomography particle and antiparticle, e.g. electron and positron, disappear into photons
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Penning e ect
Ne, Ar
Richardson e ect, thermal BaO , W, Mo, used in
emission
tv and electron
microscopes
skin e ect
Cu
pinch e ect
InSb, plasmas
Josephson e ect
Nb-Oxide-Nb
Sasaki–Shibuya e ect
n-Ge, n-Si
switchable magnetism InAs:Mn
ionization through collision with metastable atoms emission of electrons from hot metals
high current density on exterior of wire high current density on interior of wire tunnel current ows through insulator between two superconductors anisotropy of conductivity due to applied electric eld voltage switchable magnetization Ref. 594
Interactions with magnetic elds
Hall e ect Zeeman e ect
silicon; used for magnetic eld measurements
Cd
Paschen–Back e ect
atomic gases
ferromagnetism
Fe, Ni, Co, Gd
paramagnetism
Fe, Al, Mg, Mn, Cr
diamagnetism magnetostriction
water, Au, graphite, NaCl
CeB , CePd Al
magnetoelastic e ect
Fe, Ni
acoustomagnetic e ect spin valve e ect
metal alloys, anti-the stickers
metal multilayers
magneto-optical activity or int glass Faraday e ect or Faraday rotation
magnetic circular dichroism
gases
Majorana e ect
colloids
voltage perpendicular to current ow in applied magnetic eld
change of emission frequency with magnetic eld
change of emission frequency in strong magnetic elds
spontaneous magnetization; material strongly attracted by magnetic elds
induced magnetization parallel to applied eld; attracted by magnetic elds
induced magnetization opposed to applied eld; repelled by magnetic elds
change of shape or volume by applied magnetic eld
change of magnetization by tension or pressure
excitation of mechanical oscillations through magnetic eld
electrical resistance depends on spin direction of electrons with respect to applied magnetic eld
polarization angle is rotated with magnetic eld; di erent refraction index for right
and le circularly polarized light, as in magneto-optic (MO) recording
di erent absorption for right- and le -circularly polarized light; essentially the same as the previous one
speci c magneto-optic e ect
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photoelectromagnetic
InSb
e ect
current ow due to light irradiation of semiconductor in a magnetic eld
Voigt e ect
vapours
birefringence induced by applied magnetic eld
Cotton–Mouton e ect liquids
birefringence induced by applied magnetic eld
Hanle e ect
Hg
change of polarization of uorescence with magnetic eld
Shubnikov–de Haas e ect Bi
periodic change of resistance with applied magnetic eld
thermomagnetic e ects: BiSb alloys Ettinghausen e ect, Righi–Leduc e ect, Nernst e ect, magneto–Seebeck e ect
relation between temperature, applied elds and electric current
Ettinghausen–Nernst e ect Bi
appearance of electric eld in materials with temperature gradients in magnetic
elds
photonic Hall e ect
CeF
transverse light intensity depends on the applied magnetic eld Ref. 595
magnetocaloric e ect
gadolinium, GdSiGe material cools when magnetic eld is
alloys
switched o Ref. 596
cyclotron resonance
semiconductors, metals
selective absorption of radio waves in magnetic elds
magnetoacoustic e ect
semiconductors, metals
selective absorption of sound waves in magnetic elds
magnetic resonance
most materials, used selective absorption of radio waves in
for imaging in
magnetic elds
medicine for structure
determination of
molecules
magnetorheologic e ect
liquids, used in advanced car suspensions
change of viscosity with applied magnetic elds
Meissner e ect
type superconductors, expulsion of magnetic eld from used for levitation superconductors
Interactions with electric elds
polarizability
all matter
ionization, eld emission, all matter, tv Schottky e ect
paraelectricity
BaTiO
dielectricity
water
polarization changes with applied electric eld
charges are extracted at high elds
applied eld leads to polarization in same direction in opposite direction
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ferroelectricity
piezoelectricity
electrostriction
pyroelectricity
electro-osmosis or electrokinetic e ect electrowetting
electrolytic activity liquid crystal e ect electro-optical activity: Kerr e ect, Pockels e ect Freederichsz e ect, Schadt–Helfrichs e ect Stark e ect
eld ionization
Zener e ect
eld evaporation
BaTiO
spontaneous polarization below critical temperature
the quartz lighter used polarization appears with tension, stress, or
in the kitchen
pressure
platinum sponges in shape change with applied voltage Ref. 597 acids
CsNO , tourmaline, change of temperature produces charge crystals with polar separation axes; used for infrared detection
many ionic liquids liquid moves under applied electric eld
Ref. 598
salt solutions on gold wetting of surface depends on applied voltage
sulphuric acid
charge transport through liquid
watch displays
molecules turn with applied electric eld
liquids (e.g. oil), crystalline solids
material in electric eld rotates light polarization, i.e. produces birefringence
nematic liquid crystals electrically induced birefringence
hydrogen, mercury
helium near tungsten tips in eld ion microscope Si
W
colour change of emitted light in electric eld
ionization of gas atoms in strong electric elds
energy-free transfer of electrons into conduction band at high elds evaporation under strong applied electric
elds
Interactions with light
absorption
coal, graphite
transformation of light into heat or other energy forms (which ones?)Challenge 1084 n
blackness
coal, graphite
complete absorption in visible range
colour, metallic shine
ruby
absorption depending on light frequency
photostriction
PbLaZrTi
light induced piezoelectricity
photography
AgBr, AgI
light precipitates metallic silver
photoelectricity,
Cs
photoe ect
current ows into vacuum due to light irradiation
internal photoelectric e ect Si p–n junctions, solar voltage generation and current ow due to
cells
light irradiation
photon drag e ect
p-Ge
current induced by photon momentum
emissivity
all bodies
ability to emit light
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transparency
glass, quartz, diamond low re ection, low absorption, low scattering
re ectivity
metals
light bounces on surface
polarization
pulled polymer sheets light transmission depending on polarization angle
optical activity
sugar dissolved in water, quartz
rotation of polarization
birefringence
feldspar,cornea
refraction index depends on polarization direction, light beams are split into two beams
dichroism
feldspar, andalusite absorption depends on polarization
optically induced
AgCl
anisotropy, Weigert e ect
optically induced birefringence and dichroism
second harmonic generation
LiNbO , KPO
light partially transformed to double frequency
luminescence: general term GaAs, tv for opposite of incandescence
cold light emission
uorescence
CaF , X-ray
light emission during and a er light
production, light tubes, absorption or other energy input
cathode ray tubes
phosphorescence
TbCl
light emission due to light, electrical or chemical energy input, continuing long a er stimulation
electroluminescence
ZnS
emission of light due to alternating electrical eld
photoluminescence
ZnS Cu, SrAlO Eu, Dy, hyamine
light emission triggered by UV light, used in safety signs
chemoluminescence
H O , phenyl oxalate cold light emission used in light sticks for
ester, dye
divers and fun
bioluminescence
glow-worm, deep sea cold light emission in animals sh
triboluminescence
sugar
light emission during friction or crushing
thermoluminescence
quartz, feldspar
light emission during heating, used e.g. for
archaeological dating of pottery Ref. 599
bremsstrahlung
X-ray generation
radiation emission through fast deceleration of electrons
Compton e ect
momentum measurements
change of wavelength of light, esp. X-rays and gamma radiation, colliding with matter
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Čerenkov e ect
water, polymer particle light emission in a medium due to
detectors
particles, e.g. emitted by radioactive
processes, moving faster than the speed of
light in that medium
transition radiation
any material
light emission due to fast particles moving from one medium to a second with di erent refractive index
electrochromicity
wolframates
colour change with applied electric eld
scattering
gases, liquids
light changes direction
Mie scattering
dust in gases
light changes direction
Raleigh scattering
sky
light changes direction, sky is blue
Raman e ect or Smekal–Raman e ect
molecular gases
scattered light changes frequency
laser activity, superradiation
beer, ruby, He–Ne emission of stimulated radiation
sonoluminescence
air in water
light emission during cavitation
gravitoluminescence
does not exist; Challenge 1085 n why?
switchable mirror
LaH
voltage controlled change from re ection to transparency Ref. 600
radiometer e ect
bi-coloured windmills irradiation turns mill (see page )
luminous pressure
idem
irradiation turns mill directly
solar sail e ect
future satellites
motion due to solar wind
acoustooptic e ect
LiNbO
di raction of light by sound in transparent materials
photorefractive materials LiNbO , GaAs, InP light irradiation changes refractive index
Auger e ect
Auger electron
electron emission due to atomic
spectroscopy
reorganization a er ionization by X-rays
Bragg re ection
crystal structure determination
X-ray di raction by atomic planes
Mößbauer e ect
Fe, used for spectroscopy
recoil-free resonant absorption of gamma radiation
pair creation
Pb
transformation of a photon in a charged particle–antiparticle pair
photoconductivity
Se, CdS
change of resistivity with light irradiation
optoacoustic a ect, photoacoustic e ect
gases, solids
creation of sound due to absorption of pulsed light
optogalvanic e ect
plasmas
change of discharge current due to light irradiation
optical nonlinear e ects: parametric ampli cation, frequency mixing, saturable absorption, n-th harmonic generation, optical Kerr e ect, etc.
phase conjugated mirror gases activity
re ection of light with opposite phase
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Material properties solidity, impenetrability
oors, columns, ropes, at most one object per place at a given time buckets
Interactions with vacuum
Casimir e ect
metals
attraction of uncharged, conducting bodies
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Challenge 1086 e Ref. 601
All matter properties in the list can be in uenced by electric or magnetic elds or directly depend on them. is shows that the nature of all these material properties is electromagnetic. In other words, charges and their interactions are an essential and fundamental part of the structure of objects. e table shows so many di erent electromagnetic properties that the motion of charges inside each material must be complex indeed. Most e ects are the topic of solid state physics,* uid and plasma physics.
Solid state physics is by far the most important part of physics, when measured by the impact it has on society. Almost all e ects have applications in technical products, and give employment to many people. Can you name a product or business application for any randomly chosen e ect from the table?
In our mountain ascent however, we look at only one example from the above list: thermal radiation, the emission of light by hot bodies.
Earnshaw’s theorem about the impossibility of a stable equilibrium for charged particles at rest implies that the charges inside matter must be moving. For any charged particle in motion, Maxwell’s equations for the electromagnetic eld show that it radiates energy by emitting electromagnetic waves. In short, classical mechanics thus predicts that matter must radiate electromagnetic energy.
Interestingly, everybody knows from experience that this is indeed the case. Hot bodies light up depending on their temperature; the working of light bulbs thus proves that metals are made of charged particles. Incandescence, as it is called, requires charges. Actually, every body emits radiation, even at room temperature. is radiation is called thermal radiation; at room temperature it lies in the infrared. Its intensity is rather weak in everyday life; it is given by the general expression
I(T) = f T
πk ch
or I(T) = f σ T
with σ = . nW K m ,
(456)
Challenge 1087 n
Ref. 602 Challenge 1088 n
where f is a material-, shape- and temperature-dependent factor, with a value between zero and one, and is called the emissivity. e constant σ is called the Stefan–Boltzmann black body radiation constant or black body radiation constant. A body whose emissivity is given by the ideal case f = is called a black body, because at room temperature such a body also has an ideal absorption coe cient and thus appears black. (Can you see why?)
e heat radiation such a body emits is called black body radiation. By the way, which object radiates more energy: a human body or an average piece of the Sun of the same mass? Guess rst!
* Probably the best and surely the most entertaining introductory English language book on the topic is the
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W
?
Ref. 603
Challenge 1089 d Ref. 604
Physicists have a strange use of the term ‘black’. Most bodies at temperatures at which they are red hot or even hotter are excellent approximations of black bodies. For example, the tungsten in incandescent light bulbs, at around K, emits almost pure black body radiation; however, the glass then absorbs much of the ultraviolet and infrared components. Black bodies are also used to de ne the colour white. What we commonly call pure white is the colour emitted by a black body of K, namely the Sun. is de nition is used throughout the world, e.g. by the Commission Internationale d’Eclairage. Hotter black bodies are bluish, colder ones are yellow, orange or red.* e stars in the sky are classi ed in this way, as summarized on page .
Let us make a quick summary of black body radiation. Black body radiation has two important properties: rst, the emitted light power increases with the fourth power of the temperature. With this relation alone you can check the temperature of the Sun, mentioned above, simply by comparing the size of the Sun with the width of your thumb when your arm is stretched out in front of you. Are you able to do this? (Hint: use the excellent approximation that the Earth’s average temperature of about . °C is due to the Sun’s irradiation.)
e precise expression for the emitted energy density u per frequency ν can be deduced from the radiation law for black bodies discovered by Max Planck**
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u(ν, T) =
πh ν c ehν kT −
.
(457)
Page 705 Challenge 1090 ny
He made this important discovery, which we will discuss in more detail in the second
part of our mountain ascent, simply by comparing this curve with experiment. e new constant h, quantum of action or Planck’s constant, turns out to have the value . ë − Js, and is central to all quantum theory, as we will see. e other constant Planck introduced,
the Boltzmann constant k, appears as a prefactor of temperature all over thermodynamics, as it acts as a conversion unit from temperature to energy.
e radiation law gives for the total emitted energy density the expression
Challenge 1091 ny from which equation (
u(T) = T
πk ch
) is deduced using I = uc . (Why?)
(458)
one by N A
&D M
, Solid State Physics, Holt Rinehart & Winston, 1976.
* Most bodies are not black, because colour is not only determined by emission, but also by absorption of
light.
** Max Planck (1858–1947), professor of physics in Berlin, was a central gure in thermostatics. He dis-
covered and named Boltzmann’s constant k and the quantum of action h, o en called Planck’s constant. His
introduction of the quantum hypothesis gave birth to quantum theory. He also made the works of Einstein
known in the physical community, and later organized a job for him in Berlin. He received the Nobel Prize
for physics in 1918. He was an important gure in the German scienti c establishment; he also was one of
the very few who had the courage to tell Adolf Hitler face to face that it was a bad idea to re Jewish profess-
ors. (He got an outburst of anger as answer.) Famously modest, with many tragedies in his personal life, he
was esteemed by everybody who knew him.
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Figure to be inserted
F I G U R E 279 Bodies inside a oven at room temperature (left) and red hot (right)
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e second property of black body radiation is the value of the peak wavelength, i.e. the wavelength emitted with the highest intensity. is wavelength determines their colour; Challenge 1092 ny it is deduced from equation ( ) to be
λmax =
hc .
kT
=
. mm K T
but
ħνmax = .
kT = ( . ë
− J K) ë T . (459)
Challenge 1093 ny Challenge 1094 ny Challenge 1095 ny
Ref. 605 Challenge 1096 n
Either of these expressions is called Wien’s colour displacement a er its discoverer.* e colour change with temperature is used in optical thermometers; this is also the way the temperatures of stars are measured. For °C, human body temperature, it gives a peak wavelength of . µm or THz, which is therefore the colour of the bulk of the radiation emitted by every human being. ( e peak wavelength does not correspond to the peak frequency. Why?) On the other hand, following the telecommunication laws of many countries, any radiation emitter needs a licence to operate; it follows that strictly in Germany only dead people are legal, and only if their bodies are at absolute zero temperature.
Note that a black body or a star can be blue, white, yellow, orange or red. It is never green. Can you explain why?
Above, we predicted that any material made of charges emits radiation. Are you able to nd a simple argument showing whether heat radiation is or is not this classically predicted radiation?
But let us come back to the question in the section title. e existence of thermal radiation implies that any hot body will cool, even if it is le in the most insulating medium there is, namely in vacuum. More precisely, if the vacuum is surrounded by a wall, the temperature of a body in the vacuum will gradually approach that of the wall.
Interestingly, when the temperature of the wall and of the body inside have become the same, something strange happens. e e ect is di cult to check at home, but impressive photographs exist in the literature.
One arrangement in which walls and the objects inside them are at the same temperature is an oven. It turns out that it is impossible to see objects in an oven using the light coming from thermal radiation. For example, if an oven and all its contents are red hot, taking a picture of the inside of the oven (without a ash!) does not reveal anything; no contrast nor brightness changes exist that allow one to distinguish the objects from the walls or their surroundings. Can you explain the nding?
* Wilhelm Wien (b. 1864 Ga ken, d. 1824 München), East-Prussian physicist; he received the Nobel Prize for physics in 1911 for the discovery of this relation.
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In short, we are able to see each other only because the light sources we use are at a di erent temperature from us. We can see each other only because we do not live in thermal equilibrium with our environment.
A
In general, classical electrodynamics can be summarized in a few main ideas.
— e electromagnetic eld is a physical observable, as shown e.g. by compass needles. — e eld sources are the (moving) charges and the eld evolution is described by Max-
well’s evolution equations, as shown, for example, by the properties of amber, lodestone, batteries and remote controls. — e electromagnetic eld changes the motion of electrically charged objects via the Lorentz expression as, for example, shown by electric motors. — e eld behaves like a continuous quantity, a distribution of little arrows, and propagates as a wave, as shown, for example, by radios and mobile phones. — e eld can exist and move in empty space, as shown, for example, by the stars.
Page 605
As usual, the motion of the sources and the eld is reversible, continuous, conserved and deterministic. However, there is quite some fun in the o ng; even though this description is correct in everyday life, during the rest of our mountain ascent we will nd that each of the bullet points is in fact wrong. A simple example shows this.
At a temperature of zero kelvin, when matter does not radiate thermally, we have the paradoxical situation that the charges inside matter cannot be moving, since no emitted radiation is observed, but they cannot be at rest either, due to Earnshaw’s theorem. In short, the simple existence of matter – with its discrete charge values – shows that classical electrodynamics is wrong.
In fact, the overview of material properties of Table makes the same point even more strongly; classical electrodynamics can describe many of the e ects listed, but it cannot explain the origin of any of them. Even though few of the e ects will be studied in our walk – they are not essential for our adventure – the general concepts necessary for their description will be the topic of the second part of this mountain ascent, that on quantum theory.
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e description of general relativity and classical electrodynamics concludes our walk hrough classical physics. In order to see its limitations, we summarize what we have found out. In nature, we learned to distinguish and to characterize objects, radiation and spacetime. All these three can move. In all motion we distinguish the xed, intrinsic properties from the varying state. All motion happens in such a way as to minimize change.
Looking for all the xed, intrinsic aspects of objects, we nd that all su ciently small objects or particles are described completely by their mass and their electric charge. ere is no magnetic charge. Mass and electric charge are thus the only localized intrinsic properties of classical, everyday objects. Both mass and electric charge are de ned by the ac-
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celerations they produce around them. Both quantities are conserved; thus they can be added. Mass, in contrast to charge, is always positive. Mass describes the interaction of objects with their environment, charge the interaction with radiation.
All varying aspects of objects, i.e. their state, can be described using momentum and position, as well as angular momentum and orientation. All can vary continuously in amount and direction. erefore the set of all possible states forms a space, the so-called phase space. e state of extended objects is given by the states of all its constituent particles. ese particles make up all objects and somehow interact electromagnetically.
e state of a particle depends on the observer. e state is useful to calculate the change that occurs in motion. For a given particle, the change is independent of the observer, but the states are not. e states found by di erent observers are related: the relations are called the ‘laws’ of motion. For example, for di erent times they are called evolution equations, for di erent places and orientations they are called transformation relations, and for di erent gauges they are called gauge transformations. All can be condensed in the principle of least action.
We also observe the motion of a massless entity: radiation. Everyday types of radiation, such as light, radio waves and their related forms, are travelling electromagnetic waves.
ey are described by same equations that describe the interaction of charged or magnetic objects. e speed of massless entities is the maximum possible speed in nature and is the same for all observers. e intrinsic properties of radiation are its dispersion relation and its energy–angular momentum relation. e state of radiation is described by its electromagnetic eld strength, its phase, its polarization and its coupling to matter. e motion of radiation describes the motion of images.
e space-time environment is described by space and time coordinates. Space-time is also able to move, by changing its curvature. e intrinsic properties of space-time are the number of dimensions, its signature and its topology. e state is given by the metric, which describes distances and thus the local warpedness. e warpedness can oscillate and propagate, so that empty space can move like a wave.
Our environment is nite in age. It has a long history, and on large scales, all matter in the universe moves away from all other matter. e large scale topology of our environment is unclear, as is unclear what happens at its spatial and temporal limits.
Motion follows a simple rule: change is always as small as possible. is applies to matter, radiation and space-time. All energy moves in the way space-time dictates it, and space moves the way energy dictates it. is relation describes the motion of the stars, of thrown stones, of light beams and of the tides. Rest and free fall are the same, and gravity is curved space-time. Mass breaks conformal symmetry and thus distinguishes space from time.
Energy and mass speed is bound from above by a universal constant c, and energy change per time is bound from above by a universal constant c G. e speed value c is realized for the motion of massless particles. It also relates space to time. e power value c G is realized by horizons. ey are found around black holes and at the border of the universe. e value also relates space-time curvature to energy ow and thus describes the elasticity of space-time.
No two objects can be at the same spot at the same time. is is the rst statement that humans encounter about electromagnetism. More detailed investigation shows that electric charge accelerates other charges, that charge is necessary to de ne length and time
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•.
intervals, and that charges are the source of electromagnetic elds. Also light is such a eld. Light travels at the maximum possible velocity. In contrast to objects, light can in-
terpenetrate. In summary, we learned that of the two naive types of object motion, namely motion due to gravity – or space-time curvature – and motion due to the electromagnetic
eld, only the latter is genuine. Above all, classical physics showed us that motion, be it linear or rotational, be it that
of matter, radiation or space-time, is conserved. Motion is continuous. More than that, motion is similar to a continuous substance: it is never destroyed, never created, but always redistributed. Owing to conservation, all motion, that of objects, images and empty space, is predictable and reversible. Owing to conservation of motion, time and space can be de ned. In addition, we found that classical motion is also right–le symmetric. Classical physics showed us that motion is predictable: there are no surprises in nature.
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T
E
Ref. 606
Maybe nature shows no surprises, but it still provides many adventures. On the th of
March , a m sized body almost hit the Earth. It passed at a distance of only
km from our planet. On impact, it would have destroyed a region the size of
London. A few months earlier, a m sized body missed the Earth by
km; the
record for closeness so far was in , when the distance was only
km.* Several
other adventures can be predicted by classical physics, as shown in Table . Many are
problems facing humanity in the distant future, but some, such as volcanic eruptions or
asteroid impacts, could happen at any time. All are research topics.
TA B L E 51 Examples of disastrous motion of possible future importance
C
Y
End of fundamental physics
c. (around year )
Giant tsunami from volcanic eruption at Canary islands
c. -
Major nuclear material accident or weapon use
unknown
Ozone shield reduction
c.
Rising ocean levels due to greenhouse warming
c. -
End of applied physics
Explosion of volcano in Greenland, leading to long darkening of unknown sky
Several magnetic north and south poles appear, allowing solar c. storms to disturb radio and telecommunications, to interrupt electricity supplies, to increase animal mutations and to disorient migrating animals such as wales, birds and tortoises
Our interstellar gas cloud detaches from the solar systems, chan- c. ging the size of the heliosphere, and thus expose us more to aurorae and solar magnetic elds
* e web pages around http://cfa-www.harvard.edu/iau/lists/Closest.html provide more information on such events.
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C
Y
Reversal of Earth’s magnetic eld, implying a time with almost no unknown magnetic eld, with increased cosmic radiation levels and thus more skin cancers and miscarriages
Atmospheric oxygen depletion due to forest reduction and exaggerated fuel consumption
Upcoming ice age
c.
Possible collision with interstellar gas cloud assumed to be c. crossed by the Earth every million years, maybe causing mass extinctions
Explosion of Yellowstone or other giant volcano leading to year- to long volcanic winter
Possible genetic degeneration of homo sapiens due to Y chromo- c. some reduction
Africa collides with Europe, transforming the Mediterranean around ë into a lake that starts evaporating
Gamma ray burst from within our own galaxy, causing radiation between and ë damage to many living beings
Asteroid hitting the Earth, generating tsunamis, storms, darken- between and ë ing sunlight, etc.
Neighbouring star approaching, starting comet shower through destabilization of Oort cloud and thus risk for life on Earth
American continent collides with Asia
ë
Instability of solar system
ë
Low atmospheric CO content stops photosynthesis
ë
Collision of Milky Way with star cluster or other galaxy
ë
Sun ages and gets hotter, evaporating seas
ë
Ocean level increase due to Earth rotation slowing/stopping (if not evaporated before)
Temperature rise/fall (depending on location) due to Earth rotation stop
Sun runs out of fuel, becomes red giant, engulfs Earth
.ë
Sun stops burning, becomes white dwarf
.ë
Earth core solidi es, removing magnetic eld and thus Earth’s . ë cosmic radiation shield
Nearby nova (e.g. Betelgeuse) bathes Earth in annihilation radi- unknown ation
Nearby supernova (e.g. Eta Carinae) blasts over solar system unknown
Galaxy centre destabilizes rest of galaxy
unknown
Universe recollapses – if ever (see page )
ë
Matter decays into radiation – if ever (see Appendix C)
Problems with naked singularities
unknown, controversial
Vacuum becomes unstable
unknown, controversial
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•.
Despite the fascination of the predictions, we leave aside these literally tremendous issues and continue on our adventure.
T
–
We can summarize classical physics with a simple statement: nature lacks surprises because classical physics is the description of motion using the concept of the in nitely small. All concepts used so far, be they for motion, space, time or observables, assume that the in nitely small exists. Special relativity, despite the speed limit, still allows in nitely small velocities; general relativity, despite its black hole limit, still allows in nitely small force and power values. Similarly, in the description of electrodynamics and gravitation, both integrals and derivatives are abbreviations of mathematical processes that use in nitely small intermediate steps.
In other words, the classical description of nature introduces the in nitely small in the description of motion. e classical description then discovers that there are no surprises in motion. e detailed study of this question lead us to a simple conclusion: the in nitely small implies determinism.* Surprises contradict the existence of the in nitely small.
On the other hand, both special and general relativity have eliminated the existence of the in nitely large. ere is no in nitely large force, power, size, age or speed.
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“ e more important fundamental laws and facts of physical science have all been discovered, and these are now so rmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote... Our future discoveries must be looked for in the sixth place of decimals. ” Albert Michelson.**
We might think that we know nature now, as did Albert Michelson at the end of the nineteenth century. He claimed that electrodynamics and Galilean physics implied that the major laws of physics were well known. e statement is o en quoted as an example of awed predictions, since it re ects an incredible mental closure to the world around him. General relativity was still unknown, and so was quantum theory.
At the end of the nineteenth century, the progress in technology due to the use of electricity, chemistry and vacuum technology had allowed better and better machines and apparatuses to be built. All were built with classical physics in mind. In the years between and , these classical machines completely destroyed the foundations of classical physics. Experiments with these apparatuses showed that matter is made of atoms, that electrical charge comes in the smallest amounts and that nature behaves randomly. Nature does produce surprises – through in a restricted sense, as we will see. Like
* No surprises also imply no miracles. Classical physics is thus in opposition to many religions. Indeed, many religions argue that in nity is the necessary ingredient to perform miracles. Classical physics shows that this is not the case. ** From his 1894 address at the dedication ceremony for the Ryerson Physical Laboratory at the University of Chicago.
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the British Empire, the reign of classical physics collapsed. Speaking simply, classical physics does not describe nature at small scales.
But even without machines, the Victorian physicist could have predicted the situation. (In fact, many more progressive minds did so.) He had overlooked a contradiction between electrodynamics and nature, for which he had no excuse. In our walk so far we found that clocks and metre bars are necessarily made of matter and based on electromagnetism. But as we just saw, classical electrodynamics does not explain the stability of matter. Matter is made of small particles, but the relation between these particles, electricity and the smallest charges is not clear. If we do not understand matter, we do not yet understand space and time, since they are de ned using measurement devices made of matter.
Worse, the Victorian physicist overlooked a simple fact: the classical description of nature does not allow one to understand life. e abilities of living beings – growing, seeing, hearing, feeling, thinking, being healthy or sick, reproducing and dying – are all unexplained by classical physics. In fact, all these abilities contradict classical physics. Understanding matter and its interactions, including life itself, is therefore the aim of the second part of our ascent of Motion Mountain. e understanding will take place at small scales; to understand nature, we need to study particles. Indeed, the atomic structure of matter, the existence of a smallest charge and the existence of a smallest entropy makes us question the existence of the in nitely small. ere is something to explore. Doing so will lead us from surprise to surprise. To be well prepared, we rst take a break.
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B
484 J
S
, L.L. D R , K.A. M
& W.Y. T , Classical Electrody-
namics, Perseus, . An excellent text on the topic by one of its greatest masters.
See also the beautiful problem book by A
B
& J -M L -
L
, La physique en questions – électricité et magnétisme, Vuibert, . Cited on
pages and .
485 K. H K See also H. J
& al., Physical Review D 66, p.
, , or H.V. K
-
& K. Z
, Particle Astrophysics, e Institute of Physics, UK, .
& M. L
, Search for magnetic monopoles trapped in matter, Phys-
ical Review Letters 75, pp. – , . See also A.S. G
& W.P. T
,
Resource letter MM- : magnetic monopoles, American Journal of Physics 58, pp. – ,
. Cited on page .
486 R. E
, Filling station res spark cars’ recall, New Scientist, pp. – , March .
Cited on page .
487 S. D
, F. O
& F. G
, On the Kelvin electrostatic generator, European
Journal of Physics 10, pp. – , . You can also nd construction plans for it in various
places on the internet. Cited on page .
488 For an etching of Franklin’s original ringing rod, see E.P. K
, Benjamin Franklin and
lightning rods, Physics Today 59, pp. – , . Cited on page .
489 For more details on various electromagnetic units, see the standard text by J.D. J
,
Classical Electrodynamics, rd edition, Wiley, . Cited on page .
490 See the old but beautiful papers by R
C. T
& T. D S
,e
electromotive force produced by the acceleration of metals, Physical Review 8, pp. – ,
,R
C. T
& T. D S
, e mass of the electric carrier in
copper, silver and aluminium, Physical Review 9, pp. – , , and the later but much
more precise experiment by C.F. K
& G.G. S
, Inertia of the carrier of
electricity in copper and aluminum, Physical Review 66, pp. – , . (Obviously the
American language dropped the ‘i’ from aluminium during that period.) e rst of these
papers is also a review of the preceding attempts, and explains the experiment in detail. e last paper shows what had to be taken into consideration to achieve su cient precision.
Cited on page .
491 is e ect has rst been measured by S.J. B
, A new electron-inertia e ect and the
determination of m/e for the free electron in copper, Philosophical Magazine 12, p. , .
Cited on page .
492 See for example C. S
,A.A. K
, T.L. R , C. S
& H.B. E
, Decapitation of tungsten eld emitter tips during sputter sharpening,
Surface Science Letters 339, pp. L –L , . Cited on page .
493 L.I. S
& M.V. B
, Gravitational-induced electric eld near a metal, Physical
Review 151, pp. – , . F.C. W
& W.M. F
, Experimental
comparison of the gravitational force on freely falling electrons and metallic electrons, Phys-
ical Review Letters 19, pp. – , . Cited on page .
494 J. L
& M. C
, Speed of light measurement using ping, electronic pre-
print available at http://www.arxiv.org/abs/physics/
. Cited on page .
495 P
M
, Tractatus de magnete, . Cited on page .
496 R. W
& W. W
. Cited on page .
, Magnetic Orientation in Animals, Springer, Berlin,
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 1097 ny
497 e ratio of angular L to magnetic M moment is
L M
=
më e
,
(460)
where e is the electron charge and m its mass. Both L and M are measurable. e rst meas-
urements were published with a -value of , most probably because the authors expected
the value. In later experiments, de Haas found other values. Measurements by other research-
ers gave values nearer to than to , a fact that was only understood with the discovery of
spin. e original publications are A. E
& W.J. H , Proefondervinderlijk
bewijs voor het bestaan der moleculaire stroomen van Ampère, Konninklijke Akademie der
Wetenschappen te Amsterdam, Verslagen 23, p. , , and A. E
& W.J.
H , Experimental proof of the existence of Ampère’s molecular currents, Konninklijke
Akademie der Wetenschappen te Amsterdam, Proceedings 18, p. , . Cited on page .
498 S.J. B B .
, Magnetization by rotation, Physical Review 6, pp. – , , and S.J. , Magnetization by rotation, Physical Review 6, pp. – , . Cited on page
499 See J.D. J
, Classical Electrodynamics, rd edition, Wiley,
, Time Harmonic Electromagnetic Fields, McGraw–Hill,
and .
, or also R.F. H . Cited on pages
500 e http://suhep.phy.syr.edu/courses/modules/MM/Biology/biology .html website gives an introduction into brain physiology. Cited on page .
501 N. S
, Invariants of the electromagnetic eld and electromagnetic waves, Amer-
ican Journal of Physics 53, pp. – , . Cited on page .
502 A.C.
T , V c in
. Cited on page .
?, European Journal of Physics 20, pp. L –L , March
503 R.H. T , S. M & H. L , Magnetic signal due to ocean tidal ow identi ed in satellite observations, Science 299, pp. – , . e lms derived from the data can be found on the http://www.tu-bs.de/institute/geophysik/spp/publikationen.html website. Cited on page .
504 H. M
, Unipolar induction: a neglected topic in the teaching of electromag-
netism, European Journal of Physics 20, pp. – , . Cited on page .
505 On the geodynamo status, see the article by P.H. R
& G.A. G
, Geody-
namo theory and simulations, Reviews of Modern Physics 72, pp. – , . An older
article is R. J
& B. R
, Geophysical dynamics at the center of the
Earth, Physics Today pp. – , August . Cited on pages and .
506 J. Y , F. L , L.W. K
& D.Y. K , Electrokinetic microchannel battery by
means of electrokinetic and micro uidic phenomena, Journal of Micromechanics and Mi-
croengineering 13, pp. – , . Cited on page .
507 O
D. J
, A relativistic paradox seemingly violating conservation of mo-
mentum law in electromagnetic systems, European Journal of Physics 20, pp. – , .
Of course, the missing momentum goes into the electromagnetic eld. Given that the elec-
tromagnetic momentum is given by the vector potential, are you able to check whether
everything comes out right? Cited on page .
508 H. V D & E.P. W
, Classical relativistic mechanics of interacting point
particles, Physical Review 136B, pp. – , . Cited on page .
509 M D. S
& J R. T
, oughts on the magnetic vector potential, Amer-
ican Journal of Physics 64, pp. – , . Cited on pages , , and .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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510 J S
, Simple derivation of magnetic vector potentials, European Journal of
Physics 14, pp. – , . Cited on page .
511 T.T. W & C.N. Y , , Concept of nonintegrable phase factors and global formulation of gauge elds, Physical Review D 12, pp. – , Cited on page .
512 An electrodynamics text completely written with (mathematical) forms is K M
&W
L. E , Elektromagnetische Felder – mathematische und physikalische
Grundlagen, Springer, . Cited on page .
513 J. T
, Twirl those organs into place – getting to the heart of how a heart knows le
from right, Science News 156, August, . A good book on asymmetries in nature is H.
B
, Rechts oder links, Wiley–Vch, . Cited on page .
514 See for example the discussion by M.C. C right, Scienti c American 224, pp. – , March
& I.L. B , On telling le from . Cited on page .
515 W
R
, Essential Relativity – Special, General, and Cosmological, revised
nd edition, Springer Verlag, , page . ere is also the beautiful paper by M. L
B
& J.-M. L -L
, Galilean electrodynamics, Nuovo Cimento B 14, p. ,
, that explains the possibilities but also the problems appearing when trying to de ne
the theory non-relativistically. Cited on page .
516 L.-C. T , J. L & G.T. G
, e mass of the photon, Reports on Progress of Physics
68, pp. – , . Cited on page .
517 For a captivating account on the history of the ideas on light, see D
P , e Fire
Within the Eye: a Historical Essay on the Nature and Meaning of Light, Princeton University
Press, . Cited on page .
518 See the text by R
L. L & A
B. F
, e Rainbow Bridge: Rainbows
in Art, Myth, and Science, Pennsylvania State University Press, . A chapter can be found
at the http://www.nadn.navy.mil/Oceanography/RainbowBridge/Chapter_ .html website.
Cited on page .
519 e beautiful slit experiment was published by E.A. M
, E.C. C
, G.W. ’
H
, M.B.
M & C.W.J. B
, Observation of the optical
analogue of quantized conductance of a point contact, Nature 350, pp. – ,
April , and in the longer version E.A. M
, E.C. C
, G.W. ’ H
,
M.B.
M & C.W.J. B
, Observation of the optical analogue of
the quantised conductance of a point contact, Physica B 175, pp. – , . e result
was also publicized in numerous other scienti c magazines. Cited on pages and .
520 A recent measurement of the frequency of light is presented in T . U , A. H , B.
G , J. R
, M. P
, M. W & T.W. H
, Phase-coherent
measurement of the hydrogen S– S transition frequency with an optical frequency inter-
val divider chain, Physical Review Letters 79, pp. – , October . Another is
C. S
, L. J
, B. B
, L. H
, F. N , L. J
, F.
B
, O. A & A. C
, Optical frequency measurement of the S- D trans-
itions in hydrogen and deuterium: Rydberg constant and Lamb shi determinations, Phys-
ical Review Letters 82, pp. – , June . Cited on page .
521 e discoverors of two such methods were awarded the on page .
Nobel Prize for physics. Cited
522 See for example G. H
, J. G & R. W
, Why are water-seeking insects not
attracted by mirages? e polarization pattern of mirages, Naturwissenscha en 83, pp. –
, . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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523 On the birefringence of the eye, see L. B , Een eigenaardige speling der natuur, Nederlands tijdschri voor natuurkunde 67, pp. – , December . In particular, a photograph of the eye using linear polarized illumination and taken through an analyser shows a black cross inside the pupil. Cited on page .
524 e standard reference on the propagation of light is M B & E W , Principles of Optics – Electromagnetic eory of Propagation, Interference and Di raction of Light, Pergamon Press, th edition, . Cited on page .
525 An introduction to the topic of the ° halo, the ° halo, Sun dogs, and the many other arcs and bows that can be seen around the Sun, see the beautifully illustrated paper by R.
G
, Lichterscheinungen, Eiskristalle und Himmelsarchäologie, Physikalische Blät-
ter 54, pp. – , , or his book Rainbows, Halos, and Glories, Cambridge University
Press . Cited on page .
526 J
E. F
& E. J
W
, e lunar laser re ector, Scienti c American
pp. – , March . Cited on page .
527 Neil Armstrong of Apollo , Jim Lovell of Apollo and Apollo , and Jim Irwin of Apollo extensively searched for it and then made negative statements, as told in Science News
p. , & December . From the space shuttle however, which circles only a few hundred kilometres above the Earth, the wall can be seen when the Sun is low enough such that the wall appears wider through its own shadow, as explained in Science News 149, p. ,
. Cited on page .
528 M. S , M. S
& G. S
, Physical Review Letters 78, pp. –
See also the more readable paper by M
S
&G
S
trapping of optical beams: spatial solitons, Physics Today 51, pp. – , August
on page .
,. , Self. Cited
529 e rst correct explanation of the light mill was given by O
R
, On cer-
tain dimensional properties of matter in the gaseous state, Royal Society Philosophical Trans-
actions Part , . e best discussion is the one given on the web by P G , in the
frequently asked question list of the usenet news group sci.physics; it is available at the http://
www.desy.de/user/projects/Physics/light-mill.html website. Cited on page .
530 P. L
, Untersuchungen über die Druckkrä e des Lichtes, Annalen der Physik 6,
pp. – , . He was also the rst who understood that this e ect is the basis for the
change of direction of the tails of comets when they circle around the Sun. Cited on page .
531 P. G
& P. O
, Applied Physics Letters 78, p. , . Cited on page .
532 A short overview is given by M P
& L A , Optical tweezers and span-
ners, Physics World pp. – , September . e original papers by Ashkin’s group are
A. A
, J.M. D
, J.E. B
& S. C , Observation of a gradient
force optical trap for dielectric particles, Optics Letters 11, p. , , and A. A ,
J.M. D
& T. Y
, Optical trapping and manipulation of single cells using in-
frared laser beams, Nature 330, p. , . A pedagogical explanation on optical spanners,
together with a way to build one, can be found in D.N. M
, J. A , R.S. C -
, F. A
,A. V & K. D
, Beth’s experiment using optical tweezers,
American Journal of Physics 69, pp. – , , and in S.P. S
, S.R. B
,
A.L. B
, B.L. B
, E.K. B
& M. P
, Inexpensive optical tweezers
for undergraduate laboratories, American Journal of Physics 67, pp. – , page .
. Cited on
533 R. B , Physical Review 50, p. , . For modern measurements, see N.B. S
,
K. D
, L. A
& M.J. P
, Optics Letters 22, p. , , and M.E.J.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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F
, T.A. N
, N.R. H
Letters 23, p. , . Cited on page .
& H. R
-D
, Optics
534 See the Latin text by D c. . Cited on page .
F
De iride et radialibus impressionibus,
535 J. W
, Multiple rainbows from single drops of water and other liquids, American
Journal of Physics 44, pp. – , , and his How to create and observe a dozen rainbows
in a single drop of water, Scienti c American 237, pp. – , . See also K. S
, An-
gular scattering and rainbow formation in pendant drops, Journal of the Optical Society of
America 69, pp. – , . Cited on page .
536 ere are also other ways to see the green ray, for longer times, namely when a fata morgana
appears at sunset. An explanation with colour photograph is contained in M. V
,
Gespiegelt in besonderen Dü en ...– Oasen, Seeungeheuer und weitere Spielereien der Fata
Morgana, Physikalische Blätter 54, pp. – , . Cited on page .
537 is famous discovery is by B
B
& P K , Basic Color Terms: eir Uni-
versality and Evolution, University of California Press, . eir ongoing world colour sur-
vey is eagerly awaited. Of course there are also ongoing studies to nd possible exceptions;
the basic structure is solid, as shown in the conference proceedings C.L. H
&L -
M , Colour Categories in ought and Language, Cambridge University Press, .
Cited on page .
538 For a thorough discussion of the various velocities connected to wave trains, see the classic
text by L
B
, Wave Propagation and Group Velocity, Academic Press, New
York, . It expands in detail the theme discussed by A
S
, Über die
Fortp anzung des Lichtes in dispergierenden Medien, Annalen der Physik, th series, 44,
pp. – , . See also A
S
, Optik, Dietrichssche Verlagsbuchand-
lung, Wiesbaden , section . An English translation A
S
, Lec-
tures on eoretical Physics: Optics, , is also available. Cited on pages , , and .
539 Changing the group velocity in bers is now even possible on demand, as shown by M.
G
-H
, K.-Y. S & L. T
, Optically controlled slow and
fast light in optical bers using stimulated Brillouin scattering, Applied Physics Letters 87,
p.
, . ey demonstrate group velocities from . c to plus in nity and beyond,
to negative values. Cited on page .
Another experiment was carried out by S. C & S. W , Linear pulse propagation
in an absorbing medium, Physical Review Letters 48, pp. – , . See also S. C
& D. S , Answer to question . Group velocity and energy propagation, American
Journal of Physics 66, pp. – , . Another example was described in by the group
of Raymond Chiao for the case of certain nonlinear materials in R. C , P.G. K
& A.M. S
, Faster than light?, Scienti c American 269, p. , August , and
R.Y. C , A.E. K
& G. K
, Tachyonlike excitations in inverted two-
level media, Physical Review Letters 77, pp. – , . On still another experimental
set-up using anomalous dispersion in caesium gas, see L.J. W , A. K
& A.
D
, Gain-assisted superluminal light propagation, Nature 406, pp. – , July
.
540 Y.P. T .
, Paradoxes in the eory of Relativity, Plenum Press, . Cited on page
541 See the excellent explanation by K Journal of Physics 69, pp. – ,
T. M D
, Negative group velocity, American
. Cited on page .
542 e prediction of negative refraction is due to V.G. V
, e electrodynamics of
substances with simultaneously negative values of ε and µ, Soviet Physics Uspekhi 10, p. ,
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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. e explanation with di erent refraction directions was published by P.M. V
,
R.M. W
& A.P. V
, Wave refraction in negative-index media: always positive
and very inhomogeneous, Physical Review Letters 88, p.
, May . Also Fermat’s
principle is corrected, as explained in V.G. V
, About the wording of Fermat’s
principle for light propagation in media with negative refraction index, http://www.arxiv.
org/abs/cond-mat/
. Cited on page .
543 e rst example of material system with a negative refraction index were presented by
David Smith and his team. R.A. S
, D.R. S
& S. S
, Experimental
veri cation of a negative index of refraction, Science 292, p. - , . More recent ex-
amples are A.A. H
, J.B. B
& I.L. C
, Experimental observations of a
le -handed material that obeys Snell’s law, Physical Review Letters 90, p.
, , C.G.
P
, R.B. G
, K. L , B.E.C. K
& M. T
, Experi-
mental veri cation and simulation of negative index of refraction using Snell’s law, Phys-
ical Review Letters 90, p.
, . S. F
, E.N. E
& C.M.
S
, Refraction in media with a negative refractive index, Physical Review Letters
90, p.
, . Cited on page .
544 S.A. R
, Physics of negative refractive index materials, Reorts on Progress of
Physics 68, pp. – , . Cited on page .
545 J. P
, Negegative refraction makes a perfect lens, Physical Review Letters 85, p. ,
. Cited on page .
546 G. N , A. E
& H. S
, Journal de Physique I (Paris) 4, p. , . Unfor-
tunately, Nimtz himself seems to believe that he transported energy or signals faster than
light; he is aided by the o en badly prepared critics of his quite sophisticated experiments.
See A. E
& G. N , Physikalische Blätter 49, p. , Dezember , and the weak
replies in Physikalische Blätter 50, p. , April . See also A.M. S
, Journal de
Physique I (Paris) 4, p. , , A.M. S
, P.G. K
& R.Y. C , Phys-
ical Review Letters 71, pp. – , , and A. R
, P. F
, G.P. P & D.
M
, Physical Review E 48, p. , . Cited on page .
547 A summary of all evidence about the motion of the aether is given by R.S. S
,
S.W. M C
, F.C. L
& G. K
, New analysis of the interferometer obser-
vations of Dayton C. Miller, Review of Modern Physics 27, pp. – , . An older text
is H. W , Annalen der Physik 26, p. , . Cited on page .
548 e history of the concept of vacuum can be found in the book by E. G
, Much Ado
About Nothing, Cambridge University Press, , and in the extensive reference text by E -
T. W
, A History of the eories of Aether and Electricity, Volume : e
Classical eories, Volume : e Modern eories, Tomash Publishers, American Institute
of Physics , . Cited on page .
e various aether models – gears, tubes, vortices – proposed in the nineteenth century
were dropped for various reasons. Since many models used to explain electric and magnetic
elds as motion of some entities, it was concluded that the speed of light would depend
on electric or magnetic elds. One type of eld was usually described by linear motion of
the entities, the other by rotatory or twisting motion; both assignments are possible. As a
consequence, aether must be a somewhat strange uid that ows perfectly, but that resists
rotation of volume elements, as McCullogh deduced in . However, experiments show
that the speed of light in vacuum does not depend on electromagnetic eld intensity. Vor-
tices were dropped because real world vortices were found out to be unstable. All models
received their nal blow when they failed to meet the requirements of special relativity.
549 is happened to Giovanni Bellini (c. – ) the great Venetian Renaissance painter,
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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who even put this experience into writing, thus producing one of the greatest ‘ga es’ ever. If you take a photograph of the e ect with a remotely controlled camera, you can prove that your camera is holy as well. Cited on page .
550 S.R. W , How retrore ectors really work, Optics & Photonics News, pp. – , December . Cited on page .
551 W.H. E pp. – , .
& B. L
, Das Hermann–Gitter, Physik in unserer Zeit 6,
. e journal also shows a colour variation of these lattices. Cited on page
552 e eye sensitivity myth is debunked in detail by B.H. S
& D.K. L
, Some
paradoxes, errors, and resolutions concerning the spectral optimization of human vision,
American Journal of Physics 67, pp. – , . Cited on page .
553 D
R. W
, Supernormal Vision, Science News 152, pp. – , November
. See also http://www.cvs.rochester.edu/people/d_williams/d_williams.html as well as
the photographs at http://www.cvs.rochester.edu/people/~aroorda/ao_research.html of the
interior of living human eyes. eir last publication is A. R
,A. M
, P. L
& D.R. W
, Packing arrangement of the three cone classes in the primate retina,
Vision Research 41, pp. – , . Cited on page .
554 S.W. H , Strategy for far- eld optical imaging and writing without di raction limit, Phys-
ics Letters A 326, pp. – , , see also V. W
& S.W. H , Nanoscale resol-
ution in the focal plane of an optical microscope, Physical Review Letters 94, p.
,,
and V. W
, J. S
, T. S
& S.W. H , Stimulated emission depletion
microscopy on lithographic microstructures, Journal of Physics B 38, pp. S –S , .
Cited on page .
555 A.D. E
& A.W. W
, e origin of cosmic rays, European Journal of
Physics 20, pp. – , , Cited on page .
556 D. S
, Electromagnetic angular momentum and quantum mechanics, American
Journal of Physics 66, pp. – , , Cited on page .
557 A. Y
, D.M. E
& N.D. L , O -resonance conduction through atomic
wires, Science 272, pp. – , June . For aluminium, gold, lead, niobium, as
well as the in uence of chemical properties, see E S
, e signature of chem-
ical valence in the electric conduction through a single-atom contact, Nature 394, pp. –
, July . Cited on page .
558 See L. K
, A myth about capacitors in series, e Physics Teacher 26, pp. – ,
, and A.P. F
, Are the textbook writers wrong about capacitors?, e Physics
Teacher 31, pp. – , . Cited on page .
559 is problem was suggested by Vladimir Surdin. Cited on page .
560 See for example, J.M. A
, A. H
& M. R , Velocity elds
inside a conducting sphere near a slowly moving charge, American Journal of Physics 62,
pp. – , . Cited on page .
561 P
C
, Open wide, this won’t hurt a bit, New Scientist p. , February
on page .
. Cited
562 Such a claim was implicitly made by D. M
, A. R
& R. R
,
Observation of superluminal behaviors in wave propagation, Physical Review Letters 84,
p. , . An excellent explanation and rebuttal was given by W.A. R
, D.S.
T
& A.L. X
, Causal explanation for observed superluminal behavior of mi-
crowave propagation in free space, http://www.arxiv.org/abs/physics/
. Cited on
page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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563 J.E. A
, E. B , D. G
& A. G
, Is the number of photons a clas-
sical invariant?, European Journal of Physics 20, pp. – , . Cited on page .
564 If you want to see more on how the world looks for the di erent types of colour blind, have a look at the http://webexhibits.org/causesofcolor/ .html or the http://www.vischeck.com/ examples web pages. Cited on page .
565 About the life-long passion that drove Luke Howard, see the book by R
H
,
e Invention of Clouds, Macmillan . Cited on page .
566 J. L
, e electri cation of thunderstorms, Quartely Journal of the Royal Meteorolo-
gical Society 107, pp. – , . For a more recent and wider review, see E
R.
W
, e tripole structure of thunderstorms, Journal of Geophysical Research 94,
pp. – , . See also the book by the N
R
C
S,
e Earth’s Electrical Environment, Studies in Geophysics, National Academy Press, .
Cited on page .
567 B.M. S
, Physics of ball lightning, Physics Reports 224, pp. – , . See also D.
F
& J. R
, Ball lightning, Physical Review 135, pp. – , .
For more folklore on the topic, just search the world wide web. Cited on page .
568 A.V. G
& K.P. Z , Runaway breakdown and the mysteries of lightning, Phys-
ics Today 58, pp. – , May . Cited on page .
569 To learn more about atmospheric currents, you may want to have a look at the popularizing
review of the US work by E.A. B
, A.A. F & J.R. B
, e global electric
circuit, Physics Today 51, pp. – , October , or the more technical overview by E.
B
, Reviews of Geophysics (supplement) 33, p. , . Cited on page .
570 e use of Schumann resonances in the Earth–ionosphere capacitor for this research eld
is explained in K. S
& M. F
, Weltweite Ortung von Blitzen, Physik
in unserer Zeit 33, pp. – , . Cited on page .
571 J.R. D
, M.A. U , H.K. R
, M. A -D , E.L. C
, J.
J
, V.A. R
, D.M. J
, K.J. R
, V. C
&B. W
,
Energetic radiation produced by rocket-triggered lightning, Science 299, pp. – , .
Cited on page .
572 J.R. D p. ,
, A fundamental limit on electric elds in air, Geophysical Research Letters 30, . Cited on page .
573 For a recent summary, see S. P
, http://www.arxiv.org/abs/gr-qc/
. See also
T.A. A
& D.J. G
, Acceleration without radiation, American Journal of
Physics 53, pp. – , . See also A. K
& G.E. T
, Radiation from an
accelerated charge and the principle of equivalence, American Journal of Physics 37, pp. –
, . Cited on page .
574 C. A
& A. S , e radiation of a uniformly accelerated charge is beyond the
horizon: a simple derivation, American Journal of Physics 74, pp. – , . Cited on
page .
575 J. Z
, X.D. S , Y.C. L , P.G. R
, X.L. S & F. W
, Inner
core di erential motion con rmed by earthquake doublet waveform doublets, Science 309,
pp. – , . Cited on page .
576 is is deduced from the − measurements, as explained in his Nobel Prize talk by H
D
, Experiments with an isolated subatomic particle at rest, Reviews of Modern Phys-
ics 62, pp. – , , and in H D
, Is the electron a composite particle?,
Hyper ne Interactions 81, pp. – , . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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577 A good and short introduction is the paper F. R reaction, American Journal of Physics 68, pp. – ,
, e self-force and radiation . Cited on page .
578 C.G. T
, Magnetic tension and the geometry of the universe, Physical Review Letters
86, pp. – , . An overview of the topic is C.G. T
, Geometrical aspects of
cosmic magnetic elds, http://www.arxiv.org/abs/gr-qc/
. Cited on page .
579 An excellent review is E.H. B Cited on pages and .
, Levitation in Physics, Science 243, pp. – , .
580 See the article by R. T
, S. B
& B. N
, Levitation in
Ultraschallfeldern – Schwebende Tröpfchen, Physik in unserer Zeit 32, pp. – , February
. Liquid drops up to g have been levitated in this way. Cited on page .
581 F.C. M & P.Z. C
, Superconducting Levitation – Applications to Bearings and
Magnetic Transportation, Wiley & Sons, . Cited on pages and .
582 W.T. S
, Who was Earnshaw?, American Journal of Physics 27, pp. – ,
on page .
. Cited
583 e trick is to show that div E = , curl E = , thus E∇ E = and, from this, ∇ E ;
there are thus no local electric eld maxima in the absence of free charges. e same proof
works for the magnetic eld. However, bodies with dielectric constants lower than their
environment can be levitated in static electric elds. An example is gas bubbles in liquids, as
shown by T.B. J
& G.W. B , Bubble dielectrophoresis, Journal of Applied Physics
48, pp. – , . Cited on page .
584 However, it is possible to levitate magnets if one uses a combination containing diamagnets.
See A.K. G , M.D. S
, M.I. B
& L.O. H
, Magnet levitation at
your ngertips, Nature 400, pp. – , . Cited on page .
585 e rst photographs of a single ion were in W. N
, M. H
, P.E.
T
& H. D
, Localized visible Ba+ mono-ion oscillator, Physical Review A
22, pp. – , . See also D.J. W
& W.M. I
, Physics Letters A 82,
p. , , as well as F. D
& H. W
, Physical Review Letters 58, p. , .
For single atoms, see photographs in Z. H & H.J. K
, Optics Letters 1, p. ,
, F. R
, D. B
, J.L. P & W. E
, Europhysics Letters
34, p. , , D. H
, H. S
, F. S
, B. U
, R.
W
& D. M
, Europhysics Letters 34, p. , . Cited on page .
586 See for example M B
, And God said...let there be levitating strawberries,
ying frogs and humans that hover over Seattle, New Scientist pp. – , July , or C.
W , Floating frogs, Science News 152, pp. – , December , and C. W , Molecular
magnetism takes o , Physics World April , page . e experiments by Andre Geim,
Jan Kees Maan, Humberto Carmona and Peter Main were made public by P. R
,
Physics World 10, p. , . Some of the results can be found in M.V. B
& A.K.
G , Of ying frogs and levitrons, European Journal of Physics 18, pp. – , . See
also their http://www-hfml.sci.kun.nl/hfml/levitate.html website. Cited on page .
587 e well-known toy allows levitation without the use of any energy source and is called the
‘Levitron’. It was not invented by Bill Hones of Fascination Toys & Gi s in Seattle, as the
http://www.levitron.com website explains. e toy is discussed by R E , Levitation
using only permanent magnets, Physics Teacher 33, p. , April . It is also discussed
in M.V. B , e LevitronTM: an adiabatic trap for spins, Proceedings of the Royal So-
ciety A 452, pp. – , , (of Berry’s phase fame) as well as by M.D. S
, L.O.
H
& S.L. R
, Spin stabilized magnetic levitation, American Journal of
Physics 65, pp. – , , and by T.B. J , M. W
& R. G , Simple theory
for the Levitron, Journal of Applied Physics 82, pp. – , . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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588 e drill trick and the building of a Levitron are described in the beautiful lecture script
by J Z
, Physik im Alltag, Skript zur Vorlesung im WS / der Universität
Regensburg. Cited on page .
589 e prediction about quantized levitation is by S
B. H , Length quantization
in levitation of magnetic microparticles by a mesoscopic superconducting ring, Physical Re-
view Letters 74, pp. – , . e topic is discussed in more detail in S
B.
H , Magnetic levitation, suspension, and superconductivity: macroscopic and meso-
scopic, Physical Review B 53, p. , , reversed in order with S
B. H ,
Quantized levitation of superconducting multiple-ring systems, Physical Review B 53,
p. , , as well as S
B. H , Quantized levitation by multiply-connected
superconductors, LT- Proceedings, in Czechoslovak Journal of Physics 46, p. , . In
, there was not yet an experimental con rmation (Stephen Haley, private communica-
tion). Cited on page .
590 All the illusions of the ying act look as if the magician is hanging on lines, as observed by many, including the author. (Photographic ashes are forbidden, a shimmery background is set up to render the observation of the lines di cult, no ring is ever actually pulled over the magician, the aquarium in which he oats is kept open to let the shing lines pass through, always the same partner is ‘randomly’ chosen from the public, etc.) Information from eyewitnesses who have actually seen the shing lines used by David Copper eld explains the reasons for these set-ups. e usenet news group alt.magic.secrets, in particular Tilman Hausherr, was central in clearing up this issue in all its details, including the name of the company that made the suspension mechanism. Cited on page .
591 Detailed descriptions of many of these e ects can be found in the excellent overview edited
by M
A
,G
M
&S
R
, E ekte der
Physik und ihre Anwendungen, Harri Deutsch, . Cited on page .
592 R. B
, K. W
, R.A. H
& S J. P
, Picosecond
discharges and stick–slip friction at a moving meniscus of mercury in glass, Nature 391,
pp. – , January . See also Science News 153, p. , January . Cited on
page .
593 H S
&R
C
, Magnetische tunneljuncties, Nederlands tijd-
schri voor natuurkunde 64, pp. – , November . Cited on page .
594 H. O O ber
, D. C , F. M
, T. O , E. A , T. D
, Electric- eld control of ferromagnetism, Nature 408, pp.
. Cited on page .
, Y. O & K. – , - Decem-
595 is e ect was discovered by G
R
,B
T
&A S
-
, Lichtverstrooiing in een magneetveld, Nederlands tijdschri voor natuurkunde 63,
pp. – , maart . Cited on page .
596 V
P
&K
A. G
, Giant magnetocaloric e ect in
Gd (Si Ge ), Physical Review Letters 78, pp. – , , and, from the same authors,
Tunable magnetic regenerator alloys with a giant magnetocaloric e ect for magnetic refri-
geration from to
K, Applied Physics Letters 70, p. , . Cited on page .
597 J. W
, R.N. V
, D. K
, P. Z
, R. W
G
, Charge-induced reversible strain in a metal, Science 300, pp.
. Cited on page .
& H. – , April
598 A. A
, Electro-osmosis on inhomogeneously charged surfaces, Physical Review Letters
75, pp. – , . Cited on page .
599 M.J. A
, ermoluminescence Dating, Academic Press, . e precision of the
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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method is far worse that C dating, however, as shown by H. H
, ermolu-
mineszenzdatierung: eine methodologische Analyse aufgrund gesicherter Befunde, Peter Lang
Verlag, . Cited on page .
600 is e ect was discovered by J.N. H
, R. G
, J.H. R
, R.J.
W
, J.P. D
, D.G.
G
& N.J. K
, Yttrium and
lanthanum hydride lms with switchable optical properties, Nature 380, pp. – , .
A good introduction is R. G
, Schaltbare Spiegel aus Metallhydriden, Physikalische
Blätter 53, pp. – , . Cited on page .
601 See any book on thermostatics, such as L
R
Physics, Wiley, nd edition, . Cited on page .
, A Modern Course in Statistical
602 e Sun emits about ë W from its mass of ë kg, about . mW kg; a person with an average mass of kg emits about W (you can check this in bed at night), i.e. about times more. Cited on page .
603 See its http://www.cie.co.at/cie website. Cited on page .
604 P.D. J , M. N , D.E. P and its changes over the past Cited on page .
, S. M
& I.G. R , Surface air temperature
years, Reviews of Geophysics 37, pp. – , May .
605 Pictures of objects in a red hot oven and at room temperature are shown in C.H. B
,
Demons, engines and the second law, Scienti c American 255, pp. – , November .
Cited on page .
606 B M G , A Guide to the End of the World: Everything you Never Wanted to Know,
Oxford University Press, . On past disasters, see introduction by T H
, Cata-
strophes and Lesser Calamities – the Causes of Mass Extinctions, Oxford University Press,
. Cited on page .
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
I
THE BRAIN, LANGUAGE AND THE HUMAN CONDITION
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Alles was überhaupt gedacht werden kann, kann klar gedacht werden.*
“ Ludwig Wittgenstein, Tractatus, . ” our quest for increased precision in the description of all motion around us, it
s time to take a break, sit down and look back. In our walk so far, which has led us to
Investigate mechanics, general relativity and electrodynamics, we used several con-
cepts without de ning them. Examples are ‘information’, ‘memory’, ‘measurement’, ‘set’,
‘number’, ‘in nity’, ‘existence’, ‘universe’ and ‘explanation’. Each of these is a common and
important term. In this intermezzo, we take a look at these concepts and try to give some
simple, but su ciently precise de nitions, keeping them as provocative and entertaining
Challenge 1098 e as possible. For example, can you explain to your parents what a concept is?
e reason for studying de nitions is simple. We need the clari c-
ations in order to get to the top of Motion Mountain. Many have lost
their way because of lack of clear concepts. In this situation, physics
has a special guiding role. All sciences share one result: every type
of change observed in nature is a form of motion. In this sense, but
in this sense only, physics, focusing on motion itself, forms the basis
for all the other sciences. In other words, the search for the famed
‘theory of everything’ is an arrogant expression for the search for a
theory of motion. Even though the knowledge of motion is basic, its precise description does not imply a description of ‘everything’: just
Ludwig Wittgenstein
try to solve a marriage problem using the Schrödinger equation to note the di erence.
Given the basic importance of motion, it is necessary that in physics all statements on
observations be as precise as possible. For this reason, many thinkers have investigated
physical statements with particular care, using all criteria imaginable. Physics is detailed
prattle by curious people about moving things. e criteria for precision appear once we
Challenge 1099 e ask: which abilities does this prattle require? You might want to ll in the list yourself.
e abilities necessary for talking are a topic of research even today. e way that
the human species acquired the ability to chat about motion is studied by evolutionary
biologists. Child psychologists study how the ability develops in a single human being.
Physiologists, neurologists and computer scientists are concerned with the way the brain
and the senses make this possible; linguists focus on the properties of the language we use,
* ‘Everything that can be thought at all can be thought clearly.’ is and other quotes of Ludwig Wittgenstein are from the equally short and famous Tractatus logico-philosophicus, written in 1918, rst published in 1921; it has now been translated into many other languages.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
:
,
while logicians, mathematicians and philosophers of science study the general properties of statements about nature. All these elds investigate tools that are essential for the development of physics, the understanding of motion and the speci cation of the unde ned concepts listed above. e elds structure this intermezzo.
E
Ref. 607 Page 450
Ref. 608
Challenge 1100 e
A hen is only an egg’s way of making another egg.
“ Samuel Butler, Life and Habit. ” e evolution of the human species is the result of a long story that has been told in
many excellent books. A summarizing table on the history of the universe is given in the chapter on general relativity. e almost incredible chain of events that has lead to one’s own existence includes the formation of atoms, of the galaxies, the stars, the planets, the Moon, the atmosphere, the rst cells, the water animals, the land animals, the mammals, the hominids, the humans, the ancestors, the family and nally oneself.
e way the particles we are made of moved during this sequence, being blown through space, being collected on Earth, becoming organized to form people, is one of the most awe-inspiring examples of motion. Remembering this fantastic sequence of motion every now and then can be an enriching experience.
In particular, without biological evolution, we would not be able to talk about motion; only moving bodies can study moving bodies. Evolution was also the fount of childhood and curiosity. In this intermezzo we will discover that most concepts of classical physics have already been introduced by little children, in the experiences they have while growing up.
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C
Ref. 609
“Physicists also have a shared reality. Other than that, there isn’t really a lot of di erence between being a physicist and being a schizophrenic. ” Richard Bandler
During childhood, everybody is a physicist. When we follow our own memories backwards in time as far as we can, we reach a certain stage, situated before birth, which forms the starting point of human experience. In that magic moment, we sensed somehow that apart from ourselves, there is something else. e rst observation we make about the world, during the time in the womb, is thus the recognition that we can distinguish two parts: ourselves and the rest of the world. is distinction is an example – perhaps the
rst – of a large number of ‘laws of nature’ that we stumble upon in our lifetime. By discovering more and more distinctions we bring structure in the chaos of experience. We quickly nd out that the world is made of related parts, such as mama, papa, milk, Earth, toys, etc.
Later, when we learn to speak, we enjoy using more di cult words and we call the surroundings the environment. Depending on the context, we call the whole formed by oneself and the environment together the (physical) world, the (physical) universe, nature,
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 1101 n
Page 676 Challenge 1102 e
or the cosmos. ese concepts are not distinguished from each other in this walk;* they are all taken to designate the sum of all parts and their relations. ey are simply taken here to designate the whole.
e discovery of the rst distinction starts a chain of similar discoveries. We extract the numerous distinctions that are possible in the environment, in our own body and in the various types of interactions between them. e ability to distinguish is the central ability that allows us to change our view from that of the world as chaos, i.e. as a big mess, to that of the world as a system, i.e. a structured set, in which parts are related in speci c ways. (If you like precision, you may ponder whether the two choices of ‘chaos’ and ‘system’ are the only possible ones. We will return to this issue in the third part of our mountain ascent.)
In particular, the observation of the di erences between oneself and the environment goes hand in hand with the recognition that not only are we not independent of the environment, but we are rmly tied to it in various inescapable ways: we can fall, get hurt, feel warm, cold, etc. Such relations are called interactions. Interactions express the observation that even though the parts of nature can be distinguished, they cannot be isolated. In other words, interactions describe the di erence between the whole and the sum of its parts. No part can be de ned without its relation to its environment. (Do you agree?)
Interactions are not arbitrary; just take touch, smell or sight as examples. ey di er in reach, strength and consequences. We call the characteristic aspects of interactions patterns of nature, or properties of nature, or rules of nature or, equivalently, with their historical but unfortunate name, ‘laws’ of nature. e term ‘law’ stresses their general validity; unfortunately, it also implies design, aim, coercion and punishment for infringement. However, no design, aim or coercion is implied in the properties of nature, nor is infringement possible. e ambiguous term ‘law of nature’ was made popular by René Descartes ( – ) and has been adopted enthusiastically because it gave weight to the laws of the state – which were far from perfect at that time – and to those of other organizations – which rarely are. e expression is an anthropomorphism coined by an authoritarian world view, suggesting that nature is ‘governed’. We will therefore use the term as rarely as possible in our walk and it will, if we do, be always between ‘ironical’ parentheses. Nature cannot be forced in any way. e ‘laws’ of nature are not obligations for nature or its parts, they are obligations only for physicists and all other people: the patterns of nature oblige us to use certain descriptions and to discard others. Whenever one says that ‘laws govern nature’ one is talking nonsense; the correct expression is rules describe nature.
During childhood we learn to distinguish between interactions with the environment (or perceptions): some are shared with others and called observations, others are uniquely personal and are called sensations.** A still stricter criterion of ‘sharedness’ is used to divide the world into ‘reality’ and ‘imagination’ (or ‘dreams’). Our walk will show that this
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Ref. 610
* e di erences in usage can be deduced from their linguistic origins. ‘World’ is derived from old Germanic ‘wer’ – person – and ‘ald’ – old – and originally means ‘lifetime’. ‘Universe’ is from the Latin, and designates the one – ‘unum’ – which one sees turning – ‘vertere’, and refers to the starred sky at night which turns around the polar star. ‘Nature’ comes also from the Latin, and means ‘what is born’. ‘Cosmos’ is from Greek κόσµος and originally means ‘order’. ** A child that is unable to make this distinction among perceptions – and who is thus unable to lie – almost surely develops or already su ers from autism, as recent psychological research has shown.
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Ref. 611 Ref. 613
distinction is not essential, provided that we stay faithful to the quest for ever increasing precision: we will nd that the description of motion that we are looking for does not depend on whether the world is ‘real’ or ‘imagined’, ‘personal’ or ‘public’.
Humans enjoy their ability to distinguish parts, which in other contexts they also call details, aspects or entities, and enjoy their ability to associate them or to observe the relations between them. Humans call this activity classi cation. Colours, shapes, objects, mother, places, people and ideas are some of the entities that humans discover rst.
Our anatomy provides a handy tool to make e cient use of these discoveries: memory. It stores a large amount of input that is called experience a erwards. Memory is a tool used by both young and old children to organize their world and to achieve a certain security in the chaos of life.
Memorized classi cations are called concepts. Jean Piaget was the rst researcher to describe the in uence of the environment on the concepts that a child forms. Step by step, children learn that objects are localized in space, that space has three dimensions, that objects fall, that collisions produce noise, etc. In particular, Piaget showed that space and time are not a priori concepts, but result from the interactions of every child with its environment.*
Around the time that a child goes to school, it starts to understand the idea of permanence of substances, e.g. liquids, and the concept of contrary. Only at that stage does its subjective experience becomes objective, with abstract comprehension. Still later, the child’s description of the world stops to be animistic: before this step, the Sun, a brook or a cloud are alive. In short, only a er puberty does a human become ready for physics.
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Ref. 612
* An overview of the origin of developmental psychology is given by J.H. F
, e Developmental
Psychology of Jean Piaget, 1963. is work summarizes the observations by the French speaking Swiss Jean
Piaget (1896–1980), the central gure in the eld. He was one of the rst researchers to look at child devel-
opment in the same way that a physicist looks at nature: carefully observing, taking notes, making exper-
iments, extracting hypotheses, testing them, deducing theories. His astonishingly numerous publications,
based on his extensive observations, cover almost all stages of child development. His central contribution
is the detailed description of the stages of development of the cognitive abilities of humans. He showed that
all cognitive abilities of children, the formation of basic concepts, their way of thinking, their ability to talk,
etc., result from the continuous interaction between the child and the environment.
In particular, Piaget described the way in which children rst learn that they are di erent from the ex-
ternal environment, and how they then learn about the physical properties of the world. Of his many books
related to physical concepts, two especially related to the topic of this walk are J. P
, Les notions de
mouvement et de vitesse chez l’enfant, Presses Universitaires de France, 1972 and Le developpement de la
notion de temps chez l’enfant, Presses Universitaires de France, 1981, this last book being born from a sugges-
tion by Albert Einstein. ese texts should be part of the reading of every physicist and science philosopher
interested in these questions.
Piaget also describes how in children the mathematical and verbal intelligence derives from sensomo-
torial, practical intelligence, which itself stems from habits and acquired associations to construct new con-
cepts. Practical intelligence requires the system of re exes provided by the anatomical and morphological
structure of our organism. us his work shows in detail that our faculty for mathematical description of
the world is based, albeit indirectly, on the physical interaction of our organism with the world.
Some of his opinions on the importance of language in development are now being revised, notably
through the rediscovery of the work of Lev Vigotsky, who argues that all higher mental abilities, emotions,
recollective memory, rational thought, voluntary attention and self-awareness, are not innate, but learned.
is learning takes place through language and culture, and in particular through the process of talking to
oneself.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Even though everyone has been a physicist in their youth, most people remain classical physicists. In this adventure we continue, using all the possibilities of a toy with which nature provides us: the brain.
Experience is the name everyone gives to their mistakes.
“ ” Oscar Wilde, Lady Windermere’s Fan.
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Ref. 615 Ref. 616 Ref. 617 Ref. 619
Ref. 618
Denken ist bereits Plastik.*
“ ” Joseph Beuys, sculptor.
Numerous observations show that sense input is processed, i.e. classi ed, stored and retrieved in the brain. Notably, lesions of the brain can lead to the loss of part or all of these functions. Among the important consequences of these basic abilities of the brain are thought and language. All such abilities result from the construction, from the ‘hardware’ of the brain.
Systems with the ability to deduce classi cations from the input they receive are called classi ers, and are said to be able to learn. Our brain shares this property with many complex systems; the brain of many animals, but also certain computer algorithms, such as the so-called ‘neural networks’, are examples of such classi ers. Such systems are studied in several elds, from biology to neurology, mathematics and computer science. Classi-
ers have the double ability to discriminate and to associate; both are fundamental to thinking.
Machine classi ers have a lot in common with the brain. As an example, following an important recent hypothesis in evolutionary biology, the necessity to cool the brain in an e ective way is responsible for the upright, bipedal walk of humans. e brain needs a powerful cooling system to work well. In this it resembles modern computers, which usually have powerful fans or even water cooling systems built into them. It turns out that the human species has the most powerful cooling system of all mammals. An upright posture allowed the air to cool the body most e ectively in the tropical environment where humans evolved. For even better cooling, humans have also no body hair, except on their head, where it protects the brain from direct heating by the Sun.**
All classi ers are built from smallest classifying entities, sometimes large numbers of them. Usually, the smallest units can classify input into only two di erent groups. e larger the number of these entities, o en called ‘neurons’ by analogy to the brain, the more sophisticated classi cations can be produced by the classi er.*** Classi ers thus work by applying more or less sophisticated combinations of ‘same’ and ‘di erent’. e distinction by a child of red and blue objects is such a classi cation; the distinction of compact and non-compact gauge symmetry groups in quantum theory is a more elaborate classi cation, but relies on the same fundamental ability.
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* inking is already sculpture.
** e upright posture in turn allowed humans to take breath independently of their steps, a feat that many
animals cannot perform. is is turn allowed humans to develop speech. Speech in turn developed the brain.
*** A good introduction to neural nets is J. H , A. K
& R. P
, Introduction to the eory
of Neural Computation, Addison Wesley, 1991.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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In all classi ers, the smallest classifying units interact with each other. O en these interactions are channelled via connections, and the set is then called a network. In these connections, signals are exchanged, via moving objects, such as electrons or photons.
us we arrive at the conclusion that the ability of the brain to classify the physical world, for example to distinguish moving objects interacting with each other, is a consequence of the fact that it itself consists of moving objects interacting with each other. Without a powerful classi er, humans would not have become such a successful animal species. And only the motion inside our brain allows us to talk about motion in general.
Numerous researchers are identifying the parts of the brain used when di erent intellectual tasks are performed. e experiments become possible using magnetic resonance imaging and other methods. Other researchers are studying how thought processes can be modelled from the brain structure. Neurology is still making regular progress. In particular, it is steadily destroying the belief that thinking is more than a physical process.
is belief results from personal fears, as you might want to test by introspection. It will disappear as time goes by. How would you argue that thought is just a physical process?
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Page 237 Challenge 1105 n
“ ese thoughts did not come in any verbal formulation. I rarely think in words at all. A thought comes, and I may try to express it in words a erward. ” Albert Einstein
We started by stating that studying physics means to talk about motion. To talk is to transmit information. Can information be measured? Can we measure the progress of physics in this way? Is the universe made of information?
Information is the result of classi cation. A classi cation is the answer to one or to several yes–no questions. Such yes–no questions are the simplest classi cations possible; they provide the basic units of classi cation, from which all others can be built. e simplest way to measure information is therefore to count the implied yes–no questions, the bits, leading to it. Are you able to say how many bits are necessary to de ne the place where you live? Obviously, the number of bits depends on the set of questions with which we start; that could be the names of all streets in a city, the set of all coordinates on the surface of the Earth, the names of all galaxies in the universe, the set of all letter combinations in the address. What is the most e cient method you can think of? A variation of the combination method is used in computers. For example, the story of this walk required about a thousand million bits. But since the amount of information in a normal letter depends on the set of questions with which we start, it is impossible to de ne a precise measure for information in this way.
e only way to measure information precisely is to take the largest possible set of questions that can be asked about a system, and to compare it with what is known about the system. In this case, the amount of unknown information is called entropy, a concept that we have already encountered. With this approach you should able to deduce yourself whether it is really possible to measure the advance of physics.
Since categorization is an activity of the brain and other, similar classi ers, information as de ned here is a concept that applies to the result of activities by people and by
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Challenge 1106 n
other classi ers. In short, information is produced when talking about the universe – the universe itself is not the same as information. ere is a growing number of publications based on the opposite of this view; however, this is a conceptual short circuit. Any transmission of information implies an interaction; physically speaking, this means that any information needs energy for transmission and matter for storage. Without either of these, there is no information. In other words, the universe, with its matter and energy, has to exist before transmission of information is possible. Saying that the universe is made of information is as meaningful as saying that it is made of toothpaste.
e aim of physics is to give a complete classi cation of all types and examples of motion, in other words, to know everything about motion. Is this possible? Or are you able to nd an argument against this endeavour?
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e brain is my second favorite organ.
“ ” Woody Allen
Memory is the collection of records of perceptions. e production of such records is the essential aspect of observation. Records can be stored in human memory, i.e. in the brain, or in machine memory, as in computers, or in object memory, such as notes on paper. Without memory, there is no science, no life – since life is based on the records inside the DNA – and especially, no fun, as proven by the sad life of those who lose their memory.
Obviously every record is an object. But under which conditions does an object qualify as a record? A signature can be the record of the agreement on a commercial transaction. A single small dot of ink is not a record, because it could have appeared by mistake, for example by an accidental blot. In contrast, it is improbable that ink should fall on paper exactly in the shape of a signature. ( e simple signatures of physicians are obviously exceptions.) Simply speaking, a record is any object, which, in order to be copied, has to be forged. More precisely, a record is an object or a situation that cannot arise nor disappear by mistake or by chance. Our personal memories, be they images or voices, have the same property; we can usually trust them, because they are so detailed that they cannot have arisen by chance or by uncontrolled processes in our brain.
Can we estimate the probability for a record to appear or disappear by chance? Yes, we can. Every record is made of a characteristic number N of small entities, for example the number of the possible ink dots on paper, the number of iron crystals in a cassette tape, the electrons in a bit of computer memory, the silver iodide grains in a photographic negative, etc. e chance disturbances in any memory are due to internal uctuations, also called noise. Noise makes the record unreadable; it can be dirt on a signature, thermal magnetization changes in iron crystals, electromagnetic noise inside a solid state memory, etc. Noise is found in all classi ers, since it is inherent in all interactions and thus in all information processing.
It is a general property that internal uctuations due to noise decrease when the size, i.e. the number of components of the record is increased. In fact, the probability pmis for a misreading or miswriting of a record changes as
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
pmis N ,
(461)
Challenge 1108 ny Challenge 1109 n
Page 222 Challenge 1110 ny
where N is the number of particles or subsystems used for storing it. is relation appears because, for large numbers, the so-called normal distribution is a good approximation of almost any process; the width of the normal distribution, which determines the probability of record errors, grows less rapidly than its integral when the number of entities is increased. (Are you able to con rm this?)
We conclude that any good record must be made from a large number of entities. e larger the number, the less sensitive the memory is to uctuations. Now, a system of large size with small uctuations is called a (physical) bath. Only baths make memories possible. In other words, every record contains a bath. We conclude that any observation of a system is the interaction of that system with a bath. is connection will be used several times in the following, in particular in quantum theory. When a record is produced by a machine, the ‘observation’ is usually called a (generalized) measurement. Are you able to specify the bath in the case of a person looking at a landscape?
From the preceding discussion we can deduce a powerful conclusion: since we have such a good memory at our disposition, we can deduce that we are made of many small parts. And since records exist, the world must also be made of a large number of small parts. No microscope of any kind is needed to con rm the existence of molecules or similar small entities; such a tool is only needed to determine the sizes of these particles.
eir existence can be deduced simply from the observation that we have memory. (Of course, another argument proving that matter is made of small parts is the ubiquity of noise.)
A second conclusion was popularized in the late s by Leo Szilard. Writing a memory does not produce entropy; it is possible to store information into a memory without increasing entropy. However, entropy is produced in every case that the memory is erased. It turns out that the (minimum) entropy created by erasing one bit is given by
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Sper erased bit = k ln ,
(462)
Challenge 1111 n Ref. 620
and the number ln . is the natural logarithm of . Erasing thus on one hand reduces the disorder of the data – the local entropy–, but on the other hand increases the total entropy. As is well known, energy is needed to reduce the entropy of a local system. In short, any system that erases memory requires energy. For example, a logical AND gate e ectively erases one bit per operation. Logical thinking thus requires energy. It is also known that dreaming is connected with the erasing and reorganization of information. Could that be the reason that, when we are very tired, without any energy le , we do not dream as much as usual?
Entropy is thus necessarily created when we forget. is is evident when we remind ourselves that forgetting is similar to the deterioration of an ancient manuscript. Entropy increases when the manuscript is not readable any more, since the process is irreversible and dissipative.* Another way to see this is to recognize that to clear a memory, e.g. a
Ref. 621 * As Wojciech Zurek clearly explains, the entropy created inside the memory is the main reason that even Maxwell’s demon cannot reduce the entropy of two volumes of gases by opening a door between them in such a way that fast molecules accumulate on one side and slow molecules accumulate on the other. (Maxwell
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magnetic tape, we have to put energy into it, and thus increase its entropy. Conversely, writing into a memory can o en reduce entropy; we remember that signals, the entities that write memories, carry negative entropy. For example, the writing of magnetic tapes usually reduces their entropy.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
T
Ref. 623, Ref. 624
Computers are boring. ey can give only answers.
“ (Wrongly) attributed to Pablo Picasso ” e human brain is built in such a way that its uctuations cannot destroy its contents.
e brain is well protected by the skull for exactly this reason. In addition, the brain literally grows connections, called synapses, between its various neurons, which are the cells doing the signal processing. e neuron is the basic processing element of the brain, performing the basic classi cation. It can only do two things: to re and not to re. (It is possible that the time at which a neuron res also carries information; this question is not yet settled.) e neuron res depending on its input, which comes via the synapses from hundreds of other neurons. A neuron is thus an element that can distinguish the inputs it receives into two cases: those leading to ring and those that do not. Neurons are thus classi ers of the simplest type, able only to distinguish between two situations.
Every time we store something in our long term memory, such as a phone number, new synapses are grown or the connection strength of existing synapses is changed. e connections between the neurons are much stronger than the uctuations in the brain. Only strong disturbances, such as a blocked blood vessel or a brain lesion, can destroy neurons and lead to loss of memory.
As a whole, the brain provides an extremely e cient memory. Despite intense e orts, engineers have not yet been able to build a memory with the capacity of the brain in the same volume. Let us estimated this memory capacity. By multiplying the number of neurons, about ,* by the average number of synapses per neuron, about , and also by the estimated number of bits stored in every synapse, about , we arrive at a storage capacity for the brain of about
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Mrewritable
bit GB .
(463)
(One byte, abbreviated B, is the usual name for eight bits of information.) Note that evolution has managed to put as many neurons in the brain as there are stars in the galaxy, and that if we add all the synapse lengths, we get a total length of about m, which corresponds to the distance to from the Earth to the Sun. Our brain truly is astronomically complex.
In practice, the capacity of the brain seems almost without limit, since the brain frees
Ref. 622
had introduced the ‘demon’ in 1871, to clarify the limits posed by nature to the gods.) is is just another way to rephrase the old result of Leo Szilard, who showed that the measurements by the demon create more entropy than they can save. And every measurement apparatus contains a memory.
To play being Maxwell’s demon, click on the http://www.wolfenet.com/~zeppelin/maxwell.htm website. * e number of neurons seems to be constant, and xed at birth. e growth of interconnections is highest between age one and three, when it is said to reach up to new connections per second.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 1112 ny
memory every time it needs some new space, by forgetting older data, e.g. during sleep.
Note that this standard estimate of bits is not really correct! It assumes that the only
component storing information in the brain is the synapse strength. erefore it only
measures the erasable storage capacity of the brain. In fact, information is also stored in
the structure of the brain, i.e. in the exact con guration in which every cell is connected
to other cells. Most of this structure is xed at the age of about two years, but it continues
to develop at a lower level for the rest of human life. Assuming that for each of the N cells
with n connections there are f n connection possibilities, this write once capacity of the
brain can be estimated as roughly N f n f n log f n bits. For N = , n = , f = , this
gives
Mwriteonce
bit GB .
(464)
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Ref. 625 Ref. 626
Recent measurements con rmed that bilingual persons, especially early bilinguals, have a higher density of grey mass in the small parietal cortex on the le hemisphere of the brain. is is a region mainly concerned with language processing. e brain thus makes also use of structural changes for optimized storage and processing.
Incidentally, even though the brains of sperm whales and of elephants can be ve to six times as heavy as those of humans, the number of neurons and connections, and thus the capacity, is lower than for humans.
Sometimes it is claimed that people use only between % or % of their brain capacity. is myth, which goes back to the nineteenth century, would imply that it is possible to measure the actual data stored in the brain and compare it with its capacity to an impossible accuracy. Alternatively, the myth implies that the processing capacity can be measured. It also implies that nature would develop and maintain an organ with % overcapacity, wasting all the energy and material to build, repair and maintain it. e myth is wrong.
e large storage capacity of the brain also shows that human memory is lled by the environment and is not inborn: one human ovule plus one sperm have a mass of about
mg, which corresponds to about ë atoms. Obviously, uctuations make it impossible to store bits in it. In fact, nature stores only about ë bits in the genes of an ovule, using atoms per bit. In contrast, a typical brain has a mass of . to kg, containing about to ë atoms, which makes it as e cient as the ovule. e di erence between the number of bits in human DNA and those in the brain nicely shows that almost all information stored in the brain is taken from the environment; it cannot be of genetic origin, even allowing for smart decompression of stored information.
In total, all these tricks used by nature result in the most powerful classi er yet known.* Are there any limits to the brain’s capacity to memorize and to classify? With the tools that humans have developed to expand the possibilities of the brain, such as paper, writing and printing to help memory, and the numerous tools available to simplify and to abbreviate classi cations explored by mathematicians, brain classi cation is only limited by the time spent practising it.Without tools, there are strict limits, of course.
e two-millimetre thick cerebral cortex of humans has a surface of about four sheets of A paper, a chimpanzee’s yields one sheet and a monkey’s is the size of a postcard. It is
* Also the power consumption of the brain is important: even though it contains only about 2% of the body’s mass, is uses 25% of the energy taken in by food.
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estimated that the total intellectually accessible memory is of the order of
Mintellectual GB ,
(465)
though with a large experimental error. e brain is also unparalleled in its processing capacity. is is most clearly demon-
strated by the most important consequence deriving from memory and classi cation: thought and language. Indeed, the many types of thinking or language we use, such as comparing, distinguishing, remembering, recognizing, connecting, describing, deducing, explaining, imagining, etc., all describe di erent ways to classify memories or perceptions. In the end all thinking and talking directly or indirectly classify observations. But how far are computers from achieving this! To talk to a computer program, such as to the famous program Eliza or its successors that mimic a psychoanalyst, is still a disappointing experience. To understand the reasons for this slow development, we ask:
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Reserve your right to think, for even to think wrongly is better than not to think at all.
“ Hypatia of Alexandria
Ein Satz kann nur sagen, wie ein Ding ist, nicht
” was es ist.* “ Ludwig Wittgenstein, Tractatus, . ” Language possibly is the most wonderful gi of human nature. Using the ability to pro-
duce sounds and to put ink on paper, people attach certain symbols,** also called words or terms in this context, to the many partitions they specify with the help of their thinking. Such a categorization is then said to de ne a concept or notion, and is set in italic typeface in this text. A standard set of concepts forms a language.*** In other words, a (human) language is a standard way of symbolic interaction between people.**** ere are human languages based on facial expressions, on gestures, on spoken words, on whistles, on written words, and more. e use of spoken language is considerably younger than the human species; it seems that it appeared only about two hundred thousand years ago. Written language is even younger, namely only about six thousand years old. But the set of concepts
* Propositions can only say how things are, not what they are.
** A symbol is a type of sign, i.e. an entity associated by some convention to the object it refers. Following
Charles Peirce (1839–1914) – see http://www.peirce.org – the most original philosopher born in the United
States, a symbol di ers from an icon (or image) and from an index, which are also attached to objects by
convention, in that it does not resemble the object, as does an icon, and in that it has no contact with the
object, as is the case for an index.
*** e recognition that language is based on a partition of ideas, using the various di erences between
them to distinguish them from each other, goes back to the Swiss thinker Ferdinand de Saussure (1857–
1913), who is regarded as the founder of linguistics. His textbook Cours de linguistique générale, Editions
Payot, 1985, has been the reference work of the eld for over half a century. Note that Saussure, in contrast
to Peirce, prefers the term ‘sign’ to ‘symbol’, and that his de nition of the term ‘sign’ includes also the object
to which it refers.
**** For slightly di erent de nitions and a wealth of other interesting information about language, see the
beautiful book by D C
, e Cambridge Encyclopedia of Language, Cambridge University Press,
1987.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 1113 e
used, the vocabulary, is still expanding. For humans, the understanding of language begins soon a er birth (perhaps even before), the active use begins at around a year of age, the ability to read can start as early as two, and personal vocabulary continues to grow as long as curiosity is alive.
Physics being a lazy way to chat about motion, it needs language as an essential tool. Of the many aspects of language, from literature to poetry, from jokes to military orders, from expressions of encouragement, dreams, love and emotions, physics uses only a small and rather special segment. is segment is de ned by the inherent restriction to talk about motion. Since motion is an observation, i.e. an interaction with the environment that several people experience in the same way, this choice puts a number of restrictions on the contents – the vocabulary – and on the form – the grammar – of such discussions.
For example, from the de nition that observations are shared by others, we get the requirement that the statements describing them must be translatable into all languages. But when can a statement be translated? On this question two extreme points of view are possible: the rst maintains that all statements can be translated, since it follows from the properties of human languages that each of them can express every possible statement. In this view, only sign systems that allow one to express the complete spectrum of human messages form a human language. is property distinguishes spoken and sign language from animal languages, such as the signs used by apes, birds or honey bees, and also from computer languages, such as Pascal or C. With this meaning of language, all statements can be translated by de nition.
It is more challenging for a discussion to follow the opposing view, namely that precise translation is possible only for those statements which use terms, word types and grammatical structures found in all languages. Linguistic research has invested considerable e ort in the distillation of phonological, grammatical and semantic universals, as they are called, from the or so languages thought to exist today.*
e investigations into the phonological aspect, which showed for example that every language has at least two consonants and two vowels, does not provide any material for the discussion of translation.** Studying the grammatical (or syntactic) aspect, one nds that all languages use smallest elements, called ‘words’, which they group into sentences.
ey all have pronouns for the rst and second person, ‘I’ and ‘you’, and always contain nouns and verbs. All languages use subjects and predicates or, as one usually says, the three entities subject, verb and object, though not always in this order. Just check the languages you know.
On the semantic aspect, the long list of lexical universals, i.e. words that appear in all languages, such as ‘mother’ or ‘Sun’, has recently been given a structure. e linguist Anna Wierzbicka performed a search for the building blocks from which all concepts can be
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Ref. 627
* A comprehensive list with 6 800 languages (and with 41 000 language and dialect names) can be found on
the world wide website by Barbara Grimes, Ethnologue – Languages of the World, to be found at the address
http://www.ethnologue.com or in the printed book of the same name.
It is estimated that
languages have existed in the past.
Nevertheless, in today’s world, and surely in the sciences, it is o en su cient to know one’s own language
plus English. Since English is the language with the largest number of words, learning it well is a greater
challenge than learning most other languages.
** Studies explore topics such as the observation that in many languages the word for ‘little’ contains an ‘i’
(or high pitched ‘e’) sound: petit, piccolo, klein, tiny, pequeño, chiisai; exceptions are: small, parvus.
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TA B L E 52 The semantic primitives, following Anna Wierzbicka
I, you, someone, something, people
[substantives]
this, the same, one, two, all, much/many [determiners and quanti ers]
know, want, think, feel, say
[mental predicates]
do, happen
[agent, patient]
good, bad
[evaluative]
big, small
[descriptors]
very
[intensi er]
can, if (would)
[modality, irrealis]
because
[causation]
no (not)
[negation]
when, where, a er (before), under (above) [time and place]
kind of, part of
[taxonomy, partonomy]
like
[hedge/prototype]
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Ref. 628
built. She looked for the de nition of every concept with the help of simpler ones, and continued doing so until a fundamental level was reached that cannot be further reduced.
e set of concepts that are le over are the primitives. By repeating this exercise in many languages, Wierzbicka found that the list is the same in all cases. She thus had discovered universal semantic primitives. In November , the list contained the terms given in Table . Following the life-long research of Anna Wierzbicka and her research school, all these concepts exist in all languages of the world studied so far.* ey have de ned the meaning of each primitive in detail, performed consistency checks and eliminated alternative approaches. ey have checked this list in languages from all language groups, in languages from all continents, thus showing that the result is valid everywhere. In every language all other concepts can be de ned with the help of the semantic primitives.
Simply stated, learning to speak means learning these basic terms, learning how to combine them and learning the names of these composites. e de nition of language given above, namely as a means of communication that allows one to express everything one wants to say, can thus be re ned: a human language is any set of concepts that includes the universal semantic primitives.
For physicists – who aim to talk in as few words as possible – the list of semantic primitives has three facets. First, the approach is appealing, as it is similar to physics’ own aim: the idea of primitives gives a structured summary of everything that can be said, just as the atomic elements structure all objects that can be observed. Second, the list of primitives can be structured. In fact, the list of primitives can be divided into two groups: one group contains all terms describing motion (do, happen, when, where, feel,
* It is easy to imagine that this research steps on the toes of many people. A list that maintains that ‘true’, ‘good’, ‘creation’, ‘life’, ‘mother’ or ‘god’ are composite will elicit violent reactions, despite the correctness of the statements. Indeed, some of these terms were added in the 1996 list, which is somewhat longer. In addition, a list that maintains that we only have about thirty basic concepts in our heads is taken by many to be o ensive.
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Challenge 1114 d Page 157
small, etc. – probably a term from the semantic eld around light or colour should be added) and the other group contains all terms necessary to talk about abstract sets and relations (this, all, kind of, no, if, etc.). Even for linguistics, aspects of motion and logical concepts are the basic entities of human experience and human thinking. To bring the issue to a point, the semantic primitives contain the basic elements of physics and the basic elements of mathematics. All humans are thus both physicists and mathematicians.
e third point is that the list of primitives is too long. e division of the list into two groups directly suggests shorter lists; we just have to ask physicists and mathematicians for concise summaries of their respective elds. To appreciate this aim, try to de ne what ‘if ’ means, or what an ‘opposite’ is – and explore your own ways of reducing the list.
Reducing the list of primitives is also one of our aims in this adventure. We will explore the mathematical group of primitives in this intermezzo; the physical group will occupy us in the rest of our adventure. However, a shorter list of primitives is not su cient. Our goal is to arrive at a list consisting of only one basic concept. Reaching this goal is not simple, though. First, we need to check whether the set of classical physical concepts that we have discovered so far is complete. Can classical physical concepts describe all observations? e second part of our adventure is devoted to this question. e second task is to reduce the list. is task is not straightforward; we have already discovered that physics is based on a circular de nition: in Galilean physics, space and time are de ned using matter, and matter is de ned using space and time. We will need quite some e ort to overcome this obstacle. e third part of this text tells the precise story. A er numerous adventures we will indeed discover a basic concept on which all other concepts are based.
We can summarize all the above-mentioned results of linguistics in the following way. By constructing a statement made only of subject, verb and object, consisting only of nouns and verbs, using only concepts built from the semantic primitives, we are sure that it can be translated into all languages. is explains why physics textbooks are o en so boring: the authors are o en too afraid to depart from this basic scheme. On the other hand, research has shown that such straightforward statements are not restrictive: with them one can say everything that can be said.
Every word was once a poem.
“ ” Ralph Waldo Emerson*
W
?
“Concepts are merely the results, rendered permanent by language, of a previous process of comparison. ” William Hamilton
ere is a group of people that has taken the strict view on translation and on precision to the extreme. ey build all concepts from an even smaller set of primitives, namely only two: ‘set’ and ‘relation’, and explore the various possible combinations of these two concepts, studying their classi cations. Step by step, this radical group, commonly called mathematicians, came to de ne with full precision concepts such as numbers, points,
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* Ralph Waldo Emerson (1803–1882), US-American essayist and philosopher.
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curves, equations, symmetry groups and more. e construction of these concepts is summarized partly in the following and partly in Appendix D.
However, despite their precision, in fact precisely because of it, no mathematical concept talks about nature or about observations.* erefore the study of motion needs other, more useful concepts. What properties must a useful concept have? For example, what is ‘freedom’ or what is a ‘parachute’? Obviously, a useful concept implies a list of its parts, its aspects and their internal relations, as well as their relation to the exterior world.
inkers in various elds, from philosophy to politics, agree that the de nition of any concept requires:
— explicit and xed content, — explicit and xed limits, — explicit and xed domain of application.
Challenge 1115 n Challenge 1116 n
e inability to state these properties or keep them xed is o en the easiest way to distinguish crackpots from more reliable thinkers. Unclearly de ned terms, which thus do not qualify as concepts, regularly appear in myths, e.g. ‘dragon’ or ‘sphinx’, or in ideologies, e.g. ‘worker’ or ‘soul’. Even physics is not immune. For example, we will discover later that neither ‘universe’ nor ‘creation’ are concepts. Are you able to argue the case?
But the three de ning properties of any concepts are interesting in their own right. Explicit content means that concepts are built one onto another. In particular, the most fundamental concepts appear to be those that have no parts and no external relations, but only internal ones. Can you think of one? Only the last part of this walk will uncover the
nal word on the topic. e requirements of explicit limits and explicit contents also imply that all concepts
describing nature are sets, since sets obey the same requirements. In addition, explicit domains of application imply that all concepts also are relations.** Since mathematics is based on the concepts of ‘set’ and of ‘relation’, one follows directly that mathematics can provide the form for any concept, especially whenever high precision is required, as in the study of motion. Obviously, the content of the description is only provided by the study of nature itself; only then do concepts become useful.
In the case of physics, the search for su ciently precise concepts can be seen as the single theme structuring the long history of the eld. Regularly, new concepts have been proposed, explored in all their properties, and tested. Finally, concepts are rejected or adopted, in the same way that children reject or adopt a new toy. Children do this unconsciously, scientists do it consciously, using language.*** For this reason, concepts are universally intelligible.
* Insofar as one can say that mathematics is based on the concepts of ‘set’ and ‘relation’, which are based on experience, one can say that mathematics explores a section of reality, and that its concepts are derived from experience. is and similar views of mathematics are called platonism. More concretely, platonism is the view that the concepts of mathematics exist independently of people, and that they are discovered, and not created, by mathematicians.
In short, since mathematics makes use of the brain, which is a physical system, actually mathematics is applied physics. ** We see that every physical concept is an example of a (mathematical) category, i.e. a combination of objects and mappings. For more details about categories, with a precise de nition of the term, see page 650. *** Concepts formed unconsciously in our early youth are the most di cult to de ne precisely, i.e. with language. Some who were unable to de ne them, such as the Prussian philosopher Immanuel Kant (1724–
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Note that the concept ‘concept’ itself is not de nable independently of experience; a concept is something that helps us to act and react to the world in which we live. Moreover, concepts do not live in a world separate from the physical one: every concept requires memory from its user, since the user has to remember the way in which it was formed; therefore every concept needs a material support for its use and application. us all thinking and all science is fundamentally based on experience.
In conclusion, all concepts are based on the idea that nature is made of related parts. is idea leads to complementing couples such as ‘noun–verb’ in linguistics, ‘set–relation’ or ‘de nition–theorem’ in mathematics, and ‘aspect of nature–pattern of nature’ in physics. ese couples constantly guide human thinking, from childhood onwards, as developmental psychology can testify.
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W
?W
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“Alles, was wir sehen, könnte auch anders sein. Alles, was wir überhaupt beschreiben können, könnte auch anders sein. Es gibt keine Ordnung der Dinge a priori.* ” Ludwig Wittgenstein, Tractatus, .
De ning sets and de ning relations are the two fundamental acts of our thinking. is can be seen most clearly in any book about mathematics; such a book is usually divided into paragraphs labelled ‘de nition’, ‘theorem’, ‘lemma’ and ‘corollary’. e rst type of paragraph de nes concepts, i.e. de nes sets, and the other three types of paragraphs express relations, i.e. connections between these sets. Mathematics is thus the exploration of the possible symbolic concepts and their relations. Mathematics is the science of symbolic necessities.
Sets and relations are tools of classi cation; that is why they are also the tools of any bureaucrat. (See Figure .) is class of humans is characterized by heavy use of paper clips, les, metal closets, archives – which all de ne various types of sets – and by the extensive use of numbers, such as reference numbers, customer numbers, passport numbers, account numbers, law article numbers – which de ne various types of relations between the items, i.e. between the elements of the sets.
Both the concepts of set and of relation express, in di erent ways, the fact that nature can be described, i.e. that it can be classi ed into parts that form a whole. e act of grouping together aspects of experience, i.e. the act of classifying them, is expressed in formal language by saying that a set is de ned. In other words, a set is a collection of elements of our thinking. Every set distinguishes the elements from each other and from the set itself. is de nition of ‘set’ is called the naive de nition. For physics, the de nition is su cient, but you won’t nd many who will admit this. In fact, mathematicians have re-
ned the de nition of the concept ‘set’ several times, because the naive de nition does
1804) used to call them ‘a priori’ concepts (such as ‘space’ and ‘time’) to contrast them with the more clearly de ned ‘a posteriori’ concepts. Today, this distinction has been shown to be unfounded both by the study of child psychology (see the footnote on page 634) and by physics itself, so that these quali ers are therefore not used in our walk. * Whatever we see could be other than it is. Whatever we can describe at all could be other than it is. ere is no a priori order of things.
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TA B L E 53 The defining properties of a set – the ZFC axioms
T
Z
–F
–C
– Two sets are equal if and only if they have the same elements. (Axiom of extensionality)
– e empty set is a set. (Axiom of the null set)
– If x and y are sets, then the unordered pair x, y is a set. (Axiom of unordered pairs)
– If x is a set of sets, the union of all its members is a set. (Union or sum set axiom)
– e entity , ,
,
, ... is a set a – in other words, in nite collections, such
as the natural numbers, are sets. (Axiom of in nity)
– An entity de ned by all elements having a given property is a set, provided this property is reas-
onable; some important technicalities de ning ‘reasonable’ are necessary. (Axiom of separation)
– If the domain of a function is a set, so is its range. (Axiom of replacement)
– e entity y of all subsets of x is also a set, called the power set. (Axiom of the power set)
– A set is not an element of itself – plus some technicalities. (Axiom of regularity)
– e product of a family of non-empty sets is non-empty. Equivalently, picking elements from a
list of sets allows one to construct a new set – plus technicalities. (Axiom of choice)
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a. e more common formulation (though equivalent to the above) is:
, ,, ,, ,,
, ... is a set.
e entity
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Challenge 1117 n
not work well for in nite sets. A famous example is the story about sets which do not con-
tain themselves. Obviously, any set is of two sorts: either it contains itself or it does not.
If we take the set of all sets that do not contain themselves, to which sort does it belong?
To avoid problems with the concept of ‘set’,
mathematics requires a precise de nition. e
rst such de nition was given by the German
mathematician Ernst Zermelo (b.
Berlin,
d.
Freiburg i.B.) and the German–Israeli
mathematician Adolf/Abraham Fraenkel (b.
München, d. Jerusalem). Later, the so-called axiom of choice was added, in order to make it
F I G U R E 280 Devices for the definition of sets (left) and of relations (right)
possible to manipulate a wider class of in nite sets.
e result of these e orts is called the ZFC de nition.* From this basic de nition we
can construct all mathematical concepts used in physics. From a practical point of view,
it is su cient to keep in mind that for the whole of physics, the naive de nition of a
set is equivalent to the precise ZFC de nition, actually even to the simper ZF de nition.
Subtleties appear only for some special types of in nite sets, but these are not used in
physics. In short, from the basic, naive set de nition we can construct all concepts used
Ref. 629
Page 649 Challenge 1118 n
* A global overview of axiomatic set theory is given by P J. C
&R
H
, Non-
Cantorian set theory, Scienti c American 217, pp. 104–116, 1967. ose were the times when Scienti c Amer-
ican was a quality magazine.
Other types of entities, more general than standard sets, obeying other properties, can also be de ned,
and are also subject of (comparatively little) mathematical research. To nd an example, see the section
on cardinals later on. Such more general entities are called classes whenever they contain at least one set.
Can you give an example? In the third part of our mountain ascent we will meet physical concepts that are
described neither by sets nor by classes, containing no set at all. at is were the real fun starts.
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Ref. 630 Challenge 1119 d
Challenge 1120 n
in physics. e naive set de nition is far from boring. To satisfy two people when dividing a cake,
we follow the rule: I cut, you choose. e method has two properties: it is just, as everybody thinks that they have the share that they deserve, and it is fully satisfying, as everybody has the feeling that they have at least as much as the other. What rule is needed for three people? And for four?
Apart from de ning sets, every child and every brain creates links between the different aspects of experience. For example, when it hears a voice, it automatically makes the connection that a human is present. In formal language, connections of this type are called relations. Relations connect and di erentiate elements along other lines than sets: the two form a complementing couple. De ning a set uni es many objects and at the same time divides them into two: those belonging to the set and those that do not; de ning a (binary) relation uni es elements two by two and divides them into many, namely into the many couples it de nes.
Sets and relations are closely interrelated concepts. Indeed, one can de ne (mathematical) relations with the help of sets. A (binary) relation between two sets X and Y is a subset of the product set, where the product set or Cartesian product X Y is the set of all ordered pairs (x, y) with x X and y Y. An ordered pair (x, y) can easily be de ned with the help of sets. Can you nd out how? For example, in the case of the relation ‘is wife of ’, the set X is the set of all women and the set Y that of all men; the relation is given by the list all the appropriate ordered pairs, which is much smaller than the product set, i.e. the set of all possible woman–man combinations.
It should be noted that the de nition of relation just given is not really complete, since every construction of the concept ‘set’ already contains certain relations, such as the relation ‘is element of.’ It does not seem to be possible to reduce either one of the concepts ‘set’ or ‘relation’ completely to the other one. is situation is re ected in the physical cases of sets and relations, such as space (as a set of points) and distance, which also seem impossible to separate completely from each other. In other words, even though mathematics does not pertain to nature, its two basic concepts, sets and relations, are taken from nature. In addition, the two concepts, like those of space-time and particles, are each de ned with the other.
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I
Mathematicians soon discovered that the concept of ‘set’ is only useful if one can also call collections such as , , , ... , i.e. of the number and all its successors, a ‘set’. To achieve this, one property in the Zermelo–Fraenkel list de ning the term ‘set’ explicitly speci es that this collection can be called a set. (In fact, also the axiom of replacement states that sets may be in nite.) In nity is thus put into mathematics and into the tools of our thought right at the very beginning, in the de nition of the term ‘set’. When describing nature, with or without mathematics, we should never forget this fact. A few additional points about in nity should be of general knowledge to any expert on motion.
Only sets can be in nite. And sets have parts, namely their elements. When a thing or a concept is called ‘in nite’ one can always ask and specify what its parts are: for space the parts are the points, for time the instants, for the set of integers the integers, etc. An
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Challenge 1121 n Page 1146
Challenge 1122 n Ref. 631
indivisible or a nitely divisible entity cannot be called in nite.* A set is in nite if there is a function from it into itself that is injective (i.e. di erent
elements map to di erent results) but not onto (i.e. some elements do not appear as images of the map); e.g. the map n n shows that the set of integers is in nite. In nity also can be checked in another way: a set is in nite if it remains so also a er removing one element, even repeatedly. We just need to remember that the empty set is nite.
ere are many types of in nities, all of di erent sizes.** is important result was discovered by the Danish-Russian-German mathematician Georg Cantor ( – ). He showed that from the countable set of natural numbers one can construct other in nite sets which are not countable. He did this by showing that the power set P(ω), namely the set of all subsets, of a countably in nite set is in nite, but not countably in nite. Sloppily speaking, the power set is ‘more in nite’ than the original set. e real numbers R, to be de ned shortly, are an example of an uncountably in nite set; there are many more of them than there are natural numbers. (Can you show this?) However, any type of in nite set contains at least one subset which is countably in nite.
Even for an in nite set one can de ne size as the number of its elements. Cantor called this the cardinality of a set. e cardinality of a nite set is simply given by the number of its elements. e cardinality of a power set is exponentiated by the cardinality of the set. e cardinality of the set of integers is called ℵ , pronounced ‘aleph zero’, a er the
rst letter of the Hebrew alphabet. e smallest uncountable cardinal is called ℵ . e next cardinal is called ℵ etc. A whole branch of mathematics is concerned with the manipulation of these in nite ‘numbers’; addition, multiplication, exponentiation are easily de ned. For some of them, even logarithms and other functions make sense.***
e cardinals de ned in this way, including ℵn, ℵω , ℵℵℵ are called accessible, because since Cantor, people have de ned even larger types of in nities, called inaccessible. ese numbers (inaccessible cardinals, measurable cardinals, supercompact cardinals, etc.) need additional set axioms, extending the ZFC system. Like the ordinals and the cardinals, they form examples of what are called trans nite numbers.
e real numbers have the cardinality of the power set of the integers, namely ℵ . Can you show this? e result leads to the famous question: Is ℵ = ℵ or not? e statement that this be so is called the continuum hypothesis and was unproven for several generations. e surprising answer came in : the usual de nition of the concept of set is not speci c enough to x the answer. By specifying the concept of set in more detail, with additional axioms – remember that axioms are de ning properties – you can make the continuum hypothesis come out either right or wrong, as you prefer.
Another result of research into trans nites is important: for every de nition of a type of in nite cardinal, it seems to be possible to nd a larger one. In everyday life, the idea of in nity is o en used to stop discussions about size: ‘My big brother is stronger than yours.’ ‘But mine is in nitely stronger than yours!’ Mathematics has shown that questions on size do continue a erwards: ‘ e strength of my brother is the power set of that of
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* erefore, most gods, being concepts and thus sets, are either nite or, in the case where they are in nite,
they are divisible. It seems that only polytheistic world views are not disturbed by this conclusion.
** In fact, there is such a huge number of types of in nities that none of these in nities itself actually de-
scribes this number. Technically speaking, there are as many in nities as there are ordinals.
*** Many results are summarized in the excellent and delightful paperback by R R
, In nity and
the Mind – the Science and Philosophy of the In nite, Bantam, Toronto, 1983.
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Ref. 632 Challenge 1123 e
yours!’ Rucker reports that mathematicians conjecture that there is no possible nor any conceivable end to these discussions.
For physicists, a simple question appears directly. Do in nite quantities exist in nature? Or better, is it necessary to use in nite quantities to describe nature? You might want to clarify your own opinion on the issue. It will be settled during the rest of our adventure.
F
Which relations are useful to describe patterns in nature? A typical example is ‘larger stones are heavier’. Such a relation is of a speci c type: it relates one speci c value of an observable ‘volume’ to one speci c value of the observable ‘weight’. Such a one-to-one relation is called a (mathematical) function or mapping. Functions are the most speci c types of relations; thus they convey a maximum of information. In the same way as numbers are used for observables, functions allow easy and precise communication of relations between observations. All physical rules and ‘laws’ are therefore expressed with the help of functions and, since physical ‘laws’ are about measurements, functions of numbers are their main building blocks.
A function f , or mapping, is a thus binary relation, i.e. a set f = (x, y) of ordered pairs, where for every value of the rst element x, called the argument, there is only one pair (x, y). e second element y is called the value of the function at the argument x.
e set X of all arguments x is called the domain of de nition and the set Y of all second arguments y is called the range of the function. Instead of f = (x, y) one writes
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f X Y and f x y or y = f (x) ,
(466)
where the type of arrow – with initial bar or not – shows whether we are speaking about sets or about elements.
We note that it is also possible to use the couple ‘set’ and ‘mapping’ to de ne all mathematical concepts; in this case a relation is de ned with the help of mappings. A modern school of mathematical thought formalized this approach by the use of (mathematical) categories, a concept that includes both sets and mappings on an equal footing in its de nition.*
To think and talk more clearly about nature, we need to de ne more specialized concepts than sets, relations and functions, because these basic terms are too general. e most important concepts derived from them are operations, algebraic structures and numbers.
A (binary) operation is a function that maps the Cartesian product of two copies of a set X into itself. In other words, an operation w takes an ordered couple of arguments
* A category is de ned as a collection of objects and a collection of ‘morphisms’, or mappings. Morphisms
can be composed; the composition is associative and there is an identity morphism. e strange world of cat-
egory theory, sometimes called the abstraction of all abstractions, is presented in F. W
L
&S
H. S
, Conceptual Mathematics: a First Introduction to Categories, Cambridge Uni-
versity Press, 1997.
Note that every category contains a set; since it is unclear whether nature contains sets, as we will discuss
on page 681, it is questionable whether categories will be useful in the uni cation of physics, despite their
intense and abstract charm.
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x X and assigns to it a value y X:
w X X X and w (x, x) y .
(467)
Challenge 1124 n Page 1204 Page 1213 Ref. 633
Is division of numbers an operation in the sense just de ned? Now we are ready to de ne the rst of three basic concepts of mathematics. An al-
gebraic structure, also called an algebraic system, is (in the most restricted sense) a set together with certain operations. e most important algebraic structures appearing in physics are groups, vector spaces, and algebras.
In addition to algebraic structures, mathematics is based on order structures and on topological structures. Order structures are building blocks of numbers and necessary to de ne comparisons of any sort. Topological structures are built, via subsets, on the concept of neighbourhood. ey are necessary to de ne continuity, limits, dimensionality, topological spaces and manifolds.
Obviously, most mathematical structures are combinations of various examples of these three basic structure types. For example, the system of real numbers is given by the set of real numbers with the operations of addition and multiplication, the order relation ‘is larger than’ and a continuity property. ey are thus built by combining an algebraic structure, an order structure and a topological structure. Let us delve a bit into the details.
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Challenge 1125 n Challenge 1126 n
Which numbers are multiplied by six when their last digit is taken away and transferred to
“the front? ” Numbers are the oldest mathematical concept and are found in all cultures. e notion
of number, in Greek ἀριθµός, has been changed several times. Each time the aim was to
include wider classes of objects, but always retaining the general idea that numbers are
entities that can be added, subtracted, multiplied and divided.
e modern way to write numbers, as e.g. in
ë=
, is essential
for science.* It can be argued that the lack of a good system for writing down and for
calculating with numbers delayed the progress of science by several centuries. (By the
way, the same can be said for the a ordable mass reproduction of written texts.)
e simplest numbers, , , , , , ..., are usually seen as being taken directly from ex-
perience. However, they can also be constructed from the notions of ‘relation’ and ‘set’.
One of the many possible ways to do this (can you nd another?) is by identifying a nat-
ural number with the set of its predecessors. With the relation ‘successor of ’, abbreviated
S, this de nition can be written as
=, =S = , = ,
=S = =
,
and n + = S n = , ..., n .
(468)
is set, together with the binary operations ‘addition’ and ‘multiplication,’ constitutes
* However, there is no need for written numbers for doing mathematics, as shown by M
A
,
Ethnomathematics – A Multicultural View of Mathematical Ideas, Brooks/Cole, 1991.
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Page 1194
the algebraic system N = (N, +, ë, ) of the natural numbers. For all number systems the algebraic system and the set are o en sloppily designated by the same symbol. e algebraic system N is a so-called semi-ring, as explained in Appendix D. (Some authors prefer not to count the number zero as a natural number.) Natural numbers are fairly useful.
TA B L E 54 Some large numbers
N
E
Around us
,,
to ë c. c. .ë
Information c. c. c. c. c.
ë ë
number of angels that can be in one place at the same time, following omas Aquinas Ref. 634
number of times a newspaper can be folded in alternate perpendicular directions
largest number of times a paper strip has been folded in the same direction
Ref. 635
number of digits in precision measurements that will probably never be achieved
petals of common types of daisy and sun ower Ref. 636
faces of a diamond with brilliant cut
stars visible in the night sky
leaves of a tree ( m beech)
humans in the year
ants in the world
number of snow akes falling on the Earth per year
grains of sand in the Sahara desert
stars in the universe
cells on Earth
atoms making up the Earth ( ë mol)
km ë ë . ë kg m ë mol kg ë
atoms in the visible universe
photons in the visible universe
number of atoms tting in the visible universe
number of space-time points inside the visible universe
record number of languages spoken by one person words spoken on an average day by a man words spoken on an average day by a woman number of languages on Earth words of the English language (more than any other language, with the possible exception of German) number of scientists on Earth around the year words spoken during a lifetime ( / time awake, words per minute) pulses exchanged between both brain halves every second
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N
c. ë
ë ë !ë ! = .ë .ë .ë c. c. c. Parts of us
ë to
ë
ë ë ë .ë ë
E
words heard and read during a lifetime
image pixels seen in a lifetime ( ë s ë ( ms) ë (awake) ë (nerves to the brain) Ref. 637
bits of information processed in a lifetime (the above times )
printed words available in (di erent) books around the world (c. ë
books consisting of
words)
possible positions of the
Rubik’s Cube Ref. 638
possible positions of the
Rubik-like cube
possible positions of the
Rubik-like cube
possible games of chess
possible games of go
possible states in a personal computer
numbers of muscles in the human body, of which about half are in the face hairs on a healthy head neurons in the brain of a grasshopper light sensitive cells per retina ( million rods and million cones) neurons in the human brain memory bits in the human brain blinks of the eye during a lifetime (about once every four seconds when awake) breaths taken during human life heart beats during a human life letters (base pairs) in haploid human DNA bits in a compact disc humans who have ever lived cells in the human body bacteria carried in the human body
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Page 1194 Ref. 639
e system of integers Z = (..., − , − , , , , ..., +, ë, , ) is the minimal ring that is an extension of the natural numbers. e system of rational numbers Q = (Q, +, ë, , ) is the minimal eld that is an extension of the ring of the integers. ( e terms ‘ring’ and ‘ eld’ are explained in Appendix D.) e system of real numbers R = (R, +, ë, , , ) is the minimal extension of the rationals that is continuous and totally ordered. (For the de nition of continuity, see page and .) Equivalently, the reals are the minimal extension of the rationals forming a complete, totally strictly-Archimedean ordered eld. is is the historical construction – or de nition – of the integer, rational and real numbers from the natural numbers. However, it is not the only one construction possible. e most beautiful de nition of all these types of numbers is the one discovered in by John Conway, and popularized by him, Donald Knuth and Martin Kruskal.
— A number is a sequence of bits. e two bits are usually called ‘up’ and ‘down’. Examples
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11 2
1 11 1/2
0
111 3 11 1 31/2 1 1 31 /141 1/4
-1/4
-1/2
-1
1 1
-3/4
1
-3/2
-2
11 -3
111
smaller
earlier
1111 4
-4 1111
...
ω = smallest infinite
1
1=1
1=1111...
11111 1 1
ω+4
2ω
2 ω
ω e
ω π 1111 8/3
1111 1 ω-4
1 1 ω/2
11
1
1
ω/4 1
sqrt(ω)
11 2/3
11 11 2/3+2ι/3
ι = 1/ω = simplest infinitesimal
11
1 1 1111 1 1 1
ι
4ι
11
1
1 11 1
sqrt(ι) ι2
-1/3 11111
-4/5 11 11 1 1 1
11 1 1 11111 11...
-sqrt(2) -ω 1=1111 ...
1 1 -ω/2 -2ω
1 1
-ω 2
ω -e
1
1
F I G U R E 281 The surreal numbers in conventional and in bit notation
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of numbers and the way to write them are given in Figure 281.
— e empty sequence is zero.
— A nite sequence of n ups is the integer number n, and a nite sequence of n downs
is the integer −n. Finite sequences of mixed ups and downs give the dyadic rational
numbers. Examples are 1, 2, 3, − , 19/4,
, etc. ey all have denominators with
a power of 2. e other rational numbers are those that end in an in nitely repeating
string of ups and downs, such as the reals, the in nitesimals and simple in nite num-
bers. Longer countably in nite series give even more crazy numbers. e complete
class is called the class of surreal numbers.*
ere is a second way to write surreal numbers. e rst is the just mentioned sequence of bits. But in order to de ne addition and multiplication, another notation is usually used, deduced from Figure . A surreal α is de ned as the earliest number of
* e surreal numbers do not form a set since they contain all ordinal numbers, which themselves do not form a set, even though they of course contain sets. In short, ordinals and surreals are classes which are larger than sets.
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all those between two series of earlier surreals, the le and the right series:
α = a, b, c, ... A, B, C, ... with a, b, c, < α < A, B, C .
(469)
For example, we have
= , , = , =− , − , =− ,
=,
, = , ,, ,
, , , =+
,
(470)
showing that the nite surreals are the dyadic numbers m n (n and m being integers). Given two surreals α = ..., a, ... ..., A, ... with a < α < A and β = ..., b, ... ..., B, ... with b < β < B, addition is de ned recursively, using earlier, already de ned numbers, as
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α + β = ..., a + β, ..., α + b, ... ..., A + β, ..., α + B, ... .
(471)
Challenge 1127 n
is de nition is used simply because it gives the same results as usual addition for integers and reals. Can you con rm this? By the way, addition is not always commutative. Are you able to nd the exceptions, and to nd the de nition for subtraction? Multiplication is also de ned recursively, namely by the expression
αβ = ..., aβ + αb − ab, ..., Aβ + αB − AB, ... ..., aβ + αB − aB, ..., Aβ + αb − Ab, ... .
(472)
Ref. 639
Challenge 1128 n Appendix D
ese de nitions allow one to write ι = ω, and to talk about numbers such as ω , the
square root of in nity, about ω + , ω − , ω, eω and about other strange numbers shown
in Figure . However, the surreal numbers are not commonly used. More common is
one of their subsets.
e real numbers are those surreals whose length is not larger than in nity and that
do not have periodic endings with a period of length . In other words, the surreals dis-
tinguish the number .
from the number , whereas the reals do not. In fact,
between the two, there are in nitely many surreal numbers. Can you name a few?
Reals are more useful for describing nature than surreals, rst because they form a set –
which the surreals do not – and secondly because they allow the de nition of integration.
Other numbers de ned with the help of reals, e.g. the complex numbers C and the qua-
ternions H, are presented in Appendix D. A few more elaborate number systems are also
presented there.
To conclude, in physics it is usual to call numbers the elements of any set that is a semi-
ring (e.g. N), a ring (e.g. Z) or a eld (Q, R, C or H). All these concepts are de ned in
Appendix D. Since numbers allow one to compare magnitudes and thus to measure, they
play a central role in the description of observations.
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Ref. 640 Challenge 1129 n
“A series of equal balls is packed in such a way that the area of needed wrapping paper is minimal. For small numbers of balls the linear package, with all balls in one row, is the most e cient. For which number of balls is the linear ” package no longer a minimum?
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Ref. 641 Ref. 642
“Die Forderung der Möglichkeit der einfachen Zeichen ist die Forderung der Bestimmtheit des Sinnes.* ” Ludwig Wittgenstein, Tractatus, .
Several well-known physicists have repeatedly asked why mathematics is so important. For example, Niels Bohr is quoted as having said: ‘We do not know why the language of mathematics has been so e ective in formulating those laws in their most succinct form.’ Eugene Wigner wrote an o en cited paper entitled e unreasonable e ectiveness of mathematics. At the start of science, many centuries earlier, Pythagoras and his contemporaries were so overwhelmed by the usefulness of numbers in describing nature, that Pythagoras was able to organize a sect based on this connection. e members of the inner circle of this sect were called ‘learned people,’ in Greek ‘mathematicians’, from the Greek µάθηµα ‘teaching’. is sect title then became the name of the modern profession.
ese men forgot that numbers, as well as a large part of mathematics, are concepts developed precisely with the aim of describing nature. Numbers and mathematical concepts were developed right from the start to provide as succinct a description as possible.
at is one consequence of mathematics being the science of symbolic necessities. Perhaps we are being too dismissive. Perhaps these thinkers mainly wanted to express their feeling of wonder when experiencing that language works, that thinking and our brain works, and that life and nature are so beautiful. is would put the title question nearer to the well-known statement by Albert Einstein: ‘ e most incomprehensible fact about the universe is that it is comprehensible.’ Comprehension is another word for description, i.e. for classi cation. Obviously, any separable system is comprehensible, and there is nothing strange about it. But is the universe separable? As long as is it described as being made of particles and vacuum, this is the case. We will nd in the third part of this adventure that the basic assumption made at our start is built on sand. e assumption that observations in nature can be counted, and thus that nature is separable, is an approximation. e quoted ‘incomprehensibility’ becomes amazement at the precision of this approximation. Nevertheless, Pythagoras’ sect, which was based on the thought that ‘everything in nature is numbers’, was wrong. Like so many beliefs, observation will show that it was wrong.
Die Physik ist für Physiker viel zu schwer.**
“ ” David Hilbert
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* e requirement that simple signs be possible is the requirement that sense be determinate. ** Physics is much too di cult for physicists.
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Page 157 Ref. 643
Page 1049
“Die Sätze der Mathematik sind Gleichungen, also Scheinsätze. Der Satz der Mathematik drückt keinen Gedanken aus.* ” Ludwig Wittgenstein, Tractatus, . , .
Surely, mathematics is a vocabulary that helps us to talk with precision. Mathematics can be seen as the exploration of all possible concepts that can be constructed from the two fundamental bricks ‘set’ and ‘relation’ (or some alternative, but equivalent pair). Mathematics is the science of symbolic necessities. Rephrased again, mathematics is the exploration of all possible types of classi cations. is explains its usefulness in all situations where complex, yet precise classi cations of observations are necessary, such as in physics.
However, mathematics cannot express everything that humans want to communicate, such as wishes, ideas or feelings. Just try to express the fun of swimming using mathematics. Indeed, mathematics is the science of symbolic necessities; thus mathematics is not a language, nor does it contain one. Mathematical concepts, being based on abstract sets and relations, do not pertain to nature. Despite its beauty, mathematics does not allow us to talk about nature or the observation of motion. Mathematics does not tell what to say about nature; it does tell us how to say it.
In his famous lecture in Paris, the German mathematician David Hilbert** gave a list of great challenges facing mathematics. e sixth of Hilbert’s problems was to nd a mathematical treatment of the axioms of physics. Our adventure so far has shown that physics started with circular de nitions that has not yet been eliminated a er years of investigations: space-time is de ned with the help of objects and objects are de ned with the help of space and time. Being based on a circular de nition, physics is thus not modelled a er mathematics, even if many physicists and mathematicians, including Hilbert, would like it to be so. Physicists have to live with logical problems and have to walk on unsure ground in order to achieve progress. In fact, they have done so for years. If physics were an axiomatic system, it would not contain contradictions; on the other hand, it would cease to be a language and would cease to describe nature. We will return to this issue later.
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C **
Challenge 1130 n What is the largest number that can be written with four digits of 2 and no other sign?
* e propositions of mathematics are equations, and therefore pseudo-propositions. A proposition of mathematics does not express a thought. ** David Hilbert (1862 Königsberg–1943 Göttingen), professor of mathematics in Göttingen, greatest mathematician of his time. He was a central gure to many parts of mathematics, and also played an important role both in the birth of general relativity and of quantum theory. His textbooks are still in print. His famous personal credo was: ‘Wir müssen wissen, wir werden wissen.’ (We must know, we will know.) His famous Paris lecture is published e.g. in Die Hilbertschen Probleme, Akademische Verlagsgesellscha Geest & Portig, 1983. e lecture galvanized all of mathematics. (Despite e orts and promises of similar fame, nobody in the world had a similar overview of mathematics that allowed him or her to repeat the feat in the year 2000.) In his last decade he su ered the persecution of the Nazi regime; the persecution eliminated Göttingen from
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And with four 4s?
Challenge 1131 e
**
Pythagorean triplets are integers that obey a + b = c . Give at least ten examples. en show the following three properties: at least one number in a triplet is a multiple of 3; at least one number in a triplet is a multiple of 4; at least one number in a triplet is a multiple of 5.
Challenge 1132 d
**
e number n, when written in decimal notation, has a periodic sequence of digits.
e period is at most n − digits long, as for = .
.... Which other
numbers n have periods of length n − ?
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**
Challenge 1133 n Challenge 1134 n
Felix Klein was a famous professor of mathematics at Göttingen University. ere were two types of mathematicians in his department: those who did research on whatever they wanted and those for which Klein provided the topic of research. To which type did Klein belong?
Obviously, this is a variation of another famous puzzle. A barber shaves all those people who do not shave themselves. Does the barber shave himself?
Challenge 1135 n
**
Everybody knows what a magic square is: a square array of numbers, in the simplest case from 1 to 9, that are distributed in such a way that the sum of all rows, columns (and possibly all diagonals) give the same sum. Can you write down the simplest magic cube?
Ref. 644
**
e digits 0 to 9 are found on keyboards in two di erent ways. Calculators and keyboards have the 7 at the top le , whereas telephones and automatic teller machines have the digit 1 at the top le . e two standards, respectively by the International Standards Organization (ISO) and by the International Telecommunication Union (ITU, formerly CCITT), evolved separately and have never managed to merge.
** Leonhard Euler in his notebooks sometimes wrote down equations like
+ + + + + ... = − .
(473)
Challenge 1136 d Can this make sense?
the list of important science universities, without recovering its place up to this day.
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Ref. 645 Page 663
Die Grenzen meiner Sprache bedeuten die Grenzen meiner Welt.*
“ Ludwig Wittgenstein, Tractatus, .
Der Satz ist ein Bild der Wirklichkeit. Der Satz
” ist ein Modell der Wirklichkeit, so wie wir sie “uns denken.**
” Ludwig Wittgenstein, Tractatus, .
In contrast to mathematics, physics does aim at being a language. rough the description of motion it aims to express everything observed and, in particular, all examples and possibilities of change.*** Like any language, physics consists of concepts and sentences. In order to be able to express everything, it must aim to use few words for a lot of facts.**** Physicists are essentially lazy people: they try to minimize the e ort in everything they do.
e concepts in use today have been optimised by the combined e ort of many people to be as practical, i.e. as powerful as possible. A concept is called powerful when it allows one to express in a compact way a large amount of information, meaning that it can rapidly convey a large number of details about observations.
General statements about many examples of motion are called rules or patterns. In the past, it was o en said that ‘laws govern nature’, using an old and inappropriate ideology. A physical ‘law’ is only a way of saying as much as possible with as few words as possible. When saying ‘laws govern nature’ we actually mean to say ‘being lazy, we describe observations with patterns’. Laws are the epitome of laziness. Formulating laws is pure sloth. In fact, the correct expression is patterns describe nature.
Physicists have de ned the laziness necessary for their eld in much detail. In order to become a master of laziness, we need to distinguish lazy patterns from those which are not, such as lies, beliefs, statements that are not about observations, and statements that are not about motion. We do this below.
e principle of extreme laziness is the origin, among others, of the use of numbers in physics. Observables are o en best described with the help of numbers, because numbers allow easy and precise communication and classi cation. Length, velocity, angles, temperature, voltage or eld strength are of this type. e notion of ‘number’, used in every
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* e limits of my language are the limits of my world. ** A proposition is a picture of reality. A proposition is a model of reality as we imagine it. *** All observations are about change or variation. e various types of change are studied by the various sciences; they are usually grouped in the three categories of human sciences, formal sciences and natural sciences. Among the latter, the oldest are astronomy and metallurgy. en, with the increase of curiosity in early antiquity, came the natural science concerned with the topic of motion: physics. In the course of our walk it will become clear that this seemingly restrictive de nition indeed covers the whole set of topics studied in physics. In particular it includes the more common de nition of physics as the study of matter, its properties, its components and their interactions. **** A particular, speci c observation, i.e. a speci c example of input shared by others, is called a fact, or in other contexts, an event. A striking and regularly observed fact is called a phenomenon, and a general observation made in many di erent situations is called a (physical) principle. (O en, when a concept is introduced that is used with other meaning in other elds, in this walk it is preceded by the quali er ‘physical’ or ‘mathematical’ in parentheses.) Actions performed towards the aim of collecting observations are called experiments. e concept of experiment became established in the sixteenth century; in the evolution of a child, it can best be compared to that activity that has the same aim of collecting experiences: play.
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measurement, is constructed, o en unconsciously, from the notions of ‘set’ and ‘relation’, as shown above. Apart from the notion of number, other concepts are regularly de ned to allow fast and compact communication of the ‘laws’ of nature; all are ‘abbreviation tools.’ In this sense, the statement ‘the level of the Kac–Moody algebra of the Lagrangian of the heterotic superstring model is equal to one’ contains precise information, explainable to everybody; however, it would take dozens of pages to express it using only the terms ‘set’ and ‘relation.’ In short, the precision common in physics results from its quest for laziness.
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Das logische Bild der Tatsachen ist der Gedanke.*
“ Ludwig Wittgenstein, Tractatus, ” e title question is o en rephrased as: are physical concepts free of beliefs, taste or per-
sonal choices? e question has been discussed so much that it even appears in Hollywood movies. We give a short summary that can help you to distinguish honest from dishonest teachers.
Creation of concepts, in contrast to their discovery, would imply free choice between many alternative possibilities. e chosen alternative would then be due to the beliefs or tastes used. In physics (in obvious contrast to other, more ideological elds of enquiry), we know that di erent physical descriptions of observations are either equivalent or, in the opposite case, imprecise or even wrong. A description of observations is thus essentially unique: any choices of concepts are only apparent. ere is no real freedom in the de nition of physical concepts. In this property, physics is in strong contrast to artistic activity.
If two di erent concepts can be used to describe the same aspect of observations, they must be equivalent, even if the relation that leads to the equivalence is not immediately clear. In fact, the requirement that people with di erent standpoints and observing the same event deduce equivalent descriptions lies at the very basis of physics. It expresses the requirement that observations are observer independent. In short, the strong requirement of viewpoint independence makes the free choice of concepts a logical impossibility.
e conclusion that concepts describing observations are discovered rather than created is also reached independently in the eld of linguistics by the above-mentioned research on semantic primitives,** in the eld of psychology by the observations on the formation of the concepts in the development of young children, and in the eld of ethology by the observations of animal development, especially in the case of mammals. In all three elds detailed observations have been made of how the interactions between an individual and its environment lead to concepts, of which the most basic ones, such as space, time, object or interaction, are common across the sexes, cultures, races and across many animal species populating the world. Curiosity and the way that nature works leads to the same concepts for all people and even the animals; the world o ers only one possibility, without room for imagination. Imagining that physical concepts can be created at your leisure is a belief – or a useful exercise, but never successful.
* A logical picture of facts is a thought. ** Anna Wierzbicka concludes that her research clearly indicates that semantic primitives are discovered, in Ref. 628 particular that they are deduced from the fundamentals of human experience, and not invented.
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Page 1115
Physical concepts are classi cations of observations. e activity of classi cation itself
follows the patterns of nature; it is a mechanical process that machines can also perform.
is means that any distinction, i.e. any statement that A is di erent from B, is a theory-
free statement. No belief system is necessary to distinguish di erent entities in nature.
Cats and pigs can also do so. Physicists can be replaced by animals, even by machines.
Our mountain ascent will repeatedly con rm this point.
As already mentioned, the most popular physical concepts allow to describe observa-
tions as succinctly and as accurately as possible. ey are formed with the aim of having
the largest possible amount of understanding with the smallest possible amount of e ort.
Both Occam’s razor – the requirement not to introduce unnecessary concepts – and the
drive for uni cation automatically reduce the number and the type of concepts used in
physics. In other words, the progress of physical science was and is based on a programme
that reduces the possible choice of concepts as drastically as possible.
In summary, we found that physical concepts are the same for everybody and are free
of beliefs and personal choices: they are rst of all boring. Moreover, as they could stem
from machines instead of people, they are born of laziness. Despite these human analo-
gies – not meant to be taken too seriously – physical concepts are not created; they are
discovered. If a teacher tells you the opposite, he is lying.
Having handled the case of physical concepts, let us now turn to physical statements.
e situation is somewhat similar: physical statements must be lazy, arrogant and boring.
Let us see why.
“Wo der Glaube anfängt, hört die Wissenscha auf.* Ernst Haeckel, Natürliche Schöpfungsgeschichte,
”.
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Grau, treuer Freund, ist alle eorie, Und grün des Lebens goldner Baum.**
“ J.W. v. Goethe, Faust.
Physics is usually presented as an objective
” science, but I notice that physics changes and “the world stays the same, so there must be
something subjective about physics.
” Richard Bandler
Progressing through the study of motion re ects a young child’s attitude towards life. e progress follows the simple programme on the le of Table .
Adult scientists do not have much more to add, except the more fashionable terms on the right, plus several specialized professions to make money from them. e experts of step are variously called lobbyists or fund raisers; instead of calling this program ‘curiosity’, they call it the ‘scienti c method.’ ey mostly talk. Physics being the talk about motion,*** and motion being a vast topic, many people specialize in this step.
* Where belief starts, science ends. ** ‘Grey, dear friend, is all theory, and green the golden tree of life.’ Johann Wolfgang von Goethe (1749– 1832), the most in uential German poet. *** Several sciences have the term ‘talk’ as part of their name, namely all those whose name nishes in ‘-logy’,
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TA B L E 55 The ‘scientific method’
Normal description Curiosity
1. look around a lot 2. don’t believe anything told 3. choose something interesting and explore it yourself 4. make up your own mind and describe precisely what you saw 5. check if you can also describe similar situations in the same way 6. increase the precision of observation until the checks either fail or are complete 7. depending on the case, continue with step 4 or 1
Lobbyist description Scienti c method 1. interact with the world 2. forget authority 3. observe
4. use reason, build hypothesis
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5. analyse hypothesis
6. perform experiments until hypothesis is proved false or established
7. ask for more money
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e experts of step are called experimental physicists or simply experimentalists, a term derived from the Latin ‘experiri’, meaning ‘to try out’. Most of them are part of the category ‘graduate students’. e experts of steps and are called theoretical physicists or simply theoreticians.* is is a rather modern term; for example, the rst professors of theoretical physics were appointed around the start of the twentieth century. e term is derived from the Greek θεωρία meaning ‘observation, contemplation’. Finally, there are the people who focus on steps to , and who induce others to work on steps to ; they are called geniuses.
Obviously an important point is hidden in step : how do all these people know whether their checks fail? How do they recognize truth?
All professions are conspiracies against laymen.
“ ” George Bernard Shaw
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Get your facts straight, and then you can distort them at your leisure.
“ ” Mark Twain
such as e.g. biology. e ending stems from ancient Greek and is deduced from λήγηιν meaning ‘to say, to talk’. Physics as the science of motion could thus be called ‘kinesiology’ from κίνησις, meaning ‘motion’; but for historical reasons this term has a di erent meaning, namely the study of human muscular activity. e term ‘physics’ is either derived from the Greek φύσικη (τέχνη is understood) meaning ‘(the art of) nature’, or from the title of Aristotle’ works τά φυσικά meaning ‘natural things’. Both expressions are derived from φύσις, meaning ‘nature’. * If you like theoretical physics, have a look at the refreshingly candid web page by Nobel Prize winner Gerard ‘t Hoo with the title How to become a good theoretical physicist. It can be found at http://www.phys. uu.nl/~thoo /theorist.html.
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,
“ e pure truth is always a lie.
” Bert Hellinger
Lies are useful statements, as everybody learns during their youth. One reason that they
are useful is because we can draw any imaginable conclusion from them. A well-known
discussion between two Cambridge professors early in the twentieth century makes the
point. McTaggart asked: ‘If + = , how can you prove that I am the pope?’ Godfrey
Hardy: ‘If + = , then = ; subtract ; then = ; but McTaggart and the pope are two;
therefore McTaggart and the pope are one.’ As noted long ago, ex falso quodlibet. From
what is wrong, anything imaginable can be deduced. It is true that in our mountain ascent
we need to build on previously deduced results and that our trip could not be completed
if we had a false statement somewhere in our chain of arguments. But lying is such an
important activity that one should learn to perform it well.
ere are various stages in the art of lying. Many animals have been shown to de-
Ref. 610 ceive their kin. Children start lying just before their third birthday, by hiding experiences.
Adults cheat on taxes. And many intellectuals or politicians even claim that truth does
not exist.
However, in most countries, everybody must know what ‘truth’ is, since in a law court
for example, telling an untruth can lead to a prison sentence. e courts are full of experts
in lie detection. If you lie in court, you better do it well; experience shows that you might
get away with many criminal activities. In court, a lie is a statement that knowingly con-
trasts with observations.* e truth of a statement is thus checked by observation. e
check itself is sometimes called the proof of the statement. For law courts, as for physics,
truth is thus the correspondence with facts, and facts are shared observations. A ‘good’ lie
is thus a lie whose contrast with shared observations is hard to discover.
e rst way of lying is to put an emphasis on the sharedness only. Populists and po-
lemics do this regularly. (‘Every foreigner is a danger for the values of our country.’) Since
almost any imaginable opinion, however weird, is held by some group – and thus shared
– one can always claim it as true. Unfortunately, it is no secret that ideas also get shared
because they are fashionable, imposed or opposed to somebody who is generally disliked.
O en a sibling in a family has this role – remember Cassandra.** For a good lie we thus
need more than sharedness, more than intersubjectivity alone.
A good lie should be, like a true statement, really independent of the listener and the
observer and, in particular, independent of their age, their sex, their education, their civil-
ization or the group to which they belong. For example, it is especially hard – but not im-
possible – to lie with mathematics. e reason is that the basic concepts of mathematics,
be they ‘set’, ‘relation’ or ‘number’, are taken from observation and are intersubjective, so
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* Statements not yet checked are variously called speculations, conjectures, hypotheses, or – wrongly – simply
theses. Statements that are in correspondence with observations are called correct or true; statements that
contrast with observations are called wrong or false.
** e implications of birth order on creativity in science and on acceptance of new ideas has been studied in
the fascinating book by F
J. S
, Born to Rebel – Birth Order, Family Dynamics and Creative
Lives, Panthon Books, 1996. is exceptional book tells the result of a life-long study correlating the personal
situations in the families of thousands of people and their receptivity to about twenty revolutions in the
recent history. e book also includes a test in which the reader can deduce their own propensity to rebel,
on a scale from 0 to 100 %. Darwin scores 96 % on this scale.
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Page 749 Ref. 646
that statements about them are easily checked. Usually, lies thus avoid mathematics.* Secondly, a ‘good’ lie should avoid statements about observations and use interpreta-
tions instead. For example, some people like to talk about other universes, which implies talking about fantasies, not about observations. A good lie has to avoid, however, to fall in the opposite extreme, namely to make statements which are meaningless; the most destructive comment that can be made about a statement is the one used by the great Austrian physicist Wolfgang Pauli: ‘ at is not even wrong.’
irdly, a good lie doesn’t care about observations, only about imagination. Only truth needs to be empirical, to distinguish it from speculative statements. If you want to lie ‘well’ even with empirical statements, you need to pay attention. ere are two types of empirical statements: speci c statements and universal statements. For example, ‘On the
st of August I saw a green swan swimming on the northern shore of the lake of Varese’ is speci c, whereas ‘All ravens are black’ is universal, since it contains the term ‘all’. ere is a well-known di erence between the two, which is important for lying well: speci c statements cannot be falsi ed, they are only veri able, and universal statements cannot be veri ed, they are only falsi able. Why is this so?
Universal statements such as ‘the speed of light is constant’ cannot be tested for all possible cases. (Note that if they could, they would not be universal statements, but just a list of speci c ones.) However, they can be reversed by a counter-example. Another example of the universal type is: ‘Apples fall upwards.’ Since it is falsi ed by an observation conducted by Newton several centuries ago, or by everyday experience, it quali es as an (easily detectable) lie. In general therefore, lying by stating the opposite of a theory is usually unsuccessful. If somebody insists on doing so, the lie becomes a superstition, a belief, a prejudice or a doctrine. ese are the low points in the art of lying. A famous case of insistence on a lie is that of the colleagues of Galileo, who are said to have refused to look through his telescope to be convinced that Jupiter has moons, an observation that would have shaken their belief that everything turns around the Earth. Obviously these astronomers were amateurs in the art of lying. A good universal lie is one whose counterexample is not so easily spotted.
ere should be no insistence on lies in physics. Unfortunately, classical physics is full of lies. We will dispel them during the rest of our walk.
Lying by giving speci c instead of universal statements is much easier. (‘I can’t remember.’) Even a speci c statement such as ‘yesterday the Moon was green, cubic and smelled of cheese’ can never be completely falsi ed: there is no way to show with absolute certainty that this is wrong. e only thing that we can do is to check whether the statement is compatible with other observations, such as whether the di erent shape a ected the tides as expected, whether the smell can be found in air collected that day, etc. A good speci c lie is thus not in contrast with other observations.**
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* In mathematics, ‘true’ is usually speci ed as ‘deducible’ or ‘provable’; this is in fact a special case of the usual de nition of truth, namely ‘correspondence with facts’, if one remembers that mathematics studies the properties of classi cations. ** It is o en di cult or tedious to verify statements concerning the past, and the di culty increases with the distance in time. at is why people can insist on the occurrence of events which are supposed to be exceptions to the patterns of nature (‘miracles’). Since the advent of rapid means of communication these checks are becoming increasingly easy, and no miracles are le over. is can be seen in Lourdes in France, where even though today the number of visitors is much higher than in the past, no miracles have been seen
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Incidentally, universal and speci c statements are connected: the opposite of a universal statement is always a speci c statement, and vice versa. For example, the opposite of the general statement ‘apples fall upwards’, namely ‘some apples fall downwards’, is speci c. Similarly, the the speci c statement ‘the Moon is made of green cheese’ is in opposition to the universal statement ‘the Moon is solid since millions of years and has almost no smell or atmosphere.’
In other words, law courts and philosophers disagree. Law courts have no problem with calling theories true, and speci c statements lies. Many philosophers avoid this. For example, the statement ‘ill-tempered gaseous vertebrates do not exist’ is a statement of the universal type. If a universal statement is in agreement with observations, and if it is falsi able, law courts call it true. e opposite, namely the statement: ‘ill-tempered gaseous vertebrates do exist’, is of the speci c type, since it means ‘Person X has observed an ill-tempered gaseous vertebrate in some place Y at some time Z’. To verify this, we need a record of the event. If such a record, for example a photographs or testimony does not exist, and if the statement can be falsi ed by other observations, law courts call the speci c statement a lie. Even though these are the rules for everyday life and for the law, there is no agreement among philosophers and scientists that this is acceptable. Why? Intellectuals are a careful lot, because many of them have lost their lives as a result of exposing lies too openly.
In short, speci c lies, like all speci c statements, can never be falsi ed with certainty. is is what makes them so popular. Children learn speci c lies rst. (‘I haven’t eaten the jam.’) General lies, like all general statements, can always be corroborated by examples. is is the reason for the success of ideologies. But the criteria for recognizing lies, even general lies, have become so commonplace that beliefs and lies try to keep up with them. It became fashionable to use expressions such as ‘scienti c fact’ – there are no non-scienti c facts –, or ‘scienti cally proven’ – observations cannot be proven otherwise – and similar empty phrases. ese are not ‘good’ lies; whenever we encounter sentences beginning with ‘science says ...’ or ‘science and religion do ...’, replacing ‘science’ by ‘knowledge’ or ‘experience’ is an e cient way of checking whether such statements are to be taken seriously or not.* Lies di er from true statements in their emotional aspect. Speci c statements are usually boring and fragile, whereas speci c lies are o en sensational and violent. In contrast, general statements are o en daring and fragile whereas general lies are usually boring and violent. e truth is fragile. True statements require the author to stick his neck out to criticism. Researchers know that if one doesn’t stick the neck out, it can’t be an observation or a theory. (A theory is another name for one or several connected, not yet falsi ed
Ref. 647
in decades. In fact, all modern ‘miracles’ are kept alive only by consciously eschewing checks, such as the supposed
yearly liquefaction of blood in Napoli, the milk supposedly drunk by statues, the supposed healers in television evangelism, etc. Most miracles only remain because many organizations make money out of the dif-
culty of falsifying speci c statements. For example, when the British princess Diana died in a car crash in 1997, even though the events were investigated in extreme detail, the scandal press could go on almost without end about the ‘mysteries’ of the accident. * To clarify the vocabulary usage of this text: religion is spirituality plus a varying degree of power abuse. e mixture depends on each person’s history, background and environment. Spirituality is the open participation in the whole of nature. Most, maybe all, people with a passion for physics are spiritual. Most are not religious.
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universal statements about observations.)* Telling the truth does make vulnerable. For this reason, theories are o en daring, arrogant or provoking; at the same time they have to be fragile and vulnerable. For men, theories thus resemble what they think about women. Darwin’s e origin of the species, which developed daring theories, illustrates the stark contrast between the numerous boring and solid facts that Darwin collected and the daring theory that he deduced. Boredom of facts is a sign of truth.
In contrast, the witch-hunters propagating the so-called ‘intelligent design’ are examples of liars. e speci c lies they propagate, such as ‘the world was created in October
’, are sensational, whereas the general lies they propagate, such as ‘there have not been big changes in the past’, are boring. is is in full contrast with common sense. Moreover, lies, in contrast to true statements, make people violent. e worse the lie, the more violent the people. is connection can be observed regularly in the news. In other words, ‘intelligent design’ is not only a lie, it is a bad lie. A ‘good’ general lie, like a good physical theory, seems crazy and seems vulnerable, such as ‘people have free will’. A ‘good’ speci c lie is boring, such as ‘this looks like bread, but for the next ten minutes it is not’. Good lies do not induce violence. Feelings can thus be a criterion to judge the quality of lies, if we pay careful attention to the type of statement. A number of common lies are discussed later in this intermezzo.
An important aspect of any ‘good’ lie is to make as few public statements as possible, so that critics can check as little as possible. (For anybody sending corrections of mistakes in this text, the author provides a small reward.) To detect lies, public scrutiny is important, though not always reliable. Sometimes, even scientists make statements which are not based on observations. However, a ‘good’ lie is always well prepared and told on purpose; accidental lies are frowned upon by experts. Examples of good lies in science are ‘aether’, ‘UFOs’, ‘creation science’, or ‘cold fusion’. Sometimes it took many decades to detect the lies in these domains.
To sum up, the central point of the art of lying without being caught is simple: do not divulge details. Be vague. All the methods used to verify a statement ask for details, for precision. For any statement, its degree of precision allows one to gauge the degree to which the author is sticking his neck out. e more precision that is demanded, the weaker a statement becomes, and the more likely a fault will be found, if there is one. is is the main reason that we chose an increase in precision as a guide for our mountain ascent. By the way, the same method is used in criminal trials. To discover the truth, investigators typically ask all the witnesses a large number of questions, allowing as many details as possible come to light. When su cient details are collected, and the precision is high enough, the situation becomes clear. Telling ‘good’ lies is much more di cult than
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* In other words, a set of not yet falsi ed patterns of observations on the same topic is called a (physical) theory. e term ‘theory’ will always be used in this sense in this walk, i.e. with the meaning ‘set of correct general statements’. is use results from its Greek origin: ‘theoria’ means ‘observation’; its original meaning, ‘passionate and emphatic contemplation’, summarizes the whole of physics in a single word. (‘ eory’, like ‘theatre’, is formed from the root θέ, meaning ‘the act of contemplating’.) Sometimes, however, the term ‘theory’ is used – being confused with ‘hypothesis’ – with the meaning of ‘conjecture’, as in ‘your theory is wrong’, sometimes with the meaning of ‘model’, as in ‘Chern–Simons’ theory and sometimes with the meaning of ‘standard procedure’, as in ‘perturbation theory’. ese incorrect uses are avoided here. To bring the issue to a point: the theory of evolution is not a conjecture, but a set of correct statements based on observation.
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Ref. 649
telling the truth; it requires an excellent imagination.
I
?
Truth is an abyss.
Democritus
“ ” To teach superstitions as truth is a most terrible thing.
“ Hypatia of Alexandria (c. – ) ” [Absolute truth:] It is what scientists say it is
when they come to the end of their labors.
“ ” Charles Peirce
Ref. 650
Truth is a rhetorical concept.
“ ” Paul Feyerabend
Not all statements can be categorized as true or false. Statements can simply make no sense. ere are even such statements in mathematics, where they are called undecidable. An example is the continuum hypothesis. is hypothesis is undecidable because it makes a statement that depends on the precise meaning of the term ‘set’; in standard mathematical usage the term is not de ned su ciently precisely so that a truth value can be assigned to the continuum hypothesis. In short, statements can be undecidable because the concepts contained in them are not sharply de ned.
Statements can also be undecidable for other reasons. Phrases such as ‘ is statement is not true’ illustrate the situation. Kurt Gödel* has even devised a general way of constructing such statements in the domain of logic and mathematics. e di erent variations of these self-referential statements, especially popular both in the eld of logic and computer science, have captured a large public.** Similarly undecidable statements can be constructed with terms such as ‘calculable’, ‘provable’ and ‘deducible’.
In fact, self-referential statements are undecidable because they are meaningless. If the usual de nition of ‘true’, namely corresponding to facts, is substituted into the sentence ‘ is statement is not true’, we quickly see that it has no meaningful content. e most famous meaningless sentence of them all was constructed by the linguist Noam Chomsky:
Ref. 616
Colorless green ideas sleep furiously.
It is o en used as an example for the language processing properties of the brain, but nobody sensible elevates it to the status of a paradox and writes philosophical discussions about it. To do that with the title of this section is a similar waste of energy.
e main reason for the popular success of self-reference is the di culty in perceiving the lack of meaning.*** A good example is the statement:
* Kurt Gödel (1906–1978), famous Austrian logician.
** A general introduction is given in the beautiful books by R
S
: Satan, Cantor and
In nity and Other Mind-boggling Puzzles, Knopf, 1992; What is the Name of is Book? e Riddle of Dracula
and Other Logical Puzzles, Touchstone, 1986, and e Lady or the Tiger? And Other Puzzles, Times Books,
1982. Also de nitions can have no content, such as David Hilbert’s ‘smallest number that has not been
mentioned this century’ or ‘the smallest sequence of numbers that is described by more signs than this
sentence’.
*** A well-known victim of this di culty is Paulus of Tarsus. e paradox of the Cretan poet Epimenedes
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is statement is false or you are an angel.
Challenge 1138 n We can actually deduce from it that ‘you are an angel.’ Can you see how? If you want, you can change the second half and get even more interesting statements. Such examples show that statements referring to themselves have to be treated with great care when under investigation. In short, whenever you meet somebody who tries to use the self-referential construction by Kurt Gödel to deduce another statement, take a step back, or better, a few more. Self-reference, especially the type de ned by Gödel, is a hard but common path – especially amongst wannabe-intellectuals – to think, tell and write nonsense. Nothing useful can be deduced from nonsense. Well, not entirely; it does help to meet psychiatrists on a regular basis. In physics, in the other natural sciences, and in legal trials these problems do not emerge, because self-referential statements are not used.* In fact, the work of logicians con rms, o en rather spectacularly, that there is no way to extend the term ‘truth’ beyond the de nition of ‘correspondence with facts.’
“Ein Satz kann unmöglich von sich selbst aussagen, daß er wahr ist.** Ludwig Wittgenstein, Tractatus, .
”
C
Some lies are entertaining, others are made with criminal intent; some are good, others are bad.
** Challenge 1140 e ‘Yesterday I drowned.’ Is this a good or a bad lie?
Challenge 1141 ny
**
In the 1990s, so-called crop circles were formed by people walking with stilts, a piece of wood and some rope in elds of crops. Nevertheless, many pretended and others believed that these circles were made by extraterrestrial beings. Is this a good or a bad lie? Can you give a reason why this is impossible?
**
Sometimes it is heard that a person whose skin is completely covered with nest metal powder will die, due to the impossibility of the skin to breathe. Can you show that this is Challenge 1142 ny wrong?
**
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Ref. 651 Challenge 1137 n Challenge 1139 ny
(6th century ) who said ‘All Cretans lie’ is too di cult for the notoriously humour-impaired Paulus, who in his letter to Titus (chapter 1, verses 12 and 13, in the christian bible) calls Epimenedes a ‘prophet’, adds some racist comments, and states that this ‘testimony’ is true. But wait; there is a nal twist to this story.
e statement ‘All Cretans lie’ is not a paradox at all; a truth value can actually be ascribed to it, because the statement is not really self-referential. Can you con rm this? e only genuine paradox is ‘I am lying’, to which it is indeed impossible to ascribe a truth value. * Why are circular statements, like those of Galilean physics, not self-referential? ** It is quite impossible for a proposition to state that it itself is true.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
,
A famous mixture of hoax and belief premises that the Earth was created about six thousand years ago. (Some believers even use this false statement as justi cation for violence Challenge 1143 ny against non-believers.) Can you explain why the age is wrong?
**
A famous provocation: the world has been created last Saturday. Can you decide whether Challenge 1144 ny this is wrong?
**
Hundreds of hoaxes are found on the http://www.museumofhoaxes.com website. It gives an excellent introduction into the art of lying; of course it exposes only those who have been caught. Enjoy the science stories, especially those about archaeology. (Several other sites with similar content can be found on the internet.)
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Challenge 1145 e
**
In the 1990s, many so-called ‘healers’ in the Philippines made millions by suggesting patients that they were able to extract objects from their bodies without operating. Why is this not possible? (For more information on health lies, see the http://www.quackwatch. com website.)
Challenge 1146 e
**
Since the 1980s, people have claimed that it is possible to acquire knowledge simply from somebody km away, without any communication between the two people. However, the assumed ‘morphogenetic elds’ cannot exist. Why not?
**
Challenge 1147 n
It is claimed that a Fire Brigade building in a city in the US hosts a light bulb that has been burning without interruption since 1901 (at least it was so in 2005). Can this be true? Hundreds of such stories, o en called ‘urban legends,’ can be found on the http:// www.snopes.com website. However, some of the stories are not urban legends, but true, as the site shows.
**
‘ is statement has been translated from French into English’. Is the statement true, false or neither?
**
Aeroplanes have no row 13. Many tall hotels have no oor 13. What is the lie behind this Challenge 1148 ny habit? What is the truth behind it?
**
‘In the middle age and in antiquity, people believed in the at earth.’ is is a famous lie that is rarely questioned. e historian Reinhard Krüger has shown that the lie is most of all due to the writers omas Paine (1794) and Washington Irving (1928). Fact is that since Aristotle, everybody believed in a spherical Earth.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
O
Ref. 652
Knowledge is a sophisticated statement of ignorance.
“ Attributed to Karl Popper ” e collection of a large number of true statements about a type of observations, i.e. of
a large number of facts, is called knowledge. Where the domain of observations is su ciently extended, one speaks of a science. A scientist is thus somebody who collects knowledge.* We found above that an observation is classi ed input into the memory of several people. Since there is motion all around, to describe all these observations is a mammoth task. As for every large task, to a large extent the use of appropriate tools determines the degree of success that can be achieved. ese tools, in physics and in all other sciences, fall in three groups: tools for the collection of observations, tools to communicate observations and tools to communicate relations between observations. e latter group has been already discussed in the section on language and on mathematics. We just touch on the other two.
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“Every generation is inclined to de ne ‘the end of physics’ as coincident with the end of their scienti c contributions. ” Julian Schwinger**
Physics is an experimental science; it rests on the collection of observations. To realize this task e ectively, all sorts of instruments, i.e. tools that facilitate observations, have been developed and built. Microscopes, telescopes, oscilloscopes, as well as thermometers, hygrometers, manometers, pyrometers, spectrometers amongst others are familiar examples. e precision of many of these tools is being continuously improved even today; their production is a sizeable part of modern industrial activity, examples being electrical measuring apparatus and diagnostic tools for medicine, chemistry and biology. Instruments can be as small as a tip of a few tungsten atoms to produce an electron beam of a few volts, and as large as km in circumference, producing an electron beam with more than GV e ective accelerating voltage. Instruments have been built that contain and measure the coldest known matter in the universe. Other instruments can measure length variations of much less than a proton diameter over kilometre long distances. In-
Ref. 653
* e term ‘scientist’ is a misnomer peculiar to the English language. Properly speaking, a ‘scientist’ is a follower of scientism, an extremist philosophical school that tried to resolve all problems through science. For this reason, some religious sects have the term in their name. Since the English language did not have a shorter term to designate ‘scienti c persons’, as they used to be called, the term ‘scientist’ started to appear in the United States, from the eighteenth century onwards. Nowadays the term is used in all English-speaking countries – but not outside them, fortunately. ** Julian Seymour Schwinger (1918–1994), US-American infant prodigy. He was famous for his clear thinking and his excellent lectures. He worked on waveguides and synchroton radiation, made contributions to nuclear physics and developed quantum electrodynamics. For the latter he received the 1965 Nobel Prize in physics together with Tomonaga and Feynman. He was a thesis advisor to many famous physicists and wrote several excellent and in uential textbooks. Nevertheless, at the end of his life, he became strangely interested in a hoax turned sour: cold fusion.
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Ref. 654, Ref. 655 Ref. 656
struments have been put deep inside the Earth, on the Moon, on several planets, and have been sent outside the Solar system.
In this walk, instruments are not described; many good textbooks on this topic are available. Most observations collected by instruments are not mentioned here. e most important results in physics are recorded in standard publications, such as the Landolt– Börnstein series and the physics journals (Appendix E gives a general overview of information sources).
Will there be signi cant new future observations in the domain of the fundamentals of motion? At present, in this speci c domain, even though the number of physicists and publications is at an all-time high, the number of new experimental discoveries has been steadily diminishing for many years and is now fairly small. e sophistication and investment necessary to obtain new results has become extremely high. In many cases, measuring instruments have reached the limits of technology, of budgets or even those of nature. e number of new experiments that produce results showing no deviation from theoretical predictions is increasing steadily. e number of historical papers that try to enliven dull or stalled elds of enquiry are increasing. Claims of new e ects which turn out to be false, due to measurement errors, self-deceit or even fraud have become so frequent that scepticism has become a common response. Although in many domains of science, including physics, discoveries are still expected, on the fundamentals of motion the arguments just presented seem to show that new observations are only a remote possibility. e task of collecting observations on the foundations of motion (though not on other topics of physics) seems to be complete. Indeed, most observations described here were obtained before the end of the twentieth century. We are not too early with our walk.
Ref. 657
A
Measure what is measurable; make measurable what is not.
“ ” Wrongly attributed to Galileo. ?
“Scientists have odious manners, except when you prop up their theory; then you can borrow money from them. ” Mark Twain
e most practical way to communicate observations was developed a long time ago: by measurements. A measurement allows e ective communication of an observation to other times and places. is is not always as trivial as it sounds; for example, in the Middle Ages people were unable to compare precisely the ‘coldness’ of the winters of two di erent years! e invention of the thermometer provided a reliable solution to this requirement. A measurement is thus the classi cation of an observation into a standard set of observations; to put it simply, a measurement is a comparison with a standard. is de nition of a measurement is precise and practical, and has therefore been universally adopted. For example, when the length of a house is measured, this aspect of the house is classi ed into a certain set of standard lengths, namely the set of lengths de ned by multiples of a unit. A unit is the abstract name of the standard for a certain observable. Numbers and units allow the most precise and most e ective communication of measurement results.
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For all measurable quantities, practical standard units and measurement methods have been de ned; the main ones are listed and de ned in Appendix B. All units are derived from a few fundamental ones; this is ultimately due to our limited number of senses: length, time and mass are related to sight, hearing and touch. Our limited number of senses is, in turn, due to the small number of observables of nature.
We call observables the di erent measurable aspects of a system. Most observables, such as size, speed, position, etc. can be described by numbers, and in this case they are quantities, i.e. multiples of some standard unit. Observables are usually abbreviated by (mathematical) symbols, usually letters from some alphabet. For example, the symbol c commonly speci es the velocity of light. For most observables, standard symbols have been de ned by international bodies.* e symbols for the observables that describe the state of an object are also called variables. Variables on which other observables depend are o en called parameters. (Remember: a parameter is a variable constant.) For example, the speed of light is a constant, the position a variable, the temperature is o en a parameter, on which the length of an object, for example, can depend. Note that not all observables are quantities; in particular, parities are not multiples of any unit.
Today the task of de ning tools for the communication of observations can be considered complete. (For quantities, this is surely correct; for parity-type observables there could be a few examples to be discovered.) is is a simple and strong statement. Even the BIPM, the Bureau International des Poids et Mesures, has stopped adding new units.**
As a note, the greatness of a physicist can be ranked by the number of observables he has introduced. Even a great scientist such as Einstein, who discovered many ‘laws’ of nature, only introduced one new observable, namely the metric tensor for the description of gravity. Following this criterion – as well as several others – Maxwell is the most important physicist, having introduced electric and magnetic elds, the vector potential, and several other material dependent observables. For Heisenberg, Dirac and Schrödinger, the wave function describing electron motion could be counted as half an observable (as it is a quantity necessary to calculate measurement results, but not itself an observable). Incidentally, even the introduction of any term that is taken up by others is a rare event; ‘gas’, ‘entropy’ and only a few others are such examples. It has always been much more di cult to discover an observable than to discover a ‘law’; usually, observables are developed by many people cooperating together. Indeed, many ‘laws’ bear people’s names, but almost no observables.
If the list of observables necessary to describe nature is complete, does this mean that all the patterns or rules of nature are known? No; in the history of physics, observables were usually de ned and measured long before the precise rules connecting them were found. For example, all observables used in the description of motion itself, such as time, position and its derivatives, momentum, energy and all the thermodynamic quantities, were de ned before or during the nineteenth century, whereas the most precise versions
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* All mathematical symbols used in this walk, together with the alphabets from which they are taken, are listed in Appendix A on notation. ey follow international standards whenever they are de ned. e standard symbols of the physical quantities, as de ned by the International Standards Organization (ISO), the International Union of Pure and Applied Physics (IUPAP) and the International Union of Pure and Applied Chemistry (IUPAC), can be found for example in the bible, i.e. the CRC Handbook of Chemistry and Physics, CRC Press, Boca Raton, 1992. ** e last, the katal or mol s, was introduced in 1999. Physical units are presented in Appendix B.
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of the patterns or ‘laws’ of nature connecting them, special relativity and non-equilibrium thermodynamics, have been found only in the twentieth century. e same is true for all observables connected to electromagnetic interaction. e corresponding patterns of nature, quantum electrodynamics, was discovered long a er the corresponding observables. e observables that were discovered last were the elds of the strong and the weak nuclear interactions. Also, in this case, the patterns of nature were formulated much later.*
D
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An observation is an interaction with some part of nature leading to the production of a record, such as a memory in the brain, data on a tape, ink on paper, or any other xed process applied to a support. e necessary irreversible interaction process is o en called writing the record. Obviously, writing takes a certain amount of time; zero interaction time would give no record at all. erefore any recording device, including our brain, always records some time average of the observation, however short it may be.
What we call a xed image, be it a mental image or a photograph, is always the time average of a moving situation. Without time averaging, we would have no xed memories. On the other hand, any time averaging introduces a blur that hides certain details; and in our quest for precision, at a certain moment, these details are bound to become important.
e discovery of these details will begin in the second part of the walk, the one centred on quantum theory. In the third part of our mountain ascent we will discover that there is a shortest possible averaging time. Observations of that short duration show so many details that even the distinction between particles and empty space is lost. In contrast, our concepts of everyday life appear only a er relatively long time averages. e search for an average-free description of nature is one of the big challenges of our adventure.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
I
?
Nur gesetzmäßige Zusammenhänge sind denkbar.**
“ Ludwig Wittgenstein, Tractatus, .
ere is a tradition of opposition between
” adherents of induction and of deduction. In my “view it would be just as sensible for the two ends
of a worm to quarrel.
” Alfred North Whitehead
Induction is the usual term used for the act of making, from a small and nite number of experiments, general conclusions about the outcome of all possible experiments performed in other places, or at other times. In a sense, it is the technical term for sticking out one’s neck, which is necessary in every scienti c statement. Induction has been a major topic of discussion for science commentators. Frequently one nds the remark that
* Is it possible to talk about observations at all? It is many a philosopher’s hobby to discuss whether there actually is an example for an ‘Elementarsatz’ mentioned by Wittgenstein in his Tractatus. ere seems to be at least one that ts: Di erences exist. It is a simple sentence; in the third part of our walk, it will play a central role. ** Only connexions that are subject to law are thinkable.
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Challenge 1149 n
knowledge in general, and physics in particular, relies on induction for its statements. According to some, induction is a type of hidden belief that underlies all sciences but at the same time contrasts with them.
To avoid wasting energy, we make only a few remarks. e rst can be deduced from a simple experiment. Try to convince a critic of induction to put their hand into a re. Nobody who honestly calls induction a belief should conclude from a few unfortunate experiences in the past that such an act would also be dangerous in the future... In short, somehow induction works.
A second point is that physical universal statements are always openly stated; they are never hidden. e refusal to put one’s hand into a re is a consequence of the invariance of observations under time and space translations. Indeed, general statements of this type form the very basis of physics. However, no physical statement is a belief only because it is universal; it always remains open to experimental checks. Physical induction is not a hidden method of argumentation, it is an explicit part of experimental statements. In fact, the complete list of ‘inductive’ statements used in physics is given in the table on page . ese statements are so important that they have been given a special name: they are called symmetries. e table lists all known symmetries of nature; in other words, it lists all inductive statements used in physics.
Perhaps the best argument for the use of induction is that there is no way to avoid it when one is thinking. ere is no way to think, to talk or to remember without using concepts, i.e. without assuming that most objects or entities have the same properties over time. ere is also no way to communicate with others without assuming that the observations made from the other’s viewpoint are similar to one’s own. ere is no way to think without symmetry and induction. Indeed, the concepts related to symmetry and induction, such as space and time, belong to the fundamental concepts of language. e only sentences which do not use induction, the sentences of logic, do not have any content (Tractatus, . ). Indeed, without induction, we cannot classify observations at all! Evolution has given us memory and a brain because induction works. To criticize induction is not to criticize natural sciences, it is to criticize the use of thought in general. We should never take too seriously people who themselves do what they criticize in others; sporadically pointing out the ridicule of this endeavour is just the right amount of attention they deserve.
e topic could be concluded here, were it not for some interesting developments in modern physics that put two additional nails in the co n of arguments against induction. First, in physics whenever we make statements about all experiments, all times or all velocities, such statements are actually about a nite number of cases. We know today that in nities, both in size and in number, do not occur in nature. e in nite number of cases appearing in statements in classical physics and in quantum mechanics are apparent, not real, and due to human simpli cations and approximations. Statements that a certain experiment gives the same result ‘everywhere’ or that a given equation is correct for ‘all times’, always encompass only a nite number of examples. A great deal of otherwise o en instinctive repulsion to such statements is avoided in this way. In the sciences, as well as in this book, ‘all’ never means an in nite number of cases.
Finally, it is well known that extrapolating from a few cases to many is false when the few cases are independent of each other. However, this conclusion is correct if the cases are interdependent. From the fact that somebody found a penny on the street on two
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subsequent months, cannot follow that he will nd one the coming month. Induction is only correct if we know that all cases have similar behaviour, e.g. because they follow from the same origin. For example, if a neighbour with a hole in his pocket carries his salary across that street once a month, and the hole always opens at that point because of the beginning of stairs, then the conclusion would be correct. It turns out that the results of modern physics encountered in the third part of our walk show that all situations in nature are indeed interdependent, and thus we prove in detail that what is called ‘induction’ is in fact a logically correct conclusion.
In the progress of physics, the exception usually
“ ” turned out to be the general case.
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Ref. 658 Ref. 659
Der Zweck der Philosophie ist die logische Klärung der Gedanken.*
“ Ludwig Wittgenstein, Tractatus, . ” To talk well about motion means to talk precisely. Precision requires avoiding
hree common mistakes in the description of nature. First, concepts should never have a contradiction built into their de nition. For ex-
ample, any phenomenon occurring in nature evidently is a ‘natural’ phenomenon; therefore, to talk about either ‘supernatural’ phenomena or ‘unnatural’ phenomena is a mistake that nobody interested in motion should let go unchallenged; such terms contain a logical contradiction. Naturally, all observations are natural. Incidentally, there is a reward of more than a million dollars for anybody proving the opposite. In over twenty years, nobody has yet been able to collect it.
Second, concepts should not have unclear or constantly changing de nitions. eir content and their limits must be kept constant and explicit. e opposite of this is o en encountered in crackpots or populist politicians; it distinguishes them from more reliable thinkers. Physicists can also fall into the trap; for example, there is, of course, only one single (physical) universe, as even the name says. To talk about more than one universe is an increasingly frequent error.
ird, concepts should not be used outside their domain of application. It is easy to succumb to the temptation to transfer results from physics to philosophy without checking the content. An example is the question: ‘Why do particles follow the laws of nature?’ e
aw in the question is due to a misunderstanding of the term ‘laws of nature’ and to a confusion with the laws of the state. If nature were governed by ‘laws’, they could be changed by parliament. Remembering that ‘laws of nature’ simply means ‘pattern’, ‘property’ or ‘description of behaviour’, and rephrasing the question correctly as ‘Why do particles behave in the way we describe their behaviour?’ one can recognize its senselessness.
In the course of our walk, we will o en be tempted by these three mistakes. A few such situations follow, with the ways of avoiding them.
Consistency is the last refuge of the unimaginative.
“ ” Oscar Wilde
* Philosophy aims at the logical clari cation of thoughts.
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Challenge 1150 ny Page 260 Ref. 660
Challenge 1151 n Page 1018
e whole is always more than the sum of its parts.
“ Aristotle, Metaphysica, f– a. ” In the physical description of nature, the whole is always more than the sum of its parts.
Actually, the di erence between the whole and the sum of its parts is so important that it has a special name: the interaction between the parts. For example, the energy of the whole minus the sum of the energies of its parts is called the energy of interaction. In fact, the study of interactions is the main topic of physics. In other words, physics is concerned primarily with the di erence between the parts and the whole, contrary to what is o en suggested by bad journalists or other sloppy thinkers.
Note that the term ‘interaction’ is based on the general observation that anything that a ects anything else is, in turn, a ected by it; interactions are reciprocal. For example, if one body changes the momentum of another, then the second changes the momentum of the rst by the same (negative) amount. e reciprocity of interactions is a result of conservation ‘laws’. e reciprocity is also the reason that somebody who uses the term ‘interaction’ is considered a heretic by monotheistic religions, as theologians regularly point out. ey repeatedly stress that such a reciprocity implicitly denies the immutability of the deity. (Are they correct?)
e application of the de nition of interaction also settles the frequently heard question of whether in nature there are ‘emergent’ properties, i.e. properties of systems that cannot be deduced from the properties of their parts and interactions. By de nition, there are no emergent properties. ‘Emergent’ properties can only appear if interactions are approximated or neglected. e idea of ‘emergent’ properties is a product of minds with restricted horizons, unable to see or admit the richness of consequences that general principles can produce. In defending the idea of emergence, one belittles the importance of interactions, working, in a seemingly innocuous, maybe unconscious, but in fact sneaky way, against the use of reason in the study of nature. ‘Emergence’ is a belief.
e simple de nition of interaction given above sounds elementary, but it leads to surprising conclusions. Take the atomic idea of Democritus in its modern form: nature is made of vacuum and of particles. e rst consequence is the paradox of incomplete description: experiments show that there are interactions between vacuum and particles. However, interactions are di erences between parts and the whole, in this case between vacuum and particles on the one hand, and the whole on the other. We thus have deduced that nature is not made of vacuum and particles alone.
e second consequence is the paradox of overcomplete description: experiments also show that interactions happen through exchange of particles. However, we have counted particles already as basic building blocks. Does this mean that the description of nature by vacuum and particles is an overdescription, counting things twice?
We will resolve both paradoxes in the third part of the mountain ascent.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 1153 n Challenge 1154 n
You know what I like most?
“ ” Rhetorical questions.
Assume a friend tells you ‘I have seen a grampus today!’ You would naturally ask what it looks like. What answer do we expect? We expect something like ‘It’s an animal with a certain number of heads similar to a X, attached to a body like a Y, with wings like a Z, it make noises like a U and it felt like a V ’ – the letters denoting some other animal or object. Generally speaking, in the case of an object, this scene from Darwin’s voyage to South America shows that in order to talk to each other, we rst need certain basic, common concepts (‘animal’, ‘head’, ‘wing’, etc.). In addition, for the de nition of a new entity we need a characterization of its parts (‘size’, ‘colour’), of the way these parts relate to each other, and of the way that the whole interacts with the outside world (‘feel’, ‘sound’). In other words, for an object to exist, we must be able to give a list of relations with the outside world. An object exists if we can interact with it. (Is observation su cient to determine existence?)
For an abstract concept, such as ‘time’ or ‘superstring’, the de nition of existence has to be re ned only marginally: (physical) existence is the e ectiveness to describe interactions accurately. is de nition applies to trees, time, virtual particles, imaginary numbers, entropy and so on. It is thus pointless to discuss whether a physical concept ‘exists’ or whether it is ‘only’ an abstraction used as a tool for descriptions of observations. e two possibilities coincide. e point of dispute can only be whether the description provided by a concept is or is not precise.
For mathematical concepts, existence has a somewhat di erent meaning: a mathematical concept is said to exist if it has no built-in contradictions. is is a much weaker requirement than physical existence. It is thus incorrect to deduce physical existence from mathematical existence. is is a frequent error; from Pythagoras’ times onwards it was often stated that since mathematical concepts exist, they must therefore also exist in nature. Historically, this error occurred in the statements that planet orbits ‘must’ be circles, that planet shapes ‘must’ be spheres or that physical space ‘must’ be Euclidean. Today this is still happening with the statements that space and time ‘must’ be continuous and that nature ‘must’ be described by sets. In all these cases, the reasoning is wrong. In fact, the continuous attempts to deduce physical existence from mathematical existence hide that the opposite is correct: a short re ection shows that mathematical existence is a special case of physical existence.
We note that there is also a di erent type of existence, namely psychological existence. A concept can be said to exist psychologically if it describes human internal experience.
us a concept can exist psychologically even if it does not exist physically. It is easy to nd examples from the religions or from systems that describe inner experiences. Also myths, legends and comic strips de ne concepts that only exist psychologically, not physically. In our walk, whenever we talk about existence, we mean physical existence only.
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Challenge 1155 e
“Wer Wissenscha und Kunst besitzt, Hat auch Religion; Wer jene beiden nicht besitzt, Der habe Religion.* Johann Wolfgang von Goethe, Zahme Xenien,
”IX
Using the above de nition of existence, the question becomes either trivial or imprecise. It is trivial in the sense that things necessarily exist if they describe observations, since they were de ned that way. But perhaps the questioner meant to ask: Does reality exist independently of the observer?
Using the above, this question can be rephrased: ‘Do the things we observe exist independently of observation?’ A er thousands of years of extensive discussion by professional philosophers, logicians, sophists and amateurs the answer is the same: it is ‘Yes’, because the world did not change a er great-grandmother died. e disappearance of observers does not seem to change the universe. ese experimental ndings can be corroborated by inserting the de nition of ‘existence’ into the question, which then becomes: ‘Do the things we observe interact with other aspects of nature when they do not interact with people?’ e answer is evident. Recent popular books on quantum mechanics fantasize about the importance of the ‘mind’ of observers – whatever this term may mean; they provide pretty examples of authors who see themselves as irreplaceable, seemingly having lost the ability to see themselves as part of a larger entity.
Of course there are other opinions about the existence of things. e most famous is that of the Irishman George Berkeley ( – ) who rightly understood that thoughts based on observation alone, if spread, would undermine the basis of the religious organization of which he was one of the top managers. To counteract this tendency, in he published A Treatise Concerning the Principles of Human Knowledge, a book denying the existence of the material world. is reactionary book became widely known in likeminded circles (it was a time when few books were written) even though it is based on a fundamentally awed idea: it assumes that the concept of ‘existence’ and that of ‘world’ can be de ned independently. (You may be curious to try the feat.)
Berkeley had two aims when he wrote his book. First, he tried to deny the capacity of people to arrive at judgements on nature or on any other matter from their own experience. Second, he also tried to deny the ontological reach of science, i.e. the conclusions one can draw from experience on the questions about human existence. Even though Berkeley is generally despised nowadays, he actually achieved his main aim: he was the originator of the statement that science and religion do not contradict, but complement each other. By religion, Berkeley did not mean either morality or spirituality; every scientists is a friend of both of these. By religion, Berkeley meant that the standard set of beliefs that he stood for is above the deductions of reason. is widely cited statement, itself a belief, is still held dearly by many even to this day. However, when searching for the origin of motion, all beliefs stand in the way, including this one. Carrying beliefs is like carrying oversized baggage: it prevents one from reaching the top of Motion Mountain.
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* He who possesses science and art, also has religion; he who does not possess the two, better have religion.
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Challenge 1156 n Page 996
Teacher: ‘What is found between the nucleus and the electrons?’
“Student: ‘Nothing, only air.’
“ ” ” Natura abhorret vacuum.
Antiquity
In philosophical discussions ‘void’ is usually de ned as ‘non-existence’. It then becomes
a game of words to ask for a yes or no answer to the question ‘Does the void exist?’ e
expression ‘the existence of non-existence’ is either a contradiction of terms or is at least
unclearly de ned; the topic would not seem to be of great interest. However, similar ques-
tions do appear in physics, and a physicist should be prepared to notice the di erence
of this from the previous one. Does a vacuum exist? Does empty space exist? Or is the
world ‘full’ everywhere, as the more conservative biologist Aristotle maintained? In the
past, people have been killed for giving an answer that was unacceptable to authorities.
It is not obvious, but it is nevertheless important, that the modern physical concepts
of ‘vacuum’ and ‘empty space’ are not the same as the philosophical concept of ‘void’.
‘Vacuum’ is not de ned as ‘non-existence’; on the contrary, it is de ned as the absence
of matter and radiation. Vacuum is an entity with speci c observable properties, such
as its number of dimensions, its electromagnetic constants, its curvature, its vanishing
mass, its interaction with matter through curvature and through its in uence on decay,
etc. (A table of the properties of a physical vacuum is given on page .) Historically, it
took a long time to clarify the distinction between a physical vacuum and a philosophical
void. People confused the two concepts and debated the existence of the vacuum for more
than two thousand years. e rst to state that it existed, with the courage to try to look
through the logical contradiction at the underlying physical reality, were Leucippus and
Democritus, the most daring thinkers of antiquity. eir speculations in turn elicited the
reactionary response of Aristotle, who rejected the concept of vacuum. Aristotle and his
disciples propagated the belief about nature’s horror of the vacuum.
e discussion changed completely in the seventeenth century, when the rst experi-
mental method to realize a vacuum was devised by Torricelli.* Using mercury in a glass
tube, he produced the rst laboratory vacuum. Can you guess how? Arguments against
the existence of the vacuum again appeared around , when it was argued that light
needed ‘aether’ for its propagation, using almost the same arguments that had been used
two hundred years earlier, but in di erent words. However, experiments failed to detect
any of the supposed properties of this unclearly de ned concept. Experiments in the eld
of general relativity showed that a vacuum can move – though in a completely di erent
way from the way in which the aether was expected to move – that the vacuum can be
bent, but it then tends to return to its shape. en, in the late twentieth century, quantum
eld theory again argued against the existence of a true vacuum and in favour of a space
full of virtual particle–antiparticle pairs, culminating in the discussions around the cos-
mological constant.
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* Evangelista Torricelli (b. 1608 Faenza, d. 1647 Florence), Italian physicist, pupil and successor to Galileo. e (non-SI) pressure unit ‘torr’ is named a er him.
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e question ‘Does the void exist?’ is settled conclusively only in the third part of this Page 1020 walk, in a rather surprising way.
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Ref. 648
Challenge 1157 n
Ref. 649
“It is certain and evident to our senses, that in the world some things are in motion. Now whatever is moved is moved by another... If that by which it is moved be itself moved, then this also needs to be to be moved by another, and that by another again. But this cannot go on to in nity, because then there would be no rst mover and consequently, no other mover, seeing that subsequent movers move only inasmuch as they are moved by the rst mover, as the sta moves only because it is moved by the hand. erefore it is necessary to arrive at a rst mover, moved by no other; and this everyone understands to be god. omas Aquinas (c. – ) Summa ” eologiae, I, q. .
Most of the modern discussions about set theory centre on ways to de ning the term ‘set’ for various types of in nite collections. For the description of motion this leads to two questions: Is the universe in nite? Is it a set? We begin with the rst one. Illuminating the question from various viewpoints, we will quickly discover that it is both simple and imprecise.
Do we need in nite quantities to describe nature? Certainly, in classical and quantum physics we do, e.g. in the case of space-time. Is this necessary? We can say already a few things.
Any set can be nite in one aspect and in nite in another. For example, it is possible to proceed along a nite mathematical distance in an in nite amount of time. It is also possible to travel along any distance whatsoever in a given amount of mathematical time, making in nite speed an option, even if relativity is taken into account, as was explained earlier.
Despite the use of in nities, scientists are still limited. We saw above that many types of in nities exist. However, no in nity larger than the cardinality of the real numbers plays a role in physics. No space of functions or phase space in classical physics and no Hilbert space in quantum theory has higher cardinality. Despite the ability of mathematicians to de ne much larger kinds of in nities, the description of nature does not need them. Even the most elaborate descriptions of motion use only the in nity of the real numbers.
But is it possible at all to say of nature or of one of its aspects that it is indeed in nite? Can such a statement be compatible with observations? No. It is evident that every statement that claims that something in nature is in nite is a belief, and is not backed by observations. We shall patiently eliminate this belief in the following.
e possibility of introducing false in nities make any discussion on whether humanity is near the ‘end of science’ rather di cult. e amount of knowledge and the time required to discover it are unrelated. Depending on the speed with which one advances through it, the end of science can be near or unreachable. In practice, scientists have
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thus the power to make science in nite or not, e.g. by reducing the speed of progress. As scientists need funding for their work, one can guess the stand that they usually take.
In short, the universe cannot be proven to be in nite. But can it be nite? At rst sight, this would be the only possibility le . (It is not, as we shall see.) But even though many have tried to describe the universe as nite in all its aspects, no one has yet been successful. In order to understand the problems that they encountered, we continue with the other question mentioned above:
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A simple observation leads us to question whether the universe is a set. For years it has been said that the universe is made of vacuum and particles. is implies that the universe is made of a certain number of particles. Perhaps the only person to have taken this conclusion to the limit was the English astrophysicist Arthur Eddington ( – ), who wrote:
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Ref. 663
I believe there are , , , , , , , , , , , , , , , , , , , , , , , , , , protons in the universe and the same number of electrons.
Eddington was ridiculed over and over again for this statement and for his beliefs that lead up to it. His arguments were indeed based on his personal preferences for certain pet numbers. However, we should not laugh too loudly. In fact, for years almost all scientists have thought along the same line, the only di erence being that they have le the precise number unspeci ed! In fact, any other number put into the above sentence would be equally ridiculous. Avoiding specifying it is just a cowards’ way of avoiding looking at this foggy aspect of the particle description of nature.
Is there a particle number at all in nature? If you smiled at Eddington’s statement, or if you shook your head over it, it may mean that you instinctively believe that nature is not a set. Is this so? Whenever we de ne the universe as the totality of events, or as the totality of all space-time points and objects, we imply that space-time points can be distinguished, that objects can be distinguished and that both can be distinguished from each other. We thus assume that nature is separable and a set. But is this correct? e question is important. e ability to distinguish space-time points and particles from each other is o en called locality. us the universe is separable or a set if and only if our description of it is local.* And in everyday life, locality is observed without exception.
In daily life we also observe that nature is separable and a whole at the same time. It is a ‘many that can be thought as one’: in daily life nature is a set. Indeed, the basic characteristic of nature is its diversity. In the world around us we observe changes and di erences; we observe that nature is separable. Furthermore, all aspects of nature belong together: there are relations between these aspects, o en called ‘laws,’ stating that the di erent aspects of nature form a whole, usually called the universe.
* In quantum mechanics also other, less clear de nitions of locality are used. We will mention them in the second part of this text. e issue mentioned here is a di erent, more fundamental one, and not connected with that of quantum theory.
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Page 1046 Challenge 1158 n
In other words, the possibility of describing observations with the help of ‘laws’ follows from our experience of the separability of nature. e more precisely the separability is speci ed, the more precisely the ‘laws’ can be formulated. Indeed, if nature were not separable or were not a unity, we could not explain why stones fall downwards. us we are led to speculate that we should be able to deduce all ‘laws’ from the fact that nature is separable.
In addition, only the separability allows us to describe nature at all. A description is a classi cation, that is, a mapping between certain aspects of nature and certain concepts. All concepts are sets and relations. Since the universe is separable, it can be described with the help of sets and relations. Both are separable entities with distinguishable parts. A precise description is commonly called an understanding. In short, the universe is comprehensible only because it is separable.
Moreover, only the separability of the universe makes our brain such a good instrument. e brain is built from a large number of connected components, and only the brain’s separability allows it to function. In other words, thinking is only possible because nature is separable.
Finally, only the separability of the universe allows us to distinguish reference frames, and thus to de ne all symmetries at the basis of physical descriptions. And in the same way that separability is thus necessary for covariant descriptions, the unity of nature is necessary for invariant descriptions. In other words, the so-called ‘laws’ of nature are based on the experience that nature is both separable and uni able – that it is a set.
ese arguments seem overwhelmingly to prove that the universe is a set. However, these arguments apply only to everyday experience, everyday dimensions and everyday energies. Is nature a set also outside the domains of daily life? Are objects di erent at all energies, even when they are looked at with the highest precision possible? We have three open issues le : the issue of the number of particles in the universe; the circular de nition of space, time and matter; and the issue as to whether describing nature as made of particles and void is an overdescription, an underdescription, or neither. ese three issues make us doubt whether objects are countable at all energies. We will discover in the third part of our mountain ascent that this is not the case in nature. e consequences will be extensive and fascinating. As an example, try to answer the following: if the universe is not a set, what does that mean for space and time?
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Challenge 1159 n
Each progressive spirit is opposed by a thousand men appointed to guard the past.
“ Maurice Maeterlink ” Following the de nition above, existence of a concept means its usefulness to describe
interactions. ere are two common de nitions of the concept of ‘universe’. e rst is the totality of all matter, energy and space-time. But this usage results in a strange consequence: since nothing can interact with this totality, we cannot claim that the universe exists.
So let us take the more restricted view, namely that the universe is only the totality of all matter and energy. But also in this case it is impossible to interact with the universe. Can you give a few arguments to support this?
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Page 1049 Ref. 616
In short, we arrive at the conclusion that the universe does not exist. We will indeed con rm this result in more detail later on in our walk. In particular, since the universe does not exist, it does not make sense to even try to answer why it exists. e best answer might be: because of furiously sleeping, colourless green ideas.
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Ref. 665
Page 759 Page 850 Page 464 Page 463
(Gigni) De nihilo nihilum, in nihilum nil posse reverti.*
“ Persius, Satira, III, v. - . ” Anaxagoras, discovering the ancient theory that
nothing comes from nothing, decided to
“abolish the concept of creation and introduced in its place that of discrimination; he did not hesitate to state, in e ect, that all things are mixed to the others and that discrimination produces their growth. ” Anonymous fragment, Middle Ages.
e term ‘creation’ is o en heard when talking about nature. It is used in various contexts with di erent meanings.
One speaks of creation as the characterization of human actions, such as observed in an artist painting or a secretary typing. Obviously, this is one type of change. In the classi cation of change introduced at the beginning of our walk, the changes cited are movements of objects, such as the electrons in the brain, the molecules in the muscles, the material of the paint, or the electrons inside the computer. is type of creation is thus a special case of motion.
One also speaks of creation in the biological or social sense, such as in ‘the creation of life’, or ‘creation of a business’, or ‘the creation of civilization’. ese events are forms of growth or of self-organization; again, they are special cases of motion.
Physicists one o en say that a lamp ‘creates’ light or that a stone falling into a pond ‘creates’ water ripples. Similarly, they talk of ‘pair creation’ of matter and antimatter. It was one of the important discoveries of physics that all these processes are special types of motion, namely excitation of elds.
In popular writing on cosmology, ‘creation’ is also a term commonly applied, or better misapplied, to the big bang. However, the expansion of the universe is a pure example of motion, and contrary to a frequent misunderstanding, the description of the big bang contains no process that does not fall into one of the previous three categories, as shown in the chapter on general relativity. Quantum cosmology provides more reasons to support the fact that the term ‘creation’ is not applicable to the big bang. First, it turns out that the big bang was not an event. Second, it was not a beginning. ird, it did not provide a choice from a large set of possibilities. e big bang does not have any properties attributed to the term ‘creation’.
In summary, we conclude that in all cases, creation is a type of motion. ( e same applies to the notions of ‘disappearance’ and ‘annihilation’.) No other type of creation is observed in nature. In particular, the naive sense of ‘creation’, namely ‘appearance from
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* Nothing (can appear) from nothing, nothing can disappear into nothing.
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nothing’ – ex nihilo in Latin – is never observed in nature. All observed types of ‘creation’ require space, time, forces, energy and matter for their realization. Creation requires something to exist already, in order to take place. In addition, precise exploration shows that no physical process and no example of motion has a beginning. Our walk will show us that nature does not allow us to pinpoint beginnings. is property alone is su cient to show that ‘creation’ is not a concept applicable to what happens in nature. Worse still, creation is applied only to physical systems; we will discover that nature is not a system and that systems do not exist.
e opposite of creation is conservation. e central statements of physics are conservation theorems: for energy, mass, linear momentum, angular momentum, charge, etc. In fact, every conservation ‘law’ is a detailed and accurate rejection of the concept of creation. e ancient Greek idea of atoms already contains this rejection. Atomists stated that there is no creation and no disappearance, but only motion of atoms. Every transformation of matter is a motion of atoms. In other words, the idea of the atom was a direct consequence of the negation of creation. It took humanity over years before it stopped locking people in jail for talking about atoms, as had happened to Galileo.
However, there is one exception in which the naive concept of creation does apply: it describes what magicians do on stage. When a magician makes a rabbit appear from nowhere, we indeed experience ‘creation’ from nothing. At its best such magic is a form of entertainment, at its worst, a misuse of gullibility. e idea that the universe results from either of these two does not seem appealing; on second thought though, maybe looking at the universe as the ultimate entertainment could open up a fresh and more productive approach to life.
Voltaire ( – ) popularized an argument against creation o en used in the past: we do not know whether creation has taken place or not. Today the situation is di erent: we do know that it has not taken place, because creation is a type of motion and, as we will see in the third part of our mountain ascent, motion did not exist near the big bang.
Have you ever heard the expression ‘creation of the laws of nature’? It is one of the most common examples of disinformation. First of all, this expression confuses the ‘laws’ with nature itself. A description is not the same as the thing itself; everybody knows that giving their beloved a description of a rose is di erent from giving an actual rose. Second, the expression implies that nature is the way it is because it is somehow ‘forced’ to follow the ‘laws’ – a rather childish and, what is more, incorrect view. And third, the expression assumes that it is possible to ‘create’ descriptions of nature. But a ‘law’ is a description, and a description by de nition cannot be created: so the expression makes no sense at all.
e expression ‘creation of the laws of nature’ is the epitome of confused thinking. It may well be that calling a great artist ‘creative’ or ‘divine’, as was common during the Renaissance, is not blasphemy, but simply an encouragement to the gods to try to do as well. In fact, whenever one uses the term ‘creation’ to mean anything other than some form of motion, one is discarding both observations and human reason. It is one of the last pseudo-concepts of our modern time; no expert on motion should forget this. It is impossible to escalate Motion Mountain without getting rid of ‘creation’. is is not easy. We will encounter the next attempt to bring back creation in the study of quantum theory.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Every act of creation is rst of all an act of destruction.
“ ” Pablo Picasso
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Ref. 666
“In the beginning the universe was created. is has made a lot of people very angry and has been widely regarded as a bad move. ” Douglas Adams
e tendency to infer the creation of an object from its simple existence is widespread. Some people jump to this conclusion every time they see a beautiful landscape. is habit stems from the triple prejudice that a beautiful scene implies a complex description, in turn implying complex building instructions, and therefore pointing to an underlying design.
is chain of thought contains several mistakes. First, in general, beauty is not a consequence of complexity. Usually it is the opposite: indeed, the study of chaos and of selforganization demonstrated how beautifully complex shapes and patterns can be generated with extremely simple descriptions. True, for most human artefacts, complex descriptions indeed imply complex building processes; a personal computer is a good example of a complex object with a complex production process. But in nature, this connection does not apply. We have seen above that even the amount of information needed to construct a human body is about a million times smaller than the information stored in the brain alone. Similar results have been found for plant architecture and for many other examples of patterns in nature. e simple descriptions behind the apparent complexities of nature have been and are still being uncovered by the study of self-organization, chaos, turbulence and fractal shapes. In nature, complex structures derive from simple processes. Beware of anyone who says that nature has ‘in nite complexity’: rst of all, complexity is not a measurable entity, despite many attempts to quantify it. In addition, all known complex system can be described by (relatively) few parameters and simple equations. Finally, nothing in nature is in nite.
e second mistake in the argument for design is to link a description with an ‘instruction’, and maybe even to imagine that some unknown ‘intelligence’ is somehow pulling the strings of the world’s stage. e study of nature has consistently shown that there is no hidden intelligence and no instruction behind the processes of nature. An instruction is a list of orders to an executioner. But there are no orders in nature, and no executioners.
ere are no ‘laws’ of nature, only descriptions of processes. Nobody is building a tree; the tree is an outcome of the motion of molecules making it up. e genes in the tree do contain information; but no molecule is given any instructions. What seem to be instructions to us are just natural movements of molecules and energy, described by the same patterns taking place in non-living systems. e whole idea of instruction – like that of ‘law’ of nature – is an ideology, born from an analogy with monarchy or even tyranny, and a typical anthropomorphism.
e third mistake in the argument for design is the suggestion that a complex description for a system implies an underlying design. is is not correct. A complex description only implies that the system has a long evolution behind it. e correct deduction is:
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something of large complexity exists; therefore it has grown, i.e. it has been transformed through input of (moderate) energy over time. is deduction applies to owers, mountains, stars, life, people, watches, books, personal computers and works of art; in fact it applies to all objects in the universe. e complexity of our environment thus points out the considerable age of our environment and reminds us of the shortness of our own life.
e lack of basic complexity and the lack of instructions in nature con rm a simple result: there is not a single observation in nature that implies or requires design or creation. On the other hand, the variety and intensity of nature’s phenomena lls us with deep awe. e wild beauty of nature shows us how small a part of nature we actually are, both in space and in time.* We shall explore this experience in detail. We shall nd that remaining open to nature’s phenomena in all their overwhelming intensity is central to the rest of our adventure.
ere is a separation between state and church, but not yet between state and science.
“ ” Paul Feyerabend
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In theory, there is no di erence between theory and practice. In practice, there is.
“ ” Following standard vocabulary usage, a description of an observation is a list of the de-
tails. e above example of the grampus showed this clearly. In other words, a description of an observation is the act of categorizing it, i.e. of comparing, by identifying or distinguishing, the observation with all the other observations already made. A description is a classi cation. In short, to describe means to see as an element of a larger set.
A description can be compared to the ‘you are here’ sign on a city tourist map. Out of a set of possible positions, the ‘you are here’ sign gives the actual one. Similarly, a description highlights the given situation in comparison with all other possibilities. For example, the formula a = GM r is a description of the observations relating motion to gravity, because it classi es the observed accelerations a according to distance to the central body r and to its mass M; indeed such a description sees each speci c case as an example of a general pattern. e habit of generalizing is one reason for the o en disturbing dismissiveness of scientists: when they observe something, their professional training usually makes them classify it as a special case of a known phenomenon and thus keeps them from being surprised or from being exited about it.
A description is thus the opposite of a metaphor; the latter is an analogy relating an observation with another special case; a description relates an observation with a general case, such as a physical theory.
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* e search for a ‘sense’ in life or in nature is a complicated (and necessary) way to try to face the smallness of human existence.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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“Felix qui potuit rerum cognoscere causas, atque metus omnis et inexorabile fatum subjecit pedibus strepitumque acherontis avari. ” Vergilius*
“Der ganzen modernen Weltanschauung liegt die Täuschung zugrunde, daß die sogenannten Naturgesetze die Erklärungen der Naturerscheinungen seien.** ” Ludwig Wittgenstein, Tractatus, .
— Why are the leaves of most trees green? Because they absorb red and blue light. Why do they absorb those colours? Because they contain chlorophyll. Why is chlorophyll green? Because all chlorophyll types contain magnesium between four pyrrole groups, and this chemical combination gives the green colour, as a result of its quantum mechanical energy levels. Why do plants contain chlorophyll? Because this is what land plants can synthesize. Why only this? Because all land plants originally evolved from the green algae, who are only able to synthesize this compound, and not the compounds found in the blue or in the red algae, which are also found in the sea.
— Why do children climb trees, and why do some people climb mountains? Because of the sensations they experience during their activity: the feelings of achievement, the symbolic act to go upwards, the wish to get a wider view of the world are part of this type of adventure.
Ref. 667
e ‘why’-questions in the last two paragraphs show the general di erence between reasons and purposes (although the details of these two terms are not de ned in the same way by everybody). A purpose or intention is a classi cation applied to the actions of humans or animals; strictly speaking, it speci es the quest for a feeling, namely for achieving some type of satisfaction a er completion of the action. On the other hand, a reason is a speci c relation of a fact with the rest of the universe, usually its past. What we call a reason always rests outside the observation itself, whereas a purpose is always internal to it.
Reasons and purposes are the two possibilities of explanations, i.e. the two possible answers to questions starting with ‘why’. Usually, physics is not concerned with purpose or with people’s feeling, mainly because its original aim, to talk about motion with precision, does not seem to be achievable in this domain. erefore, physical explanations of facts are never purposes, but are always reasons. A physical explanation of an observation is always the description of its relation with the rest of nature.***
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* ‘Happy he who can know the causes of things and who, free of all fears, can lay the inexorable fate and the noise of Acheron to his feet.’ (Georg. 2, 490 ss.) Publius Vergilius Maro (70–19 ), the great roman poet, is author of the Aeneis. Acheron was the river crossed by those who had just died and were on their way to the Hades. ** e whole modern conception of the world is founded on the illusion that the so-called laws of nature are the explanations of natural phenomena. *** It is important to note that purposes are not put aside because they pertain to the future, but because they are inadmissible anthropomorphisms. In fact, for deterministic systems, we can equally say that the
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is means that – contrary to common opinion – a question starting with ‘why’ is accessible to physical investigation as long as it asks for a reason and not for a purpose. In particular, questions such as ‘why do stones fall downwards and not upwards?’ or ‘why do electrons have that value of mass, and why do they have mass at all?’ or ‘why does space have three dimensions and not thirty-six?’ can be answered, as these ask for the connection between speci c observations and more general ones. Of course, not all demands for explanation have been answered yet, and there are still problems to be solved. Our present trail only leads from a few answers to some of the more fundamental questions about motion.
e most general quest for an explanation derives from the question: why is the universe the way it is? e topic is covered in our mountain ascent using the two usual approaches, namely:
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Tout sujet est un; et, quelque vaste qu’il soit, il peut être renfermé dans un seul discours.*
“ Bu on, Discours sur le style. ” Studying the properties of motion, constantly paying attention to increase the accuracy
of description, we nd that explanations are generally of two types:**
— ‘It is like all such cases; also this one is described by ...’ e situation is recognized as a special case of a general behaviour.
— ‘If the situation were di erent, we would have a conclusion in contrast with observations.’ e situation is recognized as the only possible case.***
In other words, the rst approach is to formulate rules or ‘laws’ that describe larger and larger numbers of observations, and compare the observation with them. is endeavour is called the uni cation of physics – by those who like it; those who don’t like it, call it ‘reductionism’. For example, the same rule describes the ight of a tennis ball, the motion of the tides at the sea shore, the timing of ice ages, and the time at which the planet Venus ceases to be the evening star and starts to be the morning star. ese processes are all consequences of universal gravitation. Similarly, it is not evident that the same rule describes the origin of the colour of the eyes, the formation of lightning, the digestion of food and the working of the brain. ese processes are described by quantum electrodynamics.
Uni cation has its most impressive successes when it predicts an observation that has not been made before. A famous example is the existence of antimatter, predicted by Dirac
Challenge 1160 n
future is actually a reason for the present and the past, a fact o en forgotten. * Every subject is one and, however vast it is, it can be comprised in a single discourse. ** Are these the only possible ones? *** ese two cases have not to be confused with similar sentences that seem to be explanations, but that aren’t:
— ‘It is like the case of ...’ A similarity with another single case is not an explanation. — ‘If it were di erent, it would contradict the idea that ...’ A contradiction with an idea or with a theory is
not an explanation.
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Challenge 1161 n Challenge 1162 n
when he investigated the solutions of an equation that describes the precise behaviour of common matter.
e second procedure in the search for explanations is the elimination of all other imaginable alternatives in favour of the actually correct one. is endeavour has no commonly accepted name: it could be called the demarcation of the ‘laws’ of physics – by those who like it; others call it ‘anthropocentrism’, or simply ‘arrogance’.
When we discover that light travels in such a way that it takes the shortest possible time to its destination, when we describe motion by a principle of least action, or when we discover that trees are branched in such a way that they achieve the largest e ect with the smallest e ort, we are using a demarcation viewpoint.
In summary, uni cation, answering ‘why’ questions, and demarcation, answering ‘why not’ questions, are typical for the progress throughout the history of physics. We can say that the dual aspects of uni cation and demarcation form the composing and the opposing traits of physics. ey stand for the desire to know everything.
However, neither demarcation nor uni cation can explain the universe. Can you see why? In fact, apart from uni cation and demarcation, there is a third possibility that merges the two and allows one to say more about the universe. Can you nd it? Our walk will automatically lead to it later.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
P,
Das wichtigste Instrument des Wissenscha lers
“ ” ist der Papierkorb.*
e wish to achieve demarcation of the patterns of nature is most interesting when we follow the consequences of di erent rules of nature until we nd them in contradiction with the most striking observation: our own human existence. In this special case the program of demarcation is o en called the anthropic principle – from the Greek ἄνθρωπος, meaning ‘man’.
For example, if the Sun–Earth distance were di erent from what it is, the resulting temperature change on the Earth would have made impossible the appearance of life, which needs liquid water. Similarly, our brain would not work if the Moon did not circle the Earth. It is only because the Moon revolves around our planet that the Earth’s magnetic
eld is large enough to protect the Earth by deviating most of the cosmic radiation that would otherwise make all life on Earth impossible. It is only because the Moon revolves around our planet that the Earth’s magnetic eld is small enough to leave enough radiation to induce the mutations necessary for evolution. It is also well-known that if there were fewer large planets in the solar system, the evolution of humans would have been impossible. e large planets divert large numbers of comets, preventing them from hitting the Earth. e spectacular collision of comet Shoemaker–Levy- with Jupiter, the astronomical event of July , was an example of this diversion of a comet.**
Also the anthropic principle has its most impressive successes when it predicts unknown observations. e most famous example stems from the study of stars. Carbon atoms, like all other atoms except most hydrogen, helium or lithium atoms, are formed
* e most important instrument of a scientist is the waste paper basket. ** For a collection of pictures of this event, see e.g. the http://garbo.uwasa. /pc/gifslevy.html website.
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Ref. 668 Ref. 669
in stars through fusion. While studying the mechanisms of fusion in , the well-known British astrophysicist Fred Hoyle* found that carbon nuclei could not be formed from the alpha particles present inside stars at reasonable temperatures, unless they had an excited state with an increased cross-section. From the fact of our existence, which is based on carbon, Hoyle thus predicted the existence of a previously unknown excited state of the carbon nucleus. And, indeed, the excited state was found a few months later by Willy Fowler.**
In its serious form, the anthropic principle is therefore the quest to deduce the description of nature from the experimental fact of our own existence. In the popular literature, however, the anthropic principle is o en changed from a simple experimental method to deduce the patterns of nature, to its perverted form, a melting pot of absurd metaphysical ideas in which everybody mixes up their favourite beliefs. Most frequently, the experimental observation of our own existence has been perverted to reintroduce the idea of ‘design’, i.e. that the universe has been constructed with the aim of producing humans; o en it is even suggested that the anthropic principle is an explanation – a gross example of disinformation.
How can we distinguish between the serious and the perverted form? We start with an observation. We would get exactly the same rules and patterns of nature if we used the existence of pigs or monkeys as a starting point. In other words, if we would reach di erent conclusions by using the porcine principle or the simian principle, we are using the perverted form of the anthropic principle, otherwise we are using the serious form. ( e carbon- story is thus an example of the serious form.) is test is e ective because there is no known pattern or ‘law’ of nature that is particular to humans but unnecessary apes or for pigs.***
“Er wunderte sich, daß den Katzen genau an den Stellen Löcher in den Pelz geschnitten wären, wo sie Augen hätten. ” Georg Christoph Lichtenberg****
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
* Fred Hoyle (b. 1915 Bingley, Yorkshire, d. 2001), important British astronomer and astrophysicist. He was
the rst and maybe only physicist who ever made a speci c prediction – namely the existence of an excited
state of the carbon nucleus – from the simple fact that humans exist. A permanent maverick, he coined the
term ‘big bang’ even though he did not accept the evidence for it, and proposed another model, the ‘steady
state’. His most important and well-known research was on the formation of atoms inside stars. He also
propagated the belief that life was brought to Earth from extraterrestrial microbes.
** William A. Fowler (1911–1995) shared the 1983 Nobel Prize in physics with Subramanyan
Chandrasekhar for this and related discoveries.
*** ough apes do not seem to be good physicists, as described in the text by D.J. P
, Folk Physics
for Apes: the Chimpanzee’s eory of How the World Works, Oxford University Press, 2000.
**** ‘He was amazed that cats had holes cut into their fur precisely in those places where they had eyes.’
Georg Christoph Lichtenberg (1742–1799), German physicist and intellectual, professor in Göttingen, still
famous today for his extremely numerous and witty aphorisms and satires. Among others of his time,
Lichtenberg made fun of all those who maintained that the universe was made exactly to the measure of
man, a frequently encountered idea in the foggy world of the anthropic principle.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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In nature there are neither rewards nor punishments – there are consequences.
“ Ivan Illich
e world owes you nothing. It was there rst.
” Mark Twain “ ” No matter how cruel and nasty and evil you may
be, every time you take a breath you make a
“ ower happy.
” Mort Sahl
Historically, the two terms ‘cause’ and ‘e ect’ have played an important role in philosoph-
ical discussions. In particular, during the birth of modern mechanics, it was important
to point out that every e ect has a cause, in order to distinguish precise thought from
thought based on beliefs, such as ‘miracles’, ‘divine surprises’ or ‘evolution from noth-
ing’. It was equally essential to stress that e ects are di erent from causes; this distinction
avoids pseudo-explanations such as the famous example by Molière where the doctor ex-
plains to his patient in elaborate terms that sleeping pills work because they contain a
dormitive virtue.
But in physics, the concepts of cause and e ect are not used at all. at miracles do not
appear is expressed every time we use symmetries and conservation theorems. e obser-
vation that cause and e ect di er from each other is inherent in any evolution equation.
Moreover, the concepts of cause and e ect are not clearly de ned; for example, it is espe-
cially di cult to de ne what is meant by one cause as opposed to several of them, and the
same for one or several e ects. Both terms are impossible to quantify and to measure. In
other words, useful as ‘cause’ and ‘e ect’ may be in personal life for distinction between
events that regularly succeed each other, they are not necessary in physics. In physical
explanations, they play no special roles.
“Ὰγαθον καὶ ξαξόν ë ἔν καὶ ταὐτό.*
” Heraclitus
“Wenn ein Arzt hinter dem Sarg seines Patienten geht, so folgt manchmal tatsächlich die Ursache der Wirkung.**
” Robert Koch
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“Variatio delectat.***
”Cicero
A lot of mediocre discussions are going on about this topic, and we will skip them here.
What is consciousness? Most simply and concretely, consciousness means the possession
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* ‘Good and bad – one and the same.’ ** ‘When a doctor walks behind the co n of his patient, indeed the cause sometimes follows the e ect.’ *** ‘Change pleases.’ Marcus Tullius Cicero (106–43 ), important lawyer, orator and politician at the end of the Roman republic.
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of a small part of oneself that is watching what the rest of oneself is perceiving, feeling,
thinking and doing. In short, consciousness is the ability to observe oneself, and in par-
ticular one’s inner mechanisms and motivations. Consciousness is the ability of introspection. For this reason, consciousness is not a prerequisite for studying motion. Indeed, animals, plants or machines are also able to observe motion. For the same reason, con-
sciousness is not necessary to observe quantum mechanical motion. On the other hand, both the study of motion and that of oneself have a lot in common: the need to observe carefully, to overcome preconceptions, to overcome fear and the fun of doing so.
For the time being, we have put enough emphasis on the precision of concepts. Talking
about motion is also something to be deeply enjoyed. Let us see why.
“Precision and clarity obey the indeterminacy relation: their product is constant.
”
C
Ref. 672 Ref. 673
“ ” Precision is the child of curiosity.
Like the history of every person, also the history of mankind charts a long struggle to avoid the pitfalls of accepting the statements of authorities as truth, without checking the facts. Indeed, whenever curiosity leads us to formulate a question, there are always two general ways to proceed. One is to check the facts personally, the other is to ask somebody. However, the last way is dangerous: it means to give up a part of oneself. Healthy people, children whose curiosity is still alive, as well as scientists, choose the rst way. A er all, science is adult curiosity.
Curiosity, also called the exploratory drive, plays strange games with people. Starting with the original experience of the world as a big ‘soup’ of interacting parts, curiosity can drive one to nd all the parts and all the interactions. It drives not only people. It has been observed that when rats show curious behaviour, certain brain cells in the hypothalamus get active and secrete hormones that produce positive feelings and emotions. If a rat has the possibility, via some implanted electrodes, to excite these same cells by pressing a switch, it does so voluntarily: rats get addicted to the feelings connected with curiosity. Like rats, humans are curious because they enjoy it. ey do so in at least four ways: because they are artists, because they are fond of pleasure, because they are adventurers and because they are dreamers. Let us see how.
Originally, curiosity stems from the desire to interact in a positive way with the environment. Young children provide good examples: curiosity is a natural ingredient of their life, in the same way that it is for other mammals and a few bird species; incidentally, the same taxonomic distribution is found for play behaviour. In short, all animals that play are curious, and vice versa. Curiosity provides the basis for learning, for creativity and thus for every human activity that leaves a legacy, such as art or science. e artist and art theoretician Joseph Beuys ( – ) had as his own guiding principle that every creative act is a form of art. Humans, and especially children, enjoy curiosity because they feel its importance for creativity, and for growth in general.
Curiosity regularly leads one to exclaim: ‘Oh!’, an experience that leads to the second reason to be curious: relishing feelings of wonder and surprise. Epicurus (Epikuros) ( –
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Ref. 674
) maintained that this experience, θαυµάζειν, is the origin of philosophy. ese feelings, which nowadays are variously called religious, spiritual, numinous, etc., are the same as those to which rats can become addicted. Among these feelings, Rudolf Otto has introduced the now classical distinction into the fascinating and the frightening. He named the corresponding experiences ‘mysterium fascinans’ and ‘mysterium tremendum’.* Within these distinctions, physicists, scientists, children and connoisseurs take a clear stand: they choose the fascinans as the starting point for their actions and for their approach to the world. Such feelings of fascination induce some children who look at the night sky to dream about becoming astronomers, some who look through a microscope to become biologists or physicists, and so on. (It could also be that genetics plays a role in this pleasure of novelty seeking.)
Perhaps the most beautiful moments in the study of physics are those appearing a er new observations have shaken our previously held thinking habits, have forced us to give up a previously held conviction, and have engendered the feeling of being lost. When, in this moment of crisis, we nally discover a more adequate and precise description of the observations, which provide a better insight into the world, we are struck by a feeling usually called illumination. Anyone who has kept alive the memory and the taste for these magic moments knows that in these situations one is pervaded by a feeling of union between oneself and the world.** e pleasure of these moments, the adventures of the change of thought structures connected with them, and the joy of insight following them provides the drive for many scientists. Little talk and lots of pleasure is their common denominator. In this spirit, the well-known Austrian physicist Victor Weisskopf ( –
) liked to say jokingly: ‘ ere are two things that make life worth living: Mozart and quantum mechanics.’
e choice of moving away from the tremendum towards the fascinans stems from an innate desire, most obvious in children, to reduce uncertainty and fear. is drive is the father of all adventures. It has a well-known parallel in ancient Greece, where the rst men studying observations, such as Epicurus, stated explicitly that their aim was to free people from unnecessary fear by deepening knowledge and transforming people from frightened passive victims into fascinated, active and responsible beings. ose ancient thinkers started to popularize the idea that, like the common events in our life, the rarer events also follow rules. For example, Epicurus underlined that lightning is a natural phenomenon caused by interactions between clouds, and stressed that it was a natural process, i.e. a process that followed rules, in the same way as the falling of a stone or any other familiar process of everyday life.
Investigating the phenomena around them, philosophers and later scientists succeeded in freeing people from most of their fears caused by uncertainty and a lack of knowledge about nature. is liberation played an important role in the history of hu-
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* is distinction is the basis of R
O , Das Heilige – Über das Irrationale in der Idee des Göttlichen
und sein Verhältnis zum Rationalen, Beck, München, 1991. is is a new edition of the epoch-making work
originally published at the beginning of the twentieth century. Rudolf Otto (1869–1937) was one of the most
important theologians of his time.
** Several researchers have studied the situations leading to these magic moments in more detail, notably
the Prussian physician and physicist Hermann von Helmholtz (1821–1894) and the French mathematician
Henri Poincaré (1854–1912). ey distinguish four stages in the conception of an idea at the basis of such
a magic moment: saturation, incubation, illumination and veri cation.
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man culture and still pervades in the personal history of many scientists. e aim to arrive at stable, rock-bottom truths has inspired (but also hindered) many of them; Albert Einstein is a well-known example for this, discovering relativity, helping to start up but then denying quantum mechanics.
Interestingly, in the experience and in the development of every human being, curiosity, and therefore the sciences, appears before magic and superstition. Magic needs deceit to be e ective, and superstition needs indoctrination; curiosity doesn’t need either. Con-
icts of curiosity with superstitions, ideologies, authorities or the rest of society are thus preprogrammed.
Curiosity is the exploration of limits. For every limit, there are two possibilities: the limit can turn out to be real or apparent. If the limit is real, the most productive attitude is that of acceptance. Approaching the limit then gives strength. If the limit is only apparent and in fact non-existent, the most productive attitude is to re-evaluate the mistaken view, extract the positive role it performed, and then cross the limit. Distinguishing between real and apparent limits is only possible when the limit is investigated with great care, openness and unintentionality. Most of all, exploring limits need courage.
Das gelü ete Geheimnis rächt sich.*
“ ” Bert Hellinger
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Ref. 676
It is dangerous to be right in matters on which the established authorities are wrong.
“ Voltaire
Manche suchen Sicherheit, wo Mut gefragt ist,
” und suchen Freiheit, wo das Richtige keine “Wahl läßt.**
” Bert Hellinger
Most of the material in this intermezzo is necessary in the adventure to get to the top of
Motion Mountain. But we need more. Like any enterprise, curiosity also requires courage,
and complete curiosity, as aimed for in our quest, requires complete courage. In fact, it is
easy to get discouraged on this trip. e journey is o en dismissed by others as useless,
uninteresting, childish, confusing, damaging or, most o en, evil. For example, between
the death of Socrates in
and Paul ierry, Baron d’Holbach, in the eighteenth
century, no book was published with the statement ‘gods do not exist’, because of the
threats to the life of anyone who dared to make the point. Even today, this type of attitude
still abounds, as the newspapers show.
rough the constant elimination of uncertainty, both curiosity and scienti c activity
are implicitly opposed to any idea, person or organization that tries to avoid the com-
parison of statements with observations. ese ‘avoiders’ demand to live with supersti-
tions and beliefs. But superstitions and beliefs produce unnecessary fear. And fear is the
basis of all unjust authorities. One gets into a vicious circle: avoiding comparison with
* e unveiled secret takes revenge.
** ‘Some look for security where courage is required and look for freedom where the right way doesn’t leave
any choice.’ is is from the beautiful booklet by B H
, Verdichtetes, Carl-Auer Systeme Verlag,
1996.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
observation produces fear – fear keeps unjust authority in place – unjust authority avoids
comparison with observation – etc.
As a consequence, curiosity and science are fundamentally opposed to unjust author-
ity, a connection that made life di cult for people such as Anaxagoras ( –
)
in ancient Greece, Hypatia in the Christian Roman empire, Galileo Galilei in the church
state, Antoine Lavoisier in France and Albert Einstein in Germany. In the second half
of the twentieth century, victims were Robert Oppenheimer, Melba Phillips and Chand-
ler Davis in the United States and Andrei Sakharov in the Soviet Union. Each of them
tell a horrible but instructive story, as have, more recently, Fang Lizhi, Xu Liangying, Liu
Gang and Wang Juntao in China, Kim Song-Man in South Korea, Otanazar Aripov in
Uzbekistan, Ramadan al-Hadi al-Hush in Libya, Bo Bo Htun in Burma, as well as many
hundreds of others. In many authoritarian societies the antagonism between curiosity
and injustice has hindered or even completely suppressed the development of physics
and other sciences, with extremely negative economic, social and cultural consequences.
When embarking on this ascent, we need to be conscious of what we are doing. In
fact, external obstacles can be avoided or at least largely reduced by keeping the project
to oneself. Other di culties still remain, this time of personal nature. Many have tried to
embark on this adventure with some hidden or explicit intention, usually of an ideological
nature, and then have got entangled by it before reaching the end. Some have not been
prepared to accept the humility required for such an endeavour. Others were not prepared
for the openness required, which can shatter deeply held beliefs. Still others were not
ready to turn towards the unclear, the dark and the unknown, confronting them at every
occasion.
On the other hand, the dangers are worth it. By taking curiosity as a maxim, facing
disinformation and fear with all one’s courage, one achieves freedom from all beliefs. In
exchange, you come to savour the fullest pleasures and the deepest satisfaction that life
has to o er.
We thus continue our hike. At this point, the trail towards the top of Motion Mountain
is leading us towards the next adventure: discovering the origin of sizes and shapes in
nature.
And the gods said to man: ‘Take what you want, and pay the price.’
“ (Popular saying) ” It is di cult to make a man miserable while he
feels he is worthy of himself.
“ ” Abraham Lincoln
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:
,
B
607 See for example the beautiful textbook by S
C. S
& R F. H
,
Evolution: An Introduction, Oxford University Press, . For fascinating story of evolu-
tion for non-specialists, see R
F
, Life – An Unauthorized Biography, Harper
Collins, , or also M
S
, Frogs, Flies & Dandelions – the Making of
Species, Oxford University Press/ . See also S
J. G
, e Panda’s thumb,
W.W. Norton & Co., , one of the several interesting and informative books on evolution-
ary biology by the best writer in the eld. An informative overview over the results of evolu-
tion, with the many-branched family tree that it produced, is given on the http://phylogeny.
arizona.edu/tree website. About the results of evolution for human beings, see the informat-
ive text by K. K
& S. K , Der Mensch in Zahlen, Spektrum Akademischer Verlag,
nd edn., . e epochal work by C
D
, On the Origin of Species, can be
found on the web, e.g. on the http://entiso .earthlink.net/origspec.htm website. Cited on
page .
608 A simple description is M Cape, . Cited on page .
RM
, e Origin of Johnny, Jonathan
609 ere is disagreement among experts about the precise timing of this experience. Some say
that only birth itself is that moment. However, there are several standard methods to re-
call memories of early life, even of the time before birth; one is described and used by J.K.
S
, Making Sense of Su ering, Penguin, New York, , translated from the
German original. Even more impressive examples are given by N.J. M , Der Kainkom-
plex – neue Wege der systemischen Familientherapie, Integral Verlag, . Cited on page
.
610 S
O’C
, Mindreading – How We Learn to Love and Lie, Arrow, . is
interesting book describes the importance of lying in the development of a human being,
and explains the troubles those people have who cannot read other minds and thus cannot
lie, such as autistics. Cited on pages and .
611 e approach to describe observations as related parts is called structuralism; the starting
point for this movement was de Saussure’s Cours de linguistique générale (see the footnote
on page ). A number of thinkers have tried to use the same approach in philosophy,
mythology and literature theory, though with little success. An overview of the (modest)
success of structuralism in linguistics and its failure in other elds is given by L. J
,
e Poverty of Structuralism: Literature and Structuralist eory, Longman, . e author
argues that when one reduces systems to interactions, one neglects the speci c content and
properties of the elements of the system, and this approach prevents a full understanding of
the system under discussion. Cited on page .
612 For a view of the mental abilities di erent from that of Piaget (described on page ),
a presently much discussed author is the Soviet experimental psychologist Lev Vigotsky,
whose path-breaking ideas and complicated life are described, e.g., in L V
,
Mind in Society, Harvard University Press, , or in R
V &J
V
, Understanding Vigotsky: a Quest for Synthesis, Blackwell Publishers, . More
extensive material can be found in the extensive work by R
V &J
V
, e Vigotsky Reader, Blackwell, . Cited on page .
613 A somewhat unconventional source for more details is the beautiful text by B
B
, e Uses of Enchantment: the Meaning and Importance of Fairy Tales,
Knopf, . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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614 Quoted in V. H
, R. R
Joseph Beuys, Achberger Verlag,
& P. S
, Soziale Plastik – Materialien zu
, p. . Cited on page .
615 e problems appearing when one loses the ability to classify or to memorise are told in
the beautiful book by the neurologist O
S , e Man Who Mistook His Wife for
a Hat, Picador, , which collects many case studies he encountered in his work. More
astonishing cases are collected in his equally impressive text An Anthropologist on Mars, Pic-
ador, .
See also the beautiful text D
D. H
, Visual Intelligence – How We Cre-
ate What We See, W.W. Norton & Co., , and the http://ari.ss.uci.edu/cogsci/personnel/
ho man.html website associated to it. Cited on pages and .
616 For a passionate introduction to the connections between language and the brain from a
Chomskian perspective, see the bestselling book by S
P
, e Language In-
stinct – How the Mind Creates Language, Harper Perennial, . e green idea sentence is
discussed in a chapter of the book. Cited on pages , , and .
617 An introduction to neurology is J
L
, Synaptic Self: How Our Brains Become
Who We Are, Viking Press, . Cited on page .
618 An overview of the status of the research into the origin of bipedalism is given by B. W , Four legs good, two legs better, Nature 363, pp. – , June . Cited on page .
619 A good introduction into the study of classi ers is J
A. A
, An Introduc-
tion to Neural Networks, MIT Press, . An introduction to computer science is given in
J. G
B
, Computer Science, An Overview, th edition, Addison Wesley,
, or in R D
&S
H
, e Analytical Engine: An Introduc-
tion to Computer Science Using the Internet, Brooks/Cole Publishers, . Cited on page
.
620 More about the connection between entropy and computers can be found in the classic
paper by R. L
, Irreversibility and heat generation in the computing process,
IBM Journal of Research and Development 5, pp. – , , and in C.H. B
&
R. L
, e fundamental physical limits of computation, Scienti c American 253,
pp. – , . Cited on page .
621 W.H. Z
, ermodynamic cost of computation, algorithmic complexity and the in-
formation metric, Nature 341, pp. – , August . Cited on page .
622 L. S
, Über die Entropieverminderung in einem thermodynamischen System bei
Eingri en intelligenter Wesen, Zeitschri für Physik 53, p. , . is classic paper can
also be found in English translation in the collected works by Leo Szilard. Cited on page
.
623 J.J. Hop eld/ Nature 376, pp. – , . is paper by one of the fathers of the eld presents one possibility by which the timing of nerve signals, instead of the usually assumed ring frequency, could also carry information. Cited on page .
624 e details of the properties of the ring patterns of neurons are nicely described in the
article by M. M
& R. D
, A silicon neuron, Nature 354, pp. – ,
/ December , in which they show how to simulate a neuron’s electrical behaviour
using a silicon circuit. Cited on page .
625 A. M
, J.T. C
, U. N
, J. O’D
, J. A
, R.S.
F
& C.J. P , Neurolinguistics: structural plasticity in the bilingual brain,
Nature 431, p. , . Cited on page .
626 e discussion whether the brain is or is not superior to a computer is nicely summarised
by G. V
, Algorithmen, Gehirne, Computer – Was sie können und was sie nicht
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Page 785
können, Teil I und Teil II, Naturwissenscha en 78, p. , , and 78, pp. – , . Cited on page .
627 However, the language with the largest available dictionary is Dutch, with the
of the Wordenboek der Nederlandsche Taal, which appeared between and
almost
entries. Cited on page .
volumes . It has
628 e list and the remark on discovery on concepts is due to a personal communication from Anna Wierzbicka. A longer list is published in her book Semantics, Primes and Universals, Oxford University Press, . Cited on pages and .
629 W.S. H
, Foundations of Mathematics, W.B. Saunders Co., Philadelphia, . ere
is also the article by P. J. C
& R. H
, Non-Cantorian set theory, Scienti c
American 217, pp. – , . Cohen was the mathematician who in proved that
the negation of the continuum hypothesis could be added to the axioms of set theory and
still produce a consistent theory; he calls such sets non-Cantorian. Cited on page .
630 See the beautiful article by I. S . Cited on page .
, Fair shares for all, New Scientist, pp. – , June
631 e proof of the independence of the continuum hypothesis came in two parts. First, Kurt Gödel proved in that an axiom can be consistently added to ZFC set theory so that the continuum hypothesis is correct. en, in , Paul Cohen proved that an axiom can be consistently added to ZFC set theory so that the continuum hypothesis is false. Cited on page .
632 R R Toronto,
, In nity and the Mind – the Science and Philosophy of the In nite, Bantam, . Cited on page .
633 is general division of mathematics is nicely explained in the text by P
B
,
Die Architektur der Mathematik – Denken in Strukturen, Rororo, . Cited on page .
634 e issue is treated in T
A
, Summa eologica, in question of the rst
part. e complete text, several thousand pages, can be found on the http://www.newadvent.
org website. We come back to it in the part on quantum theory, in the section on the Pauli
exclusion principle. It seems that the whole question goes back to P
( )L
,
Liber Sententiarum c. . Cited on page .
635 B.C. G
, How to fold paper in half twelve times:an “impossible challenge” solved
and explained, Histrical Society of Pomona Valley, , also found at http://www.osb.net/
Pomona/ times.htm. See also http://www.sciencenews.org/
/mathtrek.asp. Cited
on page .
636 I. S
, Daisy, daisy, give me your answer, do, Scienti c American, pp. – , Janu-
ary . is pedagogical article explains how the growth of plants usually leads to owers
whose number of petals is from the Fibonacci series , , , , , , , , , , , , etc.
( e gure on page is an example.) Deviations from this ideal case are also explained.
e original work are two articles by S. D
& Y. C
, La physique des spirales
végétales, La Recherche 24, pp. – , , and Phyllotaxis as a self-organised growth pro-
cess, in Growth Patterns in Physical Sciences and Biology, edited by J.M. G
-R &
al., Plenum Press, . Cited on page .
637 H. D
, e Eye, Academic Press, . Cited on page .
638 See the http://akbar.marlboro.edu/~mahoney/cube/NxN.txt website. Cited on page .
639 An introduction to the surreal numbers is given by the article by P
S
, In nity
plus one, and other surreal numbers, Discover, pp. – , December . ere is also the
text by D. K
, Surreal Numbers: How two ex-Students Turned on to Pure Mathematics
and Found Total Happiness, Addison Wesley, , or http://www-cs-faculty.stanford.edu/
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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~knuth/sn.html. e usually quoted references on the topic include J
H. C
,
On Numbers and Games, Academic Press, , E.R. B
, J.H. C
& R.K.
G , Winning Ways for Your Mathematical Plays, Volume I: Games in General, Academic
Press, , and H. G
, An Introduction to Surreal Numbers, Cambridge University
Press, . Cited on pages and .
640 is beautiful problem is discussed by I S
, A bundling fool beats the wrap, Sci-
enti c American, pp. – , June . In four dimensions, the answer is known to lie some-
where between
and
, whereas the ve-dimensional answer is conjectured to
be ‘never’. Cited on page .
641 A. P , Niels Bohr’s Times: in Physics, Philosophy and Polity, Oxford University Press, , page . Cited on page .
642 E
W
page .
, Symmetries and Re ections, Indiana University Press, . Cited on
643 A
T
, Introduction to Modern Logic, Dover,
dren’s book by the mathematician and photographer L
land. Cited on page .
. See also the famous chil-
C
, Alice in Wonder-
644 G
W
, e layout of digits on pushbutton telephones – a review of the literature,
Tele 34, pp. – , . Cited on page .
645 A clear overview of philosophy of science, o en called epistemology, without unnecessary
detail, is given by R
B
, L’Epistémologie, Presses Universitaires de France,
. Cited on page .
646 About the di erent aspects of falsi ability of general statements it is commonplace to cite the work by the epitemologist Karl Popper ( – ), especially his long and boring book Logik der Forschung, rst published in . e reason for this boredom is that Popper’s work is simply a warming-up of Pierre Duhem’s ideas. Cited on page .
647 For a good way of making blood that lique es, see L. G
, F. R
& S. D
S
, Working bloody miracles, Nature 353, p. , . e Grand dic-
tionnaire universel du XIXe siècle, by P
L
, also contains a recipe; it was
again shown to the public in the s by Henri Broch. A wonderful and classic text is
H
H
, Miracle Mongers and their Methods, Prometheus Books, Bu alo, .
e original, written in , by the world famous magician named ‘ e Great Houdini’, is
also available on the http://etext.lib.virginia.edu/toc/modeng/public/HouMirM.html web-
site. e milk drinking Indian statues were common around the world in and .
About healers, see J
R , Flim- am!, Prometheus Books, Bu alo, New York, ,
and the exquisite book by H C
Z
, Warum ich Jesus nicht leiden kann,
Rowohlt, . Cited on page .
648 R
P
, e Road to Reality: A Complete Guide to the Laws of the Universe,
Jonathan Cape, , page . Cited on page .
649 J H
, e End of Science – Facing the Limits of Knowledge in the Twilight of the
Scienti c Age, Broadway Books, , pp. – , and chapter , note . Cited on pages
and .
650 For an opinion completely contrary to the one described here, see the book by G
J.
C
, e Limits of Mathematics, Springer Verlag, , which can also be found on
the author’s website http://www.cs.auckland.ac.nz/CDMTCS/chaitin/lm.html, along with
his other works. Chaitin has devoted most of his life to the questions discussed in the section,
especially on computability. Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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:
,
651 See the book by J. B
& J. E
York, . Cited on page .
, e Liar, Oxford University Press, New
652 is de nition (statement . ) and many other statements about science are in the beau-
tiful and rightly famous text by L
W
, Tractatus logico-philosophicus,
Edition Suhrkamp, . It gives a condensed summary of the basis of science, thought and
language in a collection of highly structured and numbered sentences. Cited on pages
and .
653 See M. D
, e Klopsteg memorial lecture, American Journal of Physics 66, pp. –
, . Cited on page .
654 Well-known books are e.g. F
K
age, . Cited on page .
, Praktische Physik, Teubner, . Au-
655 Results are described e.g. in L. B
& C. S
, Lehrbuch der Experimental-
physik, Band I, II, III und IV, W. de Gruyter. Cited on page .
656 L
-B
, edited by K.-H. H
& O. M
, Zahlenwerte
und Funktionen aus Naturwissenscha en und Technik, Neue Serie, Springer Verlag, Berlin,
. is series of more than one hundred volumes contains all important observations in
the domain of physics. Cited on page .
657 e origin of this incorrect attribution is the book by G
S
, Brecht, Leben
des Galilei – Dichtung und Wirklichkeit, Ullstein, Berlin , p. . e statement has never
been made by Galilei; this issue has been discussed at length in specialist circles, e.g. by
F. K
, "Messen was meßbar ist" - Über ein angebliches Galilei-Zitat, Berichte zur
Wissenscha geschichte 11, p. , , or on the internet by Peter Jaencke. Cited on page
.
658 e strange and sometimes dangerous consequences of beliefs can be found e.g. in M
G
, Fads and Fallacies, Dover, , and in J
R , Faith Healers, Prometh-
eus Books, . e million dollar prize for showing any paranormal or supernormal e ect
is available from his http://www.randi.org website. Cited on page .
659 See the nice collection of cranks on the http://www.crank.net website. Cited on page .
660 As an example, the opposite view on the emergence of properties is strongly defended by P.
J
, Particle physics and our everyday world, Physics Today pp. – , July . Cited
on page .
661 See page of the bibliography by J B on page .
, Charles Darwin, Pimlico, . Cited
662 A beautiful introduction to Greek philosophy is E
Z
, Outlines of the History
of Greek Philosophy, Dover, , a reprint of a book published in . Among others, it
gives a clear exposition of the philosophy of Democritus and the other presocratics. Cited
on page .
663 e famous quote is found at the beginning of chapter XI, ‘ e Physical Universe’, in A -
E
, e Philosophy of Physical Science, Cambridge, . Cited on page
.
664 G
F
, Chi l’ha detto?, Hoepli, Milano, . Cited on page .
665 See J -P D Cited on page .
, Les écoles présocratiques, Folio Essais, Gallimard, p. , .
666 For a beautiful text on fractals, see the footnote on page . Cited on page .
667 As has been pointed out by René Descartes. Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
668 e famous carbon C resonance was found by Willy Fowler, as described in E. M -
B
, G.R. B
, W.A. F
& F. H , Synthesis of the
elements in stars, Reviews of Modern Physics 29, pp. – , . Cited on page .
669 An extensive overview of the topic is given in the thick book by J
D. B
&
F
J. T
, e Anthropic Cosmological Principle, Oxford University Press, .
e term itself is due to Brandon Carter, who coined it in and presented it in a sym-
posium devoted to the th anniversary of Nicolaus Copernicus. For more literature, see
Y I. B
, Resource Letter AP- : the anthropic principle, American Journal of
Physics 59, pp. – , . Cited on page .
670 V
, Candide ou l’optimisme, . See the footnote on page . e book is so good
that it was still being seized by the US customs in , and the US post o ce refused to
transport it as late as . For more details, search for ‘banned books online’ on the world-
wide web. Cited on page .
671 e number of books on consciousness is large and the contents not always interesting, and
o en not based on fact, as shown by K R. P
&J E
, e Self and its
Brain – an Argument for Interactionism, Rutledge, . Cited on page .
672 See e.g. the Encyclopedia Britannica, Macropaedia, in the entry on animal behaviour. Cited on page .
673 A straight and informative introduction to the work and ideas of Joseph Beuys (in German)
is by R
G
, Joseph Beuys, RAAbits Kunst, Raabe Fachverlag, September .
Cited on page .
674 Two studies, one by R.P. E
& al., Dopamine D receptor (D DR) exon III poly-
morphism associated with human personality trait of novelty seeking, Nature Genetics 12,
pp. – , January , and another study by J. B
& al., Population and fa-
milial association between the D dopamine receptor gene and measures of novelty seeking,
Nature Genetics 12, pp. – , January , found that people with a special form of the D
dopamine receptor gene, or D DR, are more prone to novelty seeking than people with the
usual form. e D DR gene regulates the formation of dopamine receptors, a chemical mes-
senger in the brain that has been a candidate for some time for a substance involved in the
propensity for novelty seeking. Cited on page .
675 More can be found in J
H
, e Mathematician’s Mind – e Psychology
of Invention in the Mathematical Field, Princeton Science Library, . Cited on page .
676 For example, one needs the courage to face envy. About this topic see the classic text by
H
S
, Der Neid, , published in English as Envy: A eory of Social Be-
havior, . It is the standard work in the eld. Cited on page .
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Second Part
Q
T
?
W AI
:W I M ?
Where the existence of a minimal change is deduced, implying that motion is fuzzy, that matter is not permanent, that boxes are never tight, that matter is composed of elementary units and that light and interactions are streams of particles, thus explaining why antimatter exists, why the oor does not fall but keeps on carrying us, why particles are unlike gloves, why empty space pulls mirrors together and why the stars shine.
C
V
QUANTA OF LIGHT AND MAT TER
.
–
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 677
Natura [in operationibus suis] non facit saltus.*
“ ” th century
E Motion Mountain up to this point, we completed three legs. We rst ncountered Galileo’s mechanics, the description of motion for kids, then instein’s relativity, the description of motion for science ction enthusiasts, and
nally Maxwell’s electrodynamics, the description of motion valuable to cra smen and
businessmen.
ese three classical descriptions of motion are im-
pressive, beautiful and useful. However, they also have
a small problem: they are wrong. e reason is simple:
none of them describes life. Whenever we observe a
ower such as the one of Figure , we enjoy its bright
colours, its wild smell, its so and delicate shape or
the ne details of its symmetry. None of the three clas-
sical descriptions can explain any of these properties;
neither do they explain the impression they make on
our senses. Classical physics can describe them partly,
but it cannot explain their origins. For an explanation,
we need quantum theory. In fact we will discover that in
life, every type of pleasure is an example of quantum mo-
tion. Just try; take any example of a pleasant situation,
such as a beautiful evening sky, a waterfall, a caress or F I G U RE 282 An example of a Challenge 1163 n a happy child. Classical physics is not able to explain it: quantum system (© Ata Masafumi)
the involved colours, shapes and sizes remain mysteri-
ous.
In the beginning of physics this limitation was not seen as a shortcoming: in those
times neither senses nor material properties were imagined to be related to motion. And
of course, in older times the study of pleasure was not deemed a serious topic of invest-
Page 253 igation for a respectable researcher. However, in the meantime we know that the senses
of touch, smell and sight are rst of all detectors of motion. Without motion, no senses!
* ‘Nature [in its workings] makes no jumps.’
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Page 612 Ref. 678
In addition, all detectors are built of matter. In the chapter on electromagnetism we started to understand that all properties of matter are due to motion of charged constituents. Density, sti ness, colour and all other material properties result from the electromagnetic behaviour of the Lego bricks of matter, namely the molecules, the atoms and the electrons.
us, also matter properties are consequences of motion. In addition, we saw that these tiny constituents are not correctly described by classical electrodynamics. We even found that light itself behaves unclassically. erefore the inability of classical physics to describe matter and the senses is indeed due to its intrinsic limitations.
In fact, every failure of classical physics can be traced back to a single, fundamental discovery made in by Max Planck:
In nature, actions smaller than the value ħ = . ë − Js are not observed.
Challenge 1164 n Page 612
All experiments trying to do so invariably fail.* In other words, in nature there is always some action – like in a good movie. is existence of a minimal action, the so-called quantum principle, is in full contrast with classical physics. (Why?) However, it has passed the largest imaginable number of experimental con rmations, many of which we will encounter in this second part of our mountain ascent. Planck discovered the principle when studying the properties of incandescent light, i.e. the light emanating from hot bodies. But the quantum principle also applies to motion of matter, and even, as we will see later, to motion of space-time. Incidentally, the factor results from the historical accidents in the de nition of the constant ħ, which is read as ‘eitch-bar’. Despite the missing factor, the constant ħ is called the quantum of action or also, a er its discoverer, (reduced) Planck’s constant.
e quantum principle states that no experiment whatsoever can measure an action value smaller than ħ . For a long time, even Einstein tried to devise experiments to overcome the limit. But he failed: nature does not allow it.
Interestingly, since action in physics, like action in the lm industry, is a way to measure the change occurring in a system, a minimum action implies that there is a minimum change in nature. e quantum of action thus would be better named the quantum of change. Comparing two observations, there always is change. (What is called ‘change’ in everyday life is o en called ‘change of state’ by physicists; the content is the same.) Before we explore experiments con rming this statement, we give an introduction to some of its more surprising consequences.
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T
Page 704
Since action measures change, a minimum observable action means that two subsequent observations of the same system always di er by at least ħ . In every system, there is always something happening. As a consequence, in nature there is no rest. Everything moves, all the time, at least a little bit. Natura facit saltus. True, it is only a tiny bit, as the value of ħ is so small. But for example, the quantum of action implies that in a
Ref. 679
* In fact, the cited quantum principle is a simpli cation; the constant originally introduced by Planck was the (unreduced) constant h = πħ. e factors π and 1/2 leading to the nal quantum principle were found somewhat later, by other researchers. is somewhat unconventional, but useful didactic approach is due to Niels Bohr. Nowadays, the approach is almost never found in the literature; it might be used in a teaching text for the rst time here. About Max Planck and his accomplishments, see the footnote on page 612.
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Challenge 1165 ny
mountain, a system at rest if there is any, all atoms and all electrons are continuously buzzing around. Rest can be observed only macroscopically, and only as a long time or many particle average.
Since there is a minimum action for all observers, and since there is no rest, in nature there is no perfectly straight and no perfectly uniform motion. Forget all you have learned so far. Every object moves in straight and uniform motion only approximately, and only when observed over long distances or long times. We will see later that the more massive the object is, the better the approximation is. Can you con rm this? As a consequence, macroscopic observers can still talk about space-time symmetries. Special relativity can thus easily be reconciled with quantum theory.
Obviously, also free fall, i.e. motion along geodesics, exists only as a long time average. In this sense, general relativity, being based on the existence of freely falling observers, cannot be correct when actions of the order of ħ are involved. Indeed, the reconciliation of the quantum principle with general relativity – and thus with curved space – is a big challenge. e issues are so mind-shattering that the topic forms a separate, third, part of this mountain ascent.
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Challenge 1166 n Page 221
Have you ever wondered why leaves are green? Probably you know that they are green because they absorb blue light, of short wavelengths, and red light, of long wavelengths, and let green, medium wavelength light be re ected. How can a system lter out the small and the large, and let the middle pass through? To do so, leaves must somehow measure the wavelength. But we have seen that classical physics does not allow to measure length or time intervals, as any measurement requires a measurement unit, and classical physics does not allow to de ne units for them. On the other hand, it takes only a few lines to con rm that with the help of the quantum of action ħ (and the Boltzmann constant k, which Planck discovered at the same time), fundamental measurement units of all measurable quantities can be de ned, including length and thus wavelength. Can you nd a combination of c, G and ħ giving a length? It only will take a few minutes. When Planck found the combination, he was happy like a child; he knew straight away that he had made a fundamental discovery, even though in quantum theory did not exist yet. He even told his seven-year-old son Erwin abut it, while walking with him through the forests around Berlin. Planck explained to his son that he had made a discovery as important as universal gravity. Indeed, Planck knew that he had found the key to understanding most of the e ects which were unexplained so far. In particular, without the quantum of action, colours would not exist. Every colour is a quantum e ect.*
Planck also realized that the quantum of action allows to understand the size of all things. With the quantum of action, it was nally possible to answer the question on the maximum size of mountains, of trees and of humans. Planck knew that the quantum of action con rmed the answer Galileo had deduced already long before him: sizes are due to fundamental, minimal scales in nature. e way the quantum of action allows to understand the sizes of physical systems will be uncovered step by step in the following.
e size of objects is related to the size of atoms; in turn, the size of atoms is a direct consequence of the quantum of action. Can you deduce an approximation for the size
Challenge 1167 n * It is also possible to de ne all units using c, G and e, the electron charge. Why is this not satisfactory?
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Challenge 1168 n Challenge 1169 n
of atoms, knowing that it is given by the motion of electrons of mass me and charge e, constrained by the quantum of action? is connection, a simple formula, was discovered
in by Arthur Erich Haas, years before quantum theory was formulated; at the time,
everybody made fun of him. Nowadays, the expression is found in all textbooks.*
By determining the size of atoms, the quantum of action has an important con-
sequence: Gulliver’s travels are impossible. ere are no tiny people and no giant ones.
Classically, nothing speaks against the idea; but the quantum of action does. Can you
provide the detailed argument?
But if rest does not exist, how can shapes exist? Any shape,
also that of a ower, is the result of body parts remaining
at rest with respect to each other. Now, all shapes result
from the interactions of matter constituents, as shown most
clearly in the shape of molecules. But how can a molecule,
such as the water molecule H O, have a shape? In fact, it does
not have a xed shape, but its shape uctuates, as expected
from the quantum of action. Despite the uctuations it does
have an average shape, because di erent angles and distances
correspond to di erent energies. And again, these average
length and angle values only result because the quantum of action leads to fundamental length scales in nature. Without the quantum of action, there would be no shapes in nature.
F I G U R E 283 An artistic impression of a water molecule
As we will discover shortly, quantum e ects surround us from all sides. However, since
the minimum action is so small, its e ects on motion appear mostly, but not exclusively,
in microscopic systems. e study of such systems has been called quantum mechanics
by Max Born, one of the main gures of the eld.** Later on, the term quantum theory
became more popular. In any case, quantum physics is the description of microscopic
motion. But when is quantum theory necessary? Table shows that all processes on
atomic and molecular scale, including biological and chemical ones, involve action values
near the quantum of action. So do processes of light emission and absorption. All these
phenomena can be described only with quantum theory.
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Ref. 680
* Before the discovery of ħ, the only simple length scale for the electron was the combination e ( πε mc ) fm; this value is ten thousand times smaller than an atom. ** Max Born (b. 1882 Breslau, d. 1970 Göttingen) rst studied mathematics, then turned to physics. Professor in Göttingen, he made the city one of the world centres of physics. He developed quantum mechanics with his assistants Werner Heisenberg and Pascual Jordan, then applied it to scattering, to solid state physics, to optics and to liquids. He was the rst to understood that the state function describes a probability amplitude. He is one of the authors of the famous Born & Wolf textbook on optics; it still remains the main book of the eld. Born attracted to Göttingen the most brilliant talents of the time, receiving as visitors Hund, Pauli, Nordheim, Oppenheimer, Goeppert–Mayer, Condon, Pauling, Fock, Frenkel, Tamm, Dirac, Mott, Klein, Heitler, London, von Neumann, Teller, Wigner and dozens of others. Being Jewish, Max Born lost his job in 1933; he emigrated and became professor in Edinburgh, where he stayed for twenty years. Physics at Göttingen University never recovered from this loss. For his elucidation of the meaning of the wave function he received the 1954 Nobel prize in physics.
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‘
’
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Page 220
Quantum theory results from the existence of minimum measurable values in nature, precisely in the way that Galileo already speculated about in the seventeenth century. As mentioned in detail earlier on, it was Galileo’s insistence on these ‘piccolissimi quanti’ that got him accused. Of course, we will discover that only the idea of a smallest change leads to a precise and accurate description of nature. e term ‘quantum’ theory does not mean that all measurement values are multiples of a smallest one; this is correct only in certain special cases.
Table also shows that the term ‘microscopic’ has a di erent meaning for a physicist and for a biologist. For a biologist, a system is microscopic if it requires a microscope for its observation. For a physicist however, a system is microscopic if its characteristic action is of the order of the quantum of action. In short, for a physicist, a system is microscopic if it is not visible in a (light) microscope. To increase the confusion, some quantum physicists nowadays call their own class of microscopic systems ‘mesoscopic,’ whereas many classical, macroscopic systems are now called ‘nanoscopic’. Both names mainly help to attract attention and funding.
ere is another way to characterize the di erence between a microscopic or quantum system on one side and a macroscopic or classical system on the other. A minimum action implies that the di erence of action values S between two successive observations of the same system, spaced by a time ∆t, is limited. erefore one has
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S(t + ∆t) − S(t) = (E + ∆E)(t + ∆t) − Et = E∆t + t∆E + ∆E∆t ħ .
(474)
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Now the value of the energy E and of the time t – but not that of ∆E or of ∆t – can be set to zero if we choose a suitable observer; thus the existence of a quantum of action implies that in any system the evolution is constrained by
∆E∆t ħ ,
(475)
where E is the energy of the system and t its age. In other words, ∆E is the change of Challenge 1171 ny energy and ∆t the time between two successive observations. By a similar reasoning we
nd that for any system the position and momentum values are constrained by
∆x∆p ħ ,
(476)
where ∆p is the indeterminacy in momentum and ∆x the indeterminacy in position. ese two famous relations were called indeterminacy relations by their discoverer,
Werner Heisenberg.* e name is o en incorrectly translated into English as ‘uncertainty
* One o en hears the myth that the indeterminacy relation for energy and time has another weight than the one for momentum and position. at is wrong; it is a myth propagated by the older generation of physicists. is myth survived through many textbooks for over 70 years; just forget it, as it is incorrect. It is essential to remember that all four quantities appearing in the inequalities are quantities describing the internal properties of the system. In particular, it means that t is some time variable deduced from changes
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TA B L E 56 Some small systems in motion and the observed action values for their changes
S
&
A
M
Light
Smallest amount of light absorbed by a coloured surface
ħ
quantum
Smallest hit when light re ects from mirror
ħ
quantum
Smallest visible amount of light
c. ħ
quantum
Smallest amount of light absorbed in ower petal
c. ħ
quantum
Blackening of photographic lm
c. ħ
quantum
Photographic ash
c. ħ
classical
Electricity Electron ejected from atom Electron added to molecule Electron extracted from metal Electron motion inside microprocessor
c. − ħ c. − ħ c. − ħ c. − ħ
quantum quantum quantum quantum
Signal transport in nerves, from one molecule to the next c. ħ
quantum
Current ow in lighting bolt
c. ħ
classical
Materials science Tearing apart two neighbouring iron atoms
c. − ħ
quantum
Breaking a steel bar
c. ħ
classical
Basic process in superconductivity
ħ
quantum
Basic process in transistors
ħ
quantum
Basic process in magnetic e ects
ħ
quantum
Chemistry
Atom collisions in liquids at room temperature Shape oscillation of water molecule Shape change of molecule, e.g. in chemical reaction Single chemical reaction curling a hair
c. ħ c. − ħ c. − ħ c. − ħ
quantum quantum quantum quantum
Tearing apart two mozzarella molecules
c. ħ
quantum
Smelling one molecule
c. ħ
quantum
Burning fuel in a cylinder in an average car engine explosion c. ħ
classical
Life
Air molecule hitting ear drum
c. ħ
quantum
Smallest sound signal detectable by the ear
Challenge 1170 ny
DNA duplication step in cell division
c. ħ
quantum
Ovule fecundation
c. ħ
classical
Smallest step in molecular motor
c. ħ
quantum
Sperm motion by one cell length
c. ħ
classical
Cell division
c. ħ
classical
Fruit y’s wing beat
c. ħ
classical
Person walking one body length
c. ë ħ classical
Nuclei and stars
Nuclear fusion reaction in star
c. − ħ
quantum
Particle collision in accelerator
c. ħ
quantum
Explosion of gamma ray burster
c. ħ
classical
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relations’. However, this latter name is wrong: the quantities are not uncertain, but undetermined. Due to the quantum of action, system observables have no de nite value. ere is no way to ascribe a precise value to momentum, position and other observables of a quantum system.
Any system whose indeterminacy is of the order of ħ is a quantum system; if the indeterminacy product is much larger, the system is classical, and classical physics is su cient for its description. In other words, even though classical physics assumes that there are no measurement indeterminacies in nature, a system is classical only if its indeterminacies are large compared to the minimum possible ones. As a result, quantum theory is necessary in all those cases in which one tries to measure some quantity as precisely as possible.
e indeterminacy relations again show that motion cannot be observed to in nite precision. In other words, the microscopic world is fuzzy. is strange result has many important and many curious consequences. For example, if motion cannot be observed with in nite precision, the very concept of motion needs to be used with great care, as it cannot be applied in certain situations. In a sense, the rest of our quest is an exploration of the implications of this result. In fact, as long as space-time is at, it turns out that we can keep motion as a concept describing observations, provided we remain aware of the limitations of the quantum principle.
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Page 154 Challenge 1172 ny
e quantum of action implies short-time deviations from energy, momentum and angular momentum conservation in microscopic systems. Now, in the rst part of our mountain ascent we realized that any type of nonconservation implies the existence of surprises in nature. Well, here are some of them.
Since uniform motion does not exist in the precise meaning of the term, a system moving in one dimension only, such as the hand of a clock, always has a possibility to move a bit in the opposite direction, thus leading to incorrect readings. Indeed, quantum theory predicts that clocks have limits, and that perfect clocks do not exist. In fact, quantum theory implies that strictly speaking, one-dimensional motion does not exist.
Obviously, the limitations apply also to metre bars. us the quantum of action is responsible on one hand that the possibility to perform measurements exists, and on the other hand for the limitations of measurements.
In addition, it follows from the quantum of action that any observer must be large to be inertial or freely falling, as only large systems approximate inertial motion. An observer cannot be microscopic. If humans were not macroscopic, they could neither observe nor study motion.
Ref. 681
observed inside the system and not the external time coordinate measured by an outside clock, in the same way that the position x is not the external space coordinate, but the position characterizing the system.
Werner Heisenberg (1901–1976) was an important German theoretical physicist and an excellent table tennis and tennis player. In 1925, as a young man, he developed, with some help by Max Born and Pascual Jordan, the rst version of quantum theory; from it he deduced the indeterminacy relations. For these achievements he received the Nobel Prize for physics in 1932. He also worked on nuclear physics and on turbulence. During the second world war, he worked in the German nuclear ssion program. A er the war, he published several successful books on philosophical questions in physics and he unsuccessfully tried, with some half-hearted help by Wolfgang Pauli, to nd a uni ed description of nature based on quantum theory, the ‘world formula’.
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Challenge 1173 ny Challenge 1174 ny
Due to the nite accuracy with which microscopic motion can be observed, faster
than light motion should be possible in the microscopic domain. Quantum theory thus
predicts tachyons, at least over short time intervals. For the same reason, also motion
backwards in time should be possible over microscopic times and distances. In short, a
quantum of action implies the existence of microscopic time travel.
But there is more: the quantum of action implies that there is no permanence in nature.
Imagine a moving car suddenly disappearing for ever. In such a situation neither mo-
mentum nor energy would be conserved. e action change for such a disappearance
is large compared to ħ, so that its observation would contradict even classical physics,
as you might want to check. However, the quantum of action allows that a microscopic
particle, such as an electron, disappears for a short time, provided it reappears a erwards.
e quantum of action also implies that the vacuum is not empty. If one looks at empty
space twice in a row, the two observations being spaced by a tiny time interval, some en-
ergy will be observed the second time. If the time interval is short enough, due to the
quantum of action, matter particles will be observed. Indeed, particles can appear any-
where from nowhere, and disappear just a erwards, as the action limit requires it. In
other words, classical physics’ idea of an empty vacuum is correct only when observed
over long time scales. In summary, nature shows short time appearance and disappear-
ance of matter.
e quantum of action implies that compass needles cannot work. If we look twice in
a row at a compass needle or even at a house, we usually observe that they stay oriented
in the same direction. But since physical action has the same unit as angular momentum,
a minimum value for action also means a minimum value for angular momentum. ere-
fore, every macroscopic object has a minimum value for its rotation. In other words,
quantum theory predicts that in everyday life, everything rotates. Lack of rotation exists
only approximately, when observations are spaced by long time intervals.
For microscopic systems, the situation is more involved. If their rotation angle can
be observed, such as for molecules, they behave like macroscopic objects: their position
and their orientation are fuzzy. But for those systems whose rotation angle cannot be ob-
served, the quantum of action turns out to have somewhat di erent consequences. eir
angular momentum is limited to values which are multiples of ħ . As a result, all mi-
croscopic bound systems, such as molecules, atoms, or nuclei, contain rotational motion
and rotating components.
But there is more to come. A minimum
action implies that cages in zoos are dan-
gerous and banks are not safe. A cage is a
feature requiring a lot of energy to be over-
E
come. Mathematically, the wall of a cage m p
is an energy hill, similar to the one shown
in Figure . If a particle on one side of
0
∆x
the hill has momentum p, it is simple to F I G U R E 284 Hills are never high enough show that the particle can be observed on
the other side of the hill, at position ∆x, even if its kinetic energy p m is smaller than
the height E of the hill. In everyday life this is impossible. But imagine that the missing
momentum ∆p = mE − p to overcome the hill satis es ∆x∆p ħ . e quantum of
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•.
action thus implies that a hill of width
∆x
ħ mE − p
(477)
Challenge 1175 ny Challenge 1176 ny
Challenge 1177 ny Ref. 682
Challenge 1178 ny Challenge 1179 ny
is not an obstacle to a particle of mass m. But this is not all. Since the value of the particle
momentum p is itself undetermined, a particle can overcome the hill even if the hill is
wider than value ( ), though the broader it is the smaller the probability is. As a result,
any particle can overcome any obstacle. is e ect, for obvious reasons, is called the tun-
nelling e ect. In short, the minimum action principle implies that there are no safe boxes
in nature. Due to tunnelling, matter is not impenetrable, in contrast to everyday, classical
observation. Can you explain why lion cages work despite the quantum of action?
By the way, the quantum of action also implies that a particle with a kinetic energy
larger than the energy height of a hill can get re ected by the hill. Classically this is im-
possible. Can you explain the observation?
e minimum action principle also
implies that book shelves are dangerous.
Shelves are obstacles to motion. A book
E1
in a shelf is in the same situation as the
mass in Figure ; the mass is surrounded by energy hills hindering its escape to
m
E2
the outer, lower energy world. Now, due to
the tunnelling e ect, escape is always pos- F I G U RE 285 Leaving enclosures
sible. e same picture applies to a branch
of a tree, a nail in a wall, or to anything attached to anything else. Fixing things to each
other is never for ever. We will nd out that every example of light emission, even radio-
activity, results from this e ect. e quantum of action thus implies that decay is part of
nature. In short, there are no stable excited systems in nature. For the same reason by the
way, no memory can be perfect. (Can you con rm the deduction?) Note that decay o en
appears in everyday life, where it just has a di erent name: breaking. In fact, all cases in
which something breaks require the quantum of action for their description. Obviously,
the cause of breaking is o en classical, but the mechanism of breaking is always quantum.
Only objects that follow quantum theory can break.
Taking a more general view, also ageing and death result from the quantum of action.
Death, like ageing, is a composition of breaking processes. Breaking is a form of decay,
and is due to tunnelling. Death is thus a quantum process. Classically, death does not
exist. Might this be the reason that so many believe in immortality or eternal youth?
We will also discover that the quantum of action is the origin for the importance of
the action observable in classical physics. In fact, the existence of a minimal action is the
reason for the least action principle of classical physics.
A minimum action also implies that matter cannot be continuous, but must be com-
posed of smallest entities. Indeed, the ow of a truly continuous material would contra-
dict the quantum principle. Can you give the precise argument? Of course, at this point
of our adventure, the non-continuity of matter is no news any more. In addition, the
quantum of action implies that even radiation cannot be continuous. As Albert Einstein
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Challenge 1180 ny Challenge 1181 n
Challenge 1182 ny
stated clearly for the rst time, light is made of quantum particles. More generally, the
quantum of action implies that in nature all ows and all waves are made of microscopic
particles. e term ‘microscopic’ or ‘quantum’ is essential, as such particles do not behave
like little stones. We have already encountered several di erences and we will encounter
others shortly. For these reasons, microscopic particles should bear a special name; but
all proposals, of which quanton is the most popular, have not caught on yet.
e quantum of action has several
strange consequences for microscopic
m
particles. Take two of them with the same
mass and the same composition. Ima-
gine that their paths cross and that at
the crossing they approach each other to
small distances, as shown in Figure .
A minimum action implies that in such
a situation, if the distance becomes small
m
enough, the two particles can switch role F I G U R E 286 Identical objects with crossing paths
without anybody being able to avoid or to
ever notice it. For example, in a gas it is impossible, due to the quantum of action, to fol-
low particles moving around and to say which particle is which. Can you con rm this
deduction and specify the conditions using the indeterminacy relations? In summary, in
nature it is impossible to distinguish identical particles. Can you guess what happens in the
case of light?
But matter deserves still more atten-
tion. Imagine two particles, even two dif-
M
ferent ones, approaching each other to small distances, as shown in Figure .
m1
We know that if the approach distance
gets small, things get fuzzy. Now, if some-
thing happens in that small domain in
m2
such a way that the resulting outgoing
products have the same total momentum
m
and energy as the incoming ones, the
m3
minimum action principle makes such F I G U R E 287 Transformation through reaction
processes possible. Indeed, ruling out
such processes would imply that arbitrary small actions could be observed, thus elim-
inating nature’s fuzziness, as you might want to check by yourself. In short, a minimum
action allows transformation of matter. One also says that the quantum of action allows
particle reactions. In fact, we will discover that all kinds of reactions in nature, including
chemical and nuclear ones, are only due to the existence of the quantum of action.
But there is more. Due to the indeterminacy relations, it is impossible to give a def-
inite value to both the momentum and the position of a particle. Obviously, this is also
impossible for all the components of a measurement set-up or an observer. is implies
that initial conditions – both for a system and for the measurement set-up – cannot be
exactly duplicated. A minimum action thus implies that whenever an experiment on a
microscopic system is performed twice, the outcome will be di erent. e result would
be the same only if both the system and the observer would be in exactly the same condi-
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F I G U R E 288 How do train windows manage to show two superimposed images?
F I G U R E 289 A particle and a screen with two nearby slits
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tion in both situations. is turns out to be impossible, both due to the second principle of thermodynamics and due to the quantum principle. erefore, microscopic systems behave randomly. Obviously, there will be some average outcome; nevertheless, microscopic observations are probabilistic. Albert Einstein found this conclusion of quantum theory the most di cult to swallow, as this randomness implies that the behaviour of quantum systems is strikingly di erent from that of classical systems. But the conclusion is unavoidable: nature behaves randomly.
A good example is given by trains. Einstein used trains to develop and explain relativity. But trains are also important for quantum physics. Everybody knows that one can use a train window to look either at the outside landscape or, by concentrating on the re ected image, to observe some interesting person inside the carriage. In other words, glass re ects some of the light particles and lets some others pass through. More precisely, glass re ects a random selection of light particles, yet with constant average. Partial re ection is thus similar to the tunnelling e ect. Indeed, the partial re ection of glass for photons is a result of the quantum of action. Again, the average situation can be described by classical physics, but the precise amount of partial re ection cannot be explained without quantum theory. Without the quantum of action, train trips would be much more boring.
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Page 210
e quantum of action implies a central result about the path of particles. If a particle travels from a point to another, there is no way to say which path it has taken in between. Indeed, in order to distinguish between the two possible, but only slightly di erent paths, actions smaller than ħ would have to be measured. In particular, if a particle is sent through a screen with two su ciently close slits, it is impossible to say through which slit the particle passed to the other side. e impossibility is fundamental. Matter is predicted to show interference.
We know already a moving phenomenon for which it is not possible to say with precision which path it takes when crossing two slits: waves. All waves follow the indeterminacy relations
∆ω∆t
and ∆k∆x .
(478)
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We saw above that quantum systems follow ∆E∆t ħ and ∆p∆x ħ
(479)
As a result, one is lead to ascribe a frequency and a wavelength to quantum systems:
E = ħω
and
p
=
ħk
=
ħ
π λ
(480)
e energy–frequency relation was deduced by Albert Einstein in ; it is found to be valid in particular for all examples of light emission. e more spectacular momentum– wavelength relation was rst predicted by Louis de Broglie* in and . ( is is thus another example of a discovery that was made about twenty years too late.) In short, the quantum of action implies that matter particles behave as waves.
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Page 800 Ref. 683
In computer science, a smallest change is called a ‘bit’. e existence of a smallest change in nature thus means that computer language or information science can be used to describe nature, and in particular quantum theory. However, computer language can describe only the so ware side; the hardware side of nature is also required. e hardware of nature enters the description whenever the actual value ħ of the quantum of action must be introduced.
Exploring the analogy between nature and information science, we will discover that the quantum of action implies that macroscopic physical systems cannot be copied or ‘cloned’, as quantum theorists like to say. Nature does not allow to copy objects. Copying machines do not exist. e quantum of action makes it impossible to gather and use all information in a way that allows to produce a perfect copy. As a result, we will deduce that the precise order in which measurements are performed does play a role in experiments. When the order is important, physicists speak of lack of ‘commutation’. In short physical observables do not commute.
We will also nd out that the quantum of action implies that systems are not always independent, but can be ‘entangled’. is term, introduced by Erwin Schrödinger, describes the most absurd consequences of quantum theory. Entanglement makes everything in nature connected to everything else. Entanglement produces e ects which look (but are not) faster than light. Entanglement produces a (fake) form of non-locality. Entanglement also implies that trustworthy communication does not exist.
Don’t all these deductions look wrong or at least crazy? In fact, if you or your lawyer made any of these statements in court, maybe even under oath, you would be likely to
* Louis de Broglie (b. 1892 Dieppe, d. 1987 Paris) French physicist and professor at the Sorbonne. e energy–frequency relation had earned Albert Einstein his Nobel prize already in 1921. De Broglie expanded it to the prediction of the wave nature of the electron (and of all other quantum particles); this was the essential part of his PhD. e prediction was con rmed experimentally a few years later, in 1927. For the prediction of the wave nature of matter, de Broglie received the Nobel Prize in physics in 1929. Being an aristocrat, de Broglie never did anything else in research a er that. For example, it was Schrödinger who then wrote down the wave equation, even though de Broglie could equally have done it.
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Challenge 1183 ny
end up in prison! However, all above statements are correct, as they are all con rmed by experiment. And the surprises are by far not nished. You might have noticed that, in the preceding examples, no situation related to electricity, to the nuclear interactions or to gravity was included. In these domains the surprises are even more astonishing; the observation of antimatter, of electric current ow without resistance, of the motion inside muscles, of vacuum energy, of nuclear reactions in stars and maybe soon of boiling empty space will fascinate you as much as they have fascinated and still fascinate thousands of researchers.
In particular, the consequences of the quantum of action on the early universe are simply mind-boggling. Just try to explore for yourself its consequences for the big bang. Together, all these topics will lead us a long way towards the top of Motion Mountain.
e topics are so strange, so incredible and at the same time so numerous that quantum physics can be rightly called the description of motion for crazy scientists. In a sense, this is the generalization of the previous de nition, when we called quantum physics the description of motion related to pleasure.
Sometimes it is heard that ‘nobody understands quantum theory.’ is is wrong. In fact it is worse than wrong: it is disinformation, a habit found only in dictatorships. It is used there to prevent people from making up their own mind and from enjoying life. Quantum theory is the set of consequences that follows from the existence of a minimal action. ese consequences can be understood and enjoyed by everybody. In order to do so, our rst task on our way towards the top of Motion Mountain will be the study of our classical standard of motion: the motion of light.
Nie und nirgends hat es Materie ohne Bewegung gegeben, oder kann es sie geben.
“ ” Friedrich Engels, Anti–Dühring.*
C
Even if we accept that experiments so far do not contradict the minimum action, we still have to check that the minimum action does not contradict reason. In particular, the minimum action must also appear in all imagined experiments. is is not evident.
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Angular momentum has the same unit as action. A smallest action implies that there is a smallest angular momentum in nature. How can this be, given that some particles have Challenge 1184 n spin zero, i.e., have no angular momentum?
**
Could we have started the whole discussion of quantum theory by stating that there is a Challenge 1185 n minimum angular momentum instead of a minimum action?
**
Niels Bohr, besides propagating the idea of minimum action, was also a fan of the complementarity principle. is is the idea that certain pairs of observables of a system – such as
Ref. 684 * ‘Never and nowhere has matter existed, or can it exist without motion.’ Friedrich Engels (1820–1895) was one of the theoreticians of Marxism, o en also called Communism.
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position and momentum – have linked precision: if one observable of the pair is known to high precision, the other is necessarily known with low precision. Can you deduce the Challenge 1186 ny principle from the minimum action?
**
When electromagnetic elds play a role, the value of the action (usually) depends on the choice of the vector potential, and thus on the gauge choice. We found out in the section of electrodynamics that a suitable gauge choice can change the action value by adding or subtracting any desired amount. Nevertheless, there is a smallest action in nature. is is possible, because in quantum theory, physical gauge changes cannot add or subtract any amount, but only multiples of twice the minimum value. e addition property thus does not help to go below the minimum action.
**
(Adult) plants stop to grow in the dark. Plant needs light to grow. Without light, the reactions necessary for growth stop. Can you deduce that this is a quantum e ect, not Challenge 1187 ny explainable by classical physics?
In short, besides experiment also all imagined system con rm that nature shows a minimum action.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Alle Wesen leben vom Lichte, jedes glückliche Geschöpfe.
“ ” Friedrich Schiller, Wilhelm Tell.*
Ref. 686 Ref. 685
If all the colours of materials are quantum e ects, as just argued, it becomes especially
interesting to study the properties of light in the light of the quantum of action. If in
nature there is a minimum change, there should also be a minimum illumination. is
had been already predicted by Epicurus ( –
) in ancient Greece. He stated that
light is a stream of little particles, so that the smallest illumination would be that of a
single light particle.
In the s, Brumberg and Vavilov found a beautiful way to check this prediction
using the naked eye, despite our inability to detect single photons. In fact, the experi-
ment is so simple that it could have been performed at least a century before that; but
nobody had a su ciently daring imagination to try it. e two researchers constructed
a small shutter that could be opened for time intervals of . s. From the other side, in
a completely dark room, they illuminated the opening with extremely weak green light
* ‘All beings live of light, every happy creature.’ Friedrich Schiller (b. 1759 Marbach, d. 1805 Weimar), important German poet, playwright and historian.
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– aW at nm. At that intensity, whenever the shutter opens, on average about
photons can pass, which is just the sensitivity threshold of the eye. To perform the exper-
iment, they simply looked into the shutter repeatedly. e result is surprising but simple.
Sometimes they observed light, sometimes they did not. e result is completely random.
Brumberg and Vavilov also gave the simple explanation: due to uctuations, half of the
time the number of photons is above eye threshold, half of the time below. e uctu-
ations are random, and thus the detection is as well. is would not happen if light would
be a continuous stream; in that case, the eye would detect light at every opening of the
shutter. (At higher light intensities, the percentage of non-observations quickly decreases,
in accordance with the explanation.) Nobody knows what would have happened to the
description of light if this simple experiment had been performed years earlier.
e experiment becomes clearer when we
use devices to help us. A simple way is to start with a screen behind a prism illuminated with
glass
photographic film
white light. e light is split into colours. When
the screen is put further and further away, the white
red
illumination intensity cannot become in nitely
small, as that would contradict the quantum of
violet
action. To check this prediction, we only need
some black and white photographic lm. Everybody knows that lm is blackened by daylight
F I G U R E 290 Illumination by pure-colour light
of any colour; at medium light intensities it be-
comes dark grey and at lower intensities light grey. Looking at an extremely light grey lm
under the microscope, we discover that even under uniform illumination the grey shade
actually is a more or less dense collections of black spots. Exposed lm does not show a
homogeneous colour; on the contrary, it reacts as if light is made of small particles.
is is a general observation: whenever sensitive light detectors are constructed with
the aim to ‘see’ as accurately as possible, thus in environments as dark as possible, one
always nds that light manifests itself as a stream of light quanta. Nowadays they are
usually called photons, a term that appeared in . A low or high light intensity is simply
a small or high number of photons.
Another weak source of light
are single atoms. Atoms are tiny
spheres; when they radiate light
or X-rays, the radiation should
be emitted as spherical waves.
But in all experiments it is found
Figure to be inserted
that each atom emits only one
‘blob’ of light. One never nds F I G U R E 291 Observation of photons
that the light emitted by atoms
forms a spherical wave, as is suggested by everyday physics. If a radiation emitting atom
is surrounded by detectors, there is always only a single detector that is triggered.
Experiments in dim light thus show that the continuum description of light is not
correct. More precise measurements con rm the role of the quantum of action: every
photon leads to the same amount of change in the lm. is amount of change is the
minimal amount of change that light can produce. Indeed, if a minimum action would not
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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exist, light could be packaged into arbitrary small amounts. In other words, the classical description of light by a continuous state function A(t, x) or F(t, x), whose evolution is described by a principle of least action, is wrong, as it does not describe the observed particle e ects. Another, modi ed description is required. e modi cation has to be important only at low light intensities, since at high intensities the classical Lagrangian accurately describes all experimental observations.*
At which intensities does light cease to behave as a continuous wave? Our eye can help us to nd a limit. Human eyesight does not allow to consciously distinguish single photons, even though experiments show that the hardware of the eye is able to do this.
e faintest stars which can be seen at night produce a light intensity of about . nW m . Since the pupil of the eye is quite small, and as we are not able to see individual photons, photons must have energies smaller than − J.
In today’s laboratory experiments, recording and counting individual photons is standard practice. Photon counters are part of many spectroscopy set-ups, such as those used to measure smallest concentration of materials. For example, they help to detect drugs in human hair. All these experiments thus prove directly that light is a stream of particles, as Epicurus had advanced in ancient Greece.
is and many other experiments show that a beam of light of frequency f or angular frequency ω, which determines its colour, is accurately described as a stream of photons, each with the same energy E given by
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E = ħ πf = ħω .
(481)
Page 612
is shows that for light, the smallest measurable action is given by the quantum of action ħ. is is twice the smallest action observable in nature; the reasons and implications will unfold during the rest of our walk. In summary, colour is a property of photons. A coloured light beam is a hailstorm of the corresponding photons.
e value of Planck’s constant can be determined from measurements of black bodies or other light sources. e result
ħ = . ë − Js
(482)
Ref. 687 Challenge 1188 ny
Challenge 1189 ny
is so small that we understand why photons go unnoticed by humans. Indeed, in normal light conditions the photon numbers are so high that the continuum approximation for the electromagnetic eld is of high accuracy. In the dark, the insensitivity of the signal processing of the human eye, in particular the slowness of the light receptors, makes photon counting impossible. e eye is not far from maximum possible sensitivity though. From the numbers given above about dim stars we can estimate that humans are able to see consciously ashes of about half a dozen detected photons.
In the following, we will systematically deduce the remaining properties of photons, using the data collected in classical physics, while taking the quantum of action rmly into account. For example, photons have no (rest) mass and no electric charge. Can you con rm this? In fact, experiments can only give an upper limit for both quantities. e
* is transition from the classical case to the quantum case used to be called quantization. e concept and the ideas behind it are only of historical interest today.
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Ref. 688 present experimental upper limit for the (rest) mass of a photon is − kg. We know that light can hit objects. Since the energy and the speed of photons is known,
Challenge 1190 ny we guess that the photon momentum obeys
p
=
E c
=
ħ
π λ
or
p= ħk .
(483)
Ref. 689
Challenge 1191 ny Page 573
Challenge 1192 ny
In other words, if light is made of particles, we should be able to play billiard with them. is is indeed possible, as Arthur Compton showed in a famous experiment in . He
directed X-rays, which are high energy photons, onto graphite, a material in which electrons move almost freely. He found that whenever the electrons in the material get hit by the X-ray photons, the de ected X-rays change colour. As expected, the strength of the hit depends on the de ection angle of the photon. From the colour change and the re ection angle, Compton con rmed that the photon momentum indeed obeys the above expression. All other experiments agree that photons have momentum. For example, when an atom emits light, the atom feels a recoil; the momentum again turns out to be given by the same value ( ). In short, every photon has momentum.
e value of photon momentum respects the indeterminacy principle. In the same way that it is impossible to measure exactly both the wavelength of a wave and the position of its crest, it becomes impossible to measure both the momentum and the position of a photon. Can you con rm this? In other words, the value of the photon momentum is a direct consequence of the quantum of action.
From our study of classical physics we know that light has a property beyond its colour: light can be polarized. at is only a complicated way to say that light can turn objects it shines on. In other words, light has an angular momentum oriented along the axis of propagation. What about photons? Measurements consistently nd that each light particle carries an angular momentum given by L = ħ. It is called its helicity; to make it more clear that the quantity is similar to that found for massive particles, one also speaks of the spin of photons. Photons somehow ‘turn’ – in a sense either parallel or antiparallel to the direction of motion. Again, the magnitude of the photon helicity or spin is not a surprise; it con rms the classical relation L = E ω between energy and angular momentum that we found in the section on classical electrodynamics. Note that in contrast to intuition, the angular momentum of a single photon is xed, and thus independent of its energy. Even the photons with the highest energy have L = ħ. Of course, the value of the helicity also respects the limit given by the quantum of action. e helicity value ħ – twice the minimum ħ – has important consequences which will become clear shortly.
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La lumière est un mouvement luminaire de corps lumineux.*
“ ” Blaise Pascal
* Light is the luminary movement of luminous bodies.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
In the seventeenth century, Blaise Pascal* used this sentence to make fun about certain physicists. He ridiculed (rightly so) the blatant use of a circular de nition. Of course, he was right; in his time, the de nition was indeed circular, as no meaning could be given to any of the terms. But as usual, whenever an observation is studied with care by physicists, they give philosophers a beating. All those originally unde ned terms now have a de nite meaning. Light is indeed a type of motion; this motion can rightly be called luminary because in opposition to motion of material bodies, it has the unique property v = c; the luminous bodies, today called photons, are characterized and di erentiated from all other particles by their dispersion relation E = c p, their energy E = ħω, their spin L = ħ, the vanishing value of all other quantum numbers, and by being the quanta of the electromagnetic eld.
In short, light is a stream of photons. It is indeed a luminary movement of luminous bodies. e existence of photons is the rst example of a general property of the world on small scales: all waves and all ows in nature are made of quantum particles. Large numbers of (coherent) quantum particles – or quantons – do indeed behave as waves. We will see shortly that this is the case even for matter. e fundamental constituents of all waves are quantons. ere is no exception. e everyday, continuum description of light is thus similar in many aspects to the description of water as a continuous uid; photons are the atoms of light, and continuity is an approximation valid for large particle numbers. Small numbers of quantons o en behave like classical particles. In the old days of physics, books used to discuss at length a so-called wave–particle duality. Let us be clear from the start: quantons, or quantum particles, are neither classical waves nor classical particles. In the microscopic world, quantons are the fundamental objects.
However, a lot is still unclear. Where inside matter do these monochromatic photons come from? Even more interestingly, if light is made of quantons, all electromagnetic
elds, even static ones, must be made of photons as well. However, in static elds nothing is owing. How is this apparent contradiction solved? And what e ects does the particle aspect have on these static elds? What is the di erence between quantons and classical particles? e properties of photons thus require some more careful study. Let us go on.
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Challenge 1193 ny
First of all, we might ask: what are these photons made of? All experiments so far, performed down to the present limit of about − m, give the same answer: ‘we can’t nd anything’. at is consistent both with a vanishing mass and a vanishing size of photons; indeed, one intuitively expects any body with a nite size to have a nite mass. us, even though experiments give only an upper limit, it is consistent to claim that a photon has no size.
A particle with no size cannot have any constituents. A photon thus cannot be divided into smaller entities. For this reason people refer to photons as elementary particles. We will give some strong additional arguments for this deduction soon. (Can you nd one?)
is is a strange result. How can a photon have vanishing size, have no constituents, and still be something? is is a hard questions; the answer will appear only later on. At the
* Blaise Pascal (b. 1623 Clermont, d. 1662 Paris) important French mathematician and physicist up to the age of twenty-six; he then turned theologian and philosopher.
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•.
moment we simply have to accept the situation as it is. We therefore turn to an easier issue.
A
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“Also gibt es sie doch.
” Max Planck*
Above we saw that in order to count photons, the simplest way is to distribute them across
a large screen and then to absorb them. But this method is not fully satisfying, as it des-
troys the photons. How can one count photons without destroying them?
One way is to re ect photons on a mirror and to measure the recoil of the mirror.
is seems almost unbelievable, but nowadays the e ect is becoming measurable even
for small number of photons. For example, this e ect has to be taken into account in the
mirrors used in gravitational wave detectors, where the position of laser mirrors has to
be measured to high precision.
Another way of counting photons without destroying them uses special high quality
laser cavities. Using smartly placed atoms inside such a cavity, it is possible to count the
number of photons by the e ect they have on these atoms.
In other words, light intensity can indeed be measured without absorption. However,
the next di culty appears straight away. Measurements show that even the best light
beams, from the most sophisticated lasers, uctuate in intensity. ere are no steady
beams. is does not come as a surprise: if a light beam would not uctuate, observing it
twice in a row would yield a vanishing value for the action. However, there is a minimum
action in nature, namely ħ . us any beam and any ow in nature uctuates. But there
is more.
A light beam is described by its intensity and its phase. e change – or action – oc-
curring while a beam moves is given by the variation in the product of intensity and
phase. Experiments con rm the obvious deduction: intensity and phase of beams behave
like momentum and position of particles: they obey an indeterminacy relation. You can
deduce it yourself, in the same way we deduced Heisenberg’s relations. Using as charac-
teristic intensity I = E ω the energy per circular frequency and calling the phase φ, we
get**
∆I ∆φ ħ .
(485)
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* ‘ us they do exist a er all.’ Max Planck, in later years, said this a er standing silently, for a long time, in front of an apparatus which counted single photons by producing a click for each photon it detected. It is not a secret that for a large part of his life, Planck was not a friend of the photon concept, even though his own results were the starting point for its introduction. ** A large photon number is assumed in the expression; this is obvious, as ∆φ cannot grow beyond all bounds. e exact relations are
∆I ∆ cos φ ∆I ∆ sin φ
sin φ cos φ
(484)
where x denotes the expectation value of the observable x.
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Thermal light Photon sequence: bunching
Laser light little or no bunching
time Intensity I(t)
time Probability P(I)
Bose-Einstein
Intensity correlations
2 1
coherence time
Amplitudephase diagram
intensity
2 1 time
Poisson
Nonclassical light anti-bunching
SubPoisson
2 1
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F I G U R E 292 Various types of light
For light emitted from usual lamps, the product of the le side is much larger than the quantum of action. On the other hand, laser beams can (almost) reach the limit. Laser light beams in which the two indeterminacies strongly di er from each other are called non-classical light or squeezed light; they are used in many modern research applications. Such light beams have to be treated carefully, as the smallest disturbances transform them back into usual laser beams, where the two indeterminacies have the same value. An extreme example of non-classical light are those beams with a given, xed photon number, thus with an extremely large phase indeterminacy.
e observation of non-classical light highlights a strange consequence valid even for classical light: the number of photons in a light beam is not a de ned quantity. In general it is undetermined, and it uctuates. e number of photons at the beginning of a beam is not necessarily the same as at the end of the beam. Photons, in contrast to stones, cannot be counted precisely – as long as they move and are not absorbed. In ight it is only possible to determine an approximate number, within the limit set by indeterminacy.
e most extreme example are those light beams with an (almost) xed phase. In such
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source
beam splitter
mirrors
two identical photons
•.
beam splitter
detectors
possible light paths
F I G U R E 293 An interferometer
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Challenge 1194 ny
beams, the photon number uctuates from zero to in nity. In other words, in order to produce a coherent laser beam that approximates a pure sine wave as perfectly as possible one must build a source in which the photon number is as undetermined as possible.
e other extreme is a beam with a xed number of photons; in such a beam, the phase uctuates erratically. Most daily life situations, such as the light from incandescent lamps, lie somewhere in the middle: both phase and intensity indeterminacies are of similar magnitude.
As an aside, it turns out that in deep, dark intergalactic space, far from every star, there still are about photons per cubic centimetre. But also this number, like the number of photons in a light beam, has its measurement indeterminacy. Can you estimate it?
In summary, unlike little stones, photons are not countable. And this it not the last di erence between photons and stones.
T
Challenge 1195 ny
Where is a photon when it moves in a beam of light? Quantum theory gives a simple answer: nowhere in particular. e proof is given most spectacularly by experiments with interferometers; they show that even a beam made of a single photon can be split, be led along two di erent paths and then be recombined. e resulting interference shows that the single photon cannot be said to have taken either of the two paths. If one of the two paths is blocked, the pattern on the screen changes. In other words, the photons somehow must have taken both paths at the same time. Photons cannot be localized; they have no position.*
is impossibility of localization can be speci ed. It is impossible to localize photons in the direction transverse to the motion. It is less di cult to localize photons along the motion direction. In the latter case, the quantum of action implies that the longitudinal position is uncertain within a value given by the wavelength of the corresponding colour. Can you con rm this?
In particular, this means that photons cannot be simply visualized as short wave trains. Photons are truly unlocalizable entities speci c to the quantum world.
Now, if photons can almost be localized along their motion, we can ask the following question: How are photons lined up in a light beam? Of course, we just saw that it does not
* is conclusion cannot be avoided by saying that photons are split at the beam splitter: if one puts a detector into each arm, one nds that they never detect a photon at the same time. Photons cannot be divided.
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D1
coincidence counter
D2
F I G U R E 294 How to measure photon statistics: with an electronic coincidence counter and the variation by varying the geometrical position of a detector
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Ref. 690 Ref. 691
make sense to speak of their precise position. But are photons in a perfect beam arriving
in almost regular intervals or not?
To the shame of physicists, the study of this question was started by two astronomers,
Robert Hanbury Brown and Richard Twiss, in . ey used a simple method to meas-
ure the probability that a second photon in a light beam arrives at a given time a er a
rst one. ey simply split the beam, put one detector in the rst branch and varied the
position of a second detector in the other branch.
Hanbury Brown and Twiss found that for coherent light the clicks in the two counters,
and thus the photons, are correlated. is result is in complete contrast with classical
electrodynamics. Photons are indeed necessary to describe light. In more detail, their
experiment showed that whenever a photon hits, the probability that a second one hits
just a erwards is highest. Photons in beams are thus bunched.
Every light beam shows an upper limit time to bunching, called the coherence time. For
times larger than the coherence time, the probability for bunching is low and independ-
ent of the time interval, as shown in Figure . e coherence time characterizes every
light beam, or better, every light source. In fact, it is more intuitive to use the concept of
coherence length, as it gives a clearer image for a light beam. For thermal lamps, the co-
herence length is only a few micrometers, a small multiple of the wavelength. e largest
coherence lengths, up to over
km, are realized in research lasers. Interestingly, co-
herent light is even found in nature; several special stars have been found to emit coherent
light.
Even though the intensity of a good laser light is almost constant, laser beam photons
still do not arrive in regular intervals. Even the best laser light shows bunching, though
with di erent statistics and to a lower degree than lamp light. Light for which photons ar-
rive regularly, thus showing so-called (photon) anti-bunching, is obviously non-classical
in the sense de ned above; such light can be produced only by special experimental ar-
rangements. e most extreme example is pursued at present by several research groups;
they aim to construct light sources which emit one photon at a time, at regular time in-
tervals, as reliably as possible.
In summary, experiments force us to conclude that light is made of photons, but that
photons cannot be localized in light beams. It makes no sense to talk about the position
of a photon in general; the idea makes only sense in some special situations, and then
only approximately and as a statistical average.
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lamp
Ekin kinetic energy of emitted electrons
electrons
•.
threshold
metal plate in vacuum
frequency of lamp light ω
F I G U R E 295 The kinetic energy of electrons emitted in the photoelectric effect
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Ref. 693
Ref. 694 Ref. 695
In light of the results uncovered so far, the answer to the title question is obvious. But the issue is tricky. In school books, the photoelectric e ect is usually cited as the rst and most obvious experimental proof of the existence of photons. In , Heinrich Hertz observed that for certain metals, such as lithium or caesium, incident ultraviolet light leads to charging of the metal. Later it was shown that the light leads to the emission of electrons, and that that the energy of the ejected electrons is not dependent on the intensity of the light, but only dependent on the di erence between ħ times its frequency and a material dependent threshold energy. In , Albert Einstein predicted this result from the assumption that light is made of photons of energy E = ħω. He imagined that this energy is used partly to extract the electron over the threshold, and partly to give it kinetic energy. More photons only lead to more electrons, not to faster ones.
Einstein received the Nobel price for this explanation. But Einstein was a genius; that means he deduced the correct result by a somewhat incorrect reasoning. e (small) mistake was the prejudice that a classical, continuous light beam would produce a di erent e ect. It does not take a lot to imagine that a classical, continuous electromagnetic eld interacting with discrete matter, made of discrete atoms containing discrete electrons, leads to exactly the same result, if the motion of electrons is described by quantum theory. Several researchers con rmed this point already early in the twentieth century. e photoelectric e ect by itself does not require photons for its explanation.
Many researchers were unconvinced by the photoelectric e ect. Historically, the most important argument for the necessity of light quanta was given by Henri Poincaré. In and , at age and only a few months before his death, he published two in uential papers proving that the radiation law of black bodies, the one in which the quantum of action had been discovered by Max Planck, requires the introduction of photons. He also showed that the amount of radiation emitted by a hot body is nite only due to the quantum nature of the processes leading to light emission. A description of the processes by classical electrodynamics would lead to in nite amounts of radiated energy. ese two in uential papers convinced most of the sceptic physics researchers at the time that it was worthwhile to study quantum phenomena in more detail. Poincaré did not know about the action limit S ħ ; yet his argument is based on the observation that light of a
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Ref. 696 Ref. 697 Ref. 698
Challenge 1196 ny
given frequency always has a minimum intensity, namely one photon. Splitting such a one photon beam into two beams, e.g. using a half-silvered mirror, does produce two beams. However, there is no way to nd more than one photon in those two beams together.
Another interesting experiment requiring photons is the observation of ‘molecules of photons’. In , Jacobson et al. predicted that the de Broglie wavelength of a packet of photons could be observed. Following quantum theory it is given by the wavelength of a single photon divided by the number of photons in the packet. e team argued that the packet wavelength could be observable if one would be able to split and recombine such packets without destroying the cohesion within the packet. In , this e ect was indeed observed by de Pádua and his brazilian research group. ey used a careful set-up with a nonlinear crystal to create what they call a biphoton, and observed its interference properties, nding a reduction of the e ective wavelength by the predicted factor of two. In the meantime, packages with three and even four entangled photons have been created and observed.
Still another argument for the necessity of photons is the mentioned recoil felt by atoms emitting light. e recoil measured in these cases is best explained by the emission of a photon in a particular direction. In contrast, classical electrodynamics predicts the emission of a spherical wave, with no preferred direction.
Obviously, the observation of non-classical light, also called squeezed light, also argues for the existence of photons, as squeezed light proves that photons indeed are an intrinsic aspect of light, necessary even when no interactions with matter play a role. e same is true for the Hanbury Brown–Twiss e ect.
Finally, the spontaneous decay of excited atomic states also requires the existence of photons. A continuum description of light does not explain the observation.
In summary, the concept of photon is indeed necessary for a precise description of light, but the details are o en subtle, as the properties of photons are unusual and require a change in thinking habits. To avoid these issues, all school books stop discussing photons a er the photoelectric e ect. at is a pity; things get interesting only a er that. To savour the fascination, ponder the following issue. Obviously, all electromagnetic elds are made of photons. Photons can be counted for gamma rays, X-rays, ultraviolet light, visible light and infrared light. However, for lower frequencies, photons have not been detected yet. Can you imagine what would be necessary to count the photons emitted from a radio station?
e issue directly leads to the most important question of all:
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H
?
“Fünfzig Jahre intensiven Nachdenkens haben mich der Antwort auf die Frage ‘Was sind Lichtquanten?’ nicht näher gebracht. Natürlich
bildet sich heute jeder Wicht ein, er wisse die
Antwort. Doch da täuscht er sich.
Albert Einstein,
”*
* ‘Fi y years of intense re ection have not brought me nearer to the answer of the question ‘What are light quanta?’ Of course nowadays every little mind thinks he knows the answer. But he is wrong.’
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pocket lamps F I G U R E 296 Light crossing light
lasers or other coherent sources
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
If a light wave is made of particles, one must be able to explain each and every wave properties with the help of photons. e experiments mentioned above already hinted that this is possible only because photons are quantum particles. Let us take a more detailed look at this connection.
Light can cross other light undisturbed. is observation is not hard to explain with photons; since photons do not interact with each other, and since they are point-like, they ‘never’ hit each other. In fact, there indeed is an extremely small probability for their interaction, as will be found below, but this e ect is not observable in everyday life.
But the problems are not nished yet. If two light beams of identical frequencies and xed phase relation cross, we observe alternating bright and dark regions, so-called interference fringes. How do these interference fringes appear? Obviously, photons are not detected in the dark regions. How can this be? ere is only one possible way to answer: the brightness gives the probability for a photon to arrive at that place. e fringes imply that photons behave like little arrows. Some additional thinking leads to the following description:
( ) the probability of a photon arriving somewhere is given by the square of an arrow; ( ) the nal arrow is the sum of all arrows getting there, taking all possible paths; ( ) the arrow’s direction stays xed in space when photons move; ( ) the length of an arrow shrinks with the square of the travelled distance; ( ) photons emitted by one-coloured sources are emitted with arrows of constant length pointing in direction ω t; in other words, such monochromatic sources spit out photons with a rotating mouth. ( ) photons emitted by thermal sources, such as pocket lamps, are emitted with arrows of constant length pointing in random directions. With this model* we can explain the stripes seen in laser experiments, such as those of Figure and Figure . You can check that in some regions, the two arrows travelling through the two slits add up to zero for all times. No photons are detected there. In other regions, the arrows always add up to the maximal value. ese regions are always bright. In between regions give in between shades. Obviously, for the case of pocket lamps the
* e model gives a correct description of light with the exception that it neglects polarization.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 1197 ny Ref. 699
brightness also behaves as expected: the averages then simply add up, as in the common
region in the le case of Figure .
You might want to calculate the distance of
the lines when the source distance, the colour
and the distance to the screen is given.
two lasers or
screen
Obviously, the photon model implies that in- point sources
terference patterns are built up as the sum of a
large number of one-photon hits. Using low in- S1
tensity beams, we should therefore be able to see
how these little spots slowly build up an interfer-
ence pattern by accumulating at the bright spots
and never hitting the dark regions. at is indeed the case. All experiments have con rmed this de-
S2
scription.
It is important to stress that interference of the arrow model:
two light beams is not the result of two di erent
photons cancelling out or adding each other up. t1
e cancelling would contradict energy and mo-
mentum conservation. Interference is an e ect t2 valid for each photon separately, because each
photon is spread out over the whole set-up; each photon takes all possible paths and interferes. As
t3
Paul Dirac said, each photon interferes only with F I G U RE 297 Interference and the
itself. Interference only works because photons description of light with arrows (at one
are quantons, and not at all classical particles.
particular instant of time)
Dirac’s widely cited statement leads to a fam-
ous paradox: if a photon can interfere only with itself, how can two laser beams from two
di erent lasers show interference? e answer of quantum physics is simple but strange:
in the region where the beams interfere, there is no way to say from which source a photon
is arriving. Photons are quantons; the photons in the crossing region cannot be said to
come from a speci c source. Photons in the interference region are quantons on their
own right, which indeed interfere only with themselves. In that region, one cannot hon-
estly say that light is a ow of photons. Despite regular claims of the contrary, Dirac’s
statement is correct. at is the strange result of the quantum of action.
Waves also show di raction. To understand this phenomenon with photons, let us
start with a simple mirror and study re ection rst. Photons (like any quantum particle)
move from source to detector in all ways possible. As the discoverer of this explanation,
Richard Feynman,* likes to stress, the term ‘all’ has to be taken literally. is was not a
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* Richard (‘Dick’) Phillips Feynman (b. 1918 New York City, d. 1988), US American physicist. One of the founders of quantum electrodynamics, he discovered the ‘sum-over-histories’ reformulation of quantum theory, made important contributions to the theory of the weak interaction and of quantum gravity, and co-authored a famous physics textbook, the Feynman Lectures on Physics. He is one of those theoretical physicists who made career mainly by performing complex calculations, a fact he tried to counter at the end of his life. ough he tried to surpass the genius of Wolfgang Pauli throughout his whole life, he failed in this endeavour. He was famously arrogant, disrespectful of authorities, as well as deeply dedicated to physics and to enlarging knowledge in his domain. He also was a well known collector of surprising physical explanations
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source
screen
image
•.
mirror
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arrow sum
F I G U R E 298 Light reflected by a mirror and the corresponding arrows (at one particular instant of time)
big deal in the explanation of interference. But in order to understand a mirror we have to include all possibilities, as crazy as they seem, as shown in Figure .
As said above a source emits rotating arrows. To determine the probability that light arrives at a certain image location, we have to add up all the arrows arriving at the same time at that location. For each path, the arrow orientation at the image is shown – for convenience only – below the corresponding segment of the mirror. Depending on the length of the path, the arrow arrives with a di erent angle and a di erent length. One notes that the sum of all arrows does not vanish: light does indeed arrive at the image.
e sum also shows that the largest part of the contribution is from those paths near the middle one. Moreover, if we were to perform the same calculation for another image location, (almost) no light would get there. In other words, the rule that re ection occurs with incoming angle equal to the outgoing angle is an approximation; it follows from the arrow model of light.
In fact, a detailed calculation, with more arrows, shows that the approximation is quite precise; the errors are much smaller than the wavelength of the light used.
e proof that light does indeed take all these strange paths is given by a more specialized mirror. As show in Figure , one can repeat the experiment with a mirror which re ects only along certain stripes. In this case, the stripes were carefully chosen such that the corresponding path lengths lead to arrows with a bias to one direction, namely to the le . e arrow addition now shows that such a specialized mirror, usually called a grating, allows light to be re ected in unusual directions. And indeed, this behaviour is standard for waves: it is called di raction. In short, the arrow model for photons does allow to describe this wave property of light, provided that photons follow the mentioned
and an author of several popular texts on his work and his life. He shared the 1965 Nobel Prize in physics for his work on quantum electrodynamics.
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Challenge 1198 ny Challenge 1199 ny Challenge 1200 ny Challenge 1201 ny
Ref. 700
crazy probability scheme. Do not get upset; as said before, quantum theory is the theory
of crazy people.
You may want to check that the
arrow model, with the approxima-
source
point
tions it generates by summing over
all possible paths, automatically en-
sures that the quantum of action is
indeed the smallest action that can be observed.
All waves have a signal velocity. As a consequence, waves show re-
usual mirror
arrow sum at point vanishes
fraction when they move from one
medium into another with di erent
signal velocity. Interestingly, the naive particle picture of photons as little stones would imply that light
source
screen
image
is faster in materials with high in-
dices of refraction, the so-called
dense materials. Just try it. How-
ever, experiments show that light in dense materials moves slowly. e wave picture has no di culties explaining this observation. (Can you
striped mirror
arrow sum at image
con rm it?) Historically, this was
one of the arguments against the
F I G U R E 299 The light reflected by a badly placed mirror and by a grating
particle theory of light. However,
the arrow model of light presented above is able to explain refraction properly. It is not
di cult doing so; try it.
Waves also re ect partially from materials such as glass. is
is one of the toughest properties of waves to be explained with
photons. e issue is important, as it is one of the few e ects that
is not explained by a classical wave theory of light. However, it is
air
explained by the arrow model, as we will nd out shortly. Partial
water
re ection con rms the description of the rules ( ) and ( ) of the
arrow model. Partial re ection shows that photons indeed behave
randomly: some are re ected and other are not, without any selec-
tion criterion. e distinction is purely statistical. More about this
issue shortly. In waves, the elds oscillate in time and space. One way to show
how waves can be made of particles is to show once for all how to build up a sine wave using a large number of photons. A sine wave
F I G U R E 300 If light were made of little stones, they would move faster inside
is a coherent state of light. e way to build them up was explained water
by Glauber. In fact, to build a pure sine wave, one needs a superpos-
ition of a beam with one photon, a beam with two photons, a beam with three photons,
continuing up to a beam with an in nite number of them. Together, they give a perfect
sine wave. As expected, its photon number uctuates to the highest degree possible.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Ref. 701
If we repeat the calculation for non-ideal beams, we nd that the indeterminacy relation for energy and time is respected; every emitted wave will possess a certain spectral width. Purely monochromatic light does not exist. Similarly, no system which emits a wave at random can produce a monochromatic wave. All experiments con rm these results.
Waves can be polarized. So far, we disregarded this property. In the photon picture, polarization is the result of carefully superposing beams of photons spinning clockwise and anticlockwise. Indeed, we know that linear polarization can be seen as a result of superposing circularly polarized light of both signs, using the proper phase. What seemed a curiosity in classical optics turns out to be the fundamental explanation of quantum theory.
Photons are indistinguishable. When two photons of the same colour cross, there is no way to say, a er the crossing, which of the two is which. e quantum of action makes this impossible. e indistinguishability of photons has an interesting consequence. It is impossible to say which emitted photon corresponds to which arriving photon. In other words, there is no way to follow the path of a photon in the way we are used to follow the path of a billiard ball. Particles which behave in this way are called bosons. We will discover more details about the indistinguishability of photons in the next chapter.
In summary, we nd that light waves can indeed be built of particles. However, this is only possible under the condition that photons are not precisely countable, that they are not localizable, that they have no size, no charge and no mass, that they carry an (approximate) phase, that they carry spin, that they are indistinguishable bosons, that they can take any path whatsoever, that one cannot pinpoint their origin and that their probability to arrive somewhere is determined by the square of the sum of amplitudes for all possible paths. In other words, light can be made of particles only under the condition that these particles have extremely special, quantum properties. Only these quantum properties allow them to behave like waves, in the case that they are present in large numbers.
Quantons are thus quite di erent from usual particles. In fact, one can argue that the only (classical) particle aspects of photons are their quantized energy, momentum and spin. In all other aspects photons are not like little stones. It is more honest to say that photons are calculating devices to precisely describe observations about light. O en these calculating devices are called quantons. In summary, all waves are streams of quantons. In fact, all waves are correlated streams of quantons. at is true both for light, for any other form of radiation, as well as for matter, in all its forms.
e strange properties of quantons are the reason that earlier attempts to describe light as a stream of (classical) particles, such as the one by Newton, failed miserably, under the rightly deserved ridicule of all other scientists. Indeed, Newton upheld his idea against all experimental evidence, especially that on light’s wave properties, something a physicist should never do. Only when people accepted that light is a wave and then discovered and understood that quantum particles are di erent from classical particles was the approach successful.
e indeterminacy relations show that even a single quanton can be seen as a wave; however, whenever it interacts with the rest of the world, it behaves as a particle. In fact it is essential that all waves are made of quantons; if not, interactions would not be local, and objects, in contrast to experience, could not be localized at all. To separate between wave and particle descriptions, we can use the following criterion. Whenever matter and
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
–
light interact, it is more appropriate to describe electromagnetic radiation as a wave if the
wavelength λ obeys
λ
ħc kT
,
(486)
where k = . ë − J K is Boltzmann’s constant. If the wavelength is much smaller than the right hand side, the particle description is most appropriate. If the two sides are of the same order of magnitude, both e ects play a role.
C
?–V
Challenge 1202 ny Challenge 1203 ny
Light can move faster than c in vacuum, as well as slower than c. e quantum principle provides the details. As long as the quantum principle is obeyed, the speed of a short light
ash can di er a bit from the o cial value, though only a tiny bit. Can you estimate the allowed di erence in arrival time for a light ash from the dawn of times?
e little arrow explanation gives the same result. If one takes into account the crazy possibility that photons can move with any speed, one nds that all speeds very di erent from c cancel out. e only variation that remains, translated in distances, is the indeterminacy of about one wavelength in the longitudinal direction which we mentioned already above.
However, the most absurd results of the quantum of action appear when one studies static electric elds, such as the eld around a charged metal sphere. Obviously, such a eld must also be made of photons. How do they move? It turns out that static electric elds are built of virtual photons. In the case of static electric elds, virtual photons are longitudinally polarized, do not carry energy away, and cannot be observed as free particles. Virtual photons are photons who do not appear as free particles; they only have extremely short-lived appearances before they disappear again. In other words, photons, like any other virtual particle, are ‘shadows’ of particles that obey
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∆x∆p ħ .
(487)
Page 759
Virtual particles do not obey the indeterminacy relation, but its opposite; the opposite relation expresses their short-lived appearance. Despite their intrinsically short life, and the impossibility to detect them directly, they have important e ects. We will explore virtual particles in detail shortly.
In fact, the vector potential A allows four polarizations, corresponding to the four coordinates (t, x, y, z). It turns out that for the photons one usually talks about, the free or real photons, the polarizations in t and z direction cancel out, so that one observes only the x and y polarizations in actual experiments.
For bound or virtual photons, the situation is di erent.
– CS – more to be written – CS –
In short, static electric and magnetic elds are continuous ows of virtual photons. Virtual photons can have mass, can have spin directions not pointing along the motion path, and can have momentum opposite to their direction of motion. All these proper-
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•.
Ref. 702
Page 801 Challenge 1204 ny
ties are di erent from real photons. In this way, exchange of virtual photons leads to the attraction of bodies of di erent charge. In fact, virtual photons necessarily appear in any description of electromagnetic interactions; more about their e ects, such as the famous attraction of neutral bodies, will be discussed later on.
In summary, light can indeed move faster than light, though only in amounts allowed by the quantum of action. For everyday situations, i.e. for cases with a high value of the action, all quantum e ects average out, including light velocities di erent from c.
A di erent topic also belongs into this section. Not only the position, but also the energy of a single photon can be unde ned. For example, certain materials split one photon of energy ħω into two photons, whose two energies sum up to the original one. Quantum mechanics implies that the energy partioning is known only when the energy of one of the two photons is measured. Only at that very instant the energy of the second photon is known. Before that, both photons have unde ned energies. e process of energy xing takes place instantaneously, even if the second photon is far away. We will explain below the background of this and similar strange e ects, which seem to be faster than light but which are not. Indeed, such e ects do not transmit energy or information faster than light.
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I
Ref. 703
We saw that the quantum of action implies an indeterminacy for light intensity. at implies a similar limit for electric and magnetic elds. is conclusion was rst drawn in
by Bohr and Rosenfeld. ey started from the e ects of the elds on a test particle of mass m and charge q, which are described by
ma = q(E + v B) .
(488)
Since it is impossible to measure momentum and position of a particle, they deduced an Challenge 1205 ny indeterminacy for the electrical eld given by
∆E
=
ħ q ∆x T
,
(489)
where T is the measurement time and ∆x is the position uncertainty. Every value of an electric eld, and similarly that of every magnetic eld, is thus a ected with an indeterminacy. e physical state of the electromagnetic eld behaves like the state of matter in this aspect. is is the topic we explore now.
C
**
Can di raction be explained with photons? Newton was not able to do so. Today we can do so. Figure 301 is translationally invariant along the horizontal direction; therefore, the momentum component along this direction is also conserved: p sin α = p sin α . e photon energy E = E = E is obviously conserved.
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p1
α1
air
water p2
α2
F I G U R E 301 Diffraction and photons
α
axis F I G U R E 302 A falling pencil
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
e index of refraction n is de ned with momentum and energy as
n
=
cp E
.
(490)
As a result, the ‘law’ of refraction follows.
ere is an important issue here. e velocity of a photon v = δE δp in a light ray is not the same as the phase velocity u = E p that enters in the calculation.
**
Challenge 1206 ny
A typical e ect of the quantum ‘laws’ is the yellow colour of the lamps used for street illumination in most cities. ey emit pure yellow light of one frequency; that is the reason that no other colours can be distinguished in their light. Following classical electrodynamics, harmonics of that light frequency should also be emitted. Experiments show however that this is not the case; classical electrodynamics is thus wrong. Is this argument correct?
.
–
All great things begin as blasphemies.
“ ” George Bernard Shaw
e existence of a smallest action has numerous e ects on the motion of matter. We start with a few experimental results that show that the quantum of action is indeed the smallest action.
W
Challenge 1207 n
A simple consequence of the quantum of action is the impossibility of completely lling a glass of wine. If we call ‘full’ a glass at maximum capacity (including surface tension e ects, to make the argument precise), we immediately see that the situation requires complete rest of the liquid’s surface; however, the quantum of action forbids this. Indeed, a completely quiet surface would allow two subsequent observations which di er by less
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Challenge 1208 ny
than ħ . ere is no rest in nature. In other words, the quantum of action proves the old truth that a glass of wine is always partially empty and partially full.
e quantum of action has many similar consequences for everyday life. For example, a pencil on its tip cannot stay vertical, even if it is isolated from all disturbances, such as vibrations, air molecules and thermal motion. Are you able to con rm this? In fact, it is even possible to calculate the time a er which a pencil must have fallen over.*
C
Challenge 1211 ny
Ref. 715 Challenge 1212 ny
Rest is impossible in nature. Even at lowest temperatures, particles inside matter are in motion. is fundamental lack of rest is said to be due to the so-called zero-point uctuations. A good example are the recent measurements of Bose–Einstein condensates, systems with a small number of atoms (between ten and a few million) at lowest temperatures (around nK). ese cool gases can be observed with high precision. Using elaborate experimental techniques, Bose–Einstein condensates can be put into states for which ∆p∆x is almost exactly equal to ħ , though never lower than this value.
at leads to an interesting puzzle. In a normal object, the distance between the atoms is much larger than their de Broglie wavelength. (Are you able to con rm this?) But today it is possible to cool objects to very low temperatures. At extremely low temperatures, less than nK, the wavelength of the atoms may be larger than their separation. Can you imagine what happens in such cases?
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
N
“Otium cum dignitate.**
” Cicero, De oratore.
e impossibility of rest, like all other unexplained e ects of classical physics, is most
apparent in domains where the action is near the minimum observable one. To make the
e ects most obvious, we study the smallest amount of matter that can be isolated: a single
particle. Later on we will explore situations that cover higher numbers of particles.
Experiments show that perfect rest is never observed. e quantum of action prevents
this in a simple way. Whenever the position of a system is determined to high precision,
we need a high energy probe. Indeed, only a high energy probe has a wavelength small
enough to allow a high precision for position measurements. As a result of this high en-
ergy however, the system is disturbed. Worse, the disturbance itself is also found to be
imprecisely measurable. ere is thus no way to determine the original position even by
taking the disturbance itself into account. In short, perfect rest cannot be observed. All
Ref. 704
Challenge 1209 ny Challenge 1210 ny
* at is not easy, but neither too di cult. For an initial orientation close to the vertical, the fall time T turns out to be
T = π T ln α
(491)
where α is the starting angle, and a fall by π is assumed. Here T is the oscillation time of the pencil for small
angles. (Can you determine it?)
e indeterminacy relation for the tip of the pencil yields a minimum starting angle, because the mo-
mentum indeterminacy cannot be made as large as wanted. You should be able to provide an upper limit.
Once the angle is known, you can calculate the maximum time. ** ‘Rest with dignity.’
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
systems who have ever been observed with high precision con rm that perfect rest does not exist. Among others, this result has been con rmed for electrons, neutrons, protons, ions, atoms, molecules and crystals.
F
Die Bewegung ist die Daseinsform der Materie.
“ ” Friedrich Engels, Anti–Dühring.*
Not only is rest made impossible by the quantum of action; the same impossibility ap-
plies to any situation which does not change in time, like any constant velocity. e most
important example are ows. e quantum of action implies that no ow can be station-
ary. More precisely, a smallest action implies that all ows are made of smallest entities.
All ows are made of quantum particles. Two ows ask for direct con rmation: ows of
electricity and ows of liquids.
If electrical current would be a continuous
ow, it would be possible to observe action values as small as desired. e simplest con rm-
Current
ation of the discontinuity of current ow was
discovered only in the s: take two metal
wires on the table, laying across each other. It is
preliminary
not hard to let a current ow from one wire to
graph
the other, via the crossover, and to measure the
voltage. A curve like the one shown in Figure
is found: the current increases with voltage in regular steps.
Voltage
Many other experiments con rm the result
and leave only one conclusion: there is a smal- F I G U R E 303 Steps in the flow of electricity lest charge in nature. is smallest charge has the in metals
same value as the charge of an electron. Indeed,
electrons turn out to be part of every atom, in a complex way to be explained shortly.
In metals, quite a a number of electrons can move freely; that is the reason that metal
conduct electricity so well.
Also the ow of matter shows smallest units. We mentioned in the rst part that a
consequence of the particle structure of liquids is that even in the smoothest of pipes,
even oil or any other smooth liquid still produces noise when it ows through the pipe.
We mentioned that the noise we hear in our ears in situations of absolute silence, such as
in a snowy landscape in the mountains, is due to the granularity of matter. Depending
on the material, the smallest units of matter are called atoms, ions or molecules.
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Q
Electrons, ions, atoms and molecules are quantum particles or quantons. Like photons, they show some of the aspects of everyday particles, but show many other aspects which are di erent from what is expected from little stones. Let us have a rapid tour.
Ref. 684 * ‘Motion is matter’s way of being.’
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Figure to be included
F I G U R E 304 Matter diffracts and interferes
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Everyday matter has mass, position and momentum, orientation and angular momentum, size, shape, structure and colour. What about matter quantons? First of all, matter quantons do have mass. Single particles can be slowed down or accelerated; in addition, hits by single electrons, atoms or molecules can be detected. Experiments also show that (composed) quantons have structure, size, shape and colour. We will discuss their details below. How do they move while respecting the quantum of action?
T
–
Ref. 708
In and , the French physicist Louis de Broglie pondered over the concept of photon and the possible consequences of the quantum of action for matter particles. It dawned to him that like light quanta, streams of matter particles with the same momentum should also behave as waves. e quantum of action implies wave behaviour.
is, de Broglie reasoned, should also apply to matter. He predicted that constant matter ows should have a wavelength and angular frequency given by
λ=
πħ p
and
ω
=
E ħ
.
(492)
Ref. 709 Page 560
where p and E are the momentum and the energy of the single particles. Soon a er the prediction, experiments started to provide the con rmation of the idea. It is indeed found that matter streams can di ract, refract and interfere. Due to the small value of the wavelength, one needs careful experiments to detect the e ects. Nevertheless, one a er the other, all experiments which proved the wave properties of light have been repeated for matter beams. For example, in the same way that light di racts when passing around an edge or through a slit, matter has been found to di ract in these situations. Similarly, researchers inspired by light interferometers built matter interferometers; they work with beams of electrons, nucleons, nuclei, atoms and even large molecules. In the same way that the observations of interference of light proves the wave property of light, the interference patterns observed with these instruments show the wave properties of matter.
Like light, matter is also made of particles; like light, matter behaves as a wave when large numbers of particles with the same momentum are involved. Even though beams of large molecules behave as waves, for everyday objects, such as cars on a highway, one never makes such observations. ere are two main reasons. First, for cars on highways the involved wavelength is extremely small. Second, the speeds of cars vary too much;
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Figure to be included
F I G U R E 305 Trying to measure position and momentum
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streams of objects with the same speed for all objects – only such streams have a chance to be coherent – are extremely rare in nature.
If matter behaves like a wave, we can draw a strange conclusion. For every type of wave, the position X of its maximum and the wavelength λ cannot both be sharply de ned simultaneously; on the contrary, their indeterminacies follow the relation
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
∆λ∆X = .
(493)
Similarly, for every wave the frequency ω and the instant T of its peak amplitude cannot both be sharply de ned. eir indeterminacies are related by
∆ω∆T = .
(494)
Using the wave properties of matter we get ∆p∆X ħ and
∆E∆T ħ .
(495)
Challenge 1213 ny
ese famous relations are called Heisenberg’s indeterminacy relations. ey were discovered by the German physicist Werner Heisenberg in . ey state that there is no way to ascribe a precise momentum and position to a material system, nor a precise energy and age. e more accurately one quantity is known, the less accurately the other is.* Matter quantons – like stones, but in contrast to photons – can be localized, but only approximately.
Both indeterminacy relations have been checked experimentally in great detail. e limits are easily experienced in experiments. Some attempts are shown in Figure . In fact, every experiment proving that matter behaves like a wave is a con rmation of the indeterminacy relation, and vice versa.
As a note, Niels Bohr called the relation between two variables linked in this way complementarity. He then explored systematically all such possible pairs. You should search for such observable pairs yourself. Bohr was deeply fascinated by the existence of a complementarity principle. Bohr later extended it also to philosophical aspects. In a fam-
* e policeman stops the car being driven by Werner Heisenberg. ‘Do you know how fast you were driving?’ ‘No, but I know exactly where I am!’
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source
a
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 306 On the quantization of angular momentum
ous story, somebody asked him what was the quantity complementary to precision. He answered: ‘Clarity’.
In summary, we conclude that the quantum of action prevents position and momentum values to be exactly de ned for microscopic systems. eir values are fuzzy. Like Bohr, we will explore some additional limits on motion that follow from the quantum of action.
R
N
P
Ref. 718
Tristo quel discepolo che non avanza il suo maestro.
“ Leonardo da Vinci* ” e quantum of action also has important consequences for rotational motion. Action
and angular momentum have the same physical dimensions. It only takes a little thought to show that if matter or radiation has a momentum and wavelength related by the quantum of action, then angular momentum is xed in multiples of the quantum of action; angular momentum is thus quantized.
e argument is due to Dicke and Wittke. Just imagine a source at the centre of a circular fence, made of N steel bars spaced by a distance a = πR N, as shown in Figure . In the centre of the fence we imagine a source of matter or radiation that emits particles towards the fence in any chosen direction. e linear momentum of the particle is p = ħk = πħ λ. Outside the fence, the direction of the particle is given by the condition of positive interference. In other words, the angle θ is given by a sin θ = Mλ, where M is an integer. In this process, the fence receives a linear momentum p sin θ, or an angular momentum L = pR sin θ. Inserting all expressions one nds that the transferred
* ‘Sad is that disciple who does not surpass his master.’ e statement is painted in large letters in the Aula Magna of the University of Rome.
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angular momentum is
L = NMħ .
(496)
Ref. 710 Ref. 711 Page 722
In other words, the angular momentum of the fence is an integer multiple of ħ. Of course, this argument is only a hint, not a proof. Nevertheless, the argument is correct. e angular momentum of bodies is always a multiple of ħ. Quantum theory thus states that every object rotates in steps.
But rotation has more interesting aspects. Due to the quantum of action, in the same way that linear momentum is fuzzy, angular momentum is so as well. ere is an indeterminacy relation for angular momentum L. e complementary variable is the phase angle φ of the rotation. e indeterminacy relation can be expressed in several ways. e simplest, but also the less precise is
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∆L ∆φ ħ .
(497)
( e approximation is evident: the relation is only valid for large angular momenta; the relation cannot be valid for small values, as ∆φ by de nition cannot grow beyond π. In particular, angular momentum eigenstates have ∆L = .*)
e quantization and indeterminacy of angular momentum has important consequences. Classically speaking, the poles of the Earth are spots which do not move when observed by a non-rotating observer. erefore at those spots matter would have a de ned position and a de ned momentum. However, the quantum of action forbids this. ere cannot be a North Pole on Earth. More precisely, the idea of a rotation axis is an approximation not valid in general.
Even more interesting are the e ects of the quantum of action on microscopic particles, such as atoms, molecules or nuclei. To begin with, we note that action and angular momentum have the same units. e precision with which angular momentum can be measured depends on the precision of the rotation angle. But if a microscopic particle rotates by an angle, this rotation might be unobservable, a situation in fundamental contrast with the case of macroscopic objects. Experiments indeed con rm that many microscopic particles have unobservable rotation angles. For example, in many, but not all cases, an atomic nucleus rotated by half a turn cannot be distinguished from the unrotated nucleus.
If a microscopic particle has a smallest unobservable rotation angle, the quantum of action implies that the angular momentum of that particle cannot be zero. It must always be rotating. erefore we need to check for each particle what its smallest unobservable angle of rotation is. Physicists have checked experimentally all particles in nature and have found – depending on the particle type – the following smallest unobservable angle values: , π, π, π , π, π , π , etc.
* An exact way to state the indeterminacy relation for angular momentum is ∆L ∆φ ħ − πP(π) ,
(498)
where P(π) is the normalized probability that the angular position has the value π. For an angular moRef. 712 mentum eigenstate, one has ∆φ = π and P(π) = π. is expression has been tested experimentally.
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Figure to be included
F I G U R E 307 The Stern–Gerlach experiment
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Let us take an example. Certain nuclei have a smallest unobservable rotation angle of half a turn. at is the case for a prolate nucleus – one that looks like a rugby ball – turning around its short axis. Both the largest observable rotation and the indeterminacy are thus a quarter turn. Since the change or action produced by a rotation is the number of turns times the angular momentum, we nd that the angular momentum of this nucleus is ë ħ.
As a general result we deduce from the smallest angle values that the angular momentum of a microscopic particle can be , ħ , ħ, ħ , ħ, ħ , ħ, etc. In other words, the intrinsic angular momentum of particles, usually called their spin, is an integer multiple of ħ . Spin describes how a particle behaves under rotations. (It turns out that all spin particles are composed and contain other particles; the quantum of action thus remains the limit for rotational motion in nature.)
How can a particle rotate? At this point we do not know yet how to picture the rotation. But we can feel it. is is done in the same way we showed that light is made of rotating entities: all matter, including electrons, can be polarized. is was shown most clearly by the famous Stern–Gerlach experiment.
S
,S
G
Ref. 705
In , Otto Stern and Walter Gerlach* found that a beam of silver atoms that is extracted from an oven splits into two separate beams when it passes through an inhomogeneous magnetic eld. ere are no atoms that leave the magnetic eld in intermediate locations.
e split into two is an intrinsic property of silver atoms. e split is due to the spin value of the atoms. Silver atoms have spin ħ , and depending on their orientation in space, they are de ected either in direction of the eld or against it. e splitting of the beam is a pure quantum e ect: there are no intermediate options. is result is so peculiar that it was studied in great detail.
When one of the two beams is selected – say the ‘up’ beam – and passed through a second set-up, all atoms end up in the ‘up’ beam. e other exit, for the ‘down’ beam, remains unused in this case. e up and down beams, in contrast to the original beam, cannot be split further. is is not surprising.
But if the second set-up is rotated by π with respect to the rst, again two beams – ‘right’ and ‘le ’ – are formed; it plays no role whether the incoming beam was from the
* Otto Stern (1888–1969) and Walter Gerlach (1889–1979), both German physicists, worked together at the University in Frankfurt.
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oven or an ‘up’ beam. A partially rotated set-up yields a partial, uneven split. e number ratio depends on the angle.
We note directly that if a beam from the oven is split rst vertically and then horizontally, the result di ers from the opposite order. Splitting processes do not commute. (Whenever the order of two operations is important, physicists speak of ‘lack of commutation’.) Since all measurements are processes as well, we deduce that measurements in quantum systems do not commute in general.
Beam splitting is direction dependent. Matter beams behave almost in the same way as polarized light beams. e inhomogeneous magnetic eld acts somewhat like a polarizer.
e up and down beams, taken together, behave like a fully polarized light beam. In fact, the polarization direction can be rotated (with the help of a homogeneous magnetic eld). Indeed, a rotated beam behaves in a unrotated magnet like an unrotated beam in a rotated magnet.
e ‘digital’ split forces us to rethink the description of motion. In special relativity, the existence of a maximum speed forced us to introduce the concept of space-time, and then to re ne the description of motion. In general relativity, the maximum force obliged us to introduce the concepts of horizon and curvature, and then to re ne the description of motion. At this point, the existence of the quantum of action forces us to take two similar steps. We will introduce the new concept of Hilbert space, and then we will re ne the description of motion.
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In classical physics, a physical system is said to have momentum, position, and a axis of rotation. e quantum of action makes it impossible to continue using this language. In classical physics, the state and the measurement result coincide, because measurements can be imagined to disturb the system as little as possible. But due to a smallest action in nature, the interaction necessary to perform the measurement cannot be made arbitrarily small. For example, the Stern–Gerlach experiment shows that the measured spin orientation values – like those of any other observable – are not intrinsic values, but result from the measurement process itself.
erefere, the quantum of action forces us to distinguish three entities:
— the state of the system; — the operation of measurement; — the result of the measurement.
A general state of a quantum system is thus not described by the outcomes of a measurement. e simplest case showing this is the system made of a single particle in the Stern– Gerlach experiment. e experiment shows that a spin measurement on a general (oven) particle state sometimes gives ‘up’, sometimes ‘down’ (‘up’ might be + , ‘down’ might be − ) showing that a general state has no intrinsic properties. It was also found that feeding ‘up’ into the measurement apparatus gives ‘up’ states; thus certain special states (‘eigenstates’) do remain una ected. Finally, the experiment shows that states can be rotated by applied elds; they have an abstract direction.
ese details can be formulated in a straightforward way. Since measurements are operations that take a state as input and produce as output a measurement result and an
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•.
output state, we can say:
— States are described by small rotating arrows; in other words they are complex vectors in an abstract space. e space of all possible states is called a Hilbert space.
— Measurements are operations on the state vectors. Measurements are said to be described by (self-adjoint or) Hermitean operators (or matrices).
— Measurement results are real numbers. — Changes of viewpoint are described by unitary operators (or matrices) that transform
states and measurement operators.
As required by quantum experiments, we have thus distinguished the quantities that are not distinguished in classical physics. Once this step is completed, quantum theory follows quite simply, as we shall see.
Quantum theory describes observables as operators, thus as transformations in Hilbert space, because any measurement is an interaction with a system and thus a transformation of its state.
Quantum mechanical experiments also show that the measurement of an observable can only give as result one of the possible eigenvalues of this transformation. e resulting states are eigenvectors – the ‘special’ states just mentioned. erefore every expert on motion must know what an eigenvalue and an eigenvector is. For any linear transformation T, those special vectors ψ that are transformed into multiples of themselves,
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Tψ = λψ
(499)
are called eigenvectors, and the multiplication factor λ is called the associated eigenvalue. Experiments show that the state of the system a er the measurement is given by the eigenvector of the measured eigenvalue.
In summary, the quantum of action obliges us to distinguish between three concepts that are all mixed up in classical physics: the state of a system, the measurement on a system and the measurement result. e quantum of action forces us to change the vocabulary with which we describe nature and obliges to use more di erentiated concepts. Now follows the main step: we describe motion with these concepts. is is the description that is usually called quantum theory.
In classical physics, motion is given by the path that minimizes the action. Motion takes place in such a way that the action variation δS vanishes when paths with xed end points are compared. For quantum systems, we need to rede ne the concept of action and to nd a description of its variation that are not based on paths, as the concept of ‘path’ does not exist for quantum systems.
Instead of de ning action variations for changing paths between start and end points, one de nes it for given initial and nal states. In detail, the action variation δS between an initial and a nal state is de ned as
∫ δS = ψi δ Ldt ψf ,
(500)
where L is the Lagrangian (operator). e variation of the action is de ned in the same way as in classical physics, except that the momentum and position variables are replaced
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by the corresponding operators.*
In the classical principle of least action, the path is varied while keeping the end points
xed. is variation must be translated into the language of quantum theory. In quantum
theory, paths do not exist, because position is not a well-de ned observable. e only
variation that we can use is
δ ψi ψf .
(501)
is complex number describes the variation of the temporal evolution of the system. e variation of the action must be as small as possible when the temporal evolution
is varied, while at the same time it must be impossible to observe actions below ħ . is double condition is realized by the so-called quantum action principle
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∫ ψi δ Ldt ψf = −iħδ ψi ψf .
(502)
is principle describes all quantum motion in nature. Classically, the right hand side is zero – since ħ can then be neglected – and we then recover the minimum action principle of classical physics. In quantum theory however, the variation of the action is proportional to the variation at the end points. e intermediate situations – the ‘paths’ – do not appear. In the quantum action principle, the factor −i plays an important role. We recall that states are vectors. e factor −i implies that in the complex plane, the complex variation on the right hand side is rotated by a right angle; in this way, even if the variation at the end points is small, no action change below ħ can be observed.
To be convinced about the correctness of the quantum action principle, we proceed in the following way. We rst deduce evolution equations, we then deduce all experimental e ects given so far, and nally we deduce new e ects that we compare to experiments.
T
–
–
We can also focus on the change of states with time.
gives
i
ħ
∂ ∂t
ψ
=Hψ
.
e quantum action principle then (503)
Ref. 713 Ref. 714
is famous equation is Schrödinger’s equation of motion.** In fact, Erwin Schrödinger had found his equation in two slightly di erent ways. In his rst paper, he used a variational principle slightly di erent from the one just given. In the second paper he simply asked: how does the state evolve? He imagined the state of a quanton to behave like a
* More precisely, there is also a condition for ordering of operators in mixed products, so that the lack of commutation between operators is taken into account. We do not explore this issue here. ** Erwin Schrödinger (b. 1887 Vienna, d. 1961 Vienna) was famous for being a physicien bohémien, and always lived in a household with two women. In 1925 he discovered the equation which brought him international fame and the Nobel prize for physics in 1933. He was also the rst to show that the radiation discovered by Victor Hess in Vienna was indeed coming from the cosmos. He le Germany and then again Austria out of dislike of national socialism, and was for many years professor in Dublin. ere he published the famous and in uential book What is life?. In it, he comes close to predicting the then unknown nuclear acid DNA from theoretical insight alone.
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wave and like a particle at the same time. If the state behaves ψ like a wave, it must be
described by a function (hence he called it ‘wave function’) with amplitude W multiplied by a phase factor eikx−ωt. e state can thus be written as
ψ = ψ(t, x) = W(t, x)eikx−ωt .
(504)
At the same time, the state must also behave like a a particle. In particular, the non-
relativistic particle relation between energy and momentum E = p m + V (x) must
remain valid for these waves. Using the two relations for matter wavelength and frequency,
we thus must have
i
ħ
∂ψ ∂t
=
∆ψ m
+
V (x)ψ
=
Hψ
.
(505)
Page 754
is ‘wave’ equation for the complex eld ψ became instantly famous in
when
Schrödinger, by inserting the potential felt by an electron near a proton, explained the
energy levels of the hydrogen atom. In other words, the equation explained the discrete
colours of all radiation emitted by hydrogen. We will do this below. e frequency of the
light emitted by hydrogen gas was found to be in agreement with the prediction of the
equation to ve decimal places. e aim of describing motion of matter had arrived at a
new high point.
e most precise description of matter is found when the re-
lativistic energy–momentum relation is taken into account. We
explore this approach below. Even today, predictions of atomic
spectra are the most precise and accurate in the whole study of
nature. No other description of nature has achieved a higher ac-
curacy.
We delve a bit into the details of the description with the
Schrödinger equation ( ). e equation expresses a simple con-
nection: the classical speed of matter is the group velocity of the
eld ψ. We know from classical physics that the group velocity
is not always well de ned; in cases where the group dissolves in
several peaks the concept of group velocity is not of much use;
these are the cases in which quantum motion is much di erent Erwin Schrödinger
from classical motion, as we will soon discover.
As an example, the le corner pictures in these pages show the evolution of a wave
function – actually its modulus Ψ – for the case of an exploding two-particle system.
e Schrödinger equation makes another point: velocity and position of matter are
not independent variables and cannot be chosen at leisure. Indeed, the initial condition
of a system is given by the initial value of the wave function alone. No derivatives have
to or can be speci ed. In other words, quantum systems are described by a rst order
evolution equation, in strong contrast to classical physics.
We note for completeness that in the Schrödinger equation the wave function is in-
deed a vector, despite the apparent di erences. e scalar product of two wave func-
tions/vectors is the spatial integral of the product between complex conjugate of the rst
function and the second function. In this way, all concepts of vectors, such as unit vectors,
null vectors, basis vectors, etc. can be reproduced.
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W
?W
?
Ref. 706 Ref. 707
Page 837
In , Jean Perrin, and in the Japanese physicist Nagaoka Hantaro proposed that atoms are small solar systems. In , Niels Bohr used this idea and based his epochmaking atomic calculations on it. All thus somehow assumed that hydrogen atoms are
at. However, this is not observed. Atoms, in contrast to solar systems, are quantum systems. Atoms, like protons and
many other quantum systems, do have sizes and shapes. Atoms are spherical, molecules have more complex shapes. Quantum theory gives a simple recipe for the calculation: the shape of an atom or a molecule is due to the probability distribution of its electrons. e probability distribution is given by the square modulus Ψ of the wave function.
In other words, Schrödinger’s equation de nes the shape of molecules. at is why it was said that the equation contains all of chemistry and biology. e precise shape of matter is determined by the interactions of electrons and nuclei. We come back to the issue later.
In short, the wave aspect of quantons is responsible for all shapes in nature. For example, only the wave aspect of matter, and especially that of electrons, allows to understand the shapes of molecules and therefore indirectly the shapes of all bodies around us, from owers to people.
Obviously, the quantum of action also implies that shapes uctuate. If a long molecule is held xed at its two ends, the molecule cannot remain at rest in between. Such experiments are common today; they con rm that rest does not exist, as it would contradict the existence of a minimum action in nature.
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Z
Ref. 716
In special relativity, anything moving inertially is at rest. However, the quantum of action implies that no particle can ever be at rest. erefore, no quantum system can be at in inertial motion. at is the reason that any wave function spreads out in time. In this way, a particle is never at rest, whatever the observer may be.
Only if a particle is bound, not freely moving, one can have the situation that the density distribution is stationary in time.
Another apparent case of rest in quantum theory is called the quantum Zeno e ect. Usually, observation changes the system. However, for certain systems, observation can have the opposite e ect.
e quantum Zeno e ect was partially observed by Wayno Itano and his group in , and de nitively observed by Mark Raizen and his group, in .
– CS – more to be told – CS –
Ref. 717 In an fascinating prediction, Saverio Pascazio and his team have predicted that the quantum Zeno e ect can be used to realize X-ray tomography of objects with the lowest radiation levels imaginable.
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T
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Challenge 1214 ny
A slow ball cannot roll over a high hill, says everyday experience. More precisely, classical
physics says that if the kinetic energy T is smaller than the potential energy V the ball
would have at the top of the hill, the ball cannot roll over the hill. Quantum theory simply
states the opposite. ere is a probability to pass the hill for any energy of the ball.
Since hills in quantum theory are de-
scribed by potential barriers, and objects
by wave functions, the e ect that an object
can pass the hill is called the tunnelling ef-
V
fect. For a potential barrier of nite height, m T
any initial wave function will spread bey-
ond the barrier. e wave function will
0
∆x
even be non-vanishing at the place of the F I G U RE 308 Climbing a hill barrier. All this is di erent from everyday
experience and thus from classical mechanics.
Something new is contained in this description of hills: the assumption that all
obstacles in nature can be overcome with a nite e ort. No obstacle is in nitely di cult
to surmount. (Only for a potential of in nite height the wave function would vanish and
not spread on the other side.)
How large is the e ect? A simple calculation shows that the transmission probability
P is given by
P
T(V V
−
T) e−
w
m(V−T) ħ
(506)
where w is the width of the hill. For a system of many particles, the probability is the product of the probabilities for each particle. In the case of a car in a garage, assuming it is made of atoms of room temperature, and assuming that a garage wall has a thickness of . m and a potential height of keV= aJ for the passage of an atom, one gets a probability of nding the car outside the garage of
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P
−( ) ( )
−( ) .
(507)
Challenge 1215 ny Challenge 1216 ny
is rather small value – just try to write it down to be convinced – is the reason why it is never taken into account by the police when a car is missing. (Actually, the probability is considerably smaller; can you name at least one e ect that has been forgotten in this simple calculation?) Obviously, tunnelling can be important only for small systems, made
of a few particles, and for thin barriers, with a thickness of the order of ħ m(V − T) . Tunnelling of single atoms is observed in solids at high temperature, but is not of importance in daily life. For electrons the e ect is larger; the formula gives
w . nm aJ V − T .
(508)
At room temperature, kinetic energies are of the order of zJ; increasing temperature obviously increases tunnelling. As a result, electrons or other light particles tunnel quite
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Challenge 1217 ny
easily. Indeed, every tv tube uses tunnelling at high temperature to generate the electron beam producing the picture. e heating is the reason that TV tubes take time to switch on.
For example, the tunnelling of electrons limits the ability to reduce the size of computer memories, and thus makes it impossible to produce silicon integrated circuits with one terabyte (TB) of random access memory (RAM). Are you able to imagine why? In fact, tunnelling limits the working of any type of memory, also that of our brain. If we would be much hotter than °C, we could not remember anything!
By the way, light, being made of particles, can also tunnel through potential barriers. e best potential barriers for light are called mirrors; they have barrier heights of the order of one aJ. Tunnelling implies that light can be detected behind a mirror. ese so-called evanescent waves have indeed been detected. ey are used in several highprecision experiments.
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Ref. 721 Ref. 722
“Everything turns.
” Anonymous
Spin describes how a particle behaves under rotations. e full details of spin of electrons
were deduced from experiments by two Dutch students, George Uhleneck and Samuel
Goudsmit. ey had the guts to publish what also Ralph Kronig had suspected: that elec-
trons rotate around an axis with an angular momentum of ħ . In fact, this value is correct
for all elementary matter particles. (In contrast, radiation particles have spin values that
are integer multiples of ħ.) In particular, Uhlenbeck and Goudsmit proposed a g-value
of for the electron in order to explain the optical spectra. e factor was explained by
Llewellyn omas as a relativistic e ect a few months a erwards.
In , experimental techniques became so sensitive that the magnetic e ect of a
single electron spin attached to an impurity (in an otherwise unmagnetic material) has
been detected. Researchers now hope to improve these so-called magnetic resonance
force microscopes until they reach atomic resolution.
In , the Austrian physicist Wolfgang Pauli* discovered how to include spin in a
quantum mechanical description; instead of a state function with a single component,
one needs a state function with two components. Nowadays, Pauli’s equation is mainly of
conceptual interest, because like the one by Schrödinger, it does not comply with special
relativity. However, the idea to double the necessary components was taken up by Dirac
Ref. 723
* Wolfgang Ernst Pauli (b. 1900 Vienna, d. 1958 Zürich), when 21 years old, wrote one of the best texts on special and general relativity. He was the rst to calculate the energy levels of hydrogen with quantum theory, discovered the exclusion principle, included spin into quantum theory, elucidated the relation between spin and statistics, proved the CPT theorem and predicted the neutrino. He was admired for his intelligence and feared for his biting criticisms, which lead to his nickname ‘conscience of physics’. Despite this habit he helped many people in their research, such as Heisenberg with quantum theory, without claiming any credit for himself. He was seen by many, including Einstein, as the greatest and sharpest mind of twentieth century physics. He was also famous for the ‘Pauli e ect’, i.e. his ability to trigger disasters in laboratories, machines and his surroundings by his mere presence. As we will see shortly, one can argue that Pauli got the Nobel Prize in physics in 1945 (o cially ‘for the discovery of the exclusion principle’) for nally settling the question on the number of angels that can dance on the tip of a pin.
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when he introduced the relativistic description of the electron, and the idea is used for all other particle equations.
R
A few years a er Max Planck had discovered the quantum of action, Albert Einstein published the theory of special relativity. e rst question Planck asked himself was whether the value of the quantum of action would be independent of the observer. For this reason, he invited Einstein to Berlin. By doing this, he made the then unknown patent o ce clerk famous in the world of physicis.
e quantum of action is indeed independent of the speed of the observer. All observers nd the same minimum value. To include special relativity into quantum theory, we only need to nd the correct quantum Hamiltonian operator.
Given that the classical Hamiltonian of a free particle is given by
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H = β c m + c p with p = γmv ,
(509)
one might ask: what is the corresponding Hamilton operator? A simple answer was given, Ref. 729 only in , by L.L. Foldy and S.A. Wouthuysen. e operator is almost the same one:
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
H = β c m + c p with β =
− −
(510)
e signs of the operator β distinguishes particles and antiparticles; it has two s and
two - s to take care of the two possible spin directions. With this Hamiltonian operator,
a wave function for a particle has vanishing antiparticle components, and vice versa. e
Hamilton operator yields the velocity operator v through the same relation that is valid
in classical physics:
v
=
d dt x
=
β
p c m +c p
.
(511)
is velocity operator shows a continuum of eigenvalues from minus to plus the speed of
light. e velocity v is a constant of motion, as are p and c m + c p . e orbital angular momentum l is also de ned as in classical physics through
l=x p.
(512)
Ref. 731
Both the orbital angular momentum l and the spin σ are separate constants of motion. A particle (or antiparticle) with positive (or negative) component has a wave function with only one non-vanishing component; the other three components vanish.
But alas, the representation of relativistic motion given by Foldy and Wouthuysen is not the most simple for a generalization to particles when electromagnetic interactions are present. e simple identity between classical and quantum-mechanical description is lost when electromagnetism is included. We give below the way to solve the problem.
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M
Challenge 1218 n
Combining quantum theory with special relativity leads to a maximum acceleration value
for microscopic particles. Using the time–energy indeterminacy relation, you can deduce
that
a
mc ħ
.
(513)
Ref. 719
Up to the present, no particle has ever been observed with a higher acceleration than
this value. In fact, no particle has ever been observed with accelerations approaching this
value. We note that the acceleration limit is di erent from the acceleration limit due to
general relativity:
a
c Gm
.
(514)
In particular, the quantum limit ( ) applies to microscopic particles, whereas the general relativistic limit applies to macroscopic systems. Can you con rm that in each doChallenge 1219 e main the respective limit is the smaller of the two?
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– CS – the rest of quantum theory will appear in soon – CS –
C
Quantum theory is so full of strange results that all of it could be titled ‘curiosities’. A few of the prettier cases are given here.
**
e quantum of action implies that there are no fractals in nature. Can you con rm this Challenge 1220 ny result?
Ref. 734 Challenge 1221 ny
**
Can atoms rotate? Can an atom that falls on the oor roll under the table? Can atoms be put into high speed rotation? e answer is no to all questions, because angular momentum is quantized and because atoms are not solid objects. e macroscopic case of an object turning slower and slower until it stops does not exist in the microscopic world. Can you explain how this follows from the quantum of action?
Ref. 737 Challenge 1222 n
**
Do hydrogen atoms exist? Most types of atoms have been imaged with microscopes, photographed under illumination, levitated one by one, and even moved with needles, one by one, as the picture shows. Others have moved single atoms using laser beams to push them. However, not a single of these experiments measured hydrogen atoms. Is that a reason to doubt the existence of hydrogen atoms? Taking seriously this not-so-serious discussion can be a lot of fun.
**
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•.
Light is refracted when entering dense matter. Do matter waves behave similarly? Yes, they do. In 1995, David Pritchard showed this for sodium waves entering helium and Ref. 736 xenon gas.
Challenge 1223 ny
**
Two observables can commute for two di erent reasons: either they are very similar, such as the coordinate x and x , or they are very di erent, such as the coordinate x and the momentum py. Can you give an explanation?
**
Space and time translations commute. Why then do the momentum operator and the Challenge 1224 ny Hamiltonian not commute in general?
**
Ref. 738 With two mirrors and a few photons, it is possible to capture an atom and keep it oating between the two mirrors. is feat, one of the several ways to isolate single atoms, is now
Challenge 1225 ny standard practice in laboratories. Can you imagine how it is realized?
Ref. 739
**
For a bound system in a non-relativistic state with no angular momentum, one has the
relation
rr T
ħ m
,
(515)
where m is the reduced mass and T the kinetic energy of the components, and r the size Challenge 1226 n of the system. Can you deduce the result and check it for hydrogen?
Challenge 1227 n
**
Electrons don’t like high magnetic elds. When a magnetic eld is too high, electrons are squeezed into a small space, in the direction transversal to their motion. If this spacing becomes smaller than the Compton wavelength, something special happens. Electronpositron pairs appear from the vacuum and move in such a way as to reduce the applied magnetic eld. e corresponding eld value is called the quantum critical magnetic eld. Physicists also say that the Landau levels spacing then becomes larger than the electron rest energy. Its value is about . GT. Nevertheless, in magnetars, elds over 20 times as high have been measured. How is this possible?
**
O en one reads that the universe might have been born from a quantum uctuation. Challenge 1228 ny Does this statement make sense?
Dvipsbugw
e examples so far have shown how quantons move that are described only by mass. Now we study the motion of quantum systems that are electrically charged.
Dvipsbugw
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F I G U R E 309 The spectrum of daylight: a stacked section of the rainbow (© Nigel Sharp (NOAO), FTS, NSO, KPNO, AURA, NSF)
.
“Rem tene; verba sequentur.
”Cato
A er the description of the motion of matter and radiation, the next step is the descrip-
tion of their interactions. In other words, how do charged particles react to electromag-
netic elds and vice versa? Interactions lead to surprising e ects, most of which appear
when the problem is treated taking special relativity into account.
W
?
In the beginning of the eighteenth century the English physicist William Wollaston and again the Bavarian instrument maker Joseph Fraunhofer* noted that the rainbow lacks certain colours. ese colours appear as black lines when the rainbow is spread out suf-
ciently. Figure shows them in detail; the lines are called Fraunhofer lines today. In , Gustav Kirchho and Robert Bunsen showed that the missing colours were exactly
* Born as Joseph Fraunhofer (b. 1787 Straubing, d. 1826 München). Bavarian, orphan at 11, he learned lens polishing at that age; autodidact, he studied optics from books. He entered an optical company at age 19, ensuring the success of the business, by producing the best available lenses, telescopes, micrometers, optical gratings and optical systems of his time. He invented the spectroscope and the heliometer. He discovered and counted 476 lines in the spectrum of the Sun, today named a er him. Up to this day, Fraunhofer lines are used as measurement standards. Physicists across the world would buy their equipment from him, visit him and ask for copies of his publications. Even a er his death, his instruments remain unsurpassed. With his telescopes, in 1837 Bessel was able to measure the rst parallax of a star and in 1846 Johann Gottfried Galle discovered Neptune. Fraunhofer became professor in 1819; he died young, from the consequences of the years spent working with lead and glass powder.
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•.
those colours that certain elements emitted when heated. With a little of experimenting they managed to show that sodium, calcium, barium, nickel, magnesium, zinc, copper and iron existed on the Sun. However, they were unable to attribute of the lines they observed. In , Jules Janssen and Joseph Lockyer independently predicted that the unknown lines were from a new element; it was eventually found also on Earth, in an uranium mineral called cleveite, in . Obviously it was called ‘helium’, from the Greek word ‘helios’ – Sun. Today we know that it is the second ingredient of the Sun, in order of frequency, and of the universe, a er hydrogen. Helium, despite being so common, is rare on Earth because it is a light noble gas that does not form chemical components. Helium thus tends to rise in the atmosphere until it leaves the Earth.
Understanding the colour lines produced by each element had started to become of interest already before the discovery of helium; the interest rose even more a erwards, due to the increasing applications of colours in chemistry, physics, technology, crystallography, biology and lasers. It is obvious that classical electrodynamics cannot explain the sharp lines. Only quantum theory can explain colours.
Dvipsbugw
W
?
e simplest atom to study is the atom of hydrogen. Hydrogen gas emits light consisting
of a number of sharp spectral lines. Already in century the Swiss school teacher
Johann Balmer ( – ) had discovered that the wavelengths of visible lines follow
from the formula
λmn = R − m .
(516)
is expression was generalized by Johannes Rydberg ( – olet and infrared colours
λmn = R n − m ,
) to include the ultravi(517)
Ref. 724 Challenge 1229 ny
where n and m n are positive integers, and the so-called Rydberg constant R has the value . µm− . us quantum theory has a clearly de ned challenge here: to explain the formula and the value of R.
By the way, the transition λ for hydrogen – the shortest wavelength possible – is called the Lyman-alpha line. Its wavelength, . nm, lies in the ultraviolet. It is easily observed with telescopes, since most of the visible stars consist of excited hydrogen. e Lyman-alpha line is regularly used to determine the speed of distant stars or galaxies, since the Doppler e ect changes the wavelength when the speed is large. e record so far is a galaxy found in with a Lyman-alpha line shi ed to nm. Can you calculate the speed with which it moves away from the Earth?
ere are many ways to deduce Balmer’s formula from the minimum action. In , Schrödinger solved his equation of motion for the electrostatic potential V (r) = e πε r of a point-like proton; this famous calculation however, is long and complex. In order to understand hydrogen colours, it is not necessary to solve an equation of motion; it is su cient to compare the initial and nal state. is can be done most easily by noting that a speci c form of the action must be a multiple of ħ . is approach was developed by Einstein, Brillouin and Keller and is now named a er them. It states that
Dvipsbugw
Ref. 725 the action S of any quantum system obeys
∫ S = π
dqi pi = (ni + µi )ħ
(518)
Challenge 1230 ny
for every coordinate qi and its conjugate momentum pi. Here, ni can be or any positive integer and µi is the so-called Maslov index (an even integer) that in the case of atoms has the value for the radial and azimuthal coordinates r and θ, and for the rotation
angle φ.
Any rotational motion in a spherical potential V (r) is characterized by a constant
energy E, and constant angular momenta L and Lz. erefore the conjugate momenta for the coordinates r, θ and φ are
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pr =
m(E
−
V
(r))
−
L r
pθ =
L
− Lz sin θ
pφ = Lz .
(519)
Using these expressions in equation ( ) and setting n = nr + nθ + nφ +
result
En = n
−me ( πε ) ħ
=
−R n
. aJ n
. n
eV
.
yields* the (522)
Challenge 1232 e Challenge 1233 ny
Using the idea that a hydrogen atom emits a single photon when its electron changes from state En to Em, one gets the formula found by Balmer and Rydberg. ( is whole discussion assumes that the electrons in hydrogen atoms are in eigenstates. Can you argue why this is the case?)
e e ective radius of the electron orbit in hydrogen is given by
rn = n
ħ πε πme
=n a
n
pm .
(523)
e smallest value pm for n = is called the Bohr radius and is abbreviated a . Quantum theory thus implies that a hydrogen atom excited to the level n = is about µm in size, larger than many bacteria! is feat has indeed been achieved, even
* e calculation is straightforward. A er insertion of V (r) = e πε r into equation (519) one needs to Challenge 1231 ny perform the (tricky) integration. Using the general result
∫π
dz z
Az + Bz − C = − C +
B −A
one gets
(nr +
)ħ + L = nħ =
e πε
is leads to the energy formula (522).
m −E
.
(520) (521)
Dvipsbugw
•.
Figure to be included F I G U R E 310 The energy levels of hydrogen
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though such blown-up atoms, usually called Rydberg atoms, are extremely sensitive to Ref. 727 perturbations.
e orbital frequency of electrons in hydrogen is
and the electron speed is
fn = n
em εh
(524)
vn =
e nε h
. Mm s n
.c n
(525)
Ref. 726
As expected, the further electrons orbit the nucleus, the slower they move. is result can be checked by experiment. Exchanging the electron by a muon allows to measure the time dilation of its lifetime. e measurements coincide with the formula. We note that the speeds are slightly relativistic. However, this calculation did not take into account relativistic e ects. Indeed, precision measurements show slight di erences between the calculated energy levels and the measured ones.
R
Ref. 725
Also in the relativistic case, the EBK action has to be a multiple of ħ
expression of energy
E + mc =
p c +m c
−
e πε r
. From the relativistic (526)
Challenge 1234 ny we get the expression
pr =
mE( +
E mc
)+
me πε
r
(
+
E mc
).
(527)
We now use the expression for the dimensionless
ne structure constant α =
e πε
ħc
=
Challenge 1235 ny
R mc
. . e radial EBK action then implies that
Dvipsbugw
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(Enl + mc ) = mc
+
α
.
(n − l − + (l + ) − α )
(528)
is result is correct for point-like electrons. In reality, the electron has spin / ; the cor-
rect relativistic energy levels thus appear when we set l = j
in the above formula.
e result can be approximated by
En j
=
−R n
(
+
α n
(
j
n +
−
) + ...)
Dvipsbugw (529)
It reproduces the hydrogen spectrum to an extremely high accuracy. Only the introduction of virtual particle e ects yields an even better result. We will present this point later on.
R
–
e equation was more intelligent than I was. Paul Dirac about his equation, repeating
“ a statement made by Heinrich Hertz. ” Unfortunately, the representation of relativistic motion given by Foldy and Wouthuysen
is not the most simple for a generalization to particles in the case that electromagnetic interactions are present. e simple identity between classical and quantum-mechanical description is lost if electromagnetism is included.
Charged particles are best described by another, equivalent representation of the Hamiltonian, which was discovered much earlier, in , by the British physicist Paul Dirac.* Dirac found a neat trick to take the square root appearing in the relativistic energy operator. In this representation, the Hamilton operator is given by
HDirac = βm + α ë p
(530)
Its position operator x is not the position of a particle, but has additional terms; its velocity operator has only the eigenvalues plus or minus the velocity of light; the velocity operator is not simply related to the momentum operator; the equation of motion contains the famous ‘Zitterbewegung’ term; orbital angular momentum and spin are not separate constants of motion.
* Paul Adrien Maurice Dirac (b. 1902 Bristol, d. 1984 Tallahassee), British physicist, born as son of a Frenchspeaking Swiss immigrant. He studied electrotechnics in Bristol, then went to Cambridge, where he later became professor on the chair Newton had held before. In the years from 1925 to 1933 he published a stream of papers, of which several were worth a Nobel Prize, which he received in 1933. He uni ed special relativity and quantum theory, he predicted antimatter, he worked on spin and statistics, he predicted magnetic monopoles, he speculated on the law of large numbers etc. His introversion, friendliness and shyness, his deep insights into nature, combined with a dedication to beauty in theoretical physics, made him a legend all over the world already during his lifetime. For the latter half of his life he tried, unsuccessfully, to nd an alternative to quantum electrodynamics, of which he was the founder, as he was repelled by the problems of in nities. He died in Florida, where he lived and worked a er his retirement from Cambridge.
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•.
Page 551
So why use this horrible Hamiltonian? It is the only Hamiltonian that can be easily used for charged particles. Indeed, it is transformed to the one coupled to the electromagnetic eld by the so-called minimal coupling, i.e. by the substitution
p p − qA .
(531)
Ref. 730
that treats electromagnetic momentum like particle momentum. With this prescription, Dirac’s Hamiltonian describes the motion of charged particles interacting with an electromagnetic eld Paul Dirac A. is substitution is not possible in the Foldy–Wouthuysen Hamiltonian. In the Dirac representation, particles are pure, point-like, structureless electric charges; in the Foldy–Wouthuysen representation they acquire a charge radius and a magnetic moment interaction. (We come back to the reasons below, in the section on QED.)
In more detail, the simplest description of an electron (or any other elementary, stable, electrically charged particle of spin ) is given by the equations
Dvipsbugw
dρ dt
=
[H, ρ]
HDirac = βmc + α ë (p − qA(x, t))c + qφ(x, t) with
−i
α=
α=
i −i
α=
−
i
−
β= HMaxwell = .
− −
(532)
e rst Hamiltonian describes how charged particles are moved by electromagnetic elds, and the second describes how elds are moved by charged particles. Together, they form what is usually called quantum electrodynamics or QED for short.
As far as is known today, the relativistic description of the motion of charged matter and electromagnetic elds given by equation ( ) is perfect: no di erences between theory and experiment have ever been found, despite intensive searches and despite a high reward for anybody who would nd one. All known predictions completely correspond with the measurement results. In the most spectacular cases, the correspondence between theory and measurement is more than fourteen digits. But the precision of QED is less interesting than those of its features that are missing in classical electrodynamics. Let’s have a quick tour.
– CS – more to come here – CS –
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A
Ref. 732
Antimatter is now a household term. Interestingly, the concept was formed before any experimental evidence for it was known. Indeed, the antimatter companion of the electron was predicted in by Paul Dirac from his equation. Without knowing this prediction, Carl Anderson discovered it in and called it positron, even though ‘positon’, without the ‘r’, would have been the correct name. Anderson was studying cosmic rays and noticed that some ‘electrons’ were turning the wrong way in the magnetic eld he had applied to his apparatus. He checked everything in his machine and nally deduced that he found a particle with the same mass as the electron, but with positive electric charge.
e existence of positrons has many strange implications. Already in , before their discovery, the swedish theorist Oskar Klein had pointed out that Dirac’s equation for electrons makes a strange prediction: when an electron hits a su ciently steep potential wall, the re ection coe cient is larger than unity. Such a wall will re ect more than what is thrown at it. In , a er the discovery of the positron, Werner Heisenberg and Hans Euler explained the paradox. ey found that the Dirac equation predicts a surprising e ect: if an electric eld exceeds the critical value of
Dvipsbugw
Ec
=
mec eλe
=
me c eħ
= . EV m ,
(533)
Page 875
the vacuum will spontaneously generate electron–positron pairs, which then are separated by the eld. As a result, the original eld is reduced. is so-called vacuum polarization is also the reason for the re ection coe cient greater than unity found by Klein, since steep potentials correspond to high electric elds.
Truly gigantic examples of vacuum polarization, namely around charged black holes, will be described later on.
We note that such e ects show that the number of particles is not a constant in the microscopic domain, in contrast to everyday life. Only the di erence between particle number and antiparticle number turns out to be conserved. is topic will be expanded in the chapter on the nucleus.
Of course, the generation of electron–positron pairs is not a creation out of nothing, but a transformation of energy into matter. Such processes are part of every relativistic description of nature. Unfortunately, physicists have the habit to call this transformation ‘creation’ and thus confuse this issue somewhat. Vacuum polarization is a process transforming, as we will see, virtual photons into matter. at is not all: the same can also be done with real photons.
V
QED
Page 733
Contrary to what was said so far, there is a case where actions smaller than the minimal one do play a role. We already encountered an example: in the collision between two electrons, there is an exchange of virtual photons. We know that the exchanged virtual photon cannot be observed. Indeed, the action value S for this exchange obeys
S ħ.
(534)
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•.
In short, virtual particles are those particles that appear only as mediators in interactions; they cannot be observed. Virtual particles are intrinsically short-lived; they are the opposite of free or real particles. In a certain sense, virtual particles are particles bound both in space and time.
– CS – more to come – CS –
In summary, all virtual matter and radiation particle-antiparticles pairs together form what we call the vacuum; in addition, virtual radiation particles form static elds. Virtual particles are needed for a full description of interactions, and in particular, they are responsible for every decay process.
We will describe a few more successes of quantum theory shortly. Before we do that, we settle one important question.
Dvipsbugw
C
When is an object composite? Quantum theory gives several pragmatic answers. e rst one is somewhat strange: an object is composite when its gyromagnetic ratio is di erent than the one predicted by QED. e gyromagnetic ratio γ is de ned as the ratio between the magnetic moment M and the angular momentum L. In other terms,
M = γL .
(535)
Page 897
e gyromagnetic ratio γ is measured in s− T− =C kg and determines the energy levels of magnetic spinning particles in magnetic elds; it will reappear later in the context of magnetic resonance imaging. All candidates for elementary particles have spin . e gyromagnetic ratio for spin particles of mass m can be written as
γ
=
M ħ
=
e m
.
(536)
Challenge 1236 ny Page 897
( e expression eħ m is o en called the magneton of the particle; the dimensionless factor g/ is o en called the gyromagnetic ratio as well; this sometimes leads to confusion.) e criterion of elementarity thus can be reduced to a criterion on the value of the dimensionless number , the so-called -factor. If the -factor di ers from the value predicted by QED for point particles, about . , the object is composite. For example, a He+ helium ion has a spin and a value of . ë . Indeed, the radius of the helium ion is ë − m, obviously nite and the ion is a composite entity. For the proton, one measures a -factor of about . . Indeed, experiments yield a nite proton radius of about . fm.
Also the neutron, which has a magnetic moment despite being neutral, must therefore be composite. Indeed, its radius is approximately that of the proton. Similarly, molecules, mountains, stars and people must be composite. Following this rst criterion, the only elementary particles are leptons – i.e. electrons, muons, tauons and neutrinos –, quarks and intermediate bosons – i.e. photons, W-bosons, Z-bosons and gluons. More details on these particles will be uncovered in the chapter on the nucleus.
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Ref. 733
Another simple criterion for compositeness has just been mentioned: any object with a measurable size is composite. is criterion produces the same list of elementary particles as the rst. Indeed, this criterion is related to the previous one. e simplest models for composite structures predicts that the -factor obeys
−
=
R λC
(537)
Challenge 1237 e
where R is the radius and λC = h mc the Compton wavelength of the system. e expression is surprisingly precise for helium ions, helium , tritium ions and protons, as you might want to check.
A third criterion for compositeness is more general: any object larger than its Compton length is composite. e background idea is simple. An object is composite if one can detect internal motion, i.e. motion of some components. Now the action of any part with mass mpart moving inside a composed system of size r follows
Dvipsbugw
Spart < π r mpart c < π r m c
(538)
where m is the mass of the composite object. On the other hand, following the principle of quantum theory, this action, to be observable, must be larger than ħ . Inserting this condition, we nd that for any composite object*
r
ħ πmc
.
(539)
e right hand side di ers only by a factor π from the so-called Compton (wave)length
λ
=
h mc
.
(540)
Challenge 1239 ny
of an object. Any object larger than its own Compton wavelength is thus composite. Any object smaller than the right hand side of expression ( ) is thus elementary. Again, only leptons, including neutrinos, quarks and intermediate bosons pass the test. All other objects are composite, as the tables in Appendix C make clear. is third criterion produces the same list as the previous ones. Can you explain the reason?
Interestingly, the topic is not over yet. Even stranger statements about compositeness will appear when gravity is taken into account. Just be patient; it is worth it.
C Colours are at least as interesting in quantum theory as they are in classical electrodynamics.
**
Challenge 1238 ny * Can you nd the missing factor of 2? And is the assumption valid that the components must always be lighter than the composite?
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•.
If atoms contain orbiting electrons, the rotation of the Earth, via the Coriolis acceleration, should have an e ect on their motion. is beautiful prediction is due to Mark Silverman; Ref. 726 the e ect is so small however, that is has not been measured yet.
Ref. 735
**
Light is di racted by material gratings. Can matter be di racted by light gratings? Surprisingly, it actually can, as predicted by Dirac and Kapitza in 1937. In 1986, this was accomplished with atoms. For free electrons the feat is more di cult; the clearest con-
rmation came in 2001, when the technology advances for lasers were used to perform a beautiful measurement of the typical di raction maxima for electrons di racted by a light grating.
Dvipsbugw
Ref. 726
**
Light is totally re ected when it is directed to a dense material under an angle so large that it cannot enter it any more. Interestingly, in the case that the material is excited, the totally re ected beam can be ampli ed. is has been shown by several Russian physicists.
**
Where is the sea bluest? Sea water is blue because it absorbs red and green light. Sea water can also be of bright colour if the sea oor re ects light. Sea water is o en also green, because it o en contains small particles that scatter or absorb blue light. Most frequently, this is soil or plankton. e sea is thus especially blue if it is deep, clear and cold, so that it is low in plankton content. (Satellites determine plankton content from the ‘greenness’ of the sea.) ere is a place where the sea is deep, cold and quiet for most parts of the year: the Sargasso sea. It is o en called the bluest spot of the oceans.
T
Page 578
e great Wolfgang Pauli used to say that a er his death, the rst question he would ask the devil would be an explanation of Sommerfeld’s ne structure constant. (People used to comment that a er the devil will have explained it to him, he would think a little, and then snap ‘Wrong!’) e name ne structure constant was given by Arnold Sommerfeld to the dimensionless constant of nature given by
α=
e πε ħc
.
() .
( ).
(541)
is number rst appeared in explanations for the ne structure of certain atomic colour spectra, hence its name. Sommerfeld was the rst to understand its general importance.
e number is central to quantum electrodynamics for several reasons. First of all, it describes the strength of electromagnetism. Since all charges are multiples of the electron charge, a higher value would mean a stronger attraction or repulsion between charged bodies. e value of α thus determines the size of atoms, and thus the size of all things, as well as all colours.
Secondly, only because this number is quite a bit smaller than unity are we able to talk about particles at all. e argument is somewhat involved; it will be detailed later
Dvipsbugw
Challenge 1240 ny Ref. 740
Challenge 1241 n
on. In any case, only the small value of the ne structure constant makes it possible to distinguish particles from each other. If the number were near or larger than one, particles would interact so strongly that it would not be possible to observe or to talk about particles at all.
is leads to the third reason for the importance of the ne structure constant. Since it is a dimensionless number, it implies some yet unknown mechanism that xes its value. Uncovering this mechanism is one of the challenges remaining in our adventure. As long as the mechanism remains unknown, we do not understand the colour and size of a single thing around us.
Small changes in the strength of electromagnetic attraction between electrons and protons would have numerous important consequences. Can you describe what would happen to the size of people, to the colour of objects, to the colour of the Sun or to the workings of computers if the strength would double? And if it would drop to half the usual value over time?
Explaining the number is the most famous and the toughest challenge of modern physics since the issue appeared in the s. It is the reason for Pauli’s request to the devil. In
, during his Nobel Prize lecture, he repeated the statement that a theory that does not determine this number cannot be complete. e challenge is so tough that for the rst
years there were only two classes of physicists: those who did not even dare to take on the challenge, and those who had no clue. is fascinating story still awaits us.
e topic of the ne structure constant is so deep that it leads many astray. For example, it is o en heard that in physics it is impossible to change physical units in such a way that ħ, c and e are equal to at the same time; these voices suggest that doing so would not allow that the number . ... would keep its value. Can you show that the argument is wrong, and that doing so does not a ect the ne structure constant?
To continue with the highest e ciency on our path across quantum theory, we rst look at two important topics: the issue of indistinguishability and the issue of interpretation of its probabilities.
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B
677 G
F
, Chi l’ha detto?, Hoepli, Milano, . Cited on page .
678 e quantum of action is introduced in M P
, Über irreversible Strahlungsvor-
gänge, Sitzungsberichte der Preußischen Akademie der Wissenscha en, Berlin, pp. –
, . In this paper, Planck used the letter b for what nowadays is called h. Cited on
page .
679 Bohr explained the indivisibilty of the quantum of action in his famous Como lecture. N.
B , Atomtheorie und Naturbeschreibung, Springer, Berlin, . On page he writes:
“No more is it likely that the fundamental concepts of the classical theories will ever become
super uous for the description of physical experience. e recognition of the indivisibility
of the quantum of action, and the determination of its magnitude, not only depend on an
analysis of measurements based on classical concepts, but it continues to be the application
of these concepts alone that makes it possible to relate the symbolism of the quantum theory
to the data of experience.” He also writes: “...the fundamental postulate of the indivisibility
of the quantum of action is itself, from the classical point of view, an irrational element
which inevitably requires us to forgo a causal mode of description and which, because of
the coupling between phenomena and their observation, forces us to adopt a new mode
of description designated as complementary in the sense that any given application of
classical concepts precludes the simultaneous use of other classical concepts which in a
di erent connection are equally necessary for the elucidation of the phenomena...” and
“...the nite magnitude of the quantum of action prevents altogether a sharp distinction
being made between a phenomenon and the agency by which it is observed, a distinction
which underlies the customary concept of observation and, therefore, forms the basis of
the classical ideas of motion.” Other statements about the indivisibility of the quantum of
action can be found in N. B , Atomic Physics and Human Knowledge, Science Editions,
New York, . See also M J
, e Philosophy of Quantum Mechanics, Wiley,
rst edition, , pp. – .
For some of the rare modern publications emphasizing the quantum of action see M.B.
M
, Physics Letters A e action uncertainty principle and quantum gravity, 162,
p. , , and M.B. M
, e action uncertainty principle in continuous quantum
measurements, Physics Letters A 155, pp. – , .
Schwinger’s quantum action principle is also used in R
F.W. B
, Atoms
in Molecules - A Quantum eory, Oxford University Press, .
ere is a large number of general textbooks on quantum theory. e oldest textbooks
are obviously German or English. e choice should be taken by the reader, considering
his own preferences.
A well-known conceptual introduction is J -M
L -L
&
F
B
, Quantique – Rudiments, Masson, , translated into English
as Quantics, North-Holland, .
One of the most beautiful books is J
S
, Quantum Mechanics - Sym-
bolism of Atomic Measurements, edited by Berthold-Georg Englert, Springer Verlag, .
A modern approach with a beautiful introduction is M S
&G
W , Quantentheorie – Grundlagen und Anwendungen, Spektrum Akademischer Verlag,
.
A standard beginner’s text is C. C
-T
, B. D & F. L
, Mécanique
quantique I et II, Hermann, Paris, . It is also available in several translations.
A good text is A
P , Quantum theory – Concepts and Methods, Kluwer, .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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For a lively approach, see V
I , e Force of Symmetry, Cambridge Univer-
sity Press/ .
Other textbooks regularly appear around the world. Cited on page .
680 M B , Zur Quantenmechanik der Stoßvorgänge (vorläu ge Mitteilung), Zeitschri für Physik 37, pp. – , , M B , Quantenmechanik der Stoßvorgänge, Zeitschri für Physik 38, pp. – , . Cited on page .
681 See for example the papers by J H
, e uncertainty principle for energy
and time, American Journal of Physics 64, pp. – , , and by P B , On the
time–energy uncertainty reaction, part & , Foundations of Physics 20, pp. – , . A
classic is the paper by E
P. W
, On the time–energy uncertainty reaction, in
A
S
&E
P. W
, editors, Aspects of Quantum eory, Cambridge
University Press, . Cited on page .
682 See also the booklet by C
M
, Warum alles kaputt geht - Form und Versagen
in Natur und Technik, Forschungszentrum Karlsruhe, . Cited on page .
683 R. C
, J. B & H. H
, Characterizing quantum theory in terms of
information-theoretic constraints, http://www.arxiv.org/abs/quant-ph/
. Cited on
page .
684 e quotes on motion are found in chapter VI of F. E
, Herrn Eugen Dührings Um-
wälzung der Wissenscha , Verlag für fremdsprachliche Literatur, Moskau, . e book is
commonly called Anti–Dühring. Cited on pages and .
685 E.M. B on page .
& S.I. V
, Izvest. Akad. Nauk. Omen Ser. 7, p. , . Cited
686 R
L
page .
, e Quantum eory of Light, Oxford University Press, . Cited on
687 F. R
& D.A. B
, Single-photon detection by rod cells of the retina, Reviews of
Modern Physics 70, pp. – , . ey also mention that the eye usually works at
photon uxes between µm s (sunlight) and − µm s (starlight). e cones in the
retina detect, in colour, light intensities in the uppermost or decades, whereas the rods
detect, in black and white, the lower light intensities. Cited on page .
688 E. F
, H. K
, R.A. L
, A.T.Y. L & M. P
, New geomag-
netic limit on the photon mass and on long-range forces coexisting with electromagnetism,
Physical Review Letters 73, pp. – , . Cited on page .
689 A.H. C pp. – ,
, e scattering of X-rays as particles, American Journal of Physics 29, . Cited on page .
690 e famous paper is R. H
B
& R.Q. T , Nature 178, p. , .
ey got the idea to measure light in this way from their earlier work, which used the same
method with radiowaves: R. H
B
& R.Q. T , Nature 177, p. , ,
Cited on page .
691 Science September Cited on page .
692 L. M
, Con guration-space photon number operators in quantum optics, Physical
Review 144, pp. – , . No citations.
693 A. E
, Über einen die Erzeugung und Umwandlung des Lichtes betre enden heur-
istischen Standpunkt, Annalen der Physik 17, pp. – , . Cited on page .
694 See the summary by P.W. M
, Answer to question : What (if anything) does the
photoelectric e ect teach us?, American Journal of Physics 65, pp. – , . Cited on page
.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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695 For a detailed account, See J
J. P
, Poincaré’s proof of the quantum discon-
tinuity of nature, American Journal of Physics 63, pp. – , . e original papers are
HP
, Sur la théorie des quanta, Comptes Rendus de l’Académie des Sciences
(Paris) 153, pp. – , , as well as H
P
, Sur la théorie des quanta,
Journal de Physique (Paris) 2, pp. – , . Cited on page .
696 J. J
, G. B , I. C
& Y. Y
, Photonic de Broglie waves, Phys-
ical Review Letters 74, pp. – , . e rst measurement was published by E.J.S.
F
, C.H. M
& S. P
, Measurement of the de Broglie wavelength of
a multiphoton wave packet, Physical Review Letters 82, pp. – , . Cited on page
.
697 For the three photon state, see M.W. M
, J.S. L
& A.M. S
,
Super-resolving phase measurements with a multiphoton entangled state, Nature 429,
pp. – , , and for the four-photon state see, in the same edition, P. W
,
J.-W. P , M. A
, R. U , S. G
& A. Z
, De Broglie
wavelength of a non-local four-photon state, Nature 429, p. - , . Cited on page
.
698 For an introduction to spueezed light, see L. M
, Non-classical states of the electro-
magnetic eld, Physica Scripta T 12, pp. – , . Cited on page .
699 e famous quote on single photon interference is found on page of P.A.M. D , e Principles of Quantum Mechanics, Clarendon Press, Oxford, . It is also discussed,
somewhat confusely, in the otherwise informative article by H. P , Interference between independent photons, Reviews of Modern Physics 58, pp. – , . Cited on page .
700 e original papers on coherent states are three R.J. G
, e quantum theory of
optical coherence, Physical Review 130, pp. – , , J.R. K
, Continuous-
representation theory, I and II, Journal of Mathematical Physics 4, pp. – , , and
E.C.G. S
, Equivalence of semiclassical and quantum mechanical descriptions
of statistical light beams, Physical Review Letters 10, p. , . Cited on page .
701 R. K , J. A
& A. A
, Evolution of the modern photon, American Journal of
Physics 57, pp. – , , Cited on page .
702 W
T
, J. B
, H. Z
& N. G , Violation of bell inequalit-
ies by photons more than km apart, Physical Review Letters 81, pp. – , October
. Cited on page .
703 N. B
& L. R
, Zur Frage der Meßbarkeit der elektromagnetischen Feld-
größen, Mat.-fys. Medd. Danske Vid. Selsk. 12, p. , . e results were later published
in English language as N. B & L. R
, Field and charge measurements in
quantum electrodynamics, Physical Review 78, pp. – , . Cited on page .
704 E.I. B
, e rigid pendulum – an antique but evergreen physical model, European
Journal of Physics 20, pp. – , . Cited on page .
705 W. G
& O. S , Der experimentelle Nachweis des magnetischen Moments des
Silberatoms, Zeitschri für Physik 8, p. , . See also the pedagogical explanation by
M. H
, S. H , A. K
& A. W
, Quantum measurement and
the Stern–Gerlach experiment, American Journal of Physics 66, pp. – , . Cited on
page .
706 J. P
, Nobel Prize speech, http://www.nobel.se. H. N
, Kinetics of a system
of particles illustrating the line and the band spectrum and the phenomena of radioactivity,
Philosophical Magazine S , 7, pp. – , March . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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707 N. B , On the constitution of atoms and molecules: Introduction and Part I – binding of electrons by positive nuclei, Philosophical Magazine 26, pp. – , , On the constitution of atoms and molecules: Part II – systems containing only a single nucleus, ibid., pp. – , On the constitution of atoms and molecules: Part III, ibid., pp. – . Cited on page .
708 L. B
, Comptes rendus de l’Académie des Sciences 177, pp. – ,
on page .
. Cited
709 M. A
, O. N , J. V –A
, C. K
, G.
Z
& A.
Z
, Wave–particle duality of C molecules, Nature 401, pp. – , Octo-
ber . See also the observation for tetraphenyleprophyrin and C F by the same team,
published as L. H
& al., Wave nature of biomolecules and uorofullerenes,
Physical Review Letters 91, p.
,.
No experiment of quantum theory has been studied as much as quantum intereference.
Also the transition from interference to no interference has been epxlored, as in P. F
,
A. M
& S. P
, Mesoscopic interference, Recent Developments in Physics 3,
pp. – , . Cited on page .
710 P. C
& M.M. N , Phase and angle variables in quantum mechanics, Re-
view of Modern Physics 40, pp. – , . Cited on page .
711 e indeterminacy relation for rotational motion is well explained by W.H. L
,
Amplitude and phase uncertainty relations, Physics Letters 7, p. , . Cited on page .
712 S. F
-A
, S.M. B
, E. Y , J. L
, J. C
& M. P -
, Uncertainty principle for angular position and angular momentum, New Journal of
Physics 6, p. , . is is a freely accessible online journal. Cited on page .
713 E pp. pp.
S –, –,
, Quantisierung als Eigenwertproblem I, Annalen der Physik 79, , and Quantisierung als Eigenwertproblem II, Annalen der Physik 79, . Cited on page .
714 C.G. G , G. K & V.A. N
, From Maupertius to Schrödinger. Quantization
of classical variational principles, American Journal of Physics 67, pp. – , . Cited
on page .
715 e whole bunch of atoms behaves as one single molecule; one speaks of a Bose–Einstein
condensate. e rst observations, worth a Nobel prize, were by M.H. A
& al.,
Observation of Bose–Einstein condensation in a dilute atomic vapour, Science 269, pp. –
, , C.C. B
, C.A. S
, J.J. T
& R.G. H , Evidence of
bose-einstein condensation in an atomic gas with attractive interactions, Physical Review
Letters 75, pp. – , , K.B. D , M.-O. M , M.R. A
, N.J.
D
, D.S. D
, D.M. K & W. K
, Bose-Einstein Condensation in
a Gas of Sodium Atoms, Physical Review Letters 75, pp. – , . For a simple intro-
duction, see W. K
, Experimental studies of Bose–Einstein condensation, Physics
Today pp. – , December . Cited on page .
716 W.M. I
, D.J. H
, J.J. B
& D.J. W
, Quantum Zeno e ect,
Physical Review A 41, pp. – , . M.C. F
, B. G
-M
&
M.G. R
, Observation of the Quantum Zeno and Anti-Zeno e ects in an unstable sys-
tem, Physical Review Letters 87, p.
, , also http://www-arxiv.org/abs/quant-ph/
. Cited on page .
717 See P. F
, Z. H
, G. K
, S. P
&J. Ř
Quantum Zeno
tomography, Physical Review A, 66, p.
, . Cited on page .
718 R
H. D
&J
P. W
, Introduction to quantum theory, Addison–
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Wesley, Reading, Massachusetts, . See also S
G
ics, Jojn Wiley & Sons, . Cited on page .
, Quantum Phys-
719 G. P
, Shadows of a maximal acceleration, http://www.arxiv.org/abs/gr-qc/
.
Cited on page .
720 Y. A
& D. B , Signi cance of electromagnetic potentials in the quantum
theory, Physical Review 115, pp. – , . No citations.
721 G.E. U
& S. G
, Ersetzung der Hypothese vom unmechanischen
Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons,
Naturwissenscha en 13, pp. – , . Cited on page .
722 L. T .
, e motion of the spinning electron, Nature 117, p. , . Cited on page
723 K.
M
& E. S
. Cited on page .
, Wolfgang Pauli, Physics Today pp. – , February
724 R. P , D. S
, J. R
, J.-F. L B
& J.-P. K , ISAAC/VLT
observations of a lensed galaxy at z= . , Astronomy and Astrophysics 416, p. L , .
Cited on page .
725 A pedagogical introduction is given by L.J. C
& D.G. E , Use of the Einstein–
Brillouin–Keller action quantization, American journal of Physics 72, pp. – , .
Cited on pages and .
726 M
P. S
, And yet it moves: strange systems and subtle questions in physics,
Cambridge University Press . A beautiful book from an expert of motion. Cited on
pages and .
727 J. N 59, pp.
& al., Spectroscopy of Rydberg atoms at n – , . Cited on page .
, Physical Review Letters
728 is is explained by J.D. H , Mystery error in Gamow’s Tompkins reappears, Physics Today pp. – , May . No citations.
729 L.L. F
& S.A. W
, On the Dirac theory of spin / particles and its non-
relativistic limit, Physical Review 78, pp. – , . Cited on page .
730 L.L. F
, e electromagnetic properties of Dirac particles, Physical Review 83, pp. –
, . L.L. F
, e electron–neutron interaction, Physical Review 83, pp. – ,
. L.L. F
, Electron–neutron interaction, Review of Modern Physics 30, pp. – ,
. Cited on page .
731 J
P. C
&B
H.J. M K
, e Foldy–Wouthuysen transforma-
tion, American Journal of Physics 63, pp. – , . Cited on page .
732 H. E
& B. K
, Naturwissenscha en 23, p. , , H. E , Annalen der
Physik 26, p. , , W. H
& H. E , Folgerung aus der Diracschen e-
orie des Electrons, Zeitschri für Physik 98, pp. – , . Cited on page .
733 e g-factors for composite nuclei are explained by ... See also H D
, Is the
electron a composite particle?, Hyper ne Interactions 81, pp. – , . Cited on page .
734 J.P. W
, G. N
& I. K
, Is it possible to rotate an atom?, Optics
Communications 93, pp. – , . We are talking about atoms rotating around their
centre of mass; atoms can of course rotate around other bodies, as discussed by M.P. S -
, Circular birefringence of an atom in uniform rotation: the classical perspective,
American Journal of Physics 58, pp. – , . Cited on page .
735 For the atomic case, see P.L. G
, G.A. R & D.E. P
, Di raction of
atoms by light: the near resonant Kapitza–Dirac e ect, Physical Review Letters 56, pp. –
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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, . Many ealry experimental attempts, in particular those performed in the s,
for the di raction of electrons by light were controversial; most showed only the de ec-
tion of electrons, as explained by H. B
, Contemporary Physics 41, p. , .
Later on, he and his group performed the newest and most spectacular experiment, demon-
strating real di raction, including interference e ects; it is described in D.L. F
,
K. A
& H. B
, Observation of the Kapitza–Dirac e ect, Nature 413,
pp. – , . Cited on page .
736 J. S
, M.S. C
, C.R. E
, T.D. H
, S. W -
& D.E. P
, Index of refraction of various gases for sodium matter
waves, Physical Review Letters 74, p. - , . Cited on page .
737 e nearest to an image of a hydrogen atom is found in A. Y
, Watching an atom
tunnel, Nature 409, pp. – , . e experiments on Bose–Einstein condensates are
also candidates for images of hydrogen atoms. e company Hitachi made a fool of itself in
by claiming in a press release that its newest electron microscope could image hydrogen
atoms. Cited on page .
738 An introduction is given by P. P
& G. R
, Wie fängt man ein Atom mit einem
Photon?, Physikalische Blätter 56, pp. – , . Cited on page .
739 A.M. W
, Kinetic energy, size, and the uncertainty principle, American Journal of
Physics 42, pp. – , . Cited on page .
740 W ber
P , Exclusion principle and quantum mechanics, Nobel lecture, Decem, in Nobel Lectures, Physics, Volume , - , Elsevier, . Cited on page .
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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C
VI
PERMUTATION OF PARTICLES
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W are we able to distinguish twins from each other? Why can we distinguish hat looks alike, such as a copy from an original? Most of us are convinced that henever we compare an original with a copy, we can nd a di erence. is conviction turns out to be correct, even though it is a quantum e ect that is in contrast with classical physics.
Indeed, quantum theory has a lot to say about copies and their di erences. ink about any method that allows to distinguish objects: you will nd that it runs into trouble for Challenge 1242 n point-like particles. erefore in the quantum domain something must change about our ability to distinguish particles and objects. Let us explore the issue.
.
?
Some usually forgotten properties of objects are highlighted by studying a pretty combinRef. 741 atorial puzzle: the glove problem. It asks:
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
How many surgical gloves (for the right hand) are necessary if m doctors need to operate w patients in a hygienical way, so that nobody gets in contact with the body uids of anybody else?
e same problem also appears in other settings. For example, it also applies to condoms, men and women – and is then (o cially) called the condom problem – or to computers, interfaces and computer viruses. In fact, the term ‘condom problem’ is the term used in the books that discuss it. Obviously, the optimal number of gloves is not the product wm. In fact, the problem has three subcases.
Challenge 1243 e
Challenge 1244 e Ref. 742
Challenge 1245 e
— e simple case m = w = already provides the most important ideas needed. Are
you able to nd the optimal solution and procedure? — In the case w = and m odd or the case m = and w odd, the solution is (m + )
gloves. is is the optimal solution, as you can easily check yourself. — A solution with a simple procedure for all other cases is given by w + m gloves,
where x means the smallest integer greater than or equal to x. For example, for two
men and three women this gives only three gloves. (However, this formula does not
always give the optimal solution; better values exist in certain subcases.)
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•.
Page 1051 Page 238
Two basic properties of gloves determine the solution to the puzzle. First, gloves have two sides, an interior and an exterior one. Secondly, gloves can be distinguished from each other. Do these two properties also apply to particles? We will discuss the issue of double-sidedness in the third part of the mountain ascent. In fact, the question whether particles can be turned inside out will be of great importance for their description and their motion. In the present chapter we concentrate on the second issue, namely whether objects and particles can always be distinguished. We will nd that elementary particles do not behave like gloves in these but in an even more surprising manner. (In fact, they do behave like gloves in the sense that one can distinguish right-handed from le -handed ones.)
In everyday life, distinction of objects can be achieved in two ways. We are able to distinguish objects – or people – from each other because they di er in their intrinsic properties, such as their mass, colour, size or shape. In addition, we are also able to distinguish objects if they have the same intrinsic properties. Any game of billiard suggests that by following the path of each ball, we can distinguish it from the others. In short, objects with identical properties can also be distinguished using their state.
e state of a billiard ball is given by its position and momentum. In the case of billiard balls, the state allows distinction because the measurement error for the position of the ball is much smaller than the size of the ball itself. However, in the microscopic domain this is not the case. First of all, atoms or other microscopic particles of the same type have the same intrinsic properties. To distinguish them in collisions, we would need to keep track of their motion. But we have no chance to achieve this. Already in the nineteenth century it was shown experimentally that even nature itself is not able to do this. is result was discovered studying systems which incorporate a large number of colliding atoms of the same type: gases.
e calculation of the entropy of a simple gas, made of N simple particles of mass m moving in a volume V , gives
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
S = kln
V
(
mkT πħ
)
N
+ kN + klnα
(542)
Challenge 1246 e Ref. 743
where k is the Boltzmann constant, T the temperature and ln the natural logarithm. In this formula, the pure number α is equal to if the particles are distinguishable, and equal to N! if they are not. Measuring the entropy thus allows us to determine α and therefore whether particles are distinguishable. It turns out that only the second case describes nature. is can be checked with a simple test: only in the second case does the entropy of two volumes of identical gas add up.* e result, o en called Gibbs’ paradox,** thus proves that the microscopic components of matter are indistinguishable: in a system of
Challenge 1247 ny * Indeed, the entropy values observed by experiment are given by the so-called Sackur–Tetrode formula
S = kNln
V N
(
mkT πħ
)
+ kN
(543)
which follows when α = N! is inserted above. ** Josiah Willard Gibbs (1839–1903), US-American physicist who was, with Maxwell and Planck, one of the three founders of statistical mechanics and thermodynamics; he introduced the concepts of ensemble and of phase.
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m
m F I G U R E 311 Identical objects with crossing paths
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
microscopic particles, there is no way to say which particle is which. Indistinguishability is an experimental property of nature.*
e properties of matter would be completely di erent without indistinguishability. For example, we will discover that without it, knifes and swords would not cut. In addition, the soil would not carry us; we would fall right through it. To illuminate the issue in more detail, we explore the next question.
W
?
Take two microscopic particles with the same mass, the same composition and the same shape, such as two atoms. Imagine that their paths cross, and that they approach each other to small distances at the crossing, as shown in Figure . In a gas, both the collision of atoms or a near miss are examples. Experiments show that at small distances it is impossible to say whether the two particles have switched roles or not. is is the main reason that makes it impossible in a gas to follow particles moving around and to determine which particle is which. is impossibility is a consequence of the quantum of action.
For a path that brings two approaching particles very close to each other, a role switch requires only a small amount of change, i.e. only a small (physical) action. However, we know that there is a smallest observable action in nature. Keeping track of each particles at small distances would require action values smaller than the minimal action observed in nature. e existence of a smallest action makes it thus impossible to keep track of microscopic particles when they come too near to each other. Any description of several particles must thus take into account that a er a close encounter, it is impossible to say which is which.
In short, indistinguishability is a consequence of the existence of a minimal action in nature. is result leads straight away to the next question:
C
?
In everyday life, objects can be counted because they can be distinguished. Since quantum particles cannot be distinguished, we need some care in determining how to count them.
e rst step is the de nition of what is meant by a situation without any particle at all.
* When radioactivity was discovered, people thought that it contradicted the indistinguishability of atoms, as decay seems to single out certain atoms compared to others. But quantum theory then showed that this is not the case and that atoms do remain indistinguishable.
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•.
is seems an easy thing to do, but later on we will encounter situations where already this step runs into di culties. In any case, the rst step is thus the speci cation of the vacuum. Any counting method requires that situations without particles be clearly separated from situations with particles.
e second step is the speci cation of an observable useful for determining particle number. e easiest way is to chose one of those quantum numbers which add up under composition, such as electric charge.* Counting is then performed by measuring the total charge and dividing by the unit charge.
is method has several advantages. First of all, it is not important whether particles are distinguishable or not; it works in all cases. Secondly, virtual particles are not counted. is is a welcome state of a airs, as we will see, because for virtual particles, i.e. for particles for which E p c + m c , there is no way to de ne a particle number anyway.
e side e ect of the counting method is that antiparticles count negatively! Also this consequence is a result of the quantum of action. We saw above that the quantum of action implies that even in vacuum, particle–antiparticle pairs are observed at su ciently high energies. As a result, an antiparticle must count as minus one particle. In other words, any way of counting particles can produce an error due to this e ect. In everyday life this limitation plays no role, as there is no antimatter around us. e issue does play a role at higher energies, however. It turns out that there is no general way to count the exact number of particles and antiparticles separately; only the sum can be de ned. In short, quantum theory shows that particle counting is never perfect.
In summary, nature does provide a way to count particles even if they cannot be distinguished, though only for everyday, low energy conditions; due to the quantum of action, antiparticles count negatively, and provide a limit to the counting of particles at high energies.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
W
?
Since particles are countable but indistinguishable, there exists a symmetry of nature for systems composed of several identical particles. Permutation symmetry, also called exchange symmetry, is the property of nature that observations are unchanged under exchange of identical particles. Together with space-time symmetry, gauge symmetry and the not yet encountered renormalization symmetry, permutation symmetry forms one of the four pillars of quantum theory. Permutation symmetry is a property of composed systems, i.e. of systems made of many (identical) subsystems. Only for such systems does indistinguishability play a role.
In other words, ‘indistinguishable’ is not the same as ‘identical’. Two particles are not the same; they are more like copies of each other. On the other hand, everyday life experience shows us that two copies can always be distinguished under close inspection, so that the term is not fully appropriate either. In the microscopic domain, particles are countable and completely indistinguishable.** Particles are perfect copies of each other.
Challenge 1248 n
* In everyday life, the weight or mass is commonly used as observable. However, it cannot be used in the quantum domain, except for simple cases. Can you give at least two reasons, one from special relativity and one from general relativity? ** e word ‘indistinguishable’ is so long that many physicists sloppily speak of ‘identical’ particles nevertheless. Take care.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
We will discover shortly that permutation is partial rotation. Permutation symmetry Challenge 1249 e thus is a symmetry under partial rotations. Can you nd out why?
I
Challenge 1250 n
e indistinguishability of particles leads to important conclusions about the description
of their state of motion. is happens because it is impossible to formulate a description
of motion that includes indistinguishability right from the start. Are you able to con rm
this? As a consequence we describe a n-particle state with a state Ψ ...i... j...n which assumes that distinction is possible, as expressed by the ordered indices in the notation,
and we introduce the indistinguishability a erwards. Indistinguishability means that the
exchange of any two particles results in the same physical system.* Now, two quantum
states have the same physical properties if they di er at most by a phase factor; indistin-
guishability thus requires
Ψ ...i...j...n = eiα Ψ ... j...i...n
(544)
for some unknown angle α. Applying this expression twice, by exchanging the same couple of indices again, allows us to conclude that e iα = . is implies that
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Ψ ...i...j...n = Ψ ...j...i...n ,
(545)
in other words, a wave function is either symmetric or antisymmetric under exchange of indices. Quantum theory thus predicts that particles are indistinguishable in one of two distinct ways.** Particles corresponding to symmetric wave functions are called bosons, those corresponding to antisymmetric wave functions are called fermions.***
Experiments show that the behaviour depends on the type of particle. Photons are bosons. On the other hand, electrons, protons and neutrons are found to be fermions. Also about half of the atoms are found to behave as bosons (at moderate energies). In fact, a composite of an even number of fermions (at moderate energies) – or of any number of bosons (at any energy) – turns out to be a boson; a composite of an odd number of fermions is (always) a fermion. For example, almost all of the known molecules are bosons (electronically speaking). Fermionic molecules are rather special and even have a special name in chemistry; they are called radicals and are known for their eagerness to react and to form normal bosonic molecules. Inside the human body, too many radicals can
Ref. 744
* We therefore have the same situation that we encountered already several times: an overspeci cation of the mathematical description, here the explicit ordering of the indices, implies a symmetry of this description, which in our case is a symmetry under exchange of indices, i.e., under exchange of particles. ** is conclusion applies to three-dimensional space only. In two dimensions there are more possibilities. *** e term ‘fermion’ is derived from the name of the Italian physicist and Nobel Prize winner Enrico Fermi (b. 1901 Roma, d. 1954 Chicago) famous for his all-encompassing genius in theoretical and experimental physics. He mainly worked on nuclear and elementary particle physics, on spin and on statistics. For his experimental work he was called ‘quantum engineer’. He is also famous for his lectures, which are still published in his own hand-writing, and his brilliant approach to physical problems. Nevertheless, his highly deserved Nobel Prize was one of the few cases in which the prize was given for a discovery which turned out to be incorrect.
‘Bosons’ are named a er the Indian physicist Satyenra Nath Bose (b. 1894 Calcutta, d. 1974 Calcutta) who rst described the statistical properties of photons. e work was later expanded by Albert Einstein.
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mirrors
source
two identical photons
•.
beam splitter
detectors
possible light paths
F I G U R E 312 Two photons and interference
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 1251 ny
have adverse e ects on health; it is well known that vitamin C is important because it is e ective in reducing the number of radicals.
To which class of particles do mountains, trees, people and all other macroscopic objects belong?
T
Ref. 745
A simple experiment allows to determine the behaviour of photons. Take a source that emits two photons of identical frequency and polarization at the same time, as shown in Figure . In the laboratory, such a source can be realized with a down-converter, a material that converts a photon of frequency ω into two photons of frequency ω. Both photons, a er having travelled exactly the same distance, are made to enter the two sides of a beam splitter (for example, a half-silvered mirror). At the two exits of the beam splitter are two detectors. Experiments show that both photons are always detected together on the same side, and never separately on opposite sides. is result shows that photons are bosons. Fermions behave in exactly the opposite way; two fermions are always detected separately on opposite sides, never together on the same side.
T
If experiments force us to conclude that nobody, not even nature, can distinguish any two particles of the same type, we deduce that they do not form two separate entities, but some sort of unity. Our naive, classical sense of particle as a separate entity from the rest of the world is thus an incorrect description of the phenomenon of ‘particle’. Indeed, no experiment can track particles with identical intrinsic properties in such a way that they can be distinguished with certainty. is impossibility has been checked experimentally with all elementary particles, with nuclei, with atoms and with numerous molecules.
How does this t with everyday life, i.e. with classical physics? Photons do not worry us much here. Let us focus the discussion on matter particles. We know to be able to distinguish electrons by pointing to F I G U R E 313 Particles as localized excitations the wire in which they ow, and we can distinguish our fridge from that of our neighbour. While the quantum of action makes distinction impossible, everyday life allows it. e simplest explanation is to ima-
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gine a microscopic particle, especially an elementary one, as a bulge, i.e. as a localized excitation of the vacuum. Figure shows two such bulges representing two particles. It is evident that if particles are too near to each other, it makes no sense to distinguish them; we cannot say any more which is which.
e bulge image shows that either for large distances or for high potential walls separating them, distinction of identical particles does become possible. In such situations, measurements allowing to track them independently do exist. In other words, we can specify a limit energy at which permutation symmetry of objects or particles separated by a distance d becomes important. It is given by
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E
=
cħ d
.
(546)
Challenge 1252 ny Challenge 1253 ny
Are you able to con rm the expression? For example, at everyday temperatures we can distinguish atoms inside a solid from each other, since the energy so calculated is much higher than the thermal energy of atoms. To have fun, you might want to determine at what energy two truly identical human twins become indistinguishable. Estimating at what energies the statistical character of trees or fridges will become apparent is then straightforward.
e bulge image of particles thus purveys the idea that distinguishability exists for objects in everyday life but not for particles in the microscopic domain. To sum up, in daily life we are able to distinguish objects and thus people for two reasons: because they are made of many parts, and because we live in a low energy environment.
e energy issue immediately adds a new aspect to the discussion. How can we describe fermions and bosons in the presence of virtual particles and of antiparticles?
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I
Quantum eld theory, as we will see shortly, simply puts the bulge idea of Figure into mathematical language. A situation with no bulge is called vacuum state. Quantum eld theory describes all particles of a given type as excitations of a single fundamental eld. Particles are indistinguishable because each particle is an excitation of the same basic substrate and each excitation has the same properties. A situation with one particle is then described by a vacuum state acted upon by a creation operator. Adding a second particle is described by adding a second creation operator, and subtracting a particle by adding a annihilation operator; the latter turns out to be the adjunct of the former.
Quantum eld theory then studies how these operators must behave to describe observations.* It arrives at the following conclusions:
— Fields with half-integer spin are fermions and imply (local) anticommutation.
* Whenever the relation
[b, b†] = bb† − b†b =
(547)
holds between the creation operator b† and the annihilation operator b, the operators describe a boson. If the operators for particle creation and annihilation anticommute
d, d† = dd† + d†d =
(548)
they describe a fermion. e so de ned bracket is called the anticommutator bracket.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•.
— Fields with integer spin are bosons and imply (local) commutation. — For all elds at spacelike separations, the commutator – respectively anticommutator
– vanishes. — Antiparticles of fermions are fermions, and antiparticles of bosons are bosons. — Virtual particles behave like their real counterparts.
ese connections are at the basis of quantum eld theory. ey describe how particles are identical. But why are they? Why are all electrons identical? Quantum eld theory describes electrons as identical excitations of the vacuum, and as such as identical by construction. Of course, this answer is only partially satisfying. We will nd a better one only in the third part of our mountain ascent.
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H
?
Ref. 746
A simple but e ective experiment testing the fermion behaviour of electrons was carried out by Ramberg and Snow. ey sent an electric current of A through a copper wire for one month and looked for X-ray emission. ey did not nd any. ey concluded that electrons are always in an antisymmetric state, with a symmetric component of less than
ë−
(549)
of the total state. Electrons are always in an antisymmetric state: thus they are fermions. e reasoning behind this elegant experiment is the following. If electrons would not
always be fermions, every now and then an electron could fall into the lowest energy level of a copper atom, leading to X-ray emission. e lack of such X-rays implies that electrons are fermions to a very high accuracy. X-rays could be emitted only if they were bosons, at least part of the time. Indeed, two electrons, being fermions, cannot be in the same state: this restrition is called the Pauli exclusion principle. It applies to all fermions and is our next topic.
C
,
Ref. 748
Can classical systems be indistinguishable? ey can: large molecules are examples – provided they are made of exactly the same isotopes. Can large classical systems, made of a mole or more particles be indistinguishable? is simple question e ectively asks whether a perfect copy, or (physical) clone of a system is possible.
It could be argued that any factory for mass-produced goods, such as one producing shirt buttons or paper clips, shows that copies are possible. But the appearance is deceiving. On a microscope there is usually some di erence. Is this always the case? In , the Dutch physicist Dennis Dieks and independently, the US-American physicists Wootters and Zurek, published simple proofs that quantum systems cannot be copied. is is the famous no-cloning theorem.
A copying machine is a machine that takes an original, reads out its properties and produces a copy, leaving the original unchanged. is seems de nition seems straightforward. However, we know that if we extract information from an original, we have to interact with it. As a result, the system will change at least by the quantum of action. We thus expect that due to quantum theory, copies and originals can never be identical.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Quantum theory proves this in detail. A copying machine is described by an operator that maps the state of an original system to the state of the copy. In other words, a copying machine is linear. is linearity leads to a problem. Simply stated, if a copying machine were able to copy originals either in state A or in state B , it could not decide what to do if the state of the original were A + B . On one hand, the copy should be A + B ; on the other hand, the linearity of the copier forbids this. Indeed, a copier is a device described by an operator U that changes the starting state s c of the copy in the following way:
— If the original is in state A , a copier acts as
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UA s c= A Ac.
(550)
— If the original is in state B , a copier acts as
UB s c= B B c.
(551)
As a result of these two requirements, an original in the state A + B is treated by the
copier as
U A+B s c = A A c+ B B c .
(552)
is is in contrast to what we want, which would be
Uwanted A + B s c = ( A + B )( A c + B c) .
(553)
Ref. 749 Challenge 1254 n
In other words, a copy machine cannot copy a state completely.* is is the no-cloning theorem.
e impossibility of copying is implicit in quantum theory. If we were able to clone systems, we could to measure a variable of a system and a second variable on its copy. We would be thus able to beat the indeterminacy relation. is is impossible. Copies are and always must be imperfect.
Other researchers then explored how near to perfection a copy can be, especially in the case of classical systems. To make a long story short, these investigations show that also the copying or cloning of macroscopic systems is impossible. In simple words, copying machines do not exist. Copies can always be distinguished from originals if observations are made with su cient care. In particular, this is the case for biological clones; biological clones are identical twins born following separate pregnancies. ey di er in their nger prints, iris scans, physical and emotional memories, brain structures, and in many other aspects. (Can you specify a few more?) In short, biological clones, like identical twins, are not copies of each other.
e lack of quantum mechanical copying machines is disappointing. Such machines, or teleportation devices, could be fed with two di erent inputs, such as a lion and a goat,
* e no-cloning theorem puts severe limitations on quantum computers, as computations o en need copies of intermediate results. It also shows that faster-than-light communication is impossible in EPR experiments. In compensation, quantum cryptography becomes possible – at least in the laboratory. Indeed, the no-cloning theorem shows that nobody can copy a quantum message without being noticed. e speci c ways to use this result in cryptography are the 1984 Bennett–Brassard protocol and the 1991 Ekert protocol.
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and produce a superposition: a chimaera. Quantum theory shows that all these imaginary beings cannot be realized.
In summary, everyday life objects such as photocopies, billiard balls or twins are always distinguishable. ere are two reasons: rst, quantum e ects play no role in everyday life, so that there is no danger of unobservable exchange; secondly, perfect clones of classical systems do not exist anyway, so that there always are tiny di erences between any two objects, even if they look identical at rst sight. Gloves can always be distinguished.
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Ref. 750 Page 192
We saw above that spin is the observation that matter rays, like light rays, can be polarized. Spin thus describes how particles behave under rotations, and it proves that particles are not simple spheres shrunk to points. We also saw that spin describes a fundamental di erence between quantum systems and gloves: spin speci es the indistinguishability of quantum systems. Let us explore this connection in more detail.
e general background for the appearance of spin was clari ed by Eugene Wigner in .* He started by recapitulating that any quantum mechanical particle, if elementary, must behave like an irreducible representation of the set of all viewpoint changes. is set forms the symmetry group of at space-time, the so-called inhomogeneous Lorentz group. We have seen in the chapter on classical mechanics how this connection between elementarity and irreducibility arises. To be of physical relevance for quantum theory, representations have to be unitary. e full list of irreducible unitary representations of viewpoint changes thus provides the range of possibilities for any particle that wants to be elementary. Cataloguing the possibilities, one nds rst of all that every elementary particle is described by four-momentum – no news so far – and by an internal angular momentum, the spin. Four-momentum results from the translation symmetry of nature, and spin from its rotation symmetry. e momentum value describes how a particle behaves under translation, i.e. under position and time shi of viewpoints. e spin value describes how an object behaves under rotations in three dimensions, i.e. under orientation change of viewpoints.** As is well known, the magnitude of four-momentum is an invariant property, given by the mass, whereas its orientation in space-time is free. Similarly, the magnitude of spin is an invariant property, and its orientation has various possibilities with respect to the direction of motion. In particular, the spin of massive particles behaves di erently from that of massless particles. For massive particles, the inhomogeneous Lorentz group implies that the invariant
magnitude of spin is J(J + ) ħ, o en simply written J. Since the value speci es the magnitude of the angular momentum, it gives the representation under rotations of a given particle type. e spin magnitude J can be any multiple of , i.e. it can take the
* Eugene Wigner (b. 1902 Budapest, d. 1995 Princeton), Hungarian–US-American theoretical physicist, received the Nobel Prize for physics in 1993. He wrote over 500 papers, many about symmetry in physics. He was also famous for being the most polite physicist in the world. ** e group of physical rotations is also called SO(3), since mathematically it is described by the group of Special Orthogonal 3 by 3 matrices.
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F I G U R E 314 An argument showing why rotations by π are equivalent to no rotation at all
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Page 1174 Page 49
values , , , , , , etc. Experiments show that electrons, protons and neutrons have spin , the W and Z particles spin and helium atoms spin . In addition, the representation of spin J is J + dimensional, meaning that the spatial orientation of the spin has J + possible values. For electrons there are thus two possibilities; they are usually called ‘up’ and ‘down’.
Spin thus only takes discrete values. is is in contrast with linear momentum, whose representations are in nite dimensional and whose possible values form a continuous range.
Also massless particles are characterized by the value of their spin. It can take the same values as in the massive case. For example, photons and gluons have spin . For massless particles, the representations are one-dimensional, so that massless particles are completely described by their helicity, de ned as the projection of the spin onto the direction of motion. Massless particles can have positive or negative helicity, o en also called right-handed and le -handed. ere is no other freedom for the orientation of spin in the massless case.
e symmetry investigations lead to the classi cation of particles by their mass, their momentum and their spin. To complete the list, the remaining symmetries must be included. ese are motion inversion parity, spatial parity and charge inversion parity. Since these symmetries are parities, each elementary particle has to be described by three additional numbers, called T, P and C, each of which can take values of either + or − . Being parities, they must be multiplied to yield the value for a composed system.
A list of the values observed for all elementary particles in nature is given in Appendix C. Spin and parities together are called quantum numbers. As we will discover later on, additional interaction symmetries will lead to additional quantum numbers. But let us return to spin.
e main result is that spin / is a possibility in nature, even though it does not appear in everyday life. Spin / means that only a rotation of degrees is equivalent to one of
degrees, while one of degrees is not, as explained in Table . e mathematician Hermann Weyl used a simple image explaining this connection.
Take two cones, touching each other at their tips as well as along a line. Hold one cone and roll the other around it, as shown in Figure . When the rolling cone, a er a full
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TA B L E 57 Particle spin as representation of the rotation group
S
S
M
[ħ] unchanged a er elementary composite rotation by
M elementary
0
any angle
none a,b
mesons, nuclei, none b atoms
1/2 2 turns
1
1 turn
e, µ, τ, q, nuclei, atoms, νe , νµ, ντ molecules
none, as neutrinos have a tiny mass
W, Z
mesons, nuclei, , γ
atoms, molecules,
toasters
3/2 2/3 turn
none b
baryons, nuclei, none b atoms
2
1/2 turn
none
nuclei
‘graviton’ c
5/2 2/5 turn
none
nuclei
none
3
1/3 turn
none
nuclei d
none
etc.d etc.d
etc.d
etc.d
etc.d
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a. Whether the Higgs particle is elementary or not is still unknown.
b. Supersymmetry predicts particles in these and other boxes.
c. e graviton has not yet been observed.
d. Nuclei exist with spins values up to at least
and (in units of ħ). Ref. 752
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 751
turn around the other cone, has come back to the original position, it has rotated by some angle. If the cones are wide, the rotation angle is small. If the cones are very thin, almost like needles, the moving cone has rotated by almost degrees. A rotation of degrees is thus similar to one by degrees. If we imagine the cone angle to vary continuously, this visualization also shows that a degree rotation can be continuously deformed into a
degree one, whereas a degree rotation cannot. To sum up, the list of possible representations thus shows that rotations require the existence of spin. But why then do experiments show that all fermions have half-integer spin and that all bosons have integer spin? Why do electrons obey the Pauli exclusion principle? At rst, it is not clear what the spin has to do with the statistical properties of a particle. In fact, there are several ways to show that rotations and statistics are connected. Historically, the rst proof used the details of quantum eld theory and was so complicated that its essential ingredients were hidden. It took quite some years to convince everybody that a simple observation about belts was the central part of the proof.
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F I G U R E 315 A belt visualizing two spin 1/2 particles
F I G U R E 316 Another model for two spin 1/2 particles
α=
atpi
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F I G U R E 317 The human arm as spin 1/2 model
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T
Ref. 753 Challenge 1255 e
Challenge 1256 e Challenge 1257 e
e well-known belt trick (also called scissor trick) was o en used by Dirac to explain the features of spin / . Taking Figure , which models particles as indistinguishable excitations, it is not di cult to imagine a sort of sheet connecting them, similar to a belt connecting the two parts of the buckle, as shown in Figure . If one end of the belt is rotated by π along any axis, a twist is inserted into the belt. If the end is rotated for another π, bringing the total to π, the ensuing double twist can easily be undone without moving or rotating the ends. You need to experience this yourself in order to believe it.
In addition, if you take the two ends and simply swap positions, a twist is introduced into the belt. Again, a second swap will undo the twist.
In other words, if we take each end to represent a particle and a twist to mean a factor − , the belt exactly describes the phase behaviour of spin wave functions under exchange and under rotations. In particular, we see that spin and exchange behaviour are related.
e human body has such a belt built in: the arm. Just take your hand, put an object on it for clarity such as a cup, and turn the hand and object by π by twisting the arm. A er a second rotation the whole system will be untangled again.
e trick is even more impressive when many arms are used. You can put your two hands (if you chose the correct starting position) under the cup or you can take a friend or two who each keep a hand attached to the cup. e feat can still be performed: the whole system untangles a er two full turns.
Another demonstration is to connect two buckles with many bands or threads, like in Figure . Both a rotation by π of one end or an exchange of the two ends produces quite a tangle, even if one takes paths that ‘in between’ the bands; nevertheless, in both cases a second rotation leads back to the original situation.
ere is still another way to show the connection between rotation and exchange. Just glue any number of threads or bands, say half a metre long, to an asymmetric object. Like the arm of a human being, the bands are supposed to go to in nity and be attached there. If any of the objects, which represent the particles, is rotated by π, twists appear
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rotating the buckle either by 4π
or simply rearranging the bands gives the
other situation
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 318 The extended belt trick, modelling a spin 1/2 particle: independently of the number of bands attached, the two situations can be transformed into each other, either by rotating the central object by π or by keeping the central object fixed and moving the bands around it
Challenge 1258 e Ref. 754
in its strings. If the object is rotated by an additional turn, to a total of π, as shown in Figure , all twists and tangles can be made to disappear, without moving or turning the object. You really have to experience this in order to believe it. And the trick really works with any number of bands glued to the object.
Even more astonishing is the exchange part of the experiment. Take two particles of the type shown on the le side of Figure . If you exchange the positions of two such spin / particles, always keeping the ends at in nity xed, a tangled mess is created. But incredibly, if you exchange the objects a second time, everything untangles neatly, independently of the number of attached strings. You might want to test yourself that the behaviour is also valid with sets of three or more particles.
All these observations together form the spin statistics theorem for spin / particles: spin and exchange behaviour are related. Indeed, these almost ‘experimental’ arguments can be put into exact mathematical language by studying the behaviour of the con guration space of particles. ese investigations result in the following statements:
Objects of spin / are fermions.* Exchange and rotation of spin / particles are similar processes.
Note that all these arguments require three dimensions, because there are no tangles (or knots) in fewer dimensions.** And indeed, spin exists only in three or more spatial dimensions.
Challenge 1259 e Page 757
* A mathematical observable behaving like a spin 1/2 particle is neither a vector nor a tensor, as you may want to check. An additional concept is necessary; such an observable is called a spinor. We will introduce it later on. ** Of course, knots and tangles do exist in higher dimensions. Instead of considering knotted onedimensional lines, one can consider knotted planes or knotted higher-dimensional hyperplanes. For example, deformable planes can be knotted in four dimensions and deformable 3-spaces in ve dimensions.
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Here is a challenge. A spin / object can be modelled with one belt attached to it. If you want to model the spin behaviour with attached one-dimensional strings instead of Challenge 1260 n attached bands, what is the minimum number required?
TP
Ref. 755
Challenge 1261 n Ref. 747
Why are we able to knock on a door? Why can stones not y through tree trunks? How does the mountain we are walking on carry us? In classical physics, we avoided this issue, by taking solidity as a de ning property of matter. But doing so, we cheated: we have seen that matter consists mainly of empty space, so that we have to study the issue without any sneaky way out. e answer is now clear: penetration is made impossible by Pauli’s exclusion principle between the electrons inside atoms.
Why do electrons and other fermions obey the Pauli exclusion principle? e answer can be given with a beautifully simple argument. We know that exchanging two fermions produces a minus sign. Imagine these two fermions being, as a classical physicist would say, located at the same spot, or as a quantum physicist would say, in the same state. If that could be possible, an exchange would change nothing in the system. But an exchange of fermions must produce a minus sign for the total state. Both possibilities – no change at all as well as a minus sign – cannot be realized at the same time. ere is only one way out: two fermions must avoid to ever be in the same state.
e exclusion principle is the reason that two pieces of matter in everyday life cannot penetrate each other, but have to repel each other. For example, bells only work because of the exclusion principle. Bells would not work if the colliding pieces that produce the sound would interpenetrate. But in any example of two interpenetrating pieces the electrons in the atoms would have to be in similar states. is is forbidden. For the same reason we do not fall through the oor, even though gravity pulls us down, but remain on the surface. In other words, the exclusion principle implies that matter cannot be compressed inde nitely, as at a certain stage an e ective Pauli pressure appears, so that a compression limit ensues. For this reason for example, planets or neutron stars do not collapse under their own gravity.
e exclusion principle also answers the question about how many angels can dance on the top of a pin. (Note that angels must be made of fermions, as you might want to deduce from the information known about them.) Both theory and experiment con rm the answer already given by omas Aquinas in the Middle Ages: only one. e fermion exclusion principle could also be called ‘angel exclusion principle’. To stay in the topic, the principle also shows that ghosts cannot be objects, as ghosts are supposed to be able to traverse walls.
Whatever the interpretation, the exclusion principle keeps things in shape; without it, there would be no three-dimensional objects. Only the exclusion principle keeps the cloudy atoms of nature from merging, holding them apart. Shapes are a direct consequence of the exclusion principle. As a result, when we knock on a table or on a door, we show that both objects are made of fermions.
Since permutation properties and spin properties of fermions are so well described by the belt model, we could be led to the conclusion that these properties might really be consequence of such belt-like connections between particles and the outside world. Maybe for some reason we only observe the belt buckles, not the belts themselves. In the
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J = 0
J = 1/2
J = 1
F I G U R E 319 Some visualizations of spin representations
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Challenge 1262 ny
third part of this walk we will discover whether this idea is correct. So far, we have only considered spin particles. We will not talk much about systems
with odd spin of higher value, such as or . Such systems can be seen as being composed of spin entities. Can you con rm this?
We did not talk about lower spins than either. A famous theorem states that a positive spin value below is impossible, because the largest angle that can be measured in three dimensions is π. ere is no way to measure a larger angle;* e quantum of action makes this impossible. us there cannot be any spin value between and .
I
Under rotations, integer spin particles behave di erently from half-integer particles. Integer spin particles do not show the strange sign changes under rotations by π. In the belt imagery, integer spin particles need no attached strings. e spin particle obviously corresponds to a sphere. Models for other spin values are shown in Figure . Exploring their properties in the same way as above, we arrive at the so-called spin-statistics theorem:
Exchange and rotation of objects are similar processes. Objects of half-integer spin are fermions. ey obey the Pauli exclusion principle. Objects of integer spin are bosons.
Challenge 1263 ny You might prove by yourself that this su ces to show the following:
Composites of bosons, as well as composites of an even number of fermions
(at low energy), are bosons; composites of an uneven number of fermions are fermions.**
Challenge 1264 ny
* is is possible in two dimensions though. ** is sentence implies that spin 1 and higher can also be achieved with tails; can you nd such a representation?
Note that composite fermions can be bosons only up to that energy at which the composition breaks down. Otherwise, by packing fermions into bosons, we could have fermions in the same state.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
ese connections express basic characteristics of the three-dimensional world in which we live.
I
?
Challenge 1266 ny Ref. 756
e spin of a particle behaves experimentally like an intrinsic angular momentum, adds up like angular momentum, is conserved as part of angular momentum, is described like angular momentum and has a name synonymous with angular momentum. Despite all this, for many decades a strange myth was spread in physics courses and textbooks around the world, namely that spin is not a rotation about an axis. e myth maintains that any rotating object must have integer spin. Since half integer spin is not possible in classical physics, it is argued that such spin is not due to rotation. It is time to nish with this example of muddled thinking.
Electrons do have spin and are charged. Electrons and all other charged particles with spin do have a magnetic moment.* A magnetic moment is expected for any rotating charge. In other words, spin does behave like rotation. However, assuming that a particle consists of a continuous charge distribution in rotational motion gives the wrong value for the magnetic moment. In the early days of the twentieth century, when physicists were still thinking in classical terms, they concluded that spin particles thus cannot be rotating. is myth has survived through many textbooks. e correct deduction is that the assumption of continuous charge distribution is wrong. Indeed, charge is quantized; nobody today expects that elementary charge is continuously spread over space, as that would contradict its quantization.
Let us remember what rotation is. Both the belt trick for spin / as well as the integer spin case remind us: a rotation of one body around another is a fraction or a multiple of an exchange. What we call a rotating body in everyday life is a body continuously exchanging the positions of its parts. Rotation and exchange are the same.
Above we found that spin is exchange behaviour. Since rotation is exchange and spin is exchange, it follows that spin is rotation. Since we deduced, like Wigner, spin from rotation invariance, this consequence is not a surprise.
e belt model of a spin / particle tells us that such a particle can rotate continuously without any hindrance. In short, we are allowed to maintain that spin is rotation about an axis, without any contradiction to observations, even for spin / . e belt model helps to keep two things in mind: we must assume that in the belt model only the buckles can be observed and do interact, not the belts, and we must assume that elementary charge is pointlike and cannot be distributed.**
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W
?
When a sword is approaching dangerously, we can stop it with a second sword. Many old lms use such scenes. When a laser beam is approaching, it is impossible to fend it o
Challenge 1265 ny
* is can easily be measured in a an experiment; however, not one of the Stern–Gerlach type. Why? ** Obviously, the detailed structure of the electron still remains unclear at this point. Any angular momentum S is given classically by S = Θω; however, neither the moment of inertia Θ, connected to the rotation radius and electron mass, nor the angular velocity ω are known at this point. We have to wait quite a while, until the third part of our adventure, to nd out more.
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t x
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 320 Equivalence of exchange and rotation in space-time
Page 74
with a second beam, despite all science ction lms showing so. Banging two laser beams against each other is impossible.
e above discussion shows why. e electrons in the swords are fermions and obey the Pauli exclusion principle. Fermions make matter impenetrable. On the other hand, photons are bosons. Bosons can be in the same state; they allow penetration. Matter is impenetrable because at the fundamental level it is composed of fermions. Radiation is composed of bosons. e distinction between fermions and bosons thus explains why objects can be touched while images cannot. In the rst part of our mountain ascent we started by noting this di erence; now we know its origin.
R
Page 315 Challenge 1267 ny
e connection between rotation and antiparticles may be the most astonishing conclusion from the experiments showing the existence of spin. So far, we have seen that rotation requires the existence of spin, that spin appears when relativity is introduced into quantum theory, and that relativity requires antimatter. Taking these three statements together, the conclusion of the title is not surprising any more. Interestingly, there is a simple argument making the same point directly, without any help of quantum theory, when the belt model is extended from space alone to full space-time.
To learn how to think in space-time, let us take a particle spin , i.e. a particle looking like a detached belt buckle in three dimensions. When moving in a + dimensional spacetime, it is described by a ribbon. Playing around with ribbons in space-time, instead of belts in space, provides many interesting conclusions. For example, Figure shows that wrapping a rubber ribbon around the ngers can show that a rotation of a body by π in presence of a second one is the same as exchanging the positions of the two bodies.* Both sides of the hand transform the same initial condition, at one border of the hand, to the same nal condition at the other border. We have thus successfully extended a known result from space to space-time. Interestingly, we can also nd a smooth sequence of steps realizing this equivalence.
* Obviously, the next step would be to check the full spin 1/2 model of Figure 318 in four-dimensional Challenge 1268 ny space-time. But this is not an easy task; there is no generally accepted solution yet.
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t
t
t
t
t
x
x
x
x
F I G U R E 321 Belts in space-time: rotation and antiparticles
x
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
If you think that Figure is not a satisfying explanation, you are right. A more complete (yet equivalent) explanation is given by Figure . We assume that each particle is described by a segment; in the gure, they lie horizontally. e le most diagram shows two particles: one at rest and one being rotated by π. e deformation of the ribbons shows that this process is equivalent to the exchange in position of two particles, which is shown in the rightmost diagram. Again, one notes that the sequence that shows the equivalence between rotation and exchange requires the use of a particle–antiparticle pair. Without antiparticles, the equivalence of rotation and exchange would not hold. Rotation in space-time indeed requires antiparticles.
L
Page 849 Ref. 757
Challenge 1269 n
e topic of statistics is an important research eld in theoretical and experimental physics. In particular, researchers have searched and still are searching for generalizations of the possible exchange behaviours of particles.
In two spatial dimensions, the result of an exchange of the wave function is not described by a sign, but by a continuous phase. Two-dimensional objects behaving in this way, called anyons because they can have ‘any’ spin, have experimental importance, since in many experiments in solid state physics the set-up is e ectively two-dimensional. e fractional quantum Hall e ect, perhaps the most interesting discovery of modern experimental physics, has pushed anyons onto the stage of modern research.
Other theorists generalized the concept of fermions in other ways, introducing parafermions, parabosons, plektons and other hypothetical concepts. O.W. Greenberg has spent most of his professional life on this issue. His conclusion is that in + space-time dimensions, only fermions and bosons exist. (Can you show that this implies that the ghosts appearing in scottish tales do not exist?)
From a di erent viewpoint, the above belt model invites to study the behaviour of braids and knots. (In mathematics, a braid is a knot extending to in nity.) is fascinating part of mathematical physics has become important with the advent of string theory, which states that particles, especially at high energies, are not point-like, but extended entities.
Still another generalization of statistical behaviour at high energies is the concept of quantum group, which we will encounter later on. In all of these cases, the quest is to understand what happens to permutation symmetry in a uni ed theory of nature. A glimpse of the di culties appears already above: how can Figures , and be reconciled
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• and combined? We will settle this issue in the third part of our mountain ascent.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
B
741 See the paper by M
G
, Science ction puzzle tales, Clarkson Potter, 67,
pp. – , , or his book Aha! Insight, Scienti c American & W.H. Freeman, . e
rabbit story is from A. H
& P. L
, An algorithm to prevent the propagation
of certain diseases at minimum cost, in Interfaces Between Computer Science and Opera-
tions Research, edited by J.K. L
, A.H.G. R
K & P. V E B ,
Mathematisch Centrum, Amsterdam , whereas the computer euphemism is used by A.
O
& L. S , On curbing virus propagation, Technical memorandum, Bell Labs
. Cited on page .
742 A complete discussion of the problem can be found in chapter of I V , Computational Recreations in Mathematica, Addison Wesley, Redwood City, California, . Cited on page .
743 On Gibbs’ paradox, see your favourite text on thermodynamics or statistical mechanics. See
also W.H. Z
, Algorithmic randomness and physical entropy, Physical Review A 40,
pp. – , . Zurek shows that the Sackur–Tetrode formula can be derived from
algorithmic entropy considerations. Cited on page .
744 S.N. B , Plancks Gesetz und Lichtquantenhypothese, Zeitschri für Physik 26, pp. –
, . e theory was then expanded in A. E
, Quantentheorie des einatomigen
idealen Gases, Sitzungsberichte der Preussischen Akademie der Wissenscha en zu Berlin 22,
pp. – , , A. E
, Quantentheorie des einatomigen idealen Gases. Zweite
Abhandlung, Sitzungsberichte der Preussischen Akademie der Wissenscha en zu Berlin 23,
pp. – , , A. E
, Zur Quantentheorie des idealen Gases, Sitzungsberichte der
Preussischen Akademie der Wissenscha en zu Berlin 23, pp. – , . Cited on page .
745 C.K. H , Z.Y. O & L. M
, Measurement of subpicosecond time intervals
between two photons by interference, Physical Review Letters 59, pp. – , . Cited
on page .
746 e experiment is described in E. R
& G.A. S , Experimental limit on a small
violation of the Pauli principle, Physics Letters B 238, pp. – , . Other experimental
tests are reviewed in O.W. G
, Particles with small violations of Fermi or Bose
statistics, Physical Review D 43, pp. – , . Cited on page .
747 e issue is treated in his Summa eologica, in question of the rst part. e complete text, several thousand pages, can be found on the http://www.newadvent.org website. Cited on page .
748 e original no-cloning theorem is by D. D , Communication by EPR devices, Phys-
ics Letters A 92, pp. – , , and by W.K. W
& W.H. Z
, A single
quantum cannot be cloned, Nature 299, pp. – , . For a discussion of photon and
multiparticle cloning, see N. G & S. M
, Optimal quantum cloning machines,
Physics Review Letters 79, pp. – , . e whole topic has been presented in detail
by V. B
& M. H
, Quantum cloning, Physics World 14, pp. – , November
. Cited on page .
749 e most recent experimental and theoretical results on physical cloning are described in
A. L
-L
, C. S
, J.C. H
& D. B
, Experimental
quantum cloning of single photons, Science 296, pp. – , , D. C
&
S. P
, A classical analogue of entanglement, preprint http://www.arxiv.org/abs/
quant-ph/
, , and A. D
, A.R. P
& A. P
,
Classical no-cloning theorem, Physical Review Letters 88, p.
, . Cited on page
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
.
750 E. W
, On unitary representations of the inhomogeneous Lorentz group, Annals of
Mathematics 40, pp. – , . is famous paper summarises the work which later
brought him the Nobel Prize for physics. Cited on page .
751 W
P , e connection between spin and statistics, Physical Review 58,
pp. – , . Cited on page .
752 For a full list of isotopes, see R.B. F date, with CDROM, John Wiley & Sons,
, Table of Isotopes, Eighth Edition, 1999 Up. Cited on page .
753 is famous explanation is given, for example, on page
in C.W. M
, K.S.
T
& J.A. W
, Gravitation, Freeman, . It is called the scissor trick on
page of volume of R. P
& W. R
, Spinors and Spacetime, . It is also
cited and discussed by R. G
, Answer to question , American Journal of Physics 63,
p. , . Cited on page .
754 M.V. B
& J.M. R
, Indistinguishability for quantum particles: spin, statistics
and the geometric phase, Proceedings of the Royal Society in London A 453, pp. – ,
. See also the comments to this result by J. T
, Statistics given a spin, Nature
389, pp. – , September . eir newer results are M.V. B
& J.M. R
,
Quantum indistinguishability: alternative constructions of the transported basis, Journal of
Physics A (Letters) 33, pp. L –L , , and M.V. B
& J.M. R
, in Spin-
Statistics, eds. R. H
& G. T , (American Institute of Physics), , pp. – .
See also Michael Berry’s home page on the http://www.phy.bris.ac.uk/sta /berry_mv.html
website. Cited on page .
755 R.W. H pp. – ,
, Pauli principle in Euclidean geometry, American Journal of Physics 47, . Cited on page .
756 e point that spin can be seen as a rotation was already made by F.J. B
, On the
spin angular momentum of mesons, Physica 6, p. , , and taken up again by H C.
O
, What is spin?, American Journal of Physics 54, pp. – , . See also E.
D
& A. E
, Physikalische Zeitschri der Sowjetunion 12, p. , . Cited
on page .
757 Generalizations of bosons and fermions are reviewed in the (serious!) paper by
O.W. G
, D.M. G
& T.V. G
, (Para)bosons,
(para)fermions, quons and other beasts in the menagerie of particle statistics, at http://
www.arxiv.org/abs/hep-th/
. A newer summary is O.W. G
, eories
of violation of statistics, electronic preprint available at http://www.arxiv.org/abs/hep-th/
, . Cited on page .
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
C
VII
DETAILS OF QUANTUM THEORY AND
ELECTROMAGNETISM
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“ e fact that an adequate philosophical presentation has been so long delayed is no doubt caused by the fact that Niels Bohr
brainwashed a whole generation of theorists
Ref. 758
into thinking that the job was done y years
ago.
Murray Gell-Mann
” is this famous physical issue arousing such strong emotions? In particular,
ho is brainwashed, Gell-Mann, the discoverer of the quarks, or most of the
World’s physicists working on quantum theory who follow Niels Bohr’s* opinion?
In the twentieth century, quantum mechanics has thrown many in disarray. Indeed, it rad-
ically changed the two most basic concepts of classical physics: state and system. e state
is not described any more by the speci c values taken by position and momentum, but
by the speci c wave function ‘taken’ by the position and momentum operators.** In ad-
dition, in classical physics a system was described as a set of permanent aspects of nature;
permanence was de ned as negligible interaction with the environment. Quantum mech-
anics shows that this de nition has to be modi ed as well.
.
–
In order to clarify the implications of superpositions, we take a short walk around the strangest aspects of quantum theory. e section is essential if we want to avoid getting lost on our way to the top of Motion Mountain, as happened to quite a number of people since quantum theory appeared.
* Niels Bohr (b. 1885 Copenhagen, d. 1962 Copenhagen) made Copenhagen University into one of the centres of quantum theory, overshadowing Göttingen. He developed the description of the atom with quantum theory, for which he received the 1922 Nobel Prize in physics. He had to ee Denmark in 1943 a er the German invasion, because of his Jewish background, but returned there a er the war. ** It is equivalent, but maybe conceptually clearer, to say that the state is described by a complete set of commuting operators. In fact, the discussion is somewhat simpli ed in the Heisenberg picture. However, here we study the issue in the Schrödinger picture, using wave functions.
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W
?
e evolution equation of quantum mechanics is linear in the wave function; thus we can
imagine and try to construct systems where the state ψ is a superposition of two radically
distinct situations, such as those of a dead and of a living cat. is famous ctional animal
is called Schrödinger’s cat a er the originator of the example. Is it possible to produce it?
How would it evolve in time? We can ask the same questions about a superposition of a
state where a car is inside a closed garage with a state where the car is outside.
Such strange situations are not usually observed in everyday life. e reason for this
rareness is an important aspect of what is o en called the ‘interpretation’ of quantum
mechanics. In principle, such strange situations are possible, and the superposition of
macroscopically distinct states has actually been observed in a few cases, though not for
cats, people or cars. To get an idea of the constraints, let us specify the situation in more
detail.* e object of discussion are linear superpositions of the type ψ = aψa + bψb, where ψa and ψb are macroscopically distinct states of the system under discussion, and where a and b are some complex coe cients. States are called macroscopically distinct
when each state corresponds to a di erent macroscopic situation, i.e. when the two states
can be distinguished using the concepts or measurement methods of classical physics.
In particular, this means that the physical action necessary to transform one state into
the other must be much larger than ħ. For example, two di erent positions of any body
composed of a large number of molecules are macroscopically distinct.
A ‘strange’ situation is thus
a superposition of macro-
scopic distinct states. Let us
work out the essence of mac-
roscopic superpositions more
clearly. Given two macroscop-
no figure yet
ically distinct states ψa and ψb, a superposition of the type F I G U R E 322 Artist’s impression of a macroscopic ψ = aψa + bψb is called a pure superposition state. Since the states ψa and ψb can interfere, one also talks about a (phase) coherent superposition. In the case of a superposition of macroscopically distinct states, the scalar product ψ†aψb is obviously vanishing. In case of a coherent superposition, the coe cient product a b is di erent
from zero. is fact can also be expressed with the help of the density matrix ρ of the
system, de ned as ρ = ψ ψ†. In the present case it is given by
ρpure = ψ
ψ† = a ψa ψ†a + b ψb ψ†b + a b ψa
= (ψa , ψb)
a ab ab b
ψ†a ψ†b
.
ψ†b + a b ψb
ψ†a (554)
We can then say that whenever the system is in a pure state, its density matrix, or density functional, contains o -diagonal terms of the same order of magnitude as the diagonal
* Most what can be said about this topic has been said by two people: John von Neumann, who in the Ref. 759 nineteen-thirties stressed the di erences between evolution and decoherence, and by Hans Dieter Zeh, who Ref. 760 in the nineteen seventies stressed the importance of baths and the environment in the decoherence process.
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ones.* Such a density matrix corresponds to the above-mentioned strange situations that we do not observe in daily life.
We now have a look at the opposite situation. In contrast to the case just mentioned, a density matrix for macroscopic distinct states with vanishing o -diagonal elements, such as the two state example
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ρ = a ψa ψ†a + b ψb
= (ψa , ψb) a
b
ψ†b ψ†a ψ†b
(556)
Ref. 761 Challenge 1270 ny
describes a system which possesses no phase coherence at all. (Here, denotes the non-
commutative dyadic product or tensor product which produces a tensor or matrix start-
ing from two vectors.) Such a diagonal density matrix cannot be that of a pure state; it
describes a system which is in the state ψa with probability a and which is in the state
ψb with probability b . Such a system is said to be in a mixed state, because its state is not known, or equivalently, to be in a (phase) incoherent superposition, because interference
e ects cannot be observed in such a situation. A system described by a mixed state is al-
ways either in the state ψa or in the state ψb. In other words, a diagonal density matrix for macroscopically distinct states is not in contrast, but in agreement with everyday experi-
ence. In the picture of density matrices, the non-diagonal elements contain the di erence
between normal, i.e. incoherent, and unusual, i.e. coherent, superpositions.
e experimental situation is clear: for macroscopically distinct states, only diagonal
density matrices are observed. Any system in a coherent macroscopic superposition
somehow loses its o -diagonal matrix elements. How does this process of decoherence**
take place? e density matrix itself shows the way.
Indeed, the density matrix for a large system is used, in thermodynamics, for the
de nition of its entropy and of all its other thermodynamic quantities. ese studies show
that
S = −k tr(ρ ln ρ)
(557)
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where tr denotes the trace, i.e. the sum of all diagonal elements. We also remind ourselves that a system with a large and constant entropy is called a bath. In simple physical terms, a bath is a system to which we can ascribe a temperature. More precisely, a (physical) bath, or (thermodynamic) reservoir, is any large system for which the concept of equilibrium can be de ned. Experiments show that in practice, this is equivalent to the condition that a bath consists of many interacting subsystems. For this reason, all macroscopic quantities describing the state of a bath show small, irregular uctuations, a fact that will be of central importance shortly.
* Using the density matrix, we can rewrite the evolution equation of a quantum system:
ψ˙ = −iHψ
becomes
dρ dt
=
− i [H, ρ] ħ
.
(555)
Both are completely equivalent. ( e new expression is sometimes also called the von Neumann equation.) We won’t actually do any calculations here. e expressions are given so that you recognize them when you encounter them elsewhere. ** In many settings, decoherence is called disentanglement, as we will see below.
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Challenge 1271 n
Obviously, an everyday bath is also a bath in the physical sense: a thermodynamic bath is similar to an extremely large warm water bath, one for which the temperature does not change even if one adds some cold or warm water to it. Examples of physical baths are an intense magnetic eld, a large amount of gas, or a large solid. ( e meanings of ‘intense’ and ‘large’ of course depend on the system under study.) e physical concept of bath is thus an abstraction and a generalization of the everyday concept of bath.
It is easy to see from the de nition ( ) of entropy that the loss of o -diagonal elements corresponds to an increase in entropy. And it is known that any increase in entropy of a reversible system, such as the quantum mechanical system in question, is due to an interaction with a bath.
Where is the bath interacting with the system? It obviously must be outside the system one is talking about, i.e. in its environment. Indeed, we know experimentally that any environment is large and characterized by a temperature. Some examples are listed in Table . Any environment therefore contains a bath. We can even go further: for every experimental situation, there is a bath interacting with the system. Indeed, every system which can be observed is not isolated, as it obviously interacts at least with the observer; and every observer by de nition contains a bath, as we will show in more detail shortly. Usually however, the most important baths we have to take into consideration are the atmosphere around a system, the radiation attaining the system or, if the system itself is large enough to have a temperature, those degrees of freedom of the system which are not involved in the superposition under investigation.
Since every system is in contact with baths, every density matrix of a macroscopic superposition will lose its diagonal elements evenually. At rst sight, this direction of thought is not convincing. e interactions of a system with its environment can be made extremely small by using clever experimental set-ups; that would imply that the time for decoherence can be made extremely large. us we need to check how much time a superposition of states needs to decohere. It turns out that there are two standard ways to estimate the decoherence time: either by modelling the bath as large number of colliding particles, or by modelling it as a continuous eld.
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Challenge 1272 ny
If the bath is described as a set of particles randomly hitting the microscopic system,
it is best characterized by the e ective wavelength λeff of the particles and by the average interval thit between two hits. A straightforward calculation shows that the decoherence time td is in any case smaller than this time interval, so that
td thit = φσ ,
(558)
where φ is the ux of particles and σ the cross-section for the hit.* Typical values are given
* e decoherence time is derived by studying the evolution of the density matrix ρ(x, x′) of objects localized at two points x and x′. One nds that the o -diagonal elements follow ρ(x, x′, t) =
ρ(x, x′, )e−Λt(x−x′) , where the localization rate Λ is given by
Λ = k φσeff
(559)
where k is the wave number, φ the ux and σeff the cross-section of the collisions, i.e. usually the size of the Ref. 762 macroscopic object.
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TA B L E 58 Common and less common baths with their main properties
B
T
-W - P -
H
thit = σ φ
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
matter baths solid, liquid air laboratory vacuum
photon baths sunlight ‘darkness’ cosmic microwaves terrestrial radio waves Casimir e ect Unruh radiation of Earth
nuclear radiation baths radioactivity cosmic radiation solar neutrinos cosmic neutrinos
gravitational baths gravitational radiation
T
K K mK
K K .K .. K .. K zK
K MK .K
ëK
λeff
pm pm µm
nm µm mm
pm pm pm mm
−m
φ ms ms ms ms ms ms
ms ms
a
−s −s s
−s −s
s
.. s .. s .. s .. s
.. s
a
−s
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−s
−s
−s −s −s
.. s .. s .. s .. s
.. s
a. e cross-section σ in the case of matter and photon baths was assumed to be − m for atoms; for the macroscopic object a size of mm was used as example. For neutrino baths, ...
in Table . We easily note that for macroscopic objects, decoherence times are extremely short. Scattering leads to fast decoherence. However, for atoms or smaller systems, the situation is di erent, as expected.
A second method to estimate the decoherence time is also common. Any interaction of a system with a bath is described by a relaxation time tr. e term relaxation designates any process which leads to the return to the equilibrium state. e terms damping and friction are also used. In the present case, the relaxation time describes the return to equilibrium of the combination bath and system. Relaxation is an example of an irreversible evolution. A process is called irreversible if the reversed process, in which every component moves in opposite direction, is of very low probability.* For example, it is usual that
Ref. 763
One also nds the surprising result that a system hit by a particle of energy Ehit collapses the density matrix roughly down to the de Broglie (or thermal de Broglie) wavelength of the hitting particle. Both results together give the formula above. * Beware of other de nitions which try to make something deeper out of the concept of irreversibility, such as
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Ref. 764
a glass of wine poured into a bowl of water colours the whole water; it is very rarely observed that the wine and the water separate again, since the probability of all water and wine molecules to change directions together at the same time is rather low, a state of a airs making the happiness of wine producers and the despair of wine consumers.
Now let us simplify the description of the bath. We approximate it by a single, unspeci ed, scalar eld which interacts with the quantum system. Due to the continuity of space, such a eld has an in nity of degrees of freedom. ey are taken to model the many degrees of freedom of the bath. e eld is assumed to be in an initial state where its degrees of freedom are excited in a way described by a temperature T. e interaction of the system with the bath, which is at the origin of the relaxation process, can be described by the repeated transfer of small amounts of energy Ehit until the relaxation process is completed.
e objects of interest in this discussion, like the mentioned cat, person or car, are described by a mass m. eir main characteristic is the maximum energy Er which can be transferred from the system to the environment. is energy describes the interactions between system and environment. e superpositions of macroscopic states we are interested in are solutions of the Hamiltonian evolution of these systems.
e initial coherence of the superposition, so disturbingly in contrast with our everyday experience, disappears exponentially within a decoherence time td given by*
td
=
tr
Ehit Er
eEhit eEhit
kT kT
− +
(562)
where k is the Boltzmann constant and like above, Er is the maximum energy which can be transferred from the system to the environment. Note that one always has td tr. A er the decoherence time td is elapsed, the system has evolved from the coherent to the incoherent superposition of states, or, in other words, the density matrix has lost its
o -diagonal terms. One also says that the phase coherence of this system has been des-
troyed. us, a er a time td , the system is found either in the state ψa or in the state ψb, respectively with the probability a or b , and not any more in a coherent superposition which is so much in contradiction with our daily experience. Which nal state is
selected depends on the precise state of the bath, whose details were eliminated from the
calculation by taking an average over the states of its microscopic constituents.
Ref. 765 Challenge 1273 ny
claims that ‘irreversible’ means that the reversed process is not at all possible. Many so-called ‘contradictions’
between the irreversibility of processes and the reversibility of evolution equations are due to this mistaken
interpretation of the term ‘irreversible’.
* is result is derived as in the above case. A system interacting with a bath always has an evolution given
by the general form
dρ dt
=
−
i ħ
[H
,
ρ]
−
to
[Vj ρ, Vj†] + [Vj , ρVj†] ,
j
(560)
where ρ is the density matrix, H the Hamiltonian, V the interaction, and to the characteristic time of the interaction. Are you able to see why? Solving this equation, one nds for the elements far from the diagonal
ρ(t) = ρ e−t t . In other words, they disappear with a characteristic time to. In most situations one has a
relation of the form
t
=
tr
Ehit Er
=
thit
(561)
or some variations of it, as in the example above.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 766
e important result is that for all macroscopic objects, the decoherence time td is extremely small. In order to see this more clearly, we can study a special simpli ed case.
A macroscopic object of mass m, like the mentioned cat or car, is assumed to be at the
same time in two locations separated by a distance l, i.e. in a superposition of the two
corresponding states. We further assume that the superposition is due to the object mov-
ing as a quantum mechanical oscillator with frequency ω between the two locations; this
is the simplest possible system that shows superpositions of an object located in two dif-
ferent positions. e energy of the object is then given by Er = mω l , and the smallest transfer energy Ehit = ħω is the di erence between the oscillator levels. In a macroscopic situation, this last energy is much smaller than kT, so that from the preceding expression
we get
td = tr
Ehit Er kT
= tr
ħ mkTl
=
tr
λT l
(563)
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Ref. 767 Ref. 768
Ref. 769 Ref. 770
in which the frequency ω has disappeared. e quantity λT = ħ mkT is called the thermal de Broglie wavelength of a particle.
It is straightforward to see that for practically all macroscopic objects the typical deco-
herence time td is extremely short. For example, setting m = g, l = mm and T = K we get td tr = . ë − . Even if the interaction between the system and the environment would be so weak that the system would have as relaxation time the age of the universe, which is about ë s, the time td would still be shorter than ë − s, which is over a million times faster than the oscillation time of a beam of light (about fs for green
light). For Schrödinger’s cat, the decoherence time would be even shorter. ese times
are so short that we cannot even hope to prepare the initial coherent superposition, let
alone to observe its decay or to measure its lifetime.
For microscopic systems however, the situation is di erent. For example, for an electron in a solid cooled to liquid helium temperature we have m = . ë − kg, and typically l = nm and T = K; we then get td tr and therefore the system can stay in a coherent superposition until it is relaxed, which con rms that for this case coherent e ects
can indeed be observed if the system is kept isolated. A typical example is the behaviour
of electrons in superconducting materials. We will mention a few more below.
In the rst actual measurement of decoherence times was published by the Paris
team around Serge Haroche. It con rmed the relation between the decoherence time
and the relaxation time, thus showing that the two processes have to be distinguished
at microscopic scale. In the meantime, other experiments con rmed the decoherence
process with its evolution equation, both for small and large values of td tr. A particularly beautiful experiment has been performed in , where the disappearance of two-slit
interference for C molecules was observed when a bath interacts with them.
C
,
In summary, both estimates of decoherence times tell us that for most macroscopic objects, in contrast to microscopic ones, both the preparation and the survival of superpositions of macroscopically di erent states is made practically impossible by the interaction with any bath found in their environment, even if the usual measure of this interaction, given by the friction of the motion of the system, is very small. Even if a macroscopic
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•
system is subject to an extremely low friction, leading to a very long relaxation time, its decoherence time is still vanishingly short.
Our everyday environment is full of baths. erefore, coherent superpositions of macroscopically distinct states never appear in everyday life. In short, we cannot be dead and alive at the same time.
We also arrive at a second conclusion: decoherence results from coupling to a bath in the environment. Decoherence is a statistical or thermodynamic e ect. We will return to this issue below.
W
?W
?
In classical physics, a system is a part of nature which can be isolated from its environment. However, quantum mechanics tells us that isolated systems do not exist, since interactions cannot be made vanishingly small. e results above allow us to de ne the concept of system with more accuracy. A system is any part of nature which interacts incoherently with its environment. In other words, an object is a part of nature interacting with its environment only through baths.
In particular, a system is called microscopic or quantum mechanical and can described by a wave function ψ whenever
— it is almost isolated, with tevol = ħ ∆E < tr, and Ref. 771 — it is in incoherent interaction with its environment.
In short, a microscopic system interacts incoherently and weakly with its environment. In contrast, a bath is never isolated in the sense just given, because its evolution time is always much larger than its relaxation time. Since all macroscopic bodies are in contact with baths – or even contain one – they cannot be described by a wave function. In particular, it is impossible to describe any measuring apparatus with the help of a wave function.
We thus conclude that a macroscopic system is a system with a decoherence time much shorter than any other evolution time of its constituents. Obviously, macroscopic systems also interact incoherently with their environment. us cats, cars and television news speakers are all macroscopic systems.
A third possibility is le over by the two de nitions: what happens in the situation in which the interactions with the environment are coherent? We will encounter some examples shortly. Following this de nition, such situations are not systems and cannot be described by a wave function. For example, it can happen that a particle forms neither a macroscopic nor a microscopic system! In these situations, when the interaction is coherent, one speaks of entanglement; such a particle or set of particles is said to be entangled with its environment.
Entangled, coherently interacting systems are separable, but not divisible. In quantum theory, nature is not found to be made of isolated entities, but is still made of separable entities. e criterion of separability is the incoherence of interaction. Coherent superpositions imply the surprising consequence that there are systems which, even though they look divisible, are not. Entanglement poses a limit to divisibility. All surprising properties of quantum mechanics, such as Schrödinger’s cat, are consequences of the classical prejudice that a system made of two or more parts must necessarily be divisible into two
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space
collapse
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t1
t2
t3
t4
slit
screen
space
F I G U R E 323 Quantum mechanical motion: an electron wave function (actually its module squared) from the moment it passes a slit until it hits a screen
subsystems. But coherent superpositions, or entangled systems, do not allow division. Whenever we try to divide indivisible systems, we get strange or incorrect conclusions, such as apparent faster-than-light propagation, or, as one says today, non-local behaviour. Let us have a look at a few typical examples.
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Challenge 1274 n Page 577 Ref. 772
I
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?–A
E
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[Mr. Du y] lived a little distance away from his body ...
“ James Joyce, A Painful Case ” A er we explored non-locality in general relativity, we now study it in quantum mech-
anics. We rst look at the wave function collapse for an electron hitting a screen a er passing a slit. Following the description just deduced, the process proceeds schematically as depicted in Figure . A lm of the same process can be seen in the lower le corners on the pages following page . e situation is surprising: due to the short decoherence time, in a wave function collapse the maximum of the function changes position at extremely high speed. In fact, the maximum moves faster than light. But is it a problem?
A situation is called acausal or non-local if energy is transported faster than light. Using Figure you can determine the energy velocity involved, using the results on signal propagation. e result is a value smaller than c. A wave function maximum moving faster than light does not imply energy moving faster than light.
In other words, quantum theory has speeds greater than light, but no energy speeds greater than light. In classical electrodynamics, the same happens with the scalar and the
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space detector 2
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detector 1 collapse
time F I G U R E 324 Bohm’s Gedanken experiment
Page 306 Ref. 773, Ref. 774
vector potentials if the Coulomb gauge is used. We have also encountered speeds faster than that of light in the motion of shadows and in many other observations. Any physicist now has two choices: he can be straight, and say that there is no non-locality in nature; or he can be less straight, and claim there is. In the latter case, he has to claim that even classical physics is non-local. However, this never happens. On the other hand, there is a danger in this more provoking usage: a small percentage of those who say that the world is non-local a er a while start to believe that there really are faster-than-light e ects in nature. ese people become prisoners of their muddled thinking; on the other hands, muddled thinking helps to get more easily into newspapers. In short, even though the de nition of non-locality is not unanimous, here we stick to the stricter one, and de ne non-locality as energy transport faster than light.
An o en cited Gedanken experiment that shows the pitfalls of non-locality was proposed by Bohm* in the discussion around the so-called Einstein–Podolsky–Rosen paradox. In the famous EPR paper the three authors try to nd a contradiction between quantum mechanics and common sense. Bohm translated their rather confused paper into a clear Gedanken experiment. When two particles in a spin state move apart, measuring one particle’s spin orientation implies an immediate collapse also of the other particle’s spin, namely in the exactly opposite direction. is happens instantaneously over the whole separation distance; no speed limit is obeyed. In other words, entanglement seems to lead to faster-than-light communication.
Again, in Bohm’s experiment, no energy is transported faster than light. No nonlocality is present, despite numerous claims of the contrary by certain authors. e two
* David Joseph Bohm (1917–1992) American–British physicist. He codiscovered the Aharonov–Bohm effect; he spent a large part of his later life investigating the connections between quantum physics and philosophy.
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Page 580 Ref. 775
Ref. 776
entangled electrons belong to one system: assuming that they are separate only because the wave function has two distant maxima is a conceptual mistake. In fact, no signal can be transmitted with this method; the decoherence is a case of prediction which looks like a signal without being one. We already discussed such cases in the section on electrodynamics.
Bohm’s experiment has actually been performed. e rst and most famous realization was realized in by Alain Aspect; he used photons instead of electrons. Like all latter tests, it has fully con rmed quantum mechanics.
In fact, experiments such as the one by Aspect con rm that it is impossible to treat either of the two particles as a system by itself; it is impossible to ascribe any physical property, such as a spin orientation, to either of them alone. ( e Heisenberg picture would express this restriction even more clearly.)
e mentioned two examples of apparent non-locality can be dismissed with the remark that since obviously no energy ux faster than light is involved, no problems with causality appear. erefore the following example is more interesting. Take two identical atoms, one in an excited state, one in the ground state, and call l the distance that separates them. Common sense tells that if the rst atom returns to its ground state emitting a photon, the second atom can be excited only a er a time t = l c has been elapsed, i.e. a er the photon has travelled to the second atom.
Surprisingly, this conclusion is wrong. e atom in its ground state has a non-zero probability to be excited at the same moment in which the rst is de-excited. is has been shown most simply by Gerhard Hegerfeldt. e result has even been con rmed experimentally.
More careful studies show that the result depends on the type of superposition of the two atoms at the beginning: coherent or incoherent. For incoherent superpositions, the intuitive result is correct; the counter-intuitive result appears only for coherent superpositions. Again, a careful discussion shows that no real non-locality of energy is involved.
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Coherent superposition, or entanglement, is such a surprising phenomenon that many aspects have been and still are being explored.
Ref. 766 Ref. 784
**
In a few cases, the superposition of di erent macroscopic states can actually be observed by lowering the temperature to su ciently small values and by carefully choosing suitably small masses or distances. Two well-known examples of coherent superpositions are those observed in gravitational wave detectors and in Josephson junctions. In the
rst case, one observes a mass as heavy as kg in a superposition of states located at di erent points in space: the distance between them is of the order of − m. In the second case, in superconducting rings, superpositions of a state in which a macroscopic current of the order of pA ows in clockwise direction with one where it ows in counterclockwise direction have been produced.
** Ref. 778 Superpositions of magnetization in up and down direction at the same time have also be
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observed for several materials.
Challenge 1275 n
**
Some people wrongly state that an atom that is in a superposition of states centred at di erent positions has been photographed. ( is lie is even used by some sects to attract believers.) Why is this not true?
Ref. 780 Ref. 781
**
Since the 1990s, the sport of nding and playing with new systems in coherent macroscopic superpositions has taken o across the world. e challenges lie in the clean experiments necessary. Experiments with single atoms in superpositions of states are among the most popular ones.
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** Ref. 782 In 1997, coherent atom waves were extracted from a cloud of sodium atoms.
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**
Macroscopic objects usually are in incoherent states. is is the same situation as for light. e world is full of ‘macroscopic’, i.e. incoherent light: daylight, and all light from lamps, from re and from glow-worms is incoherent. Only very special and carefully constructed sources, such as lasers or small point sources, emit coherent light. Only these allow to study interference e ects. In fact, the terms ‘coherent’ and ‘incoherent’ originated in optics, since for light the di erence between the two, namely the capacity to interfere, had been observed centuries before the case of matter.
Coherence and incoherence of light and of matter manifest themselves di erently, since matter can stay at rest but light cannot and because light is made of bosons, but matter is made of fermions. Coherence can be observed easily in systems composed of bosons, such as light, sound in solids, or electron pairs in superconductors. Coherence is less easily observed in systems of fermions, such as systems of atoms with their electron clouds. However, in both cases a decoherence time can be de ned. In both cases coherence in many particle systems is best observed if all particles are in the same state (superconductivity, laser light) and in both cases the transition from coherent to incoherent is due to the interaction with a bath. A beam is thus incoherent if its particles arrive randomly in time and in frequency. In everyday life, the rarity of observation of coherent matter superpositions has the same origin as the rarity of observation of coherent light.
**
We will discuss the relation between the environment and the decay of unstable systems later on. e phenomenon is completely described by the concepts given here.
**
Challenge 1276 ny Can you nd a method to measure the degree of entanglement? Can you do so for a system made of many particles?
** e study of entanglement leads to a simple conclusion: teleportation contradicts correla-
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Challenge 1277 ny tion. Can you con rm the statement?
Ref. 786 Challenge 1278 n
**
Some people say that quantum theory could be used for quantum computing, by using coherent superpositions of wave functions. Can you give a general reason that makes this aim very di cult, even without knowing how such a quantum computer might work?
Page 638 Challenge 1279 ny
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Measurements in quantum mechanics are disturbing. ey lead to statements in which
probabilities appear. at is puzzling. For example, we speak about the probability of nd-
ing an electron at a certain distance from the nucleus of an atom. Statements like this
belong to the general type ‘when the observable A is measured, the probability to nd the
outcome a is p.’ In the following we will show that the probabilities in such statements
are inevitable for any measurement, because, as we will show, any measurement and any
observation is a special case of decoherence or disentanglement process. (Historically
however, the process of measurement was studied before the more general process of
decoherence. at explains in part why the topic is so confused in many peoples’ minds.)
What is a measurement? As already mentioned in the intermezzo a measurement is
any interaction which produces a record or a memory. (In everyday life, any e ect is a re-
cord; but this is not true in general. Can you give some examples and counter-examples?)
Measurements can be performed by machines; when they are performed by people, they
are called observations. In quantum theory, the action of measurement is not as straight-
forward as in classical physics. is is seen most strikingly when a quantum system, such
as a single electron, is rst made to pass a di raction slit, or better – in order to make its
wave aspect become apparent – a double slit and then is made to hit a photographic plate,
in order to make also its particle aspect appear. Experiment shows that the blackened
dot, the spot where the electron has hit the screen, cannot be determined in advance.
( e same is true for photons or any other particle.) However, for large numbers of elec-
trons, the spatial distribution of the black dots, the so-called di raction pattern, can be
calculated in advance with high precision.
e outcome of experiments on microscopic systems thus forces us to use probabilities
for the description of microsystems. We nd that the probability distribution p(x) of the
spots on the photographic plate can be calculated from the wave function ψ of the electron
at the screen surface and is given by p(x) = ψ†(x)ψ(x) . is is in fact a special case of
the general rst property of quantum measurements: the measurement of an observable A
for a system in a state ψ gives as result one of the eigenvalues an, and the probability Pn
to get the result an is given by
Pn = φ†nψ ,
(564)
Page 1207 Ref. 783
where φn is the eigenfunction of the operator A corresponding to the eigenvalue an. Experiments also show a second property of quantum measurements: a er the meas-
urement, the observed quantum system is in the state φn corresponding to the measured eigenvalue an. One also says that during the measurement, the wave function has collapsed from ψ to φn. By the way, both properties can also be generalized to the more
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Page 637
general cases with degenerate and continuous eigenvalues.
At rst sight, the sort of probabilities encountered in
quantum theory are di erent from the probabilities we en-
counter in everyday life. Roulette, dice, pachinko machines, the direction in which a pencil on its tip falls, have been meas-
gravity
ured experimentally to be random (assuming no cheating by
the designer or operators) to a high degree of accuracy. ese
systems do not puzzle us. We unconsciously assume that the
random outcome is due to the small, but uncontrollable vari-
ations of the starting conditions every time the experiment is repeated.*
But microscopic systems seem to be di erent. e two measurement properties just mentioned express what phys-
F I G U R E 325 A system showing probabilistic behaviour
icists observe in every experiment, even if the initial conditions are taken to be exactly
the same every time. But why then is the position for a single electron, or most other
observables of quantum systems, not predictable? In other words, what happens during
the collapse of the wave function? How long does it take? In the beginning of quantum
theory, there was the perception that the observed unpredictability is due to the lack of
information about the state of the particle. is lead many to search for so-called ‘hidden
variables’. All these attempts were doomed to fail, however. It took some time for the sci-
enti c community to realize that the unpredictability is not due to the lack of information
about the state of the particle, which is indeed described completely by the state vector ψ.
In order to uncover the origin of probabilities, let us recall the nature of a measure-
ment, or better, of a general observation. Any observation is the production of a record.
e record can be a visual or auditive memory in our brain, or a written record on pa-
per, or a tape recording, or any such type of object. As explained in the intermezzo, an
object is a record if it cannot have arisen or disappeared by chance. To avoid the in u-
ence of chance, all records have to be protected as much as possible from the outer world;
e.g. one typically puts archives in earthquake safe buildings with re protection, keeps
documents in a safe, avoids brain injury as much as possible, etc.
On top of this, records have to be protected from their internal uctuations. ese in-
ternal uctuations are due to the many components any recording device is made of. But
if the uctuations were too large, they would make it impossible to distinguish between
the possible contents of a memory. Now, uctuations decrease with increasing size of a
system, typically with the square root of the size. For example, if a hand writing is too
small, it is di cult to read if the paper gets brittle; if the magnetic tracks on tapes are
too small, they demagnetize and loose the stored information. In other words, a record is
rendered stable against internal uctuations by making it of su cient size. Every record
thus consists of many components and shows small uctuations.
erefore, every system with memory, i.e. every system capable of producing a record,
contains a bath. In summary, the statement that any observation is the production of a
record can be expressed more precisely as: Any observation of a system is the result of an
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* To get a feeling for the limitations of these unconscious assumptions, you may want to read the already mentioned story of those physicists who built a machine that could predict the outcome of a roulette ball Ref. 81 from the initial velocity imparted by the croupier.
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the quantum mechanical system
apparatus, e.g. eye, ear, or machine, with memory, i.e. coupled to a bath
H
description of its possible states
H int
determined by the type of measurement
tr
friction, e.g. due to heat flow
F I G U R E 326 The concepts used in the description of measurements
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Ref. 779
interaction between that system and a bath in the recording apparatus.*
But we can say more. Obviously, any observation measuring a physical quantity uses
an interaction depending on that same quantity. With these seemingly trivial remarks,
we can describe in more detail the process of observation, or, as it is usually called in the
quantum theory, the measurement process.
Any measurement apparatus, or detector, is characterized by two main aspects: the interaction it has with the microscopic system, and the bath it contains to produce the
record. Any description of the measurement process thus is the description of the evolu-
tion of the microscopic system and the detector; therefore one needs the Hamiltonian for
the particle, the interaction Hamiltonian, and the bath properties, such as the relaxation
time. e interaction speci es what is measured and the bath realizes the memory.
We know that only classical thermodynamic systems can be irreversible; quantum
systems are not. We therefore conclude: a measurement system must be described classically: otherwise it has no memory and is not a measurement system: it produces no
record! Memory is a classical e ect. (More precisely, it is an e ect that only appears in
the classical limit.) Nevertheless, let us see what happens if one describes the measure-
ment system quantum mechanically. Let us call A the observable which is measured in the experiment and its eigenfunctions φn. We describe the quantum mechanical system under observation – o en a particle – by a state ψ. is state can always be written as ψ = ψp ψother = n cn φn ψother, where ψother represents the other degrees of freedom of the particle, i.e. those not described – spanned, in mathematical language – by the operator A corresponding to the observable we want to measure. e numbers cn = φ†nψp give the expansion of the state ψp, which is taken to be normalized, in terms of the basis φn. For example, in a typical position measurement, the functions φn would be the position eigenfunctions and ψother would contain the information about the momentum, the spin and all other properties of the particle.
How does the system–detector interaction look like? Let us call the state of the apparatus before the measurement χstart; the measurement apparatus itself, by de nition, is a device which, when it is hit by a particle in the state φnψother, changes from the state χstart to the state χn. One then says that the apparatus has measured the eigenvalue an
* Since baths imply friction, we can also say: memory needs friction.
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•
corresponding to the eigenfunction φn of the operator A. e index n is thus the record of the measurement; it is called the pointer index or variable. is index tells us in which
state the microscopic system was before the interaction. e important point, taken from
our previous discussion, is that the states χn, being records, are macroscopically distinct, precisely in the sense of the previous section. Otherwise they would not be records, and
the interaction with the detector would not be a measurement.
Of course, during measurement, the apparatus sensitive to φn changes the part ψother of the particle state to some other situation ψother,n, which depends on the measurement and on the apparatus; we do not need to specify it in the following discussion.* Let us have
an intermediate check of our reasoning. Do apparatuses as described here exist? Yes, they
do. For example, any photographic plate is a detector for the position of ionizing particles.
A plate, and in general any apparatus measuring position, does this by changing its mo-
mentum in a way depending on the measured position: the electron on a photographic
plate is stopped. In this case, χstart is a white plate, φn would be a particle localized at spot n, χn is the function describing a plate blackened at spot n and ψother,n describes the momentum and spin of the particle a er it has hit the photographic plate at the spot n.
Now we are ready to look at the measurement process itself. For the moment, let us
disregard the bath in the detector. In the time before the interaction between the particle
and the detector, the combined system was in the initial state ψi given simply by
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ψi = ψp χstart = cnφnψother χstart .
n
(567)
A er the interaction, using the just mentioned characteristics of the apparatus, the com-
bined state ψa is
ψa = cnφnψother,n χn .
(568)
n
is evolution from ψi to ψa follows from the evolution equation applied to the particle detector combination. Now the state ψa is a superposition of macroscopically distinct states, as it is a superposition of distinct macroscopic states of the detector. In our example
ψa could correspond to a superposition of a state where a spot on the le upper corner is blackened on an otherwise white plate with one where a spot on the right lower corner
of the otherwise white plate is blackened. Such a situation is never observed. Let us see
* How does the interaction look like mathematically? From the description we just gave, we speci ed the nal state for every initial state. Since the two density matrices are related by
ρf = TρiT†
(565)
Challenge 1280 ny we can deduce the Hamiltonian from the matrix T. Are you able to see how? By the way, one can say in general that an apparatus measuring an observable A has a system interaction
Hamiltonian depending on the pointer variable A, and for which one has
[H + Hint, A] = .
(566)
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why. e density matrix ρa of this situation, given by
ρa = ψa ψ†a = cn cm(φnψother,n χn) (φmψother,m χm)† ,
n,m
(569)
contains non-diagonal terms, i.e. terms for n m, whose numerical coe cients are dif-
ferent from zero. Now let’s take the bath back in.
From the previous section we know the e ect of a bath on such a macroscopic super-
position. We found that a density matrix such as ρa decoheres extremely rapidly. We as-
sume here that the decoherence time is negligibly small, in practice thus instantaneous,*
so that the o -diagonal terms vanish, and only the nal, diagonal density matrix ρf , given by
ρf = cn (φnψother,n χn) (φnψother,n χn )†
(570)
n
has experimental relevance. As explained above, such a density matrix describes a mixed state and the numbers Pn = cn = φ†nψp give the probability of measuring the value an and of nding the particle in the state φnψother,n as well as the detector in the state χn. But this is precisely what the two properties of quantum measurements state.
We therefore nd that describing a measurement as an evolution of a quantum system
interacting with a macroscopic detector, itself containing a bath, we can deduce the two
properties of quantum measurements, and thus the collapse of the wave function, from
the quantum mechanical evolution equation. e decoherence time td of the previous section becomes the time of collapse in the case of a measurement:
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tcollapse = td < tr
(571)
We thus have a formula for the time the wave function takes to collapse. e rst experiRef. 785 mental measurements of the time of collapse are appearing and con rm these results.
H
Obviously a large number of people are not satis ed with the arguments just presented. ey long for more mystery in quantum theory. e most famous approach is the idea
that the probabilities are due to some hidden aspect of nature which is still unknown to humans. But the beautiful thing about quantum mechanics is that it allows both conceptual and experimental tests on whether such hidden variables exist without the need of knowing them.
Clearly, hidden variables controlling the evolution of microscopic system would contradict the result that action values below ħ cannot be detected. is minimum observable action is the reason for the random behaviour of microscopic systems.
Historically, the rst argument against hidden variables was given by John von Neu-
* Note however, that an exactly vanishing decoherence time, which would mean a strictly in nite number of degrees of freedom of the environment, is in contradiction with the evolution equation, and in particular with unitarity, locality and causality. It is essential in the whole argument not to confuse the logical consequences of a extremely small decoherence time with those of an exactly vanishing decoherence time.
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mann.*
– CS – to be written – CS –
Ref. 787
Ref. 788 Challenge 1281 e
An additional no-go theorem for hidden variables was published by Kochen and
Specker in , (and independently by Bell in ). It states that non-contextual hid-
den variables are impossible, if the Hilbert space has a dimension equal or larger than
three. e theorem is about non-contextual variables, i.e. about hidden variables inside
the quantum mechanical system. e Kochen–Specker theorem thus states that there is
no non-contextual hidden variables model, because mathematics forbids it. is result
essentially eliminates all possibilities, because usual quantum mechanical systems have
dimensions much larger than three.
But also common sense eliminates hidden variables, without any recourse to mathem-
atics, with an argument o en overlooked. If a quantum mechanical system had internal
hidden variables, the measurement apparatus would have zillions of them.** And that
would mean that it could not work as a measurement system.
Of course, one cannot avoid noting that about contextual hidden variables, i.e. vari-
ables in the environment, there are no restricting theorems; indeed, their necessity was
shown earlier in this section.
Obviously, despite these results, people have also looked for experimental tests on hid-
den variables. Most tests are based on the famed Bell’s equation, a beautifully simple rela-
tion published by John Bell*** in the s.
e starting idea is to distinguish quantum theory and locally realistic theories using
hidden variables by measuring the polarizations of two correlated photons. Quantum
theory says that the polarization of the photons is xed only at the time it is measured,
whereas local realistic theories say that it is xed already in advance. e correct descrip-
tion can be found by experiment.
Imagine the polarization is measured at two distant points A and B, each observer can
measure or − in each of his favourite direction. Let each observer choose two directions,
and , and call their results a , a , b and b . Since the measurement results all are either
or − , the value of the speci c expression (a + a )b + (a − a )b has always the value
.
Imagine you repeat the experiment many times, assuming that the hidden variables
appear statistically. You then can deduce (a special case of) Bell’s inequality for two hidden
variables
(a b ) + (a b ) + (a b ) − (a b )
(572)
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where the expressions in brackets are the averages of the measurement products over a
* János von Neumann (b. 1903 Budapest, d. 1957 Washington DC) Hungarian mathematician. One of the greatest and clearest minds of the twentieth century, he settled already many questions, especially in applied mathematics and quantum theory, that others still struggle with today. He worked on the atomic and the hydrogen bomb, on ballistic missiles, and on general defence problems. In another famous project, he build the rst US-American computer, building on his extension of the ideas of Konrad Zuse. ** Which leads to the de nition: one zillion is . *** John Stewart Bell (1928–1990), theoretical physicist who worked mainly on the foundations of quantum theory.
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large number of samples. is result holds independently of the directions of the involved polarizers.
On the other hand, if the polarizers and at position A and the corresponding ones at position B are chosen with angles of π , quantum theory predicts that the result is
(a b ) + (a b ) + (a b ) − (a b ) =
(573)
which is in complete contradiction with the hidden variable result. So far, all experimental checks of Bell’s equation have con rmed standard quantum
mechanics. No evidence for hidden variables has been found. is is not really surprising, since the search for such variables is based on a misunderstanding of quantum mechanics or on personal desires on how the world should be, instead of relying on experimental evidence.
Another measurable contradiction between quantum theory and locally realistic theories has been predicted by Greenberger, Horn and Zeilinger. Experiments trying to check the result are being planned. No deviation from quantum theory is expected.
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Geometric demonstramus quia facimus; si physics demonstrare possemus, faceremus.
“ Giambattista Vico* ” From the arguments presented here we draw a number of conclusions which we need
for the rest of our mountain ascent. Note that these conclusions are not yet shared by all physicists! e whole topic is still touchy.
— Probabilities do not appear in measurements because the state of the quantum system is unknown or fuzzy, but because the detailed state of the bath in the environment is unknown. Quantum mechanical probabilities are of statistical origin and are due to baths in the environment (or the measurement apparatus). e probabilities are due to the large number of degrees of freedom contained in any bath. ese large numbers make the outcome of experiments unpredictable. If the state of the bath were known, the outcome of an experiment could be predicted. e probabilities of quantum theory are ‘thermodynamic’ in origin. In other words, there are no fundamental probabilities in nature. All probabilities in nature are due to decoherence; in particular, all probabilities are due to the statistics of the many particles – some of which may be virtual – that are part of the baths in the environment. Modifying well-known words by Albert Einstein, ‘nature really does not play dice.’ We therefore called ψ the wave function instead of ‘probability amplitude’, as is o en done. State function would be an even better name.
— Any observation in everyday life is a special case of decoherence. What is usually called the ‘collapse of the wave function’ is a decoherence process due to the interaction with
* ‘We are able to demonstrate geometrical matters because we make them; if we could prove physical matters we would be able to make them.’ Giovanni Battista Vico (b. 1668 Napoli, d. 1744 Napoli) important Italian philosopher and thinker. In this famous statement he points out a fundamental distinction between mathematics and physics.
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Challenge 1282 n Page 687
the baths present in the environment or in the measuring apparatus. Because humans are warm-blooded and have memory, humans themselves are thus measurement apparatuses. e fact that our body temperature is °C is thus the reason that we see only a single world, and no superpositions. (Actually, there are more reasons; can you name a few?) — A measurement is complete when the microscopic system has interacted with the bath in the measuring apparatus. Quantum theory as a description of nature does not require detectors; the evolution equation describes all examples of motion. However, measurements do require the existence of detectors. Detectors, being machines that record observations, have to include a bath, i.e. have to be classical, macroscopic objects. In this context one speaks also of a classical apparatus. is necessity of the measurement apparatus to be classical had been already stressed in the very early stages of quantum theory. — All measurements, being decoherence processes that involve interactions with baths, are irreversible processes and increase entropy. — A measurement is a special case of quantum mechanical evolution, namely the evolution for the combination of a quantum system, a macroscopic detector and the environment. Since the evolution equation is relativistically invariant, no causality problems appear in measurements; neither do locality problems and logical problems appear. — Since both the evolution equation and the measurement process does not involve quantities other than space-time, Hamiltonians, baths and wave-functions, no other quantity plays a role in measurement. In particular, no human observer nor any consciousness are involved or necessary. Every measurement is complete when the microscopic system has interacted with the bath in the apparatus. e decoherence inherent in every measurement takes place even if nobody is looking. is trivial consequence is in agreement with the observations of everyday life, for example with the fact that the Moon is orbiting the Earth even if nobody looks at it.* Similarly, a tree falling in the middle of a forest makes noise even if nobody listens. Decoherence is independent of human observation, of the human mind and of human existence. — In every measurement the quantum system interacts with the detector. Since there is a minimum value for the magnitude of action, every observation in uences the observed.
erefore every measurement disturbs the quantum system. Any precise description of observations must also include the description of this disturbance. In the present section the disturbance was modelled by the change of the state of the system from ψother to ψother,n. Without such a change of state, without a disturbance of the quantum system, a measurement is impossible. — Since the complete measurement is described by quantum mechanics, unitarity is and remains the basic property of evolution. ere are no non-unitary processes in quantum mechanics. — e description of the collapse of the wave function as a decoherence process is an explanation exactly in the sense in which the term ‘explanation’ was de ned in the intermezzo; it describes the relation between an observation and all the other aspects of reality, in this case the bath in the detector or the environment. e collapse of the
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* e opposite view is sometimes falsely attributed to Niels Bohr. e Moon is obviously in contact with Challenge 1283 n many radiation baths. Can you list a few?
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
wave function has been both calculated and explained. e collapse is not a question of ‘interpretation’, i.e. of opinion, as unfortunately o en is suggested.* — It is not useful to speculate whether the evolution for a single quantum measurement could be determined if the state of the environment around the system were known. Measurements need baths. But a bath, being irreversible, cannot be described by a wave function, which behaves reversibly.** Quantum mechanics is deterministic. Baths are probabilistic. — In summary, there is no irrationality in quantum theory. Whoever uses quantum theory as argument for superstitions, irrational behaviour, new age beliefs or ideologies is guilty of disinformation. e statement by Gell-Mann at the beginning of this chapter is thus such an example. Another is the following well-known but incorrect statement by Richard Feynman:
Ref. 790
...nobody understands quantum mechanics.
Nobel Prizes in physics obviously do not prevent infection by ideology. On the other hand, the process of decoherence allows a clear look on various interesting issues.
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Challenge 1284 n
Space and time di er. Objects are localized in space but not in time. Why is this the case? Most bath–system interactions are mediated by a potential. All potentials are by de nition position dependent. erefore, every potential, being a function of the position x, commutes with the position observable (and thus with the interaction Hamiltonian). e decoherence induced by baths – except if special care is taken – thus rst of all destroys the non-diagonal elements for every superposition of states centred at di erent locations. In short, objects are localized because they interact with baths via potentials.
For the same reason, objects also have only one spatial orientation at a time. If the system–bath interaction is spin-dependent, the bath leads to ‘localization’ in the spin variable. is happens for all microscopic systems interacting with magnets. As a result, macroscopic superpositions of magnetization are almost never observed. Since electrons, protons and neutrons have a magnetic moment and a spin, this conclusion can even be extended: everyday objects are never seen in superpositions of di erent rotation states because their interactions with baths are spin-dependent.
As a counter-example, most systems are not localized in time, but on the contrary exist for very long times, because practically all system–bath interactions do not commute with time. In fact, this is the way a bath is de ned to begin with. In short, objects are permanent
because they interact with baths. Are you able to nd an interaction which is momentum-dependent? What is the con-
sequence for macroscopic systems? In other words, in contrast to general relativity, quantum theory produces a distinction
between space and time. In fact, we can de ne position as the observable that commutes
Ref. 789
* is implies that the so-called ‘many worlds’ interpretation is wishful thinking. e conclusion is conrmed when studying the details of this religious approach. It is a belief system, not based on facts.
** is very strong type of determinism will be very much challenged in the last part of this text, in which it will be shown that time is not a fundamental concept, and therefore that the debate around determinism looses most of its interest.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•
with interaction Hamiltonians. is distinction between space and time is due to the properties of matter and its interactions. We could not have deduced this distinction in general relativity.
A
?
Ref. 791 Page 1016
Are humans classical apparatuses? Yes, they are. Even though several prominent physicists claim that free will and probabilities are related, a detailed investigation shows that this in not the case. Our senses are classical machines in the sense described above: they record observations by interaction with a bath. Our brain is also a classical apparatus: the neurons are embedded in baths. Quantum probabilities do not play a determining role in the brain.
Any observing entity needs a bath and a memory to record its observations. at means that observers have to be made of matter; an observer cannot be made of radiation. Our description of nature is thus severely biased: we describe it from the standpoint of matter. at is a bit like describing the stars by putting the Earth at the centre of the universe. Can we eliminate this basic anthropomorphism? We will nd out as we continue.
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Ref. 792
W
,
?
Physics means talking about observations of nature. Like any observation, also measurements produce information. It is thus possible to translate much (but not all) of quantum theory into the language of information theory. In particular, the existence of a minimal change in nature implies that the information about a physical system can never be complete, that information transport has its limits and that information can never be fully trusted. e details of these studies form a fascinating way to look at the microscopic world. e studies become even more interesting when the statements are translated into the language of cryptology. Cryptology is the science of transmitting hidden messages that only the intended receiver can decrypt. In our modern times of constant surveillance, cryptology is an important tool to protect personal freedom.*
e quantum of action implies that messages can be sent in an (almost) safe way. Listening to a message is a measurement process. Since there is a smallest action, one can detect whether somebody has tried to listen to a sent message. A man in the middle attack – somebody who pretends to be the receiver and then sends a copy of the message to the real, intended receiver – can be avoided by using entangled systems as signals to transmit the information. Quantum cryptologists therefore usually use communication systems based on entangled photons.
e major issue of quantum cryptology is the key distribution problem. All secure communication is based on a secret key that is used to decrypt the message. Even if the communication channel is of the highest security – like entangled photons – one still has
* Cryptology consists of the eld of cryptography, the art of coding messages, and the eld of cryptoana-
lysis, the art of deciphering encrypted messages. For a good introduction to cryptology, see the text by
A
B
,J S
& K -D
W
, Moderne Ver-
fahren der Kryptographie, Vieweg 1995.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Page 1048
to nd a way to send the communication partner the secret key necessary for the decryption of the messages. Finding such methods is the main aspect of quantum cryptology, a large research eld. However, close investigation shows that all key exchange methods are limited in their security. In short, due to the quantum of action, nature provides limits on the possibility of sending encrypted messages. e statement of these limits is (almost) equivalent to the statement that change in nature is limited by the quantum of action.
e quantum of action provides a limit to secure information exchange. is connection also allows to brush aside several incorrect statements o en found in the media. Stating that ‘the universe is information’ or that ‘the universe is a computer’ is as devoid of reason as saying that the universe is an observation, a measurement apparatus, a clockwork or a chewing-gum dispenser. Any expert of motion should beware of these and similarly shy statements; people who use them either deceive themselves or try to deceive others.
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Ref. 793
e wave function of the universe is is frequently invoked in discussions about quantum theory. Many deduce conclusions from it, for example on the irreversibility of time, on the importance of initial conditions, on changes required to quantum theory and much more. Are these arguments correct?
e rst thing to clarify is the meaning of ‘universe’. As explained above, the term can have two meanings: either the collection of all matter and radiation, or this collection plus all of space-time. en we recall the meaning of ‘wave function’: it describes the state of a system. e state distinguishes two otherwise identical systems; for example, position and velocity distinguish two otherwise identical ivory balls on a billiard table. Alternatively and equivalently, the state describes changes in time.
Does the universe have a state? If we take the wider meaning of universe, obviously it does not. Talking about the state of the universe is a contradiction: by de nition, the concept of state, de ned as the non-permanent aspects of an object, is applicable only to parts of the universe.
We then can take the narrower sense of ‘universe’ – the sum of all matter and radiation only – and ask the question again. To determine its state, we need a possibility to measure it: we need an environment. But the environment of matter and radiation is space-time only; initial conditions cannot be determined since we need measurements to do this, and thus an apparatus. An apparatus is material system with a bath attached to it; there is no such system outside the universe.
In short, quantum theory does not allow for measurements of the universe; therefore the universe has no state. Beware of anybody who claims to know something about the wave function of the universe. Just ask him: If you know the wave function of the universe, why aren’t you rich?
Despite this conclusion, several famous physicists have proposed evolution equations for the wave function of the universe. ( e best-known is the Wheeler–DeWitt equation.) It seems a silly point, but the predictions of these equations have not been compared to experiments; the arguments just given even make this impossible in principle. e pursuits in this direction, so interesting they are, must therefore be avoided if we want to safely reach the top of Motion Mountain.
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ere are many more twists to this story. One possibility is that space-time itself, even without matter, is a bath. is speculation will be shown to be correct later on and seems to allow speaking of the wave function of all matter. But then again, it turns out that time is unde ned at the scales where space-time would be an e ective bath; this means that the concept of state is not applicable there.
A lack of ‘state’ for the universe is a strong statement. It also implies a lack of initial conditions! e arguments are precisely the same. is is a tough result. We are so used to think that the universe has initial conditions that we never question the term. (Even in this text the mistake might appear every now and then.) But there are no initial conditions of the universe.
We can retain as result, valid even in the light of the latest results of physics: the universe has no wave function and no initial conditions, independently of what is meant by ‘universe’. But before we continue to explore the consequences of quantum theory for the whole universe, we study in more detail the consequences for our everyday observations.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
.
–,
Homo sum, humani nil a me alienum puto.*
“ ”Terence
Now that we can look at quantum e ects without ideological baggage, let us have some serious fun in the world of quantum theory. e quantum of action has important consequences for biology, chemistry, technology and science ction. We will only explore a cross-section of these topics, but it will be worth it.
B
A special form of electromagnetic motion is of importance to humans: life. We mentioned at the start of quantum theory that life cannot be described by classical physics. Life is a quantum e ect. Let us see why.
Living beings can be described as objects showing metabolism, information processing, information exchange, reproduction and motion. Obviously, all these properties follow from a single one, to which the others are enabling means:
Living beings are objects able to reproduce.**
From your biology lessons you might remember the some properties of heredity. Reproduction is characterized by random changes from one generation to the next. e statistics of mutations, for example Mendel’s ‘laws’ of heredity, and the lack of intermediate states, are direct consequences of quantum theory. In other words, reproduction and growth are quantum e ects.
Challenge 1285 n
* ‘I am a man and nothing human is alien to me.’ Terence is Publius Terentius Afer (c. 190–159 ), the important roman poet. He writes this in his play Heauton Timorumenos, verse 77. ** However, there are examples of objects which reproduce and which nobody would call living. Can you
nd some examples, together with a sharper de nition?
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
In order to reproduce, living beings must be able to move in self-directed ways. An object able to perform self-directed motion is called a machine. All self-reproducing beings are machines.
Since reproduction and growth is simpler the smaller the system is, most living beings are extremely small machines for the tasks they perform, especially when compared to human made machines. is is the case even though the design of human machines has considerably fewer requirements: human-built machines do not need to be able to reproduce; as a result, they do not need to be made of a single piece of matter, as all living beings have to. But despite all the restrictions nature has to live with, living beings hold many miniaturization world records:
Page 639 Challenge 1286 n
— e brain has the highest processing power per volume of any calculating device so far. Just look at the size of chess champion Gary Kasparov and the size of the computer against which he played.
— e brain has the densest and fastest memory of any device so far. e set of compact discs (CDs) or digital versatile discs (DVDs) that compare with the brain is many thousand times larger.
— Motors in living beings are many orders of magnitude smaller than human-built ones. Just think about the muscles in the legs of an ant.
— e motion of living beings beats the acceleration of any human-built machine by orders of magnitude. No machine moves like a grasshopper.
— Living being’s sensor performance, such as that of the eye or the ear, has been surpassed by human machines only recently. For the nose this feat is still far away. Nevertheless, the sensor sizes developed by evolution – think also about the ears or eyes of a common y – is still unbeaten.
— Living beings that y, swim or crawl – such as fruit ies, plankton or amoebas – are still thousands of times smaller than anything built by humans.
— Can you spot more examples?
e superior miniaturization of living beings is due to their continuous strife for e cient construction. In the structure of living beings, everything is connected to everything: each part in uences many others. Indeed, the four basic processes in life, namely metabolic, mechanical, hormonal and electrical, are intertwined in space and time. For example, breathing helps digestion; head movements pump liquid through the spine; a single hormone in uences many chemical processes. Furthermore, all parts in living systems have more than one function. For example, bones provide structure and produce blood; ngernails are tools and shed chemical waste.
e miniaturization, the reproduction, the growth and the functioning of living beings all rely on the quantum of action. Let us see how.
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R
Life is a sexually transmitted disease.
“ ” Anonymous
All the astonishing complexity of life is geared towards reproduction. Reproduction is the ability of an object to build other objects similar to itself. Quantum theory told us that
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 327 A quantum machine (© Elmar Bartel)
Page 778 Page 838
only a similar object is possible, as an exact copy would contradict the quantum of action, as we found out above.
Since reproduction requires mass increase, reproducing objects show both metabolism and growth. In order that growth leads to an object similar to the original, a construction plan is necessary. is plan must be similar to the plan used by the previous generation. Organizing growth with a construction plan is only possible if nature is made of smallest entities which can be assembled following that plan.
We can thus deduce that reproduction implies that matter is made of smallest entities. If matter were not made of smallest entities, there would be no way to realize reproduction. Reproduction thus requires quantum theory. Indeed, without the quantum of action there would be no DNA molecules and there would be no way to inherit our own properties – our own construction plan – to children.
Passing on a plan requires that living beings have ways to store information. Living beings must have some built-in memory. We know already that a system with memory must be made of many particles. ere is no other way to store information. e large number of particles is mainly necessary to protect the information from the in uences of the outside world.
Our own construction plan, made of what biologists call genes, is stored in DNA molecules. Reproduction is thus rst of all a transfer of parent’s genes to the next generation. We will come back to the details below. We rst have a look on how our body moves itself and its genes around.
Q
Living beings are machines. How do these machines work? From a physical point of view, we need only a few sections of our walk so far to describe them: universal gravity and QED. Simply stated, life is an electromagnetic process taking place in weak gravity.*But the details of this statement are tricky and interesting. Table gives an overview of motion
* In fact, also the nuclear interactions play some role for life: cosmic radiation is one source for random mutations, which are so important in evolution. Plant growers o en use radioactive sources to increase Ref. 797 mutation rates. But obviously, radioactivity can also terminate life.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
TA B L E 59 Motion and motors in living beings
M
E
M
Growth Construction
collective molecular processes in cell ion pumps growth
gene turn-on and turn-o
linear molecular motors
ageing
linear molecular motors
materialtypes and properties (poly-material transport through muscles saccharides, lipids, proteins, nucleic acids, others)
forces and interactions between bio-cell membrane pumps molecules
Functioning
Defence Reproduction
details of metabolism (respiration, di-muscles, ion pumps gestion)
energy ow in biomolecules
thermodynamics of whole living sys-muscles tem and of its parts
muscle working
linear molecular motors
nerve signalling
ion motion, ion pumps
brain working
ion pumps
illnesses
cell motility, chemical pumps
viral infection of a cell
rotational molecular motors for RNA transport
the immune system
cell motility, linear molecular motors
information storage and retrieval
cell division sperm motion courting evolution
linear molecular motors inside cells, sometimes rotational motors, as in viruses
linear molecular motors inside cells
rotational molecular motors
muscles, brain, linear molecular motors
muscles, linear molecular motors
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processes in living beings. Interestingly, all motion in living beings can be summarized in a few classes by asking for the motor driving it.
e nuclear interactions are also implicitly involved in several other ways. ey were necessary to form the materials – carbon, oxygen, etc. – required for life. Nuclear interactions are behind the main mechanism for the burning of the Sun, which provides the energy for plants, for humans and for all other living beings (except a few bacteria in inaccessible places).
Summing up, the nuclear interactions play a role in the appearance and in the in destruction of life; but they play no (known) role for the actions of particular living beings.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•
Nature only needs few small but powerful devices to realize all motion types used by living beings. Given the long time that living systems have been around, these devices are extremely e cient. In fact, ion pumps, chemical pumps, rotational and linear molecular motors are all specialized molecular motors. Ion and chemical pumps are found in membranes and transport matter. Rotational and linear motor move structures against membranes. In short, all motion in living beings is due to molecular motors. Even though there is still a lot to be learned about them, what is known already is spectacular enough.
H
?–M
How do our muscles work? What is the underlying motor? One of the beautiful results of
modern biology is the elucidation of this issue. It turns out that muscles work because they
contain molecules which change shape when supplied with energy. is shape change is
repeatable. A clever combination and repetition of these molecular shape changes is then
used to generate macroscopic motion. ere are three basic classes of molecular motors:
linear motors, rotational motors and pumps.
Linear motors are at the basis of muscle motion; other linear motors separate genes
during cell division. ey also move organelles inside cells and displace cells through the
body during embryo growth, when wounds heal, or in other examples of cell motility. A
typical molecular motor consumes around to ATP molecules per second, thus
about to aW. e numbers are small; however, we have to take into account that
Challenge 1287 n the power white noise of the surrounding water is nW. In other words, in every mo-
lecular motor, the power of the environmental noise is eight to nine orders of magnitude
higher than the power consumed by the motor. e ratio shows what a fantastic piece of
machinery such a motor is.
Page 74 We encountered rotational motors already above. Nature uses them to rotate the cilia
Ref. 800 of many bacteria as well as sperm tails. Researchers have also discovered that evolution
produced molecular motors which turn around DNA helices like a motorized bolt would
turn around a screw. Such motors are attached at the end of some viruses and insert the
DNA into virus bodies when they are being built by infected cells, or extract the DNA from
Ref. 798 the virus a er it has infected a cell. Another rotational motor, the smallest known so far
– nm across and nm high – is ATP synthase, a protein that synthesizes most ATP in
cells.
e ways molecules produce movement in linear motors was uncovered during the
Ref. 799
s. e results then started a wave of research on all other molecular motors found
in nature. All molecular motors share a number of characteristic properties. ere are
no temperature gradients involved, as in car engines, no electrical currents, as in elec-
trical motors, and no concentration gradients, as found in chemically induced motion.
e central part of linear molecular motors is a combination of two protein molecules,
namely myosin and actin. Myosin changes between two shapes and literally walks along
actin. It moves in regular small steps. e motion step size has been measured with beau-
tiful experiments to always be an integer multiple of . nm. A step, usually forward, but
Ref. 799 sometimes backwards, results whenever an ATP (adenosine triphosphate) molecule, the
standard biological fuel, hydrolyses to ADP (adenosine diphosphate), thus releasing its
energy. e force generated is about to pN; the steps can be repeated several times
a second. Muscle motion is the result of thousand of millions of such elementary steps
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 328 Myosin and actin: the building bricks of a simple linear molecular motor
Ref. 801
taking place in concert. How do molecular motors work? ese motors are are so small that the noise due to
the molecules of the liquid around them is not negligible. But nature is smart: with two tricks it takes advantage of Brownian motion and transforms it into macroscopic molecular motion. Molecular motors are therefore also called Brownian motors. e transformation of disordered molecular motion into ordered macroscopic motion is one of the great wonders of nature. e rst trick of nature is the use of an asymmetric, but periodic potential, a so-called ratchet.* e second trick of nature is a temporal variation of the potential, together with an energy input to make it happen. e most important realizations are shown in Figure .
e periodic potential variation allows that for a short time the Brownian motion of the moving molecule – typically µm s – a ects its position. en the molecule is xed again. In most of these short times of free motion, the position will not change. But if the position does change, the intrinsic asymmetry of the ratchet shape ensures that in most cases the molecule advances in the preferred direction. en the molecule is xed again, waiting for the next potential change. On average, the myosin molecule will thus move in one direction. Nowadays the motion of single molecules can be followed in special experimental set-ups. ese experiments con rm that muscles use such a ratchet mechanism.
e ATP molecule adds energy to the system and triggers the potential variation through the shape change it induces in the myosin molecule. at is how our muscles work.
Another well-studied linear molecular motor is the kinesin–microtubule system which carries organelles from one place to the other within a cell. As in the previous example, also in this case chemical energy is converted into unidirectional motion. Researchers were able to attach small silica beads to single molecules and to follow their motion. Using laser beams, they could even apply forces to these single molecules. Kinesin was found to move with around nm s, in steps lengths which are multiples of
nm, using one ATP molecule at a time, and exerting a force of about pN.
* It was named by Walt Disney a er by Ratchet Gearloose, the famous inventor from Duckburg.
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U(t1)
•
Fixed position
U(t2)
Brownian motion can take place
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U(t3)
Most probable next fixed position if particle moved
F I G U R E 329 Two types of Brownian motors: switching potential (left) and tilting potential (right)
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Quantum ratchets also exist as human built systems, such as electrical ratchets for electron motion or optical ratchets that drive small particles. Extensive experimental research is going on in the eld.
C e physics of life is still not fully explored yet.
** Challenge 1288 n How would you determine which of two identical twins is the father of a baby?
Challenge 1289 n
**
Can you give at least ve arguments to show that a human clone, if there will ever be one, is a completely di erent person that the original? In fact, the rst cloned cat, born in 2002, looked completely di erent from the ‘original’ (in fact, its mother). e fur colour and its patch pattern were completely di erent from that of the mother. Analogously, identical human twins have di erent nger prints, iris scans, blood vessel networks, intrauterine experiences, among others.
Challenge 1290 n
**
Many molecules found in living beings, such as sugar, have mirror molecules. However, in all living beings only one of the two sorts is found. Life is intrinsically asymmetric. How can this be?
**
How is it possible that the genetic di erence between man and chimpanzee is regularly given as about 1 %, whereas the di erence between man and woman is one chromosome
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TA B L E 60 Approximate number of living species
L
D
E
.
.
Viruses Prokaryotes (‘bacteria’) Fungi Protozoa Algae Plants Nematodes Crustaceans Arachnids Insects Molluscs Vertebrates Others
4 000
ë
4000
ë
72 000
ë
40 000
ë
40 000
ë
270 000
ë
25 000
ë
40 000
ë
75 000
ë
950 000
ë
70 000
ë
45 000
ë
115 000
ë
ë
ë
.ë
ë
ë
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ë
ë
ë
ë
ë
ë
ë
ë
Total
.ë
.ë
ë
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 1291 n in 46, in other words, about 2.2 %?
**
What is the longest time a single bacterium has survived? Not 5000 years as the bacteria found in Egyptian mummies, not even 25 million years as the bacteria resurrected from the intestines in insects enclosed in amber. It is much longer, namely an astonishing 250 million years. is is the time that bacteria discovered in the 1960s by Hans P ug in (low-radioactivity) salt deposits in Fulda (Germany) have hibernated there before being brought back to life in the laboratory. e result has been recently con rmed by the discovery of another bacterium in a North-American salt deposit in the Salado formation.
Ref. 410
**
In 1967, a TV camera was deposited on the Moon. Unknown to everybody, it contained a small patch of Streptococcus mitis. ree years later, the camera was brought back to Earth. e bacteria were still alive. ey had survived for three years without food, water or air. Life can be resilient indeed.
Ref. 802
**
In biology classi cations are extremely useful. is is in full contrast to the situation in physics. Table 60 gives an overview of the magnitude of the task. is wealth of material can be summarized in one graph, shown in Figure 330.
Newer research seems to suggest some slight changes to the picture. So far however, there still is only a single root to the tree.
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Bacteria
Archaea
Green non-sulfur
Animals
bacteria
Methano-
Gram-positive bacteria
Purple bacteria
Methano- microbiales bacteriales
Thermo-
extreme Halophiles
Cyanobacteria Flavobacteria and relatives
proteus
Pyrodictum
Methanococcales Thermococcales
Eucarya
Ciliates Green plants Fungi Flagellates
Microsporidia
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Thermotogales
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 330 A modern version of the evolutionary tree
** Challenge 1292 r How did life start?
** Challenge 1293 n Could life have arrived to Earth from outer space?
Challenge 1294 ny
**
Life is not a clearly de ned concept. e de nition used above, the ability to reproduce, has its limits when applied to old animals, to a hand cut o by mistake, to sperm or to ovules. It also gives problems when trying to apply it to single cells. Is the de nition of life as ‘self-determined motion in the service of reproduction’ more appropriate? Or is the de nition of living beings as ‘what is made of cells’ more precise?
**
Also growth is a type of motion. Some is extremely complex. Take the growth of acne. It requires a lack of zinc, a weak immune system, several bacteria, as well as the help of Demodex brevis, a mite (a small insect) that lives in skin pores. With a size of . mm, somewhat smaller than the full stop at the end of this sentence, this and other animals living on the human face can be observed with the help of a strong magnifying glass.
Humans have many living beings on board. For example, humans need bacteria to live. It is estimated that 90% of the bacteria in the human mouth alone are not known yet; only 500 species have been isolated so far. ese useful bacteria help us as a defence against the more dangerous species.
**
Mammals have a narrow operating temperature. In contrast to machines, humans funcChallenge 1295 d tion only if the internal temperature is within a narrow range. Why? Does this require-
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
ment also apply to extraterrestrials – provided they exist?
** How did the rst cell arise? is question is still open. However, researchers have found several substances that spontaneously form closed membranes in water. Such substances also form foams. It might well be that life formed in foam.
T
Page 586 Ref. 803
Pleasure is a quantum e ect.
What is mind but motion in the intellectual sphere?
“ Oscar Wilde ( – ) e Critic as Artist. ” e reason is simple. Pleasure comes from the senses. All
senses measure. And all measurements rely on quantum theory. e human body, like an
expensive car, is full of sensors. Evolution has build these sensors in such a way that they
trigger pleasure sensations whenever we do with our body what we are made for.
Of course, no scientist will admit that he studies pleasure. So he says that he studies
the senses. e eld is fascinating and still evolving; here we can only have a quick tour
of the present knowledge.
Among the most astonishing aspects of our body sensors is their sensitivity. e ear is
so sensitive and at the same time so robust against large signals that the experts are still
studying how it works. No known sensor can cover an energy range of ; the detected
intensity ranges from pW m (some say pW m ) to W m , the corresponding air
pressure variations from µPa to Pa. e lowest intensity is that of a W sound
source heard at a distance of
km, if no sound is lost in between.
Audible sound wavelengths span from m ( Hz) to mm ( kHz). In this range,
the ear is able to distinguish at least pitches with its
to
hair cells. But
the ear is also able to distinguish from Hz using a special pitch sharpening mech-
anism.
e eye is a position dependent photon detector. Each eye contains around million
separate detectors on the retina. eir spatial density is the highest possible that makes
sense, given the diameter of the lens of the eye. ey give the eye a resolving power of ′
and the capacity to detect down to incident photons in . s, or absorbed photons
in the same time. ere are about million less sensitive colour detectors, the cones,
whose distribution we have seen earlier on. e di erent chemicals in the three cone types
(red, green, blue) lead to di erent sensor speeds; this can be checked with the simple test
shown in Figure . e images of the eye are only sharp if the eye constantly moves in
small random motions. If this motion is stopped, for example with chemicals, the images
produced by the eye become unsharp.
e eye also contains million highly sensitive general light intensity detectors, the
rods. is sensitivity di erence is the reason that at night all cats are grey. Until recently,
human built light sensors with the same sensitivity as rods had to be helium cooled, be-
cause technology was not able to build sensors at room temperature as sensitive as the
human eye.
e touch sensors are distributed over the skin, with a surface density which varies
from one region to the other. It is lowest on the back and highest in the face and on the
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 331 The different speed of the eye’s colour sensors, the cones, lead to a strange effect when this picture (in colour version) is shaken right to left in weak light
Ref. 804
tongue. ere are separate sensors for pressure, for deformation, for vibration, for tickling, for heat, for coldness, and for pain. Some react proportionally to the stimulus intensity, some di erentially, giving signals only when the stimulus changes.
e taste mechanisms of tongue are only partially known. Since , the tongue is known to produces six taste signals* – sweet, salty, bitter, sour, proteic and fat – and the mechanisms are just being unravelled. ( e sense for proteic, also called umami, has been discovered in , by Ikeda Kikunae; the sense for ‘fat’ has been discovered only in .) Democritus imagined that taste depends on the shape of atoms. Today it is known that sweet taste is connected with certain shape of molecules. Despite all this, no sensor with a distinguishing ability of the same degree as the tongue has yet been built by humans.
Ref. 805 Challenge 1297 r
Page 528
e nose has about di erent smell receptors; through combinations it is estimated
that it can smell about
di erent smells. Together with the ve signals that the sense
of taste can produce, the nose also produces a vast range of taste sensations. It protects
against chemical poisons, such as smoke, and against biological poisons, such as faecal
matter. In contrast, arti cial gas sensors exist only for a small range of gases. Good arti -
cial taste and smell sensors would allow to check wine or cheese during their production,
thus making its inventor extremely rich.
e human body also contains orientation sensors in the ear, extension sensors in each
muscle, pain sensors almost all over the skin and inside the body, heat sensors and cold-
ness sensors on the skin and in other places. Other animals feature additional types of
sensors. Sharks can feel electrical elds, snakes have sensors for infrared; both are used
to locate prey. Pigeons, trout and sharks can feel magnetic elds, and use this sense for
navigation. Many birds and certain insects can see UV light. Bats are able to hear ultra-
sound up to kHz and more. Whales and elephants can detect and localize infrasound
signals.
* Taste sensitivity is not separated on the tongue into distinct regions; this is an incorrect idea that has been copied from book to book for over a hundred years. You can perform a falsi cation by yourself, using sugar Challenge 1296 n or salt grains.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 806
In summary, the sensors with which nature provides us are state of the art; their sensitivity and ease of use is the highest possible. Since all sensors trigger pleasure or help to avoid pain, nature obviously wants us to enjoy life with the most intense pleasure possible. Studying physics is one way to do this.
ere are two things that make life worth living: Mozart and quantum mechanics.
“ ” Victor Weisskopf*
T
Page 212
“ ere is no such thing as perpetual tranquillity of mind while we live here; because life itself is but motion, and can never be without desire, nor without fear, no more than without sense. ” omas Hobbes ( – ) Leviathan.
e main unit processing all these signals, the brain, is another of the great wonders of nature. e human brain has the highest complexity of all brains known;** its processing power and speed is orders of magnitude larger than any device build by man.
We saw already how electrical signals from the sensors are transported into the brain. In the brain, the arriving signals are classi ed and stored, sometimes for a short time, sometimes for a long time. e details of the various storage mechanisms, essentially taking place in the structure and the connection strength between brain cells, were elucidated by modern neuroscience. e remaining issue is the process of classi cation. For certain low level classi cations, such as colours or geometrical shapes for the eye or sound harmonies for the ear, the mechanisms are known. But for high-level classi cations, such as the ones used in conceptual thinking, the aim is not yet achieved. It is not well known how to describe the processes of reading, understanding and talking in terms of signal motions. Research is still in full swing and will remain so for the largest part of the twenty-
rst century. In the following we have a look at a few abilities of our brain, of our body and of other
bodies which are important for the study of motion.
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L’horologe fait de la réclame pour le temps.***
“ ” Georges Perros
* Victor Friedrich Weisskopf (b. 1908 Vienna, d. 2002 Cambridge), acclaimed theoretical physicist who worked with Einstein, Born, Bohr, Schrödinger and Pauli. He catalysed the development of quantum electrodynamics and nuclear physics. He worked on the Manhattan project but later in life intensely campaigned against the use of nuclear weapons. During the cold war he accepted the membership in the Soviet Academy of Sciences. He was professor at MIT and for many years director of CERN, in Geneva. He wrote several successful physics textbooks. e author heard him making the above statement in Geneva, in 1981, during one of his lectures. ** is is not in contrast with the fact that one or two whale species have brains with a slightly larger mass.
e larger mass is due to the protection these brains require against the high pressures which appear when whales dive (some dive to depths of km). e number of neurons in whale brains is considerably smaller than in human brains. *** Clocks are ads for time.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 1298 n Ref. 807
Most clocks used in everyday life are electromagnetic. (Do you know an exception?) Any clock on the wall, be it mechanical, quartz controlled, radio or solar controlled, or of any other type, is based on electromagnetic e ects. ere are even clocks of which we do not even know how they work. Just look at singing. We know from everyday experience that humans are able to keep the beat to within a few per cent for a long time. Also when we sing a musical note we reproduce the original frequency with high accuracy. In many movements humans are able to keep time to high accuracy, e.g. when doing sport or when dancing. (For shorter or longer times, the internal clocks are not so precise.)
In addition, all clocks are limited by quantum mechanics, including the simple pendulum. Let us explore the topic.
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Page 492 Challenge 1299 d
Die Zukun war früher auch besser.*
“ ” Karl Valentin, German writer.
In general relativity, we found that clocks do not exist, because there is no unit of time that can be formed using the constants c and G. Clocks, like any measurement standard, need matter and non-gravitational interactions to work. is is the domain of quantum theory. Let us see what the situation is in this case.
First of all, the time operator, or any operator proportional to it, is not an observable. Indeed, the time operator is not Hermitean, as any observable must be. In other words, there is no physical observable whose value is proportional to time. On the other hand, clocks are quite common; for example, the Sun or Big Ben work to everybody’s satisfaction. Nature thus encourages us to look for an operator describing the position of the hands of a clock. However, if we look for such an operator we nd a strange result. Any quantum system having a Hamiltonian bounded from below – having a lowest energy – lacks a Hermitean operator whose expectation value increases monotonically with time.
is result can be proven rigorously. In other words, quantum theory states that time cannot be measured.
at time cannot be measured is not really a surprise. e meaning of this statement is that every clock needs to be wound up a er a while. Take a mechanical pendulum clock. Only if the weight driving it can fall forever, without reaching a bottom position, can the clock go on working. However, in all clocks the weight has to stop when the chain end is reached or when the battery is empty. In other words, in all real clocks the Hamiltonian is bounded from below.
In short, quantum theory says that any clock can only be approximate. Quantum theory shows that exact clocks do not exist in nature. Obviously, this result can be of importance only for high precision clocks. What happens if we try to increase the precision of a clock as much as possible?
High precision implies high sensitivity to uctuations. Now, all clocks have a motor inside that makes them work. A high precision clock thus needs a high precision motor. In all clocks, the position of the motor is read out and shown on the dial. e quantum
* Also the future used to be better in the past.
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Challenge 1300 ny Challenge 1301 ny
of action implies that a precise clock motor has a position indeterminacy. e clock pre-
cision is thus limited. Worse, like any quantum system, the motor has a small, but nite
probability to stop or to run backwards for a while.
You can check this prediction yourself. Just have a look at a clock when its battery is
almost empty, or when the weight driving the pendulum has almost reached the bottom
position. It will start doing funny things, like going backwards a bit or jumping back
and forward. When the clock works normally, this behaviour is only strongly reduced in
amount; however, it is still possible, though with low probability. is is true even for a
sundial.
In other words, clocks necessarily have to be macroscopic in order to work properly.
A clock must be large as possible, in order to average out its uctuations. Astronomical
systems are examples. A good clock must also be well-isolated from the environment,
such as a freely ying object whose coordinate is used as time variable, as is done in
certain optical clocks.
How big is the problem we have thus discovered? What is the actual error we make
when using clocks? Given the various limitations due to quantum theory, what is the
ultimate precision of a clock?
To start with, the indeterminacy relation provides the limit that the mass M of a clock
must be larger than
M
ħ cτ
(574)
Challenge 1302 e which is obviously always ful lled in everyday life. But we can do better. Like for a pendulum, we can relate the accuracy τ of the clock to its maximum reading time T. e idea
Ref. 808 was rst published by Salecker and Wigner. ey argued that
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M
ħT c ττ
(575)
Challenge 1303 e
where T is the time to be measured. You might check that this directly requires that any clock must be macroscopic.
Let us play with the formula by Salecker and Wigner. One way to rephrase it is the following. ey showed that for a clock which can measure a time t, the size l is connected to the mass m by
l
ħt m
.
(576)
Ref. 810
How close can this limit be achieved? It turns out that the smallest clocks known, as well as the clocks with most closely approach this limit are bacteria. e smallest bacteria, the mycoplasmas, have a mass of about ë − kg, and reproduce every min, with a precision of about min. e size predicted from expression ( ) is between . µm and
. µm. e observed size of the smallest mycoplasmas is . µm. e fact that bacteria can come so close to the clock limit shows us again what a good engineer evolution has been.
Note that the requirement by Salecker and Wigner is not in contrast with the possibility to make the oscillator of the clock very small; people have built oscillators made of a
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Ref. 809
single atom. In fact, such oscillations promise to be the most precise human built clocks.
In the real world, the expression can be stated even more strictly. e whole mass M
cannot be used in the above limit. For clocks made of atoms, only the binding energy
between atoms can be used. is leads to the so-called standard quantum limit for clocks;
it limits their frequency ν by
δν ν
=
∆E Etot
(577)
where ∆E = ħ T is the energy indeterminacy stemming from the nite measuring time T and Etot = N Ebind is the total binding energy of the atoms in the metre bar. However, the quantum limit has not been achieved for clocks, even though experiments are getting near to it.
In summary, clocks exist only in the limit of ħ being negligible. In practice, the errors made by using clocks and metre bars can be made as small as required; it su ces to make the clocks large enough. We can thus continue our investigation into the details of matter without much worry. Only in the third part of our mountain ascent, where the precision requirements will be higher and general relativity will limit the size of physical systems, things will get much more interesting: the impossibility to build clocks will then become a central issue.
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Ref. 812 Ref. 811
Among many things, living beings process information. Also computers do this, and like computers, all living beings need a clock to work well. Every clock needs is made up of the same components. It needs an oscillator determining the rhythm and a mechanism to feed the oscillator with energy. A clock also needs an oscillation counter, i.e. a mechanism that reads out the clock signal; a means of signal distribution throughout the system is required, synchronizing the processes attached to it. In addition, a clock needs a reset mechanism. If the clock has to cover many time scales, it needs several oscillators with di erent oscillation frequencies and a way to reset their relative phases.
Even though physicists know the details of technical clock building fairly well, we still do not know many parts of biological clocks. Most oscillators are chemical systems; some, like the heart muscle or the timers in the brain, are electrical systems. e general elucidation of chemical oscillators is due to Ilya Prigogine; it has earned him a Nobel Prize for chemistry in . But not all the chemical oscillators in the human body are known yet, not to speak of the counter mechanisms. For example, a –minute cycle inside each human cell has been discovered only in , and the oscillation mechanism is not yet fully clear. (It is known that a cell fed with heavy water ticks with –minute instead of
–minute rhythm.) It might be that the daily rhythm, the circadian clock, is made up of or reset by of these –minute cycles, triggered by some master cells in the human body. e clock reset mechanism for the circadian clock is also known to be triggered by daylight; the cells in the eye who perform this have been pinpointed only in . e light signal is processed by the superchiasmatic nucleus, two dedicated structures in the brain’s hypothalamus. e various cells in the human body react di erently depending on the phase of this clock.
e clock with the longest cycle in the human body controls ageing. A central mech-
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 813
anism for this clock seems to be the number of certain molecules attached to the DNA of the human chromosomes. At every division, one molecule is lost. When the molecules are all lost, the cell dies. Research into the mechanisms and the exceptions to this process (cancer cells, sexual cells) is ongoing.
e basis of the monthly period in women is equally interesting and complex. e most fascinating clocks are those at the basis of conscious time. Of these, the brain’s stopwatch or interval timer, has been most intensely studied. Only recently was its mechanism uncovered by combining data on human illnesses, human lesions, magnetic resonance studies, and e ects of speci c drugs. e basic mechanism takes place in the striatum in the basal ganglia of the brain. e striatum contains thousands of timer cells with di erent periods. ey can be triggered by a ‘start’ signal. Due to their large number, for small times of the order of one second, every time interval has a di erent pattern across these cells. e brain can read these patterns and learn them. In this way we can time music or speci c tasks to be performed, for example, one second a er a signal.
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Challenge 1304 ny
For length measurements, the situations is similar to that for time measurements. e limit by Salecker and Wigner can also be rewritten for length measurement devices. Are you able to do it?
In general relativity we found that we need matter for any length measurement. Quantum theory, our description of matter, again shows that metre sticks are only approximately possible, but with errors which are negligible if the device is macroscopic.
W
,
?
“Future: that period of time in which our a airs prosper, our friends are true, and our happpiness is assured. ” Ambrose Bierce
If due to the quantum of action perfect clocks do not exist, is determinism still the correct description of nature?
We have seen that predictions of the future are made di cult by nonlinearities and the divergence of from similar conditions; we have seen that many particles make it di cult to predict the future due to the statistical nature of their initial conditions; we have seen that quantum theory makes it o en hard to fully determine initial states; we have seen that not-trivial space-time topology can limit predictability; nally, we will discover that black hole and similar horizons can limit predictability due to their one-way transmission of energy, mass and signals.
Nevertheless, we also learned that all these limitations can be overcome for limited time intervals; in practice, these time intervals can be made so large that the limitations do not play a role in everyday life. In summary, in quantum theory both determinism and time remain applicable, as long as we do not extend it to in nite space and time. When extremely large dimensions and intervals need to be taken into account, quantum theory cannot be applied alone; in those cases, general relativity needs to be taken into account.
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Ref. 814 Challenge 1305 ny
I prefer most of all to remember the future.
“ ” Salvador Dalì
e decoherence of superposition of macroscopically distinct states plays an important role in another common process: the decay of unstable systems or particles. Decay is any spontaneous change. Like the wave aspect of matter, decay is a process with no classical counterpart. True, decay, including the ageing of humans, can be followed in classical physics; however, its origin is a pure quantum e ect.
Experiments show that the prediction of decay, like that of scattering of particles, is only possible on average, for a large number of particles, never for a single one. ese results con rm the quantum origin of the process. In every decay process, the superposition of macroscopically distinct states, which in this case are those of a decayed and an undecayed particle, is made to decohere rapidly by the interaction with the environment, and the ‘environment’ vacuum is su cient to induce the decoherence. As usual, the details of the states involved are unknown for a single system and make any prediction for a single system impossible.
Decay is in uenced by the environment, even in the case that it is ‘only’ the vacuum. is is true for all systems, including radioactive nuclei. e statement can be con rmed by experiment. By enclosing a part of space between two conducting plates, one can change the degrees of freedom of the vacuum contained between them. Putting an electromagnetically unstable particle, such as an excited atom, between the plates, indeed changes the lifetime of the particle. Can you explain why this method is not useful to lengthen the lifespan of humans? What is the origin of decay? Decay is always due to tunnelling. With the language of quantum electrodynamics, we can rephrase the answer: decay is motion induced by the vacuum uctuations. Vacuum uctuations are random. e experiment between the plates con rms the importance of the environment uctuations for the decay process. For a system consisting of a large number N of identical particles, the decay is described by
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N˙
=
−N τ
where
τ=
π ħ
ψfinal Hint ψfinal
.
(578)
e decay is thus essentially an exponential one, independently of the details of the physical process. In addition, the decay time τ depends on the interaction and on the square modulus of the transition matrix element. is result was named the golden rule by Fermi,* because it works so well despite being an approximation whose domain of applicability is not easy to specify.
In practice, decay follows an exponential law. Experiments failed to see a deviation from this behaviour for over half a century. On the other hand, quantum theory shows
* Originally, the golden rule is an expression from the christian bible, namely the sentence ‘Do to others what you want them to do to you’.
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Ref. 815 Ref. 816
that decay is exponential only in certain special systems. A calculation that takes into account higher order terms predicts deviations from exponential decay for completely isolated systems: for short times, the decay rate should vanish; for long times, the decay rate should follow an algebraic – not an exponential – dependence on time, in some cases even with superimposed oscillations. Only a er an intense experimental search deviations for short times have nally been observed. e observation of deviations at long times are rendered impossible by the ubiquity of thermal noise. eory shows that the exponential decay so regularly found in nature results only when the environment is noisy, the system made of many particles, or both. Since this is usually the case, the exceptional exponential decay becomes the (golden) rule in usual observations.
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Ref. 817
Utere tempore.*
Ovidius
“ ” As shown by perception research, what humans call ‘present’ has a duration of a few tenths
of a second. is leads us to ask whether the physical present might have a duration as
well.
Every observation, like every photograph, implies a time average: observations average
interactions over a given time. For a photograph, the duration is given by the shutter time;
for a measurement, the average is de ned by the set-up used. Whatever this set-up might
be, the averaging time is never zero.
We thus need to ask whether the result of an observation will change if the observation
time is shortened as much as possible, or if the observations will simply approach some
limit situation. In everyday life, we are used to imagine that shortening the time taken
to measure the position of a point object as much as possible will approach the ideal of
a particle xed at a given point in space. When Zeno discussed ight of an arrow, he
assumed that this is possible.
Quantum theory has brought us so many surprises that the question should be stud-
ied carefully. We already know that the quantum of action makes rest an impossibility.
However, the issue here is di erent: we are asking whether we can say that a system is
at a given spot at a given time. In order to determine this, we could use a photographic
camera whose shutter time can be reduced at will. What would we nd? When the shut-
ter time approaches the oscillation period of light, the sharpness of the image would de-
crease; in addition, the colour of the light would be in uenced by the shutter motion. We
can increase the energy of the light used, but the smaller wavelengths only shi the prob-
lem. At extremely small wavelengths, matter becomes transparent, and shutters cannot be
realized any more. Quantum theory does not con rm the naive expectation that shorter
shutter times lead to sharper images. In other words, the quantum aspects of the world
show us that there is no way in principle to approach the limit that Zeno was discussing.
Whenever one reduces shutter times as much as possible, observations become unsharp.
is counter-intuitive result is due to the quantum of action: through the indeterminacy
relation, the smallest action prevents that moving objects are at a xed position at a given
* ‘Use your time.’ Tristia 4, 3, 83
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time. Zeno’s discussion was based on an extrapolation of classical physics into domains where it is not valid any more. ere is no ‘point-like’ instant of time that describes the present. e present is always an average over a non-vanishing interval of time.
W
?
Zeno was thus wrong in assuming that motion is a sequence of speci c positions in space. Quantum theory implies that motion is not the change of position with time. e investigation of the issue showed that this statement is only an approximation for low energies or for long observation times.
How then can we describe motion in quantum theory? Quantum theory shows that motion is the low energy approximation of quantum evolution. Quantum evolution assumes that space and time measurements of su cient precision can be performed. We know that for any given observation energy, we can build clocks and metre bars with much higher accuracy than required, so that quantum evolution is applicable in all cases.
Motion is an approximation of quantum evolution. Obviously, this pragmatic description of motion rests on the assumption that for any
observation energy we can nd a still higher energy used by the measurement instruments to de ne space and time. We deduce that if a highest energy would exist in nature, we would get into big trouble, as quantum theory would then break down. As long as energy has no limits, all problems are avoided, and motion remains a sequence of quantum observables or quantum states, whichever you prefer.
e assumption of energy without limit works extremely well; it lies at the basis of the whole second part of the mountain ascent, even though it is rather hidden. In the third and nal part, we will discover that there indeed is a maximum energy in nature, so that we will need to change our approach. However, this energy value is so huge that it does not bother us at all at this point of our exploration. But it will do so later on.
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Page 691
Consciousness is our ability to observe what is going on in our mind. is activity, like any type of change, can itself be observed and studied. Obviously, consciousness takes place in the brain. If it were not, there would be no way to keep it connected with a given person. We know that each brain moves with over one million kilometres per hour through the cosmic background radiation; we also observe that consciousness moves along with it.
e brain is a quantum system; it is based on molecules and electrical currents. e changes in consciousness that appear when matter is taken away from the brain – in operations or accidents – or when currents are injected into the brain – in accidents, experiments or misguided treatments – have been described in great detail by the medical profession. Also the observed in uence of chemicals on the brain – from alcohol to hard drugs – makes the same point. e brain is a quantum system.
Magnetic resonance imaging can detect which parts of the brain work when sensing, remembering or thinking. Not only is sight, noise and thought processed in the brain; we can follow the processing on computer screens. e other, more questionable experi-
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Challenge 1306 ny Challenge 1307 ny
mental method, positron tomography, works by letting people swallow radioactive sugar. It con rms the ndings on the location of thought and on its dependence on chemical fuel. In addition, we already know that memory depends on the particle nature of matter. All these observations depend on the quantum of action.
Not only the consciousness of others, also your own consciousness is a quantum process. Can you give some arguments?
In short, we know that thought and consciousness are examples of motion. We are thus in the same situation as material scientists were before quantum theory: they knew that electromagnetic elds in uence matter, but they could not say how electromagnetism was involved in the build-up of matter. We know that consciousness is made from the signal propagation and signal processing in the brain; we know that consciousness is an electrochemical process. But we do not know yet the details of how the signals make up consciousness. Unravelling the workings of this fascinating quantum system is the aim of neurological science. is is one of the great challenges of twenty- rst century science.
It is sometimes claimed that consciousness is not a physical process. Every expert of motion should be able to convincingly show the opposite, even though the details are not clear yet. Can you add arguments to the ones given here?
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Studying nature can be one of the most intense pleasures of life. All pleasure is based on the ability to observe motion. Our human condition is central to this ability. In our adventure so far we found that we experience motion only because we are of nite size, only because we are made of a large but nite number of atoms, only because we have a nite but moderate temperature, only because we are a mixture of liquids and solids, only because we are electrically neutral, only because we are large compared to a black hole of our same mass, only because we are large compared to our quantum mechanical wavelength, only because we have a limited memory, only because our brain forces us to approximate space and time by continuous entities, and only because our brain cannot avoid describing nature as made of di erent parts. If any of these conditions were not ful lled we would not observe motion; we would have no fun studying physics.
In addition, we saw that we have these abilities only because our forefathers lived on Earth, only because life evolved here, only because we live in a relatively quiet region of our galaxy, and only because the human species evolved long a er than the big bang.
If any of these conditions were not ful lled, or if we were not humans (or animals), motion would not exist. In many ways motion is thus an illusion, as Zeno of Elea had claimed. To say the least, the observation of motion is due to the limitations of the human condition. A complete description of motion and nature must take this connection into account. Before we do that, we explore a few details of this connection.
C e fascination of quantum e ects is still lasting, despite over
years of intense studies.
** Challenge 1308 n Are ghost images in TV sets, o en due to spurious re ections, examples of interference?
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** Challenge 1309 r What happens when two monochromatic electrons overlap?
**
e sense of smell is quite complex. For example, the substance that smells most badly to humans is skatole or 3-methylindole. is is the molecule to which the human nose is most sensitive. Skatole makes faeces smell bad; it is a result of haemoglobin entering the digestive tract through the bile. In contrast to humans, skatole attracts ies; it is also used by some plants for the same reason.
On the other hand, small levels of skatole do not smell bad to humans. It is also used by the food industry in small quantities to give smell and taste to vanilla ice cream.
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Challenge 1310 n
**
It is worth noting that human senses detect energies of quite di erent magnitudes. e eyes can detect light energies of about aJ, whereas the sense of touch can detect only energies as large as about µJ. Is one of the two systems relativistic?
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 818
**
Compared to all primates, the human eye is special: it is white, thus allowing others to see the direction in which one looks. Comparison with primates shows that the white colour has evolved to allow more communication between individuals.
Challenge 1311 n
**
e high sensitivity of the ear can be used to hear light. To do this, take an empty ml jam glass. Keeping its axis horizontal, blacken the upper half of the inside with a candle.
e lower half should remain transparent. A er doing this, close the jam glass with its lid, and drill a 2 to mm hole into it. If you now hold the closed jam glass with the hole to your ear, keeping the black side up, and shining into it from below with a W light bulb, something strange happens: you hear a Hz sound. Why?
**
Most senses work already before birth. It is well-known that playing the violin to a pregnant mother every day during the pregnancy has an interesting e ect. Even if nothing is told about it to the child, it will become a violin player later on.
Ref. 819
**
ere is ample evidence that not using the senses is damaging. People have studied what happens when in the rst years of life the vestibular sense – the one used for motion detection and balance restoration – is not used enough. Lack of rocking is extremely hard to compensate later in life. Also dangerous is the lack of use of the sense of touch. Babies, like all small mammals, that are generally and systematically deprived of these experiences tend to violent behaviour during the rest of their life.
**
Nature has indeed invented pleasure as a guide for a human behaviour. All of biology builds on chemistry and on materials science. Let’s have a short overview of both elds.
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Figure to be included
F I G U R E 332 Atoms and dangling bonds
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DNA
Ref. 828 Page 1181
It is an old truth that Schrödinger’s equation contains all of chemistry.* With quantum theory, for the rst time people were able to calculate the strengths of chemical bonds, and what is more important, the angle between them. Quantum theory thus explains the shape of molecules and thus indirectly, the shape of all matter.
To understand molecules, the rst step is to understand atoms. e early quantum theorists, above all Niels Bohr, spent a lot of energy in understanding their structure. e main result is that atoms are structured, though spherical electron clouds.
A er more than thirty years of work by the brightest physicists in Göttingen and Copenhagen, it was found that electrons in atoms form various layers around the central nucleus. e layers are numbered from the inside by a number called the principal quantum number, usually written n.
Quantum theory shows that the rst layer has room for two electrons, the second for , the third for and the general n-th shell for n electrons. A way to picture this connection is shown in Figure . It is called the periodic table of the elements. e standard way to show the table is shown in Appendix C.
– CS – more to be added – CS –
Ref. 829 Ref. 830 Challenge 1312 e
Page 897
When atoms approach each other, they can form one or several bonds. e preferred distance of these bonds, the angles between them, are due to the structure of the atomic electron clouds.
Do you remember those funny pictures of school chemistry about orbitals and dangling bonds? Well, dangling bonds can now be seen. Several groups were able to image them using scanning force or scanning tunnelling microscopes.
e angles between the bonds explain why the angle of tetrahedral skeletons ( arctan = . °) are so common in molecules. For example, the H-O-H angle in water molecules is °.
At the centre of each atom cloud is the nucleus, which contains almost all the atomic mass. e nucleus consists of protons and neutrons. e structure of nuclei is even more complex than that of electron clouds. We explore it in a separate chapter later on.
Ref. 820 * e precise statement is: the Dirac equation contains all of chemistry. e relativistic e ects that distinguish the two equations are necessary, for example, to understand why gold is yellow and does not like to react or why mercury is liquid.
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n = 1
1 H
2 He
n = 2
7 N
8 O
6 C
3 Li
4 Be
9 F
5 B
10 Ne
n = 3
25 Mn
26 Fe
24 Cr
15 P
16 S
27 Co
23 V
14 Si
11 Na
12 Mg
17 Cl
28 Ni
22 Ti
13 Al
18 Ar
29 Cu
21 Sc
30 Zn
n = 4 61 Pm
64 Gd
65 Tb
63 Eu
43 Tc
44 Ru
66 Dy
62 Sm
42 Mo
33 As
34 Se
45 Rh
67 Ho
41 Nb
32 Ge
19 K
20 Ca
35 Br
46 Pd
60 Nd
40 Zr
31 Ga
36 Kr
47 Ag
69 Tm
59 Pr
39 Y
48 Cd
70 Yb
58 Ce
71 Lu
68 Er
n = 5 93 Np
96 Cm
97 Bk
95 Am
75 Re
76 Os
98 Cf
94 Pu
74 W
51 Sb
52 Te
77 Ir
99 Es
73 Ta
50 Sn
37 Rb
38 Sr
53 I
78 Pt
92 U
72 Hf
49 In
54 Xe
79 Au
101 Md
91 Pa
57 La
80 Hg
102 No
90 Th
103 Lr
100 Fm
n = 6
105 Ha
107 Bh
108 Hs
106 Sg
83 Bi
84 Po
109 Mt
82 Pb
55 Cs
56 Ba
85 At
104 Rf
81 Tl
86 Rn
111 Uuu
89 Ac
112 Uub
110 Ds
n = 7
115 Uup 116 Uuh
114 Uuq
87 Fr
88 Ra
117 Uus
113 Uut 118 Uuo
n = 8
119 Uun 120 Udn
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s p d
f
shell of last electron
F I G U R E 333 An unusual form of the periodic table of the elements
shell electron number increases clockwise
R
D
Probably the most fascinating molecule is human deoxyribonucleic acid. e nucleic acids where discovered in by the Swiss physician Friedrich Miescher ( – ) in white blood cells. In he published an important study showing that the molecule is contained in spermatozoa, and discusses the question if this substance could be related
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Figure missing
F I G U R E 334 Several ways to picture DNA
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to heredity. With his work, Miescher paved the way to a research eld that earned many colleagues Nobel Prizes (though not for himself).
DNA is a polymer, as shown in Figure , and is among the longest molecules known. Human DNA molecules, for example, can be up to cm in length. It consists of a double helix of sugar derivates, to which four nuclei acids are attached in irregular order.
At the start of the twentieth century it became clear that Desoxyribonukleinsäure (DNS) or deoxyribonucleic acid (DNA) in English – was precisely what Erwin Schrödinger had predicted to exist in his book What Is Life? As central part of the chromosomes contained the cell nuclei, DNA is responsible for the storage and reproduction of the information on the construction and functioning of Eukaryotes. e information is coded in the ordering of the four nucleic acids. DNA is the carrier of hereditary information. DNA determines in great part how the single cell we all once have been grows into the complex human machine we are as adults. For example, DNA determines the hair colour, predisposes for certain illnesses, determines the maximum size one can grow to, and much more. Of all known molecules, human DNA is thus most intimately related to human existence. e large size of the molecules is the reason that understanding its full structure and its full contents is a task that will occupy scientists for several generations to come.
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Muscles produce motion through electrical stimulation. Can technical systems do the same? ere is a candidate. So-called electroactive polymers change shape when they are activated with electrical current or with chemicals. ey are lightweight, quiet and simple to manufacture. However, the rst arm wrestling contest between human and arti cial muscles held in 2005 was won by a teenage girl. e race to do better is ongoing.
**
A cube of sugar does not burn. However, if you put some cigarette ash on top of it, it burns. Why?
**
Challenge 1313 ny Why do organic materials burn at much lower temperature than inorganic materials?
**
An important aspect of life is death. When we die, conserved quantities like our energy,
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•
momentum, angular momentum and several other quantum numbers are redistributed. ey are redistributed because conservation means that nothing is lost. What does all
Challenge 1314 ny this imply for what happens a er death?
**
Chemical reactions can be slow but still dangerous. Spilling mercury on aluminium will lead to an amalgam that reduces the strength of the aluminium part a er some time. at is the reason that bringing mercury thermometers on aeroplanes is strictly forbidden.
M
Did you know that one cannot use a boiled egg as a toothpick?
“ Karl Valentin ” It was mentioned several times that the quantum of action explains all properties of mat-
ter. Many researchers from physics, chemistry, metallurgy, engineering, mathematics and biology have cooperated in the proof of this statement. In our mountain ascent we have little time to explore this vast topic. Let us walk through a selection.
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We do not fall through the mountain we are walking on. Some interaction keeps us from
falling through. In turn, the continents keep the mountains from falling through them.
Also the liquid magma in the Earth’s interior keeps the continents from sinking. All
these statements can be summarized. Atoms do not penetrate each other. Despite being
mostly empty clouds, atoms keep a distance. All this is due to the Pauli principle between
electrons. the fermion character of electrons avoids that atoms interpenetrate. At least on
Earth.
Not all oors keep up due to the fermion character of electrons. Atoms are not impen-
etrable at all pressures. ey can collapse, and form new types of oors. Some oors are
so exciting to study that people have spent their whole life to understand why they do not
fall, or when they do, how it happens: the surfaces of stars.
In most stars, the radiation pressure of the light plays only a minor role. Light pressure
does play a role in determining the size of red giants, such as Betelgeuse; but for average
stars, light pressure is negligible.
In most stars, such as in the Sun, the gas pressure takes the role which the incompress-
ibility of solids and liquids has for planets. e pressure is due to the heat produced by
the nuclear reactions.
e next star type appears whenever light pressure, gas pressure and the electronic
Pauli pressure cannot keep atoms from interpenetrating. In that case, atoms are com-
pressed until all electrons are pushed into the protons. Protons then become neut-
rons, and the whole star has the same mass density of atomic nuclei, namely about
. ë kg m . A drop weighs about
tons. In these so-called neutron stars, the
oor – or better, the size – is also determined by Pauli pressure; however, it is the Pauli
pressure between neutrons, triggered by the nuclear interactions. ese neutron stars are
all around km in radius.
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TA B L E 61 The types of rocks and stones
T
P
Igneous rocks (magmatites)
formed from magma, 95% of all rocks
Sedimenary rocks (sedimentites)
o en with fossils
Metamorphic rocks transformed by
(metamorphites)
heat and pressure
Meteorites
from the solar system
S volcanic or extrusive
plutonic or intrusive clastic
biogenic
precipitate foliated
non-foliated (grandoblastic or hornfelsic) rock meteorites
E
basalt (ocean oors, Giant’s causeway), andesite, obsidian granite, gabbro
shale, siltstone, sandstone limestone, chalk, dolostone halite, gypsum
slate, schist, gneiss (Himalayas) marble, skarn, quartzite
iron meteorites
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Page 476
If the pressure increases still further the star becomes a black hole, and never stops collapsing. Black holes have no oor at all; they still have a constant size though, determined by the horizon curvature.
e question whether other star types exist in nature, with other oor forming mechanisms – such as quark stars – is still a topic of research.
R
If a geologist takes a stone his his hands, he is usually able to give, within an error of a few percent, the age of the stone. e full story forms a large part of geology, but the general lines should be known to every physicist.
Every stone arrives in your hand through the rock cycle. e rock cycle is a process that transforms magma from the interior of the Earth into igneous rocks, through cooling and crystallization. Igneous rocks, such as basalt, can transform through erosion, transport and deposition into sedimentary rocks. Either of these two rock types can be transformed through high pressures or temperatures into metamorphic rocks, such as marble. Finally, most rocks are generally – but not always – transformed back into magma.
e full rock cycle takes around to million years. For this reason, rocks that are older that this age much less common on Earth. Any stone is the product of erosion of one of the rock types. A geologist can usually tell, simply by looking at it, the type of rock it belongs to; if he sees the original environment, he can also give the age, without any laboratory.
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Challenge 1315 n Page 897
Quantum theory showed us that all obstacles have only nite potential heights. at leads to a question: Is it possible to look through matter? For example, can we see what is hidden inside a mountain? To be able to do this, we need a signal which ful ls two conditions: it must be able to penetrate the mountain, and it must be scattered in a material-dependent way. Table gives an overview of the possibilities.
We see that many signals are able to penetrate a mountain. However, only sound or radio waves provide the possibility to distinguish di erent materials, or to distinguish solids from liquids and from air. In addition, any useful method requires a large number of signal sources and of signal receptors, and thus a large amount of cash. Will there ever be a simple method allowing to look into mountains as precisely as X-rays allow to study human bodies? For example, will it ever be possible to map the interior of the pyramids? A motion expert like the reader should be able to give a de nite answer.
One of the high points of twentieth century physics was the development of the best method so far to look into matter with dimensions of about a metre or less: magnetic resonance imaging. We will discuss it later on.
e other modern imaging technique, ultrasound imaging, is getting more and more criticized. It is much used for prenatal diagnostics of embryos. However, studies have found that ultrasound produces extremely high levels of audible sound to the baby, especially when the ultrasound is switched on or o , and that babies react negatively to this loud noise.
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Ref. 821
You might have already imagined what adventures would be possible if you could be invisible for a while. Some years ago, a team of Dutch scientists found a material than can be switched from mirror mode to transparent mode using an electrical signal. is seems a rst step to realize the dream to become invisible at will.
Nature shows us how to be invisible. An object is invisible if it has no surface, no absorption and small size. In short, invisible objects are either small clouds or composed of them. Most atoms and molecules are examples. Homogeneous non-absorbing gases also realize these conditions. at is the reason that air is (usually) invisible. When air is not homogeneous, it can be visible, e.g. above hot surfaces.
In contrast to gases, solids or liquids do have surfaces. Surfaces are usually visible, even if the body is transparent, because the refractive index changes there. For example, quartz can be made so transparent that one can look through km of it; pure quartz is thus more transparent than usual air. Still, objects made of pure quartz are visible to the eye due to the index change at the surface. Quartz can be invisible only when submerged in liquids with the same refractive index.
In other words, to become invisible, we must transform ourselves into a di use cloud of non-absorbing atoms. On the way to become invisible, we would loose all memory and all genes, in short, we would loose all our individuality. But an individual cannot be made of gas. An individual is de ned through its boundary. ere is no way that we can be invisible; a reversible way to perform the feat is also impossible. In summary, quantum theory shows that only the dead can be invisible.
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TA B L E 62 Signals penetrating mountains and other matter
S
P
-A
-M
-
-U -
matter di usion of water or c. km liquid chemicals di usion of gases c. km
electromagnetism
sound, explosions, seismic waves ultrasound
.− m
c. m c. m
c. l mm
infrasound and earthquakes
static magnetic elds
km
km
medium medium
high high high medium
static electric elds electrical currents
electromagnetic
sounding ( . Hz to
Hz)
radio waves
m
low
no use
m to mm small
mm and THz waves below mm mm
infrared visible light X-rays
c. m c. cm a few metre
.m . µm µm
muons created by up to
.m
cosmic radiation c. m
weak interactions neutrino beams light years zero
strong interactions cosmic radiation radioactivity
m to km mm to m
gravitation
change of
m
gravitational
acceleration
medium medium high small
very weak
low
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
mapping hydrosystems studying vacuum systems
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oil and ore search, structure mapping in rocks medical imaging, acoustic microscopy mapping of Earth crust and mantle cable search, cable fault localization, search for structure inside soil and rocks via changes of the Earth’s magnetic eld
soil and rock investigations, search for tooth decay soil and rock investigations in deep water and on land
soil radar (up to MW), magnetic imaging, research into solar interior see through clothes, envelopes and teeth Ref. 822 mapping of soil over m imaging of many sorts medicine, material analysis, airports
nding caves in pyramids, imaging truck interiors
studies of Sun
airports oil & ore search
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TA B L E 63 Matter at lowest temperatures
P
T
L
-E
Solid
conductor
Liquid Gas
insulator bosonic
fermionic bosonic
fermionic
superconductivity antiferromagnet ferromagnet diamagnet Bose–Einstein condensation, i.e. super uidity pairing, then BEC, i.e. super uidity Bose–Einstein condensation
pairing, then Bose–Einstein condensation
lead chromium, MnO iron
He
He
Rb, Li, He, K
K, Li
Na, H,
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Page 847
e low-temperature behaviour of matter has numerous experimental and theoretical aspects. e rst issue is whether matter is always solid at low temperatures. e answer is no. All phases exist at low temperatures, as shown in Table .
Concerning the electric properties of matter at lowest temperatures, the present status is that matter is either insulating or superconducting. Finally, one can ask about the magnetic properties of matter at low temperatures. We know already that matter can not be paramagnetic at lowest temperatures. It seems that matter is either ferromagnetic, diamagnetic or antiferromagnetic at lowest temperatures.
More about super uidity and superconductivity will be told below.
C Materials science is not a central part of this walk. A few curiosities can give a taste of it.
**
Ref. 823 Challenge 1316 ny
Challenge 1317 n
What is the maximum height of a mountain? is question is of course of interest to all
climbers. Many e ects limit the height. e most important is the fact that under heavy pressure, solids become liquid. For example, on Earth this happens at about km. is
is quite a bit more than the highest mountain known, which is the volcano Mauna Kea in Hawaii, whose top is about . km above the base. On Mars gravity is weaker, so that mountains can be higher. Indeed the highest mountain on Mars, Olympus mons, is km
high. Can you nd a few other e ects limiting mountain height?
Challenge 1318 r
**
Do you want to become rich? Just invent something that can be produced in the factory, is cheap and can substitute duck feathers in bed covers, sleeping bags or in badminton shuttlecocks. Another industrial challenge is to nd an arti cial substitute for latex, and a
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
third one is to nd a substitute for a material that is rapidly disappearing due to pollution: cork.
** Challenge 1319 ny What is the di erence between solids, liquids and gases?
**
What is the di erence between the makers of bronze age knifes and the builders of the Ei el tower? Only their control of dislocation distributions.
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** Challenge 1320 n Quantum theory shows that tight walls do not exist. Every material is penetrable. Why?
**
Quantum theory shows that even if tight walls would exist, the lid of such a box can never Challenge 1321 n be tightly shut. Can you provide the argument?
**
Quantum theory predicts that heat transport at a given temperature is quantized. Can Challenge 1322 ny you guess the unit of thermal conductance?
**
Ref. 824 Ref. 825
Robert Full has shown that van der Waals forces are responsible for the way that geckos walk on walls and ceilings. e gecko, a small reptile with a mass of about 100 g, uses an elaborate structure on its feet to perform the trick. Each foot has 500 000 hairs each split in up to 1000 small spatulae, and each spatula uses the van der Waals force (or alternatively, capillary forces) to stick to the surface. As a result, the gecko can walk on vertical glass walls or even on glass ceilings; the sticking force can be as high as N per foot.
e same mechanism is used by jumping spiders (Salticidae). For example, Evarcha arcuata have hairs at their feet which are covered by hundred of thousands of setules. Again. the van der Waals force in each setule helps the spider to stick on surfaces.
Researchers have copied these mechanisms for the rst time in 2003, using microlithography on polyimide, and hope to make durable sticky materials in the future.
**
Millimetre waves or terahertz waves are emitted by all bodies at room temperature. Modern camera systems allow to image them. In this way, it is possible to see through clothes.
is ability could be used in future to detect hidden weapons in airports. But the development of a practical and a ordable detector which can be handled as easily as a binocular is still under way. e waves can also be used to see through paper, thus making it unnecessary to open letters in order to read them. Secret services are exploiting this technique. A third application of terahertz waves might be in medical diagnostic, for example for the search of tooth decay. Terahertz waves are almost without side e ects, and thus superior to X-rays. e lack of low-priced quality sources is still an obstacle to their application.
**
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 826
Does the melting point of water depend on the magnetic eld? is surprising claim was made in 2004 by Inaba Hideaki and colleagues. ey found a change of . mK T. It is known that the refractive index and the near infrared spectrum of water is a ected by magnetic elds. Indeed, not everything about water might be known yet.
**
Plasmas, or ionized gases, are useful for many applications. Not only can they be used for heating or cooking and generated by chemical means (such plasmas are variously called re or ames) but they can also be generated electrically and used for lighting or deposition of materials. Electrically generated plasmas are even being studied for the disinfection of dental cavities.
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Challenge 1323 ny
**
It is known that the concentration of CO in the atmosphere between 1800 and 2005 has increased from 280 to 380 parts per million. (How would you measure this?) It is known without doubt that this increase is due to human burning of fossil fuels, and not to natural sources such as the oceans or volcanoes. ere are three arguments. First of all, there was a parallel decline of the C C ratio. Second, there was a parallel decline of the C C ratio. Finally, there was a parallel decline of the oxygen concentration. All three measurements independently imply that the CO increase is due to the burning of fuels, which are low in C and in C, and at the same time decrease the oxygen ratio. Natural sources do not have these three e ects. Since CO is a major greenhouse gas, the data implies that humans are also responsible for a large part of the temperature increase during the same period.
Ref. 827
**
e technologies to produce perfect crystals, without grain boundaries or dislocations, are an important part of modern industry. Perfectly regular crystals are at the basis of the integrated circuits used in electronic appliances, are central to many laser and telecommunication systems and are used to produce synthetic jewels.
Synthetic diamonds have now displaced natural diamonds in almost all applications. In the last years, methods to produce large, white, jewel-quality diamonds of ten carats and more are being developed. ese advances will lead to a big change in all the domains that depend on these stones, such as the production of the special surgical knives used in eye lens operation.
** Challenge 1324 ny How can a small plant pierce through tarmac?
Q
“I were better to be eaten to death with a rust than to be scoured to nothing with perpetual motion. William Shakespeare ( – ) King Henry ”IV.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Quantum e ects do not appear only in microscopic systems. Several quantum e ects are important in modern life; transistors, lasers, superconductivity and a few other e ects and systems are worth knowing.
M
–
Ref. 837
We are used to think that friction is inevitable. We even learned that friction was an inevitable result of the particle structure of matter. I should come to the surprise of every physicist that motion without friction is possible.
In Gilles Holst and Heike Kamerlingh Onnes discovered that at low temperatures, electric currents can ow with no resistance, i.e., with no friction, through lead. e observation is called superconductivity. In the century a er that, many metals, alloys and ceramics have been found to show the same behaviour.
e condition for the observation of motion without friction is that quantum e ects play an essential role. at is the reason for the requirement of low temperature in such experiments. Nevertheless, it took over years to reach a full understanding of the effect. is happened in , when Bardeen, Cooper and Schrie er published their results. At low temperatures, electron behaviour is dominated by an attractive interaction that makes them form pairs – today called Cooper pairs – that are e ective bosons. And bosons can all be in the same state, thus e ectively moving without friction.
For superconductivity, the attractive interaction between electrons is due to the deformation of the lattice. Two electrons attract each other in the same way as two masses attract each other due to deformation of the space-time mattress. However, in the case of solids, the deformations are quantized. With this approach, Bardeen, Cooper and Schrieffer explained the lack of electric resistance of superconducting materials, their complete diamagnetism (µr = ), the existence of an energy gap, the second-order transition to normal conductivity at a speci c temperature, and the dependence of this temperature on the mass of the isotopes. Last but not least, they received the Nobel Prize in .*
Another type of motion without friction is super uidity. Already in , Pyotr Kapitsa had predicted that normal helium ( He), below a transition observed at the temperature of . K, would be a super uid. In this domain, the uid moves without friction through tubes. (In fact, the uid remains a mixture of a super uid component and a normal component.) Helium is even able, a er an initial kick, to ow over obstacles, such as glass walls, or to ow out of bottles. He is a boson, so no pairing is necessary for it to ow without friction. is research earned Kapitsa a Nobel Prize in .
e explanation of superconductivity also helped for fermionic super uidity. In , Richardson, Lee, and Oshero found that even He is super uid at temperatures below . mK. He is a fermion, and requires pairing to become super uid. In fact, below . mK, He is even super uid in two di erent ways; one speaks of phase A and phase
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* For John Bardeen (1908–1991), this was his second, a er he had got the rst Nobel Prize in 1956, shared with William Shockley and Walter Brattain, for the discovery of the transistor. e rst Nobel Prize was a problem for Bardeen, as he needed time to work on superconductivity. In an example to many, he reduced the tam-tam around himself to a minimum, so that he could work as much as possible on the problem of superconductivity. By the way, Bardeen is topped by Frederick Sanger and by Marie Curie. Sanger rst won a Nobel Prize in chemistry in 1958 by himself and then won a second one shared with Walter Gilbert in 1980; Marie Curie rst won one with her husband and a second one by herself, though in two di erent elds.
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Figure missing
F I G U R E 335 Nuclear magnetic resonance shows that vortices in superfluid He-B are quantized.
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Ref. 831
B.* In this case, the theoreticians had been faster. e theory for superconductivity
through pairing had been adapted to super uids already in – before any data were available – by Bohr, Mottelson and Pines. is theory was then adapted again by Anthony Leggett.** e attractive interaction between He atoms turns out to be the spin-spin interaction.
In super uids, like in ordinary uids, one can distinguish between laminar and turbulent ow. e transition between the two regimes is mediated by the behaviour of vortices. But in super uids, vortices have properties that do not appear in normal uids. In the super uid He-B phase, vortices are quantized: vortices only exist in integer multiples of the elementary circulation h m He. Present research is studying how these vortices behave and how they induce the transition form laminar to turbulent ows.
In recent years, studying the behaviour of gases at lowest temperatures has become very popular. When the temperature is so low that the de Broglie wavelength is comparable to the atom-atom distance, bosonic gases form a Bose-Einstein condensate. e rst one were realized in by several groups; the group around Eric Cornell and Carl Wieman used Rb, Rand Hulet and his group used Li and Wolfgang Ketterle and his group used Na. For fermionic gases, the rst degenerate gas, K, was observed in by the group around Deborah Jin. In , the same group observed the rst gaseous fermi condensate, a er the potassium atoms paired up.
Q
Ref. 832
In , the Spanish physicist J.L. Costa–Krämer and his colleagues performed a simple experiment. ey put two metal wires on top of each other on a kitchen table and attached a battery, a kΩ resistor and a storage oscilloscope to them. en they measured the electrical current while knocking on the table. In the last millisecond before the wires detach, the conductivity and thus the electrical current diminished in regular steps of a µA, as can easily be seen on the oscilloscope. is simple experiment could have beaten, if it had been performed a few years earlier, a number of enormously expensive experiments which discovered this quantization at costs of several million euro each, using complex set-ups and extremely low temperatures.
In fact, quantization of conductivity appears in any electrical contact with a small cross-
* ey received the Nobel Prize in 1996 for this discovery. ** Aage Bohr and Ben Mottelson received the Nobel Prize in 1975, Anthony Leggett in 2003.
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Challenge 1325 ny
section. In such situations the quantum of action implies that the conductivity can only
be a multiple of e ħ
Ω. Can you con rm this result?
Note that electrical conductivity can be as small as required; only quantized electrical
conductivity has the minimum value of e ħ.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
T
H
Ref. 834 Ref. 833
Ref. 836
e fractional quantum Hall e ect is one of the most intriguing discoveries of materials
science. e e ect concerns the ow of electrons in a two-dimensional surface. In ,
Robert Laughlin predicted that in this system one should be able to observe objects with
electrical charge e . is strange and fascinating prediction was indeed veri ed in .
e story begins with the discovery by Klaus von Klitzing of the quantum Hall ef-
fect. In , Klitzing and his collaborators found that in two-dimensional systems at low
temperatures – about K – the electrical conductance S is quantized in multiples of the
quantum of conductance
S
=
n
e ħ
.
(579)
Ref. 835 Ref. 834
e explanation is straightforward: it is the quantum analogue of the classical Hall effect, which describes how conductance varies with applied magnetic eld. Von Klitzing received the Nobel Prize for physics for the discovery, since the e ect was completely unexpected, allows a highly precise measurement of the ne structure constant and also allows one to build detectors for the smallest voltage variations measurable so far.
Two years later, it was found that in extremely strong magnetic elds the conductance could vary in steps one third that size. Shortly a erwards, even stranger numerical fractions were also found. In a landmark paper, Robert Laughlin explained all these results by assuming that the electron gas could form collective states showing quasiparticle excitations with a charge e . is was con rmed years later and earned him a Nobel price as well. We have seen in several occasions that quantization is best discovered through noise measurements; also in this case, the clearest con rmation came from electrical current noise measurements. How can we imagine these excitations?
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– CS – explanation to be inserted – CS –
What do we learn from this result? Systems in two dimensions have states which follow di erent rules than systems in three dimensions. Can we infer something about quarks from this result? Quarks are the constituents of protons and neutrons, and have charges e and e . At this point we need to stand the suspense, as no answer is possible; we come back to this issue later on.
L
-
Ref. 853
Photons are vector bosons; a lamp is thus a vector boson launcher. All lamps fall into one of three classes. Incandescent lamps use emission from a hot solid, gas discharge lamps use excitation of atoms, ions or molecules through collision, and solid state lamps generate (cold) light through recombination of charges in semiconductors.
Most solid state lights are light emitting diodes. e large progress in brightness of
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TA B L E 64 A selection of lamps L
H
L-
L-
-
(2003) (2003) (2003)
Incandescent lamps tungsten wire light bulbs, halogen lamps
lm W
. cent lm
h
Gas discharge lamps
oil lamps, candle neon lamps mercury lamps metal halogenide lamps (ScI or ‘xenon’, NaI, DyI , HoI , TmI ) sodium low pressure lamps sodium high pressure lamps
lm W lm W
lm W lm W
... cent lm ... h ... cent lm ... h
... cent lm ... h ... cent lm ... h
Recombination lamps re y
light emitting diodes He-Ne laser
Ideal white lamp
Ideal coloured lamp
lm W lm W
c. lm W
lm W
c. h
cent lm
h
cent lm h
n.a.
n.a.
n.a.
n.a.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 1326 n
light emitting diodes could lead to a drastic reduction in future energy consumption, if their cost is lowered su ciently. Many engineers are working on this task. Since the cost is a good estimate for the energy needed for production, can you estimate which lamp is the most friendly to the environment?
Nobody thought much about lamps, until Albert Einstein and a few other great physicists came along, such as eodore Maiman, Hermann Haken and several others that got the Nobel Prize with their help. In , Einstein showed that there are two types of sources of light – or of electromagnetic radiation in general – both of which actually ‘create’ light. He showed that every lamp whose brightness is turned up high enough will change behaviour when a certain intensity threshold is passed. e main mechanism of light emission then changes from spontaneous emission to stimulated emission. Nowadays such a special lamp is called a laser. ( e letters ‘se’ in laser are an abbreviation of ‘stimulated emission’.) A er a passionate worldwide research race, in Maiman was the rst to build a laser emitting visible light. (So-called masers emitting microwaves were already known for several decades.) In summary, Einstein and the other physicists showed that lasers are lamps which are su ciently turned up. Lasers consist of some light producing and amplifying material together with a mechanism to pump energy into it.
e material can be a gas, a liquid or a solid; the pumping process can use electrical current or light. Usually, the material is put between two mirrors, in order to improve
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Ref. 860 Page 251
Ref. 854 Ref. 861
the e ciency of the light production. Common lasers are semiconductor lasers (essentially highly pumped LEDs or light emitting diodes), He–Ne lasers (highly pumped neon lamps), liquid lasers (essentially highly pumped re ies) and ruby lasers (highly pumped luminescent crystals).
Lasers produce radiation in the range from microwaves and extreme ultraviolet. ey have the special property of emitting coherent light, usually in a collimated beam. erefore lasers achieve much higher light intensities than lamps, allowing their use as tools. In modern lasers, the coherence length, i.e. the length over which interference can be observed, can be thousands of kilometres. Such high quality light is used e.g. in gravitational wave detectors.
People have become pretty good at building lasers. Lasers are used to cut metal sheets up to cm thickness, others are used instead of knives in surgery, others increase surface hardness of metals or clean stones from car exhaust pollution. Other lasers drill holes in teeth, measure distances, image biological tissue or grab living cells. Most materials can be used to make lasers, including water, beer and whiskey.
Some materials amplify light so much that end mirrors are not necessary. is is the case for nitrogen lasers, in which nitrogen, or simply air, is used to produce a UV beam. Even a laser made of a single atom (and two mirrors) has been built; in this example, only eleven photons on average were moving between the two mirrors. Quite a small lamp. Also lasers emitting light in two dimensions have been built. ey produce a light plane instead of a light beam.
Lasers have endless applications. Lasers read out data from compact discs (CDs), are used in the production of silicon integrated circuits, and transport telephone signals; we already encountered lasers that work as loudspeakers. e biggest advances in recent years came from the applications of femtosecond laser pulses. Femtosecond pulses generate high-temperature plasmas in the materials they propagate, even in air. Such short pulses can be used to cut material without heating it, for example to cut bones in heart operations. Femtosecond lasers have been used to make high resolution hologram of human heads within a single ash. Recently such lasers have been used to guide lightning along a predetermined path and seem promising candidates for laser ligtning rods. A curious application is to store information in ngernails (up to Mbit for a few months) using such lasers, in a way not unlike that used in recordable compact discs (CD-R).
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C
?
In , Dirac made the famous statement already mentioned above:*
Each photon interferes only with itself. Interference between two di erent photons never occurs.
Ref. 855
O en this statement is misinterpreted as implying that two separate photon sources cannot interfere. It is almost unbelievable how this false interpretation has spread through the literature. Everybody can check that this statement is incorrect with a radio: two distant radio stations transmitting on the same frequency lead to beats in amplitude, i.e. to
* See the famous, beautiful but di cult textbook P.A.M. D Clarendon Press, Oxford, 1930, page 9.
, e Principles of Quantum Mechanics,
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Ref. 855
wave interference. ( is should not to be confused with the more common radio interference, with usually is simply a superposition of intensities.) Radio transmitters are coherent sources of photons, and any radio receiver shows that two such sources can indeed interfere.
In , interference of two di erent sources has been demonstrated with microwave beams. Numerous experiments with two lasers and even with two thermal light sources have shown light interference from the ies onwards. Most cited is the experiment by Magyar and Mandel; they used two ruby lasers emitting light pulses and a rapid shutter camera to produce spatial interference fringes.
However, all these experimental results do not contradict the statement by Dirac. Indeed, two photons cannot interfere for several reasons.
— Interference is a result of space-time propagation of waves; photons appear only when the energy–momentum picture is used, mainly when interaction with matter takes place. e description of space-time propagation and the particle picture are mutually exclusive – this is one aspect of the complementary principle. Why does Dirac seem to mix the two in his statement? Dirac employs the term ‘photon’ in a very general sense, as quantized state of the electromagnetic eld. When two coherent beams are superposed, the quantized entities, the photons, cannot be ascribed to either of the sources. Interference results from superposition of two coherent states, not of two particles.
— Interference is only possible if one cannot know where the detected photon comes from. e quantum mechanical description of the eld in a situation of interference never allows to ascribe photons of the superposed eld to one of the sources. In other words, if you can say from which source a detected photon comes from, you cannot observe interference.
— Interference between two beams requires a xed phase between them, i.e. an uncertain particle number; in other words, interference is only possible if the photon number for each of the two beams is unknown.
A better choice of words is to say that interference is always between two (indistinguishable) states, or if one prefers, between two possible (indistinguishable) histories, but never between two particles. In summary, two di erent electromagnetic beams can interfere, but not two di erent photons.
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?
Do coherent electron sources exist? Yes, as it is possible to make holograms with electron beams.* However, electron coherence is only transversal, not longitudinal. Transversal coherence is given by the possible size of wavefronts with xed phase. e limit of this size is given by the interactions such a state has with its environment; if the interactions are weak, matter wave packets of several metres of size can be produced, e.g. in particle colliders, where energies are high and interaction with matter is low.
Actually, the term transversal coherence is a fake. e ability to interfere with oneself is not the de nition of coherence. Transversal coherence only expresses that the source size is small. Both small lamps (and lasers) can show interference when the beam is split
Ref. 856 * In 2002, the rst holograms have been produced that made use of neutron beams.
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Figure yet to be included F I G U R E 336 An electron hologram
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and recombined; this is not a proof of coherence. Similarly, monochromaticity is not a proof for coherence either.
A state is called coherent if it possesses a well-de ned phase throughout a given domain of space or time. e size of that region or of that time interval de nes the degree of coherence. is de nition yields coherence lengths of the order of the source size for small ‘incoherent’ sources. Nevertheless, the size of an interference pattern, or the distance d between its maxima, can be much larger than the coherence length l or the source size s.
In summary, even though an electron can interfere with itself, it cannot interfere with a second one. Uncertain electron numbers are needed to see a macroscopic interference pattern. at is impossible, as electrons (at usual energies) carry a conserved charge.
– CS – sections on transistors and superconductivity to be added – CS –
C Many challenges in applied quantum physics remain, as quantum e ects seem to promise to realize many age-old technological dreams.
** Challenge 1327 d Is it possible to make A4-size, thin and exible colour displays for an a ordable price?
** Challenge 1328 r Will there ever be desktop laser engravers for 2000 euro?
** Challenge 1329 r Will there ever be room-temperature superconductivity?
** Challenge 1330 n Will there ever be teleportation of everyday objects?
** One process that quantum physics does not allow is telephathy. An unnamed space agency found out in the Apollo 14 mission, when, during the ight to the moon, cosmonaut Edgar Mitchell tested telepathy as communication means. Unsurprisingly, he found
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Figure will be added in the future
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F I G U R E 337 Ships in a swell
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 838 that it was useless. It is not clear why NASA spent so much money for an experiment that could have been performed for 1000 times less cost also in other ways.
** Challenge 1331 d Will there ever be applied quantum cryptology?
** Will there ever be printable polymer electronic circuits, instead of lithographically patChallenge 1332 d terned silicon electronics as is common now?
** Challenge 1333 r Will there ever be radio-controlled ying toys in the size of insects?
.
–
T central concept the quantum theory introduces in the description of nature is he idea of virtual particles. Virtual particles are short-lived particles; they owe heir existence exclusively to the quantum of action. Because of the quantum of action, they do not need to follow the energy-mass relation that special relativity requires of normal, real particles. Virtual particles can move faster than light and can move backward in time. Despite these strange properties, they have many observable e ects.
S,
C
Ref. 841
When two parallel ships roll in a big swell, without even the slightest wind blowing, they will attract each other. is e ect was well known up to the nineteenth century, when many places still lacked harbours. Shipping manuals advised captains to let the ships be pulled apart using a well-manned rowing boat.
Waves induce oscillations of ships because a ship absorbs energy from the waves.
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When oscillating, the ship also emits waves. is happens mainly towards the two sides of the ship. As a result, for a single ship, the wave emission has no net e ect on its position. Now imagine that two parallel ships oscillate in a long swell, with a wavelength much larger than the distance between the ships. Due to the long wavelength, the two ships will oscillate in phase. e ships will thus not be able to absorb energy from each other. As a result, the energy they radiate towards the outside will push them towards each other.
e e ect is not di cult to calculate. e energy of a rolling ship is
E=m hα
(580)
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where α is the roll angle amplitude, m the mass of the ship and = , m s the accelera-
tion due to gravity. e metacentric height h is the main parameter characterizing a ship,
especially a sailing ship; it tells with what torque the ship returns to the vertical when
inclined by an angle α. Typically, one has h = . m.
When a ship is inclined, it will return to the vertical by a damped oscillation. A damped
oscillation is characterized by a period T and a quality factor Q. e quality factor is the
number of oscillations the system takes to reduce its amplitude by a factor e = . . If the
quality factor Q of an oscillating ship and its oscillation period T are given, the radiated
power W is
W=
π
E QT
.
(581)
We saw above that radiation pressure is W c, where c is the wave propagation velocity. For water waves, we have the famous relation
c= T . π
(582)
Assuming that for two nearby ships each one completely absorbs the power emitted from the other, we nd that the two ships are attracted towards each other following
ma = m
π
hα QT
.
(583)
Ref. 842
Inserting typical values such as Q = . , T = s, α = . rad and a ship mass of tons, we get about . kN. Long swells thus make ships attract each other. e intensity of the attraction is comparatively small and can indeed be overcome with a rowing boat. On the other hand, even the slightest wind will damp the oscillation amplitude and have other e ects that will avoid the observation of the attraction.
Sound waves or noise in air can have the same e ect. It is su cient to suspend two metal plates in air and surround them by loudspeakers. e sound will induce attraction (or repulsion) of the plates, depending on whether the sound wavelength cannot (or can) be taken up by the other plate.
In , the Dutch physicist Hendrik Casimir made one of the most spectacular predictions of quantum theory: he predicted a similar e ect for metal plates in vacuum. Casimir, who worked at the Dutch Electronics company Philips, wanted to understand why it was
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•
so di cult to build television tubes. Television screens are made by deposing small neut-
ral particles on glass, but Casimir observed that the particles somehow attracted each
other. Casimir got interested in understanding how neutral particles interact. During
these theoretical studies he discovered that two neutral mirrors (or metal plates) would
attract each other even in complete vacuum. is is the famous Casimir e ect. Casimir
also determined the attraction strength between a sphere and a plate, and between two
spheres. In fact, all conducting bodies attract each other in vacuum, with a force depend-
ing on their geometry.
In all these situations, the role of the sea is taken by the zero-point uctuations of the
electromagnetic eld, the role of the ships by the mirrors. Casimir understood that the
space between two parallel mirrors, due to the geometrical constraints, had di erent zero-
point uctuations that the free vacuum. Like two ships, the result would be the attraction
of the mirrors.
Casimir predicted that the attraction for two mirrors of mass m and surface A is given
by
ma A
=
π
ħc d
.
(584)
Ref. 843 Ref. 844 Ref. 845
Challenge 1334 n Ref. 847 Ref. 848
Challenge 1335 n
e e ect is a pure quantum e ect; in classical electrodynamics, two neutral bodies do not attract. e e ect is small; it takes some dexterity to detect it. e rst experimental con rmation was by Derjaguin, Abrikosova and Lifshitz in ; the second experimental con rmation was by Marcus Sparnaay, Casimir’s colleague at Philips, in . Two beautiful high-precision measurements of the Casimir e ect were performed in by Lamoreaux and in by Mohideen and Roy; they con rmed Casimir’s prediction with a precision of % and % respectively.*
In a cavity, spontaneous emission is suppressed, if it is smaller than the wavelength of the emitted light! is e ect has also been observed. It con rms the old saying that spontaneous emission is emission stimulated by the zero point uctuations.
e Casimir e ect thus con rms the existence of the zero-point uctuations of the electromagnetic eld. It con rms that quantum theory is valid also for electromagnetism.
e Casimir e ect between two spheres is proportional to r and thus is much weaker than between two parallel plates. Despite this strange dependence, the fascination of the Casimir e ect led many amateur scientists to speculate that a mechanism similar to the Casimir e ect might explain gravitational attraction. Can you give at least three arguments why this is impossible, even if the e ect had the correct distance dependence?
Like the case of sound, the Casimir e ect can also produce repulsion instead of attraction. It is su cient that one of the two materials be perfectly permeable, the other a perfect conductor. Such combinations repel each other, as Timothy Boyer discovered in
. e Casimir e ect bears another surprise: between two metal plates, the speed of light
changes and can be larger than c. Can you imagine what exactly is meant by ‘speed of light’ in this context?
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Ref. 846 * At very small distances, the dependence is not d , but d .
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TB
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Page 55 Page 219
It implies that there is a speci c energy density that can be described to the vacuum. is seems obvious. However, the statement has a dramatic consequence: space-time cannot be continuous!
e reasoning is simple. If the vacuum were continuous, we could make use of the Banach–Tarski paradox and split, without any problem, a ball of vacuum into two balls of vacuum, each with the same volume. In other words, one ball with energy E could not be distinguished from two balls of energy E. is is impossible.
e Gedanken experiment tells us something important. In the same way that we used the argument to show that chocolate (and any other matter) cannot be continuous, we can now deduce that the vacuum cannot be either. However, we have no details yet. In the same way that matter turned out to possess an intrinsic scale, we can guess that this happens also to the vacuum. Vacuum has an intrinsic scale; it is not continuous. We will have to wait for the third part of the text to nd out more. ere, the structure of the vacuum will turn out to be even more interesting than that of matter.
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Page 756
In the , the measurements of the spectrum of hydrogen had yielded another e ect due to virtual particles. Willis Lamb ( –) found that the S energy level in atomic hydrogen lies slightly above the P level. is is in contrast to the calculation performed above, where the two levels are predicted to have the same energy. In reality, they have an energy di erence of . MHz or . µeV. is discovery had important consequences for the description of quantum theory and yielded Lamb a share of the Nobel Prize.
e reason for the di erence is an unnoticed approximation performed in the simple solution above. ere are two equivalent ways to explain it. One is to say that the calculation neglects the coupling terms between the Dirac equation and the Maxwell equations.
is explanation lead to the rst calculations of the Lamb shi , around the year . e other explanation is to say that the calculation neglects virtual particles. In particular, the calculation neglects the virtual photons emitted and absorbed during the motion of the electron around the nucleus. is is the explanation in line with the general vocabulary of quantum electrodynamics. QED is perturbative approach to solve the coupled Dirac and Maxwell equations.
T QED L
– CS – section on the QED Lagrangian to be added – CS –
I
e electromagnetic interaction is exchange of virtual photons. So how can the interaction be attractive? At rst sight, any exchange of virtual photons should drive the electrons from each other. However, this is not correct. e momentum of virtual photons does not have to be in the direction of its energy ow; it can also be in opposite direc-
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tion.* Obviously, this is only possible within the limits provided by the indeterminacy principle.
V e strangest result of quantum eld theory is the energy density of the vacuum.
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– CS – More to be written – CS –
M
Challenge 1336 n
Page 871 Challenge 1337 n
Mirrors also work when in motion; in contrast, walls that produce echoes do not work at all speeds. Walls do not produce echoes if one moves faster than sound. However, mirrors always produce an image. is observation shows that the speed of light is the same for any observer. Can you detail the argument?
Mirrors also di er from tennis rackets. We saw that mirrors cannot be used to change the speed of the light they hit, in contrast to what tennis rackets can do with balls. is observation shows that the speed of light is also a limit velocity. In short, the simple existence of mirrors is su cient to derive special relativity.
But there are more interesting things to be learned from mirrors. We only have to ask whether mirrors work when they undergo accelerated motion. is issue yields a surprising result.
In the s, quite a number of researchers found that there is no vacuum for accelerated observers. is e ect is called Fulling–Davies–Unruh e ect or sometimes the dynamical Casimir e ect. As a consequence, a mirror in accelerated motion re ects the
uctuations it encounters and re ects them. In short, an accelerated mirror emits light. Unfortunately, the intensity is so weak that it has not been measured up to now. We will explore the issue in more detail below. Can you explain why accelerated mirrors emit light, but not matter?
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P
When virtual particles are taken into account, light beams can ‘bang’ onto each other. is result is in contrast to classical electrodynamics. Indeed, QED shows that the virtual
electron-positron pairs allow photons to hit each other. And such pairs are found in any light beam.
However, the cross-section is small. When two beams cross, most photons will pass undisturbed. e cross-section A is approximately
A
α π
(
ħ me
c
)
(
ħω mec
)
(585)
for the case that the energy ħω of the photon is much smaller than the rest energy mec of the electron. is value is about orders of magnitude smaller than what was meas-
urable in ; the future will show whether the e ect can be observed for visible light.
* One of the most beautiful booklets on quantum electrodynamics which makes this point remains the text
by R
F
, QED: the Strange eory of Light and Matter, Penguin Books, 1990.
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Ref. 849 Challenge 1338 ny
However, for high energy photons these e ects are observed daily in particle accelerators.
In these cases one observes not only interaction through virtual electron–antielectron
pairs, but also through virtual muon–antimuon pairs, virtual quark–antiquark pairs, and
much more.
Everybody who consumes science ction knows that matter and antimatter annihilate
and transform into pure light. In more detail, a matter particle and an antimatter particle
annihilate into two or more photons. More interestingly, quantum theory predicts that
the opposite process is also possible: photons hitting photons can produce matter!
In , this was also con rmed experimentally. At the Stanford particle accelerator,
photons from a high energy laser pulse were bounced o very fast electrons. In this way,
the re ected photons acquired a large energy, when seen in the inertial frame of the ex-
perimenter. e original pulse, of nm or . eV green light, had a peak power density
of W m , about the highest achievable so far. at is a photon density of m
and an electric eld of V m, both of which were record values at the time. When
this laser pulse was re ected o a . GeV electron beam, the returning photons had an
energy of . GeV and thus had become intense gamma rays. ese gamma rays then
collided with still incoming green photons and produced electron–positron pairs by the
reaction
γ . + n γgreen e+ + e−
(586)
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for which both nal particles were detected by special apparatuses. e experiment thus showed that light can hit light in nature, and above all, that doing so can produce matter.
is is the nearest one can get to the science ction idea of light swords or of laser swords banging onto each other.
I
?
Ref. 850
Page 242 Ref. 851
If the vacuum is a sea of virtual photons and particle–antiparticle pairs, vacuum could be suspected to act as a bath. In general, the answer is negative. Quantum eld theory works because the vacuum is not a bath for single particles. However, there is always an exception. For dissipative systems made of many particles, such as electrical conductors, the vacuum can act as a viscous uid. Irregularly shaped, neutral, but conducting bodies can emit photons when accelerated, thus damping such type of motion. is is due to the Fulling–Davies–Unruh e ect, also called the dynamical Casimir e ect, as described above. e damping depends on the shape and thus also on the direction of the body’s motion.
In , Gour and Sriramkumar even predicted that Brownian motion should also appear for an imperfect, i.e. partly absorbing mirror placed in vacuum. e uctuations of the vacuum should produce a mean square displacement
d = ħ mt
(587)
increasing linearly with time; however, the extremely small displacements produced this way seem out of experimental reach so far. But the result is not a surprise. Are you able Challenge 1339 ny to give another, less complicated explanation for it?
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R
–
?
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
– CS – section on renormalization to be added – CS –
Sometimes it is claimed that the in nities appearing in quantum electrodynamics in
the intermediate steps of the calculation show that the theory is incomplete or wrong.
However, this type of statement would imply that classical physics is also incomplete or
wrong, on the ground that in the de nition of the velocity v with space x and time t,
namely
v=
dx dt
= lim
∆t
∆x ∆t
= lim
∆t
∆x ∆t
,
(588)
Ref. 852
one gets an in nity as intermediate step. Indeed, dt being vanishingly small, one could argue that one is dividing by zero. Both arguments show the di culty to accept that the result of a limit process can be a nite quantity even if in nite quantities appear in it. e parallel with the de nition of the velocity is closer than it seems; both ‘in nities’ stem from the assumption that space-time is continuous, i.e. in nitely divisible. e in nities necessary in limit processes for the de nition of di erentiation, of integration or for the renormalization scheme appear only when space-time is approximated as a complete set, or as physicists say, as a ‘continuous’ set.
On the other hand, the conviction that the appearance of an in nity might be a sign of incompleteness of a theory was an interesting development in physics. It shows how uncomfortable many physicists had become with the use of in nity in our description of nature. Notably, this was the case for Dirac himself, who, a er having laid in his youth the basis of quantum electrodynamics, has tried for the rest of his life to nd a way, without success, to change the theory so that in nities are avoided.*
Renormalization is a procedure that follows from the requirement that continuous space-time and gauge theories must work together. In particular, it follows form the requirement that the particle concept is consistent, i.e. that perturbation expansions are possible.
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Motion is an interesting topic, and when a curious person asks a question about it, most of the time quantum electrodynamics is needed for the answer. Together with gravity, quantum electrodynamics explains almost all of our everyday experience, including numerous surprises. Let us have a look at some of them.
Challenge 1340 n
**
ere is a famous riddle asking how far the last card (or the last brick) of a stack can hang over the edge of a table. Of course, only gravity, no glue or any other means is allowed to keep the cards on the table. A er you solved the riddle, can you give the solution in case that the quantum of action is taken into account?
* Not long a er his death, his wish has been ful lled, although in a di erent manner that he envisaged. e third part of this mountain ascent will show the way out of the issue.
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table
cards
or
bricks
l
h
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 338 What is the maximum possible value of h/l?
Ref. 857
Challenge 1341 n Page 875
**
Quantum electrodynamics explains why there are only a nite number of di erent atom types. In fact, it takes only two lines to prove that pair production of electron–antielectron pairs make it impossible that a nucleus has more than about 137 protons. Can you show this? e e ect at the basis of this limit, the polarization of the vacuum, also plays a role in much larger systems, such as charged black holes, as we will see shortly.
Ref. 858
**
Taking 91 of the 92 electrons o an uranium atom allows researchers to check whether the innermost electron still is described by QED. e electric eld near the uranium nucleus,
EV m is near the threshold for spontaneous pair production. e eld is the highest constant eld producible in the laboratory, and an ideal testing ground for precision QED experiments. e e ect of virtual photons is to produce a Lamb shi ; even in these extremely high elds, the value ts with the predictions.
**
Is there a critical magnetic eld in nature, like there is a critical electric eld, limited by Challenge 1342 ny spontaneous pair production?
Page 601 Challenge 1343 ny
**
In classical physics, the eld energy of a point-like charged particle, and hence its mass, was predicted to be in nite. QED e ectively smears out the charge of the electron over its Compton wavelength, so that in the end the eld energy contributes only a small correction to its total mass. Can you con rm this?
**
Microscopic evolution can be pretty slow. Light, especially when emitted by single atoms, is always emitted by some metastable state. Usually, the decay times, being induced by the vacuum uctuations, are much shorter than a microsecond. However, there are metastable atomic states with a lifetime of ten years: for example, an ytterbium ion in the F state achieves this value, because the emission of light requires an octupole transition, in
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Ref. 859 which the angular momentum changes by ħ; this is an extremely unlikely process.
**
Microscopic evolution can be pretty fast. Can you imagine how to deduce or to measure Challenge 1344 n the speed of electrons inside atoms? And inside metals?
**
Take a horseshoe. e distance between the two ends is not xed, since otherwise their position and velocity would be known at the same time, contradicting the indeterminacy relation. Of course, this reasoning is also valid for any other solid object. In short, both quantum mechanics and special relativity show that rigid bodies do not exist, albeit for di erent reasons.
**
Challenge 1345 n Ref. 862
Ref. 863 Ref. 864
Have you ever admired a quartz crystal or some other crystalline material? e beautiful shape and atomic arrangement has formed spontaneously, as a result of the motion of atoms under high temperature and pressure, during the time that the material was deep under the Earth’s surface. e details of crystal formation are complex and interesting.
For example, are regular crystal lattices energetically optimal? is simple question leads to a wealth of problems. We might start with the much simpler question whether a regular dense packing of spheres is the most dense possible. Its density is π , i.e. a bit over 74 %. Even though this was conjectured to be the maximum possible value already in 1609 by Johannes Kepler, the statement was proven only in 1998 by Tom Hales. e proof is di cult because in small volumes it is possible to pack spheres up to almost 78 %. To show that over large volumes the lower value is correct is a tricky business.
Next, does a regular crystal of solid spheres, in which the spheres do not touch, have the lowest possible entropy? is simple problem has been the subject of research only in the 1990s. Interestingly, for low temperatures, regular sphere arrangements indeed show the largest possible entropy. At low temperatures, spheres in a crystal can oscillate around their average position and be thus more disordered than if they were in a liquid; in the liquid state the spheres would block each other’s motion and would not allow to show disorder at all.
is many similar results deduced from the research into these so-called entropic forces show that the transition from solid to liquid is – at least in part – simply a geometrical e ect. For the same reason, one gets the surprising result that even slightly repulsing spheres (or atoms) can form crystals and melt at higher temperatures. ese are beautiful examples of how classical thinking can explain certain material properties, using from quantum theory only the particle model of matter.
But the energetic side of crystal formation provides other interesting questions. Quantum theory shows that it is possible that two atoms repel each other, while three attract each other. is beautiful e ect was discovered and explained by Hans–Werner Fink in 1984. He studied rhenium atoms on tungsten surfaces and showed, as observed, that they cannot form dimers – two atoms moving closeby – but readily form trimers.
is is an example contradicting classical physics; the e ect is impossible if one pictures atoms as immutable spheres, but becomes possible when one remembers that the electron clouds around the atoms rearrange depending on their environment.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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F I G U R E 339 Some snow flakes (© Furukawa Yoshinori)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 1346 n Ref. 865 Ref. 866
Page 255
For an exact study of crystal energy, the interactions between all atoms have to be included. e simplest question is to determine whether a regular array of alternatively charged spheres has lower energy than some irregular collection. Already such simple questions are still topic of research; the answer is still open.
Another question is the mechanism of face formation in crystals. Can you con rm that crystal faces are those planes with the slowest growth speed, because all fast growing planes are eliminated? e ner details of the process form a complete research eld in itself.
However, not always the slowest growing planes win out. Figure 339 shows some wellknown exceptions. Explaining such shapes is possible today, and Furukawa Yoshinori is one of the experts in the eld, heading a dedicated research team. Indeed, there remains the question of symmetry: why are crystals o en symmetric, such as snow akes, instead of asymmetric? is issue is a topic of self-organization, as mentioned already in the section of classical physics. It turns out that the symmetry is an automatic result of the way molecular systems grow under the combined in uence of di usion and nonlinear processes. e details are still a topic of research.
Ref. 867
**
A similar breadth of physical and mathematical problems are encountered in the study of liquids and polymers. e ordering of polymer chains, the bubbling of hot water, the motion of heated liquids and the whirls in liquid jets show complex behaviour that can be explained with simple models. Turbulence and self-organization will be a fascinating research eld for many years to come.
**
e ways people handle single atoms with electromagnetic elds is a beautiful example of modern applied technologies. Nowadays it is possible to levitate, to trap, to excite, to photograph, to deexcite and to move single atoms just by shining light onto them. In 1997, the Nobel Prize in physics has been awarded to the originators of the eld.
Ref. 868
**
In 1997, a Czech group built a quantum version of the Foucault pendulum, using the super uidity of helium. In this beautiful piece of research, they cooled a small ring of uid helium below the temperature of . K, below which the helium moves without friction. In such situations it thus can behave like a Foucault pendulum. With a clever arrangement,
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•
it was possible to measure the rotation of the helium in the ring using phonon signals, and to show the rotation of the Earth.
Challenge 1347 n Ref. 869
**
If an electrical wire is su ciently narrow, its electrical conductance is quantized in steps of e ħ. e wider the wire, the more such steps are added to its conductance. Can you explain the e ect? By the way, quantized conductance has also been observed for light and for phonons.
Ref. 870
**
An example of modern research is the study of hollow atoms, i.e. atoms missing a number of inner electrons. ey have been discovered in 1990 by J.P. Briand and his group. ey appear when a completely ionized atom, i.e. one without any electrons, is brought in contact with a metal. e acquired electrons then orbit on the outside, leaving the inner shells empty, in stark contrast with usual atoms. Such hollow atoms can also be formed by intense laser irradiation.
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In the past, the description of motion with formulae was taken rather seriously. Before computers appeared, only those examples of motion were studied which could be described with simple formulae. It turned out that Galilean mechanics cannot solve the three-body problem, special relativity cannot solve the two-body problem, general relativity the one-body problem and quantum eld theory the zero-body problem. It took some time to the community of physicists to appreciate that understanding motion does not depend on the description by formulae, but on the description by clear equations based on space and time.
** Challenge 1348 n Can you explain why mud is not clear?
Challenge 1349 n
**
Photons not travelling parallel to each other attract each other through gravitation and thus de ect each other. Could two such photons form a bound state, a sort of atom of light, in which they would circle each other, provided there were enough empty space for this to happen?
Challenge 1350 n
**
Can the universe ever have been smaller than its own Compton wavelength? In fact, quantum electrodynamics, or QED, provides a vast number of curiosities and
every year there is at least one interesting new discovery. We now conclude the theme with a more general approach.
H
?–T
In our quest, we have encountered motion of many sorts. erefore, the following test – not to be taken too seriously – is the ultimate physics test, allowing to check your under-
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 1351 n
Challenge 1352 n Challenge 1353 n
Challenge 1354 n Challenge 1355 n Challenge 1356 n Challenge 1357 n
standing and to compare it with that of others. Imagine that you are on a perfectly frictionless surface and that you want to move to
its border. How many methods can you nd to achieve this? Any method, so tiny its e ect may be, is allowed.
Classical physics provided quite a number of methods. We saw that for rotating ourselves, we just need to turn our arm above the head. For translation motion, throwing a shoe or inhaling vertically and exhaling horizontally are the simplest possibilities. Can you list at least six additional methods, maybe some making use of the location of the surface on Earth? What would you do in space?
Electrodynamics and thermodynamics taught us that in vacuum, heating one side of the body more than the other will work as motor; the imbalance of heat radiation will push you, albeit rather slowly. Are you able to nd at least four other methods from these two domains?
General relativity showed that turning one arm will emit gravitational radiation unsymmetrically, leading to motion as well. Can you nd at least two better methods?
Quantum theory o ers a wealth of methods. Of course, quantum mechanics shows that we actually are always moving, since the indeterminacy relation makes rest an impossibility. However, the average motion can be zero even if the spread increases with time. Are you able to nd at least four methods of moving on perfect ice due to quantum e ects?
Materials science, geophysics, atmospheric physics and astrophysics also provide ways to move, such as cosmic rays or solar neutrinos. Can you nd four additional methods?
Self-organization, chaos theory and biophysics also provide ways to move, when the inner workings of the human body are taken into account. Can you nd at least two methods?
Assuming that you read already the section following the present one, on the e ects of semiclassical quantum gravity, here is an additional puzzle: is it possible to move by accelerating a pocket mirror, using the emitted Unruh radiation? Can you nd at least two other methods to move yourself using quantum gravity e ects? Can you nd one from string theory?
If you want points for the test, the marking is simple. For students, every working method gives one point. Eight points is ok, twelve points is good, sixteen points is very good, and twenty points or more is excellent.For graduated physicists, the point is given only when a back-of-the-envelope estimate for the ensuing momentum or acceleration is provided.
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e shortest possible summary of quantum electrodynamics is the following: matter is made of charged particles which interact through photon exchange in the way described by Figure .
No additional information is necessary. In a bit more detail, quantum electrodynamics starts with elementary particles, characterized by their mass, their spin and their charge, and with the vacuum, essentially a sea of virtual particle–antiparticle pairs. Interactions between charged particles are described as the exchange of virtual photons, and decay is described as the interaction with the virtual photons of the vacuum.
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matter: q = e s = 1/2
radiation: q = 0 s = 1
interaction: α = 1/137.0359...
Σ E=0 Σ p=0 Σ s=0 Σ q=0
F I G U R E 340 QED as perturbation theory in space-time
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All physical results of QED can be calculated by using the single diagram of Figure .
As QED is a perturbative theory, the diagram directly describes the rst order e ects and
its composites describe e ects of higher order. QED is a perturbative theory.
QED describes all everyday properties of matter and radiation. It describes the divis-
ibility down to the smallest constituents, the isolability from the environment and the
impenetrability of matter. It also describes the penetrability of radiation. All these proper-
ties are due to electromagnetic interactions of constituents and follow from Figure .
Matter is divisible because the interactions are of nite strength, matter is separable be-
cause the interactions are of nite range, and matter is impenetrable because interactions
among the constituents increase in intensity when they approach each other, in particu-
lar because matter constituents are fermions. Radiation is divisible into photons, and is
penetrable because photons are bosons and rst order photon-photon interactions do
not exist.
Both matter and radiation are made of elementary constituents. ese elementary
constituents, whether bosons or fermions, are indivisible, isolable, indistinguishable, and
point-like.
To describe observations, it is necessary to use quantum electrodynamics in all those
situations for which the characteristic dimensions d are of the order of the Compton
wavelength
d
λC
=
h mc
.
(589)
In situations where the dimensions are of the order of the de Broglie wavelength, or equi-
valently, where the action is of the order of the Planck value, simple quantum mechanics
is su cient:
d
λdB
=
h mv
.
(590)
For larger dimensions, classical physics will do.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Together with gravity, quantum electrodynamics explains almost all observations of motion on Earth; QED uni es the description of matter and radiation in daily life. All objects and all images are described by it, including their properties, their shape, their transformations and their other changes. is includes self-organization and chemical or biological. In other words, QED gives us full grasp of the e ects and the variety of motion due to electromagnetism.
O
QED
Ref. 872 Ref. 873
Ref. 875 Ref. 874 Ref. 876
Even though QED describes motion without any discrepancy from experiment, that does not mean that we understand every detail of every example of electric motion. For example, nobody has described the motion of an animal with QED yet.* In fact, there is beautiful and fascinating work going on in many branches of electromagnetism.
Atmospheric physics still provides many puzzles and regularly delivers new, previously unknown phenomena. For example, the detailed mechanisms at the origin of aurorae are still controversial; and the recent unexplained discoveries of discharges above clouds should not make one forget that even the precise mechanism of charge separation inside clouds, which leads to lightning, is not completely clari ed. In fact, all examples of electri cation, such as the charging of amber through rubbing, the experiment which gave electricity its name, are still poorly understood.
Materials science in all its breadth, including the study of solids, uids, and plasmas, as well as biology and medicine, still provides many topics of research. In particular, the twenty- rst century will undoubtedly be the century of the life sciences.
e study of the interaction of atoms with intense light is an example of present research in atomic physics. Strong lasers can strip atoms of many of their electrons; for such phenomena, there are not yet precise descriptions, since they do not comply to the weak eld approximations usually assumed in physical experiments. In strong elds, new e ects take place, such as the so-called Coulomb explosion.
But also the skies have their mysteries. In the topic of cosmic rays, it is still not clear how rays with energies of eV are produced outside the galaxy. Researchers are intensely trying to locate the electromagnetic elds necessary for their acceleration and to understand their origin and mechanisms.
In the theory of quantum electrodynamics, discoveries are expected by all those who study it in su cient detail. For example, Dirk Kreimer has found that higher order interaction diagrams built using the fundamental diagram of Figure contain relations to the theory of knots. is research topic will provide even more interesting results in the near future.
Relations to knot theory appear because QED is a perturbative description, with the vast richness of its nonperturbative e ects still hidden. Studies of QED at high energies, where perturbation is not a good approximation and where particle numbers are not conserved, promise a wealth of new insights. We will return to the topic later on.
High energies provide many more questions. So far, the description of motion was based on the idea that measurable quantities can be multiplied and added. is always
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* On the other hand, there is beautiful work going on how humans move their limbs; it seems that humans Ref. 871 move by combining a small set of fundamental motions.
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A collapsing house
•. length l
height h
figure to be inserted
neutron beam
F I G U R E 341 The weakness of gravitation
silicon mirrors
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Challenge 1358 r
happens at one space-time point. In mathematical jargon, observables form a local algebra. us the structure of an algebra contains, implies and follows from the idea that local properties lead to local properties. We will discover later on that this basic assumption is wrong at high energies.
We de ned special relativity using v c, general relativity using L M G c and quantum theory using S ħ . How can we de ne electromagnetism in one statement?
is is not known yet. Many other open issues of more practical nature have not been mentioned. Indeed, by far the largest numbers of physicists get paid for some form of applied QED. However, our quest is the description of the fundamentals of motion. So far, we have not achieved it. For example, we still need to understand motion in the realm of atomic nuclei. But before we do that, we take a rst glimpse of the strange issues appearing when gravity and quantum theory meet.
.
–
Ref. 877 Challenge 1359 ny
Gravitation is a weak e ect. Every seaman knows it: storms are the worst part of his life,
not gravity. Nevertheless, including gravity into quantum mechanics yields a list of im-
portant issues.
Only in
it became possible to repeat Galileo’s experiment with atoms: indeed,
single atoms fall like stones. In particular, atoms of di erent mass fall with the same acce-
leration, within the experimental precision of a part in million.
In the chapter on general relativity we already mentioned that light frequency changes
with height. But for matter wave functions, gravity also changes their phase. Can you ima-
gine why? e e ect was rst con rmed in with the help of neutron interferometers,
where neutron beams are brought to interference a er having climbed some height h
at two di erent locations. e experiment is shown schematically in Figure ; it fully
con rmed the predicted phase di erence
δφ
=
m hl ħv
(591)
where l is the distance of the two climbs and v and m are the speed and mass of the
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Ref. 878 Ref. 880
neutrons. ese beautifully simple experiments have con rmed the formula within experimental errors.*
In the s, similar experiments have even been performed with complete atoms. ese set-ups allow to build interferometers so sensitive that local gravity can be measured with a precision of more than eight signi cant digits.
C
S
Ref. 881
In , the rst observation of actual quantum states due to gravitational energy was
performed. Any particle above the oor should feel the e ect of gravity.
In a few words, one can say that because the experimenters managed to slow down
neutrons to the incredibly small value of m s, using grazing incidence on a at plate
they could observe how neutrons climbed and fell back due to gravity with speeds below
a few cm s. Obviously, the quantum description is a bit more involved. e lowest energy level for
neutrons due to gravity is . ë − J, or . peV. To get an impression of it smallness, we can compare it to the value of . ë − J or . eV for the lowest state in the hydrogen atom.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
A
Ref. 903
Despite its weakness, gravitation provides many puzzles. Most famous are a number of curious coincidences that can be found when quantum mechanics and gravitation are combined. ey are usually called ‘large number hypotheses’ because they usually involve large dimensionless numbers. A pretty, but less well known version connects the Planck length, the cosmic horizon, and the number of baryons:
(Nb)
R lPl
=
t tPl
(592)
in which Nb = and t = . ë a were used. ere is no known reason why the number of baryons and the horizon size R should be related in this way. is coincidence
is equivalent to the one originally stated by Dirac,** namely
mp
ħ Gct
.
(594)
Ref. 879 Ref. 904 Page 1044
* Due to the in uence of gravity on phases of wave functions, some people who do not believe in bath induced decoherence have even studied the in uence of gravity on the decoherence process of usual quantum systems in at space-time. Predictably, the calculated results do not reproduce experiments. ** e equivalence can be deduced using Gnbmp = t , which, as Weinberg explains, is required by several cosmological models. Indeed, this can be rewritten simply as
m R mPl RPl = c G .
(593)
Together with the de nition of the baryon density nb = Nb R one gets Dirac’s large number hypothesis,
substituting protons for pions. Note that the Planck time and length are de ned as ħG c and ħG c and are the natural units of length and time. We will study them in detail in the third part of the mountain ascent.
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•.
Ref. 905
where mp is the proton mass. is approximate equality seems to suggest that certain microscopic properties, namely the mass of the proton, is connected to some general properties of the universe as a whole. is has lead to numerous speculations, especially since the time dependence of the two sides di ers. Some people even speculate whether relations ( ) or ( ) express some long-sought relation between local and global topological properties of nature. Up to this day, the only correct statement seems to be that they are coincidences connected to the time at which we happen to live, and that they should not be taken too seriously.
I
?
Challenge 1360 ny Page 407
One might think that gravity does not require a quantum description. We remember that we stumbled onto quantum e ects because classical electrodynamics implies, in stark contrast with reality, that atoms decay in about . ns. Classically, an orbiting electron would emit radiation until it falls into the nucleus. Quantum theory is necessary to save the situation.
When the same calculation is performed for the emission of gravitational radiation by orbiting electrons, one nds a decay time of around s. (True?) is extremely large value, trillions of times longer than the age of the universe, is a result of the low emission of gravitational radiation by rotating masses. erefore, the existence of atoms does not require a quantum theory of gravity.
Indeed, quantum gravity is unnecessary in every single domain of everyday life. However, quantum gravity is necessary in domains which are more remote, but also more fascinating.
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Page 471 Ref. 882
“Die Energie der Welt ist constant. Die Entropie der Welt strebt einem Maximum zu.* ” Rudolph Clausius
We have already encountered the famous statement by Clausius, the father of the term ‘entropy’. Strangely, for over hundred years nobody asked whether there actually exists a theoretical maximum for entropy. is changed in , when Jakob Bekenstein found the answer while investigating the consequences gravity has for quantum physics. He found that the entropy of an object of energy E and size L is bound by
S
E
L
kπ ħc
(595)
for all physical systems. In particular, he deduced that (nonrotating) black holes saturate Challenge 1361 n the bound, with an entropy given by
S
=
kc Għ
A
=
kG ħc
πM
(596)
* e energy of the universe is constant. Its entropy tends towards a maximum.
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Challenge 1362 n
Ref. 884 Challenge 1363 n
where A is now the area of the horizon of the black hole. It is given by A = πR = π( GM c ) . In particular, the result implies that every black hole has an entropy. Black
holes are thus disordered systems described by thermostatics. Black holes are the most disordered systems known.*
As an interesting note, the maximum entropy also gives a memory limit for memory chips. Can you nd out how?
What are the di erent microstates leading to this macroscopic entropy? It took many years to convince physicists that the microstates have to do with the various possible states of the horizon itself, and that they are due to the di eomorphism invariance at this boundary. As Gerard ’t Hoo explains, the entropy expression implies that the number of degrees of freedom of a black hole is about (but not exactly) one per Planck area of the horizon.
If black holes have entropy, they must have a temperature. What does this temperature mean? In fact, nobody believed this conclusion until two unrelated developments con rmed it within a few months. All these results were waiting to be discovered since the s, even though, incredibly, nobody had thought about them for over years.
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Ref. 885
M
–
F
–D
–U
Independently, Stephen Fulling in , Paul Davies in and William Unruh in made the same theoretical discovery while studying quantum theory: if an inertial observer observes that he is surrounded by vacuum, a second observer accelerated with respect to the rst does not: he observes black body radiation. e radiation has a spectrum corresponding to the temperature
T=a
ħ πkc
.
(597)
Ref. 886
e result means that there is no vacuum on Earth, because any observer on its surface can maintain that he is accelerated with . m s , thus leading to T = zK! We can thus measure gravity, at least in principle, using a thermometer. However, even for the largest practical accelerations the temperature values are so small that it is questionable whether the e ect will ever be con rmed experimentally. But if it will, it will be a great experiment.
When this e ect was predicted, people studied the argument from all sides. For example, it was then found that the acceleration of a mirror leads to radiation emission! Mirrors are thus harder to accelerate than other bodies of the same mass.
When the acceleration is high enough, also matter particles can be detected. If a particle counter is accelerated su ciently strongly across the vacuum, it will start count-
Ref. 883
* e precise discussion that black holes are the most disordered systems in nature is quite subtle. It is summarized by Bousso. Bousso claims that the area appearing in the maximum entropy formula cannot be taken naively as the area at a given time, and gives four arguments why this should be not allowed. However, all four arguments are wrong in some way, in particular because they assume that lengths smaller than the Planck length or larger than the universe’s size can be measured. Ironically, he brushes aside some of the arguments himself later in the paper, and then deduces an improved formula, which is exactly the same as the one he criticizes rst, just with a di erent interpretation of the area A. In short, the expression of black hole entropy is the maximum entropy for a physical system with surface A.
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•.
ing particles! We see that the di erence between vacuum and matter becomes fuzzy at large energies.
For completeness, we mention that also an observer in rotational motion detects radiation following expression ( ).
B
’
Ref. 887
In , the English physicist Stephen Hawking, famous for the courage with which he
ghts a disease which forces him into the wheelchair, surprised the world of general
relativity with a fundamental theoretical discovery. He found that if a virtual particle–
antiparticle pair appeared in the vacuum near the horizon, there is a nite chance that
one particle escapes as a real particle, while the virtual antiparticle is captured by the
black hole. e virtual antiparticle is thus of negative energy, and reduces the mass of
the black hole. e mechanism applies both to fermions and bosons. From far away this
e ect looks like the emission of a particle. Hawking’s detailed investigation showed that
the e ect is most pronounced for photon emission. In particular, Hawking showed that
black holes radiate as black bodies.
Black hole radiation con rms both the result on black hole entropy by Bekenstein
and the e ect for observers accelerated in vacuum found by Fulling, Davies and Unruh.
When all this became clear, a beautiful Gedanken experiment was published by William
Unruh and Robert Wald, showing that the whole result could have been deduced already
years earlier!
Shameful as this delay of the discovery is
for the community of theoretical physicists, the story itself remains beautiful. It starts in
space station with dynamo
the early s, when Robert Geroch stud-
ied the issue shown in Figure . Imagine
a mirror box full of heat radiation, thus full
rope
of light. e mass of the box is assumed to be negligible, such as a box made of thin aluminium paper. We lower the box, with all
box filled with light
its contained radiation, from a space station towards a black hole. On the space station,
horizon
lowering the weight of the heat radiation al-
lows to generate energy. Obviously, when the
black
box reaches the black hole horizon, the heat
support
hole
radiation is red-shi ed to in nite wavelength.
shell
At that point, the full amount of energy ori-
ginally contained in the heat radiation has
been provided to the space station. We can F I G U R E 342 A Gedanken experiment allowing now do the following: we can open the box to deduce the existence of black hole radiation
on the horizon, let drop out whatever is still
inside, and wind the empty and massless box back up again. As a result, we have com-
pletely converted heat radiation into mechanical energy. Nothing else has changed: the
black hole has the same mass as beforehand.
But this result contradicts the second principle of thermodynamics! Geroch concluded
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’
that something must be wrong. We must have forgotten an e ect which makes this process impossible.
In the s, Unruh and Wald showed that black hole radiation is precisely the forgotten e ect that puts everything right. Because of black hole radiation, the box feels buoyancy, so that it cannot be lowered down to the horizon. It oats somewhat above it, so that the heat radiation inside the box has not yet zero energy when it falls out of the opened box. As a result, the black hole does increase in mass and thus in entropy. In summary, when the empty box is pulled up again, the nal situation is thus the following: only part of the energy of the heat radiation has been converted into mechanical energy, part of the energy went into the increase of mass and thus of entropy of the black hole.
e second principle of thermodynamics is saved. Well, it is only saved if the heat radiation has precisely the right energy density at the horizon and above. Let us have a look. e centre of the box can only be lowered up to a hovering distance d above the horizon, where the acceleration due to gravity is = c GM. e energy E gained by lowering the box is
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E = mc
−m
d = mc (
−
dc GM
)
(598)
e e ciency of the process is η = E mc . To be consistent with the second law of thermodynamics, this e ciency must obey
η
=
E mc
=
−
TBH T
e
(599)
We thus nd a black hole temperature TBH given by the hovering distance d. at hovering distance d is roughly given by the size of the box. e box size in turn must be at least the wavelength of the thermal radiation; in rst approximation, Wien’s relation gives
d ħc kT. A precise calculation, rst performed by Hawking, gives the result
TBH =
ħc πkGM
=
ħc πk
R
=
ħ πkc
surf
with
surf =
c GM
(600)
Challenge 1364 ny Ref. 888
Challenge 1365 ny
where R and M are the radius and the mass of the black hole. It is either called the blackhole temperature or Bekenstein-Hawking temperature. As an example, a black hole with the mass of the Sun would have the rather small temperature of nK, whereas a smaller black hole with the mass of a mountain, say kg, would have a temperature of GK.
at would make quite a good oven. All known black hole candidates have masses in the range from a few to a few million solar masses. e radiation is thus extremely weak, the reason being that the emitted wavelength is of the order of the black hole radius, as you might want to check. e radiation emitted by black holes is o en also called BekensteinHawking radiation.
Black hole radiation is thus so weak that we can speak of an academic e ect. It leads to a luminosity that increases with decreasing mass or size as
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•.
TA B L E 65 The principles of thermodynamics and those of horizon mechanics
P
T
H
Zeroth principle the temperature T is con-the surface gravity a is con-
stant in a body at equilib-stant on the horizon
rium
First principle
energy is conserved: dE = energy is conserved:
TdS − pdV + µdN
d(mc
)=
ac πG
dA
+
ΩdJ
+
Φdq
Second principle entropy never decreases: surface area never de-
dS
creases: dA
ird principle T = cannot be achieved a = cannot be achieved
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LM
R
or
L = nAσ T
=
n
c G
ħ M
π ë
(601)
Page 611 Ref. 889
Challenge 1366 ny
where σ is the Stefan–Boltzmann or black body radiation constant, n is the number of particle degrees of freedom that can be radiated; if only photons are radiated, we have n = . (For example, if neutrinos were massless, they would be emitted more frequently than photons.)
Black holes thus shine, and the more the smaller they are. is is a genuine quantum e ect, since classically, black holes, as the name says, cannot emit any light. Even though the e ect is academically weak, it will be of importance later on. In actual systems, many other e ect around black holes increase the luminosity far above this value; indeed, black holes are usually brighter than normal stars, due to the radiation emitted by the matter falling into them. But that is another story. Here we are only treating isolated black holes, surrounded only by vacuum.
Due to the emitted radiation, black holes gradually lose mass. erefore their theoretical lifetime is nite. A calculation shows that it is given by
t=M
πG ħc
M . ë − s kg
(602)
Ref. 890
as function of their initial mass M. For example, a black hole with mass of g would have a lifetime of . ë − s, whereas a black hole of the mass of the Sun, . ë kg, would have a lifetime of about years. Obviously, these numbers are purely academic. In any case, black holes evaporate. However, this extremely slow process for usual black holes determines their lifetime only if no other, faster process comes into play. We will present a few such processes shortly. Hawking radiation is the weakest of all known e ects. It is not masked by stronger e ects only if the black hole is non-rotating, electrically neutral and with no matter falling into it from the surroundings.
So far, none of these quantum gravity e ects has been con rmed experimentally, as the values are much too small to be detected. However, the deduction of a Hawking temperature has been beautifully con rmed by a theoretical discovery of Unruh, who found that there are con gurations of uids in which sound waves cannot escape, so-called ‘silent
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’
holes’. Consequently, these silent holes radiate sound waves with a temperature satisfying the same formula as real black holes. A second type of analogue system, namely optical Ref. 892 black holes, are also being investigated.
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G
Ref. 893
Ref. 894 Ref. 895 Challenge 1367 n
In , a much more dramatic radiation e ect than black hole radiation was predicted for charged black holes by Damour and Ru ni. Charged black holes have a much shorter lifetime than just presented, because during their formation a second process takes place. In a region surrounding them the electric eld is larger than the so-called vacuum polarization value, so that large numbers of electron-positron pairs are produced, which then almost all annihilate. is process e ectively reduces the charge of the black hole to a value for which the eld is below critical everywhere, while emitting large amounts of high energy light. It turns out that the mass is reduced by up to % in a time of the order of seconds. at is quite shorter than years. is process thus produces an extremely intense gamma ray burst.
Such gamma ray bursts had been discovered in the late s by military satellites which were trying to spot nuclear explosions around the world through their gamma ray emission. e satellites found about two such bursts per day, coming from all over the sky. Another satellite, the Compton satellite, con rmed that they were extragalactic in origin, and that their duration varied between a sixtieth of a second and about a thousand seconds. In , the Italian-Dutch BeppoSAX satellite started mapping and measuring gamma ray bursts systematically. It discovered that they were followed by an a erglow in the X-ray domain of many hours, sometimes of days. In a erglow was discovered also in the optical domain. e satellite also allowed to nd the corresponding X-ray, optical and radio sources for each burst. ese measurements in turn allowed to determine the distance of the burst sources; red-shi s between . and . were measured. In it also became possible to detect optical bursts corresponding to the gamma ray ones.*
All this data together show that the gamma ray bursts have energies ranging from W to W. e larger value is about one hundredth of the brightness all stars of
the whole visible universe taken together! Put di erently, it is the same amount of energy that is released when converting several solar masses into radiation within a few seconds. In fact, the measured luminosity is near the theoretical maximum luminosity a body can have. is limit is given by
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L < LPl =
c G
=
.
ë
W,
(603)
Challenge 1368 e Ref. 895
as you might want to check yourself. In short, the sources of gamma ray bursts are the biggest bombs found in the universe. In fact, more detailed investigations of experimental data con rm that gamma ray bursts are ‘primal screams’ of black holes in formation.
With all this new data, Ru ni took up his model again in and with his collaborators showed that the gamma ray bursts generated by the annihilation of electronpositrons pairs created by vacuum polarization, in the region they called the dyadosphere,
* For more about this fascinating topic, see the http://www.aip.de/~jcg/grb.html website by Jochen Greiner.
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•.
F I G U R E 343 A selection of gamma ray bursters observed in the sky
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Ref. 896 Ref. 898
Page 484
have a luminosity and a duration exactly as measured, if a black hole of about a few up to solar masses is assumed. Charged black holes therefore reduce their charge and mass
through the vacuum polarization and electron positron pair creation process. ( e process reduces the mass because it is one of the few processes which is reversible; in contrast, most other attempts to reduce charge on a black hole, e.g. by throwing in a particle with the opposite charge, increase the mass of the black hole and are thus irreversible.) e le over remnant then can lose energy in various ways and also turns out to be responsible for the a erglow discovered by the BeppoSAX satellite. Among others, Ru ni’s team speculates that the remnants are the sources for the high energy cosmic rays, whose origin had not been localized so far. All these exciting studies are still ongoing.
Recent studies distinguish two classes of gamma ray bursts. Short gamma ray bursts, with a duration between a millisecond and two seconds, di er signi cantly in energy and spectrum from long gamma ray bursts, with an average length of ten seconds, a higher energy content and a so er energy spectrum. It is o en speculated that short bursts are due to merging neutron stars or merging black holes, whereas long bursts are emitted, as just explained, when a black hole is formed in a supernova or hypernova explosion. It also seems that gamma ray bursts are not of spherical symmetry, but that the emission takes place in a collimated beam. is puts the energy estimates given above somewhat into question. e details of the formation process are still subject to intense exploration.
Other processes leading to emission of radiation from black holes are also possible. Examples are matter falling into the black hole and heating up, matter being ejected from rotating black holes through the Penrose process, or charged particles falling into a black hole. ese mechanisms are at the origin of quasars, the extremely bright quasi-stellar sources found all over the sky. ey are assumed to be black holes surrounded by matter, in the development stage following gamma ray bursters. e details of what happens in quasars, the enormous voltages (up to V) and magnetic elds generated, as well as their e ects on the surrounding matter are still object of intense research in astrophysics.
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M
Once the concept of entropy of a black hole was established, people started to think about black holes like about any other material object. For example, black holes have a matter density, which can be de ned by relating their mass to a ctitious volume de ned by
πR . is density is given by
ρ= M
c πG
(604)
Challenge 1369 e
and can be quite low for large black holes. For the highest black holes known, with
million solar masses or more, the density is of the order of the density of air. Nevertheless,
even in this case, the density is the highest possible in nature for that mass.
By the way, the gravitational acceleration at the horizon is still appreciable, as it is given
by
surf = M
c G
=
c R
(605)
Challenge 1370 ny which is still km s for an air density black hole. Obviously, the black hole temperature is related to the entropy S by its usual de nition
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T
=
∂S ∂E
ρ
=
∂S ∂(Mc
)
ρ
(606)
Challenge 1371 ny
Challenge 1372 ny Challenge 1373 n
Ref. 899 Ref. 900 Ref. 889 Ref. 897
All other thermal properties can be deduced by the standard relations from thermostatics. In particular, it looks as if black holes are the matter states with the largest possible
entropy. Can you con rm this statement? It also turns out that black holes have a negative heat capacity: when heat is added, they
cool down. In other words, black holes cannot achieve equilibrium with a bath. is is not a real surprise, since any gravitationally bound material system has negative speci c heat. Indeed, it takes only a bit of thinking to see that any gas or matter system collapsing under gravity follows dE dR and dS dR . at means that while collapsing, the energy and the entropy of the system shrink. (Can you nd out where they go?) Since temperature is de ned as T = dS dE, temperature is always positive; from the temperature increase dT dR < during collapse one deduces that the speci c heat dE dT is negative.
Black holes, like any object, oscillate when slightly perturbed. ese vibrations have also been studied; their frequency is proportional to the mass of the black hole.
Nonrotating black holes have no magnetic eld, as was established already in the s by Russian physicists. On the other hand, black holes have something akin to a nite electrical conductivity and a nite viscosity. Some of these properties can be understood if the horizon is described as a membrane, even though this model is not always applicable. In any case, one can study and describe macroscopic black holes like any other macroscopic material body. e topic is not closed.
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•.
H
?
When a nonrotating and uncharged black hole loses mass by radiating Hawking radiation, eventually its mass reaches values approaching the Planck mass, namely a few micrograms. Expression ( ) for the lifetime, applied to a black hole of Planck mass, yields a value of over sixty thousand Planck times. A surprising large value. What happens in those last instants of evaporation?
A black hole approaching the Planck mass at some time will get smaller than its own Compton wavelength; that means that it behaves like an elementary particle, and in particular, that quantum e ects have to be taken into account. It is still unknown how these
nal evaporation steps take place, whether the mass continues to diminish smoothly or in steps (e.g. with mass values decreasing as n when n approaches zero), how its internal structure changes, whether a stationary black hole starts to rotate (as the author predicts), how the emitted radiation deviates from black body radiation. ere is still enough to study. However, one important issue has been settled.
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T
Challenge 1374 e Ref. 901
When the thermal radiation of black holes was discovered, one question was hotly debated for many years. e matter forming a black hole can contain lots of information; e.g., imagine the black hole formed by a large number of books collapsing onto each other. On the other hand, a black hole radiates thermally until it evaporates. Since thermal radiation carries no information, it seems that information somehow disappears, or equivalently, that entropy increases.
An incredible number of papers have been written about this problem, some even claiming that this example shows that physics as we know it is incorrect and needs to be changed. As usual, to settle the issue, we need to look at it with precision, laying all prejudice aside. ree intermediate questions can help us nding the answer.
— What happens when a book is thrown into the Sun? When and how is the information
radiated away? — How precise is the sentence that black hole radiate thermal radiation? Could there be
a slight deviation?
— Could the deviation be measured? In what way would black holes radiate information? You might want to make up your own mind before reading on.
Let us walk through a short summary. When a book or any other highly complex – or low entropy – object is thrown into the Sun, the information contained is radiated away.
e information is contained in some slight deviations from black hole radiation, namely in slight correlations between the emitted radiation emitted over the burning time of the Sun. A short calculation, comparing the entropy of a room temperature book and the information contained in it, shows that these e ects are extremely small and di cult to measure.
A clear exposition of the topic was given by Don Page. He calculated what information would be measured in the radiation if the system of black hole and radiation together would be in a pure state, i.e. a state containing speci c information. e result is simple. Even if a system is large – consisting of many degrees of freedom – and in pure state, any smaller subsystem nevertheless looks almost perfectly thermal. More speci cally, if a total system has a Hilbert space dimension N = nm, where n and m n are the dimensions
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’
Challenge 1375 ny
of two subsystems, and if the total system is in a pure state, the subsystem m would have
an entropy Sm given by
Sm =
− m + mn n k=n+ k
(607)
which is approximately given by
Sm
=
ln m −
m n
for
m
.
(608)
Ref. 902
To discuss the result, let us think of n and m as counting degrees of freedom, instead of Hilbert space dimensions. e rst term in equation ( ) is the usual entropy of a mixed state. e second term is a small deviation and describes the amount of speci c information contained in the original pure state; inserting numbers, one nds that it is extremely small compared to the rst. In other words, the subsystem m is almost indistinguishable from a mixed state; it looks like a thermal system even though it is not.
A calculation shows that the second, small term on the right of equation ( ) is indeed su cient to radiate away, during the lifetime of the black hole, any information contained in it. Page then goes on to show that the second term is so small that not only it is lost in measurements; it is also lost in the usual, perturbative calculations for physical systems.
e question whether any radiated information could be measured can now be answered directly. As Don Page showed, even measuring half of the system only gives about / bit of that information. It is necessary to measure the complete system to measure all the contained information. In summary, at a given instant, the amount of information radiated by a black hole is negligible when compared with the total black hole radiation, and is practically impossible to detect by measurements or even by usual calculations.
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Page 485
A black hole is a macroscopic object, similar to a star. Like all objects, it can interact with its environment. It has the special property to swallow everything that falls into them.
is immediately leads us to ask if we can use this property to cheat around the usual everyday ‘laws’ of nature. Some attempts have been studied in the section on general relativity and above; here we explore a few additional ones.
**
Challenge 1376 ny
Apart from the questions of entropy, we can look for methods to cheat around conservation of energy, angular momentum, or charge. Every Gedanken experiment comes to the same conclusions. No cheats are possible; in addition, the maximum number of degrees of freedom in a region is proportional to the surface area of the region, and not to its volume. is intriguing result will keep us busy for quite some time.
** A black hole transforms matter into antimatter with a certain e ciency. us one might
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•.
Challenge 1377 ny look for departures from particle number conservation. Are you able to nd an example?
Ref. 906
**
Black holes de ect light. Is the e ect polarization dependent? Gravity itself makes no di erence of polarization; however, if virtual particle e ects of QED are included, the story might change. First calculations seem to show that such a e ect exists, so that gravitation might produce rainbows. Stay tuned.
Challenge 1378 ny
**
If lightweight boxes made of mirrors can oat in radiation, one gets a strange consequence: such a box might self-accelerate in free space. In a sense, an accelerated box could oat on the Fulling–Davies–Unruh radiation it creates by its own acceleration.
Are you able to show the following: one reason why this is impossible is a small but di erence between gravity and acceleration, namely the absence of tidal e ects. (Other reasons, such as the lack of perfect mirrors, also make the e ect impossible.)
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Ref. 907
**
In 2003, Michael Kuchiev has made the spectacular prediction that matter and radiation with a wavelength larger than the diameter of a black hole is partly re ected when it hits a black hole. e longer the wavelength, the more e cient the re ection would be. For stellar or even bigger black holes, only photons or gravitons are predicted to be re ected. Black holes are thus not complete trash cans. Is the e ect real? e discussion is still ongoing.
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Q
Let us take a conceptual step at this stage. So far, we looked at quantum theory with gravitation; now we have a glimpse at quantum theory of gravitation.
If we focus on the similarity between the electromagnetic eld and the gravitational ‘ eld,’ we should try to nd the quantum description of the latter. Despite attempts by many brilliant minds for almost a century, this approach was not successful.* Let us see why.
T
B
A short calculation shows that an electron circling a proton due to gravity alone, without Challenge 1379 ny electrostatic attraction, would do so at a gravitational Bohr radius of
rgr.B.
=
ħ G me mp
=
.
ë
m
(609)
Challenge 1380 ny
which is about a thousand times the distance to the cosmic horizon. In fact, even in the normal hydrogen atom there is not a single way to measure gravitational e ects. (Are you able to con rm this?) But why is gravity so weak? Or equivalently, why are the universe and normal atoms so much smaller than a gravitational Bohr atom? At the present point
* Modern approaches take another direction, as explained in the third part of the mountain ascent.
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of our quest these questions cannot be answered. Worse, the weakness of gravity even means that with high probability, future experiments will provide little additional data helping to decide among competing answers. e only help is careful thought.
D
-
Ref. 908 Page 796 Challenge 1381 ny
If the gravitational eld evolves like a quantum system, we encounter all issues found in other quantum systems. General relativity taught us that the gravitational eld and spacetime are the same. As a result, we may ask why no superpositions of di erent macroscopic space-times are observed.
e discussion is simpli ed for the simplest case of all, namely the superposition, in a vacuum region of size l, of a homogeneous gravitational eld with value and one with value ′. As in the case of a superposition of macroscopic distinct wave functions, such a superposition decays. In particular, it decays when particles cross the volume. A short calculation yields a decay time given by
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td
=
(
kT πm
)
(
nl − ′)
,
(610)
Challenge 1382 ny
where n is the particle number density, kT their kinetic energy and m their mass. Inserting typical numbers, we nd that the variations in gravitational eld strength are extremely small. In fact, the numbers are so small that we can deduce that the gravitational
eld is the rst variable which behaves classically in the history of the universe. Quantum gravity e ects for space-time will thus be extremely hard to detect.
In short, matter not only tells space-time how to curve, it also tells it to behave with class. is result calls for the following question.
D
?
Ref. 909
Quantum theory says that everything that moves is made of particles. What kind of particles are gravitational waves made of? If the gravitational eld is to be treated quantum mechanically like the electromagnetic eld, its waves should be quantized. Most properties of these quanta can be derived in a straightforward way.
e r dependence of universal gravity, like that of electricity, implies that the particles have vanishing mass and move at light speed. e independence of gravity from electromagnetic e ects implies a vanishing electric charge.
e observation that gravity is always attractive, never repulsive, means that the eld quanta have integer and even spin. Vanishing spin is ruled out, since it implies no coupling to energy. To comply with the property that ‘all energy has gravity’, S = is needed. In fact, it can be shown that only the exchange of a massless spin particle leads, in the classical limit, to general relativity.
e coupling strength of gravity, corresponding to the ne structure constant of electromagnetism, is given either by
αG
=
G ħc
=
.
ë
− kg−
or by
αG
=
Gmm ħc
=
m mPl
=
E EPl
(611)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•.
Challenge 1383 n
However, the rst expression is not a pure number; the second expression is, but depends
on the mass one inserts. ese di culties re ect the fact that gravity is not properly speak-
ing an interaction, as became clear in the section on general relativity. It is o en argued
that m should be taken as the value corresponding to the energy of the system in ques-
tion. For everyday life, typical energies are eV, leading to a value αG
. Gravity
is indeed weak compared to electromagnetism, for which αem = . .
If all this is correct, virtual eld quanta would also have to exist, to explain static grav-
itational elds.
However, up to this day, the so-called graviton has not yet been detected, and there is
in fact little hope that it ever will. On the experimental side, nobody knows yet how to
build a graviton detector. Just try! On the theoretical side, the problems with the coupling
constant probably make it impossible to construct a renormalizable theory of gravity; the
lack of renormalization means the impossibility to de ne a perturbation expansion, and
thus to de ne particles, including the graviton. It might thus be that relations such as
E = ħω or p = ħ πλ are not applicable to gravitational waves. In short, it may be that
the particle concept has to be changed before applying quantum theory to gravity. e
issue is still open at this point.
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S-
e indeterminacy relation for momentum and position also applies to the gravitational eld. As a result, it leads to an expression for the indeterminacy of the metric tensor in a region of size L, which is given by
∆
lPl L
,
(612)
Challenge 1384 ny
where lPl = ħG c is the Planck length. Can you deduce the result? Quantum theory
thus shows that like the momentum or the position of a particle, also the metric tensor
is a fuzzy observable.
But that is not all. Quantum theory is based on the principle that actions below ħ
cannot be observed. is implies that the observable values for the metric in a region
of size L are bound by
ħG cL
.
(613)
Page 998
Can you con rm this? e result has far-reaching consequences. A minimum value for the metric depending inversely on the region size implies that it is impossible to say what happens to the shape of space-time at extremely small dimensions. In other words, at extremely high energies, the concept of space-time itself becomes fuzzy. John Wheeler introduced the term space-time foam to describe this situation. e term makes clear that space-time is not continuous nor a manifold in those domains. But this was the basis on which we built our description of nature so far! We are forced to deduce that our description of nature is built on sand. is issue will form the start of the third part of our mountain ascent.
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N
Page 773
Gravity has another important consequence for quantum theory. To count and de ne particles, quantum theory needs a de ned vacuum state. However, the vacuum state cannot be de ned when the curvature radius of space-time, instead of being larger than the Compton wavelength, becomes comparable to it. In such highly curved space-times, particles cannot be de ned. e reason is the impossibility to distinguish the environment from the particle in these situations: in the presence of strong curvatures, the vacuum is full of spontaneously generated matter, as black holes show. Now we just saw that at small dimensions, space-time uctuates wildly; in other words, space-time is highly curved at small dimensions or high energies. In other words, strictly speaking particles cannot be de ned; the particle concept is only a low energy approximation! We will explore this strange conclusion in more detail in the third part of our mountain ascent.
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N
Ref. 910
Ref. 912 Ref. 911
e end of the twentieth century has brought several unexpected but strong results in the semiclassical quantum gravity.
In Ford and Roman found that worm holes, which are imaginable in general relativity, cannot exist if quantum e ects are taken into account. ey showed that macroscopic worm holes require unrealistically large negative energies. (For microscopic worm holes the issue is still unclear.)
In it was found by Kay, Radzikowski and Wald that closed time-like curves do not exist in semiclassical gravity; there are thus no time machines in nature.
In Pfenning and Ford showed that warp drive situations, which are also imaginable in general relativity, cannot exist if quantum e ects are taken into account. ey also require unrealistically large negative energies.
N
is short excursion into the theory of quantum gravity showed that a lot of trouble is waiting. e reason is that up to now, we deluded ourselves. In fact, it was more than that: we cheated. We carefully hid a simple fact: quantum theory and general relativity contradict each other. at was the real reason that we stepped back to special relativity before we started exploring quantum theory. In this way we avoided all problems, as quantum theory does not contradict special relativity. However, it does contradict general relativity. e issues are so dramatic, changing everything from the basis of classical physics to the results of quantum theory, that we devote the beginning of the third part only to the exploration of the contradictions. ere will be surprising consequences on the nature of space-time, particles and motion. But before we study these issues, we complete the theme of the present, second part of the mountain ascent, namely the essence of matter and interactions.
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B
758 Gell-Mann wrote this for the Nobel Conference. M. G -M , What are the Build-
ing Blocks of Matter?, in D. H & O. P
, editors, e Nature of the Physical Uni-
verse, New York, Wiley, , p. . Cited on page .
759 See e.g. the reprints of his papers in the standard collection by J
A. W
&
W
H. Z
, Quantum eory and Measurement, Princeton University Press,
. Cited on page .
760 H.D. Z , On the interpretation of measurement in quantum theory, Foundations of Physics 1, pp. – , . Cited on page .
761 L
R
, A Modern Course in Statistical Physics, Wiley, nd edition,
lent introduction into thermodynamics. Cited on page .
. An excel-
762 E. J & H.D. Z , e emergence of classical properties through interactions with the
environment, Zeitschri für Physik B 59, pp. – , . See also E
J , Decoher-
ence and the appearance of a classical world in quantum theory, Springer Verlag, . Cited
on page .
763 M. T
, Apparent wave function collapse, Foundation of Physics Letters 6, pp. –
, . Cited on page .
764 e decoherence time is bound from above by the relaxation time. See A.O. C
&
A.J. L
, In uence of damping on quantum interference: an exactly soluble model,
Physical Review A 31, , pp. – . is is the main reference about e ects of decoher-
ence for a harmonic oscillator. e general approach to relate decoherence to the in uence
of the environment is due to Niels Bohr, and has been pursued in detail by Hans Dieter Zeh.
Cited on page .
765 G. L
, On the generators of quantum dynamical subgroups, Communications in
Mathematical Physics 48, pp. – , . Cited on page .
766 W
H. Z
, Decoherence and the transition from quantum to classical, Physics
Today pp. – , October . An easy but somewhat confusing article. His reply to the
numerous letters of response in Physics Today, April , pp. – , and pp. – , exposes
his ideas in a clearer way and gives a taste of the heated discussions on this topic. Cited on
pages and .
767 J B
, explained this regularly in the review talks he gave at the end of his life,
such as the one the author heard in Tokyo in . Cited on page .
768 e rst decoherence measurement was performed by M. B
& al., Observing pro-
gressive decoherence of the “meter” in a quantum measurement, Physical Review Letters 77,
pp. – , December . Cited on page .
769 Later experiments con rming the numerical predictions from decoherence were published
by C. M
, D.M. M
, B.E. K & D.J. W
, A “Schrödinger cat”
superposition state of an atom, Science 272, pp. – , , W.P. S
, Quantum
physics: engineering decoherence, Nature 403, pp. – , , C.J. M , B.E. K ,
Q.A. T
, C.A. S
, D. K
, W.M. I
, C. M
&
D.J. W
, Decoherence of quantum superpositions through coupling to engineered
reservoirs, Nature 403, pp. – , . See also the summary by W.T. S
, G. A -
& F. H
, Dekohärenz in o enen Quantensystemen, Physik Journal 1, pp. – ,
November . Cited on page .
770 L. H
, K. H
, B. B
, A. Z
& M. A
,
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Decoherence of matter waves by thermal emission of radiation, Nature 427, pp. – , . Cited on page .
771 K. B 25a, pp.
, Quantenmechanik und Objektivierbarkeit, Zeitschri für Naturforschung – , . Cited on page .
772 See for example D. S , Physics Today p. , September . Cited on page .
773 D
B , Quantum eory, Prentice-Hall, , pp. – . Cited on page .
774 A. E
, B. P
& N. R , Can quantum-mechanical description of real-
ity be considered complete?, Physical Review 48, pp. – , . Cited on page .
775 A. A
, J. D
& G. R
, Experimental tests of Bell’s inequalities using
time-varying analyzers, Physical Review Letters 49, pp. – , , Cited on page .
776 G.C. H
, Causality problems for Fermi’s two-atom system, Physical Review
Letters 72, pp. – , . Cited on page .
777 D
Z. A
No citations.
, Quantum Mechanics and Experience, Harvard University Press, .
778 On the superposition of magnetization in up and down directions there are numerous pa-
pers. Recent experiments on the subject of quantum tunnelling in magnetic systems are
described in D.D. A
, J.F. S
, G. G
, D.P. D V
& D.
L , Macroscopic quantum tunnelling in magnetic proteins, Physical Review Letters 88,
pp. – , , and in C. P
& al., Macroscopic quantum tunnelling e ects
of Bloch walls in small ferromagnetic particles, Europhysics Letters 19, pp. – , .
Cited on page .
779 e prediction that quantum tunnelling could be observable when the dissipative interac-
tion with the rest of the world is small enough was made by Leggett; the topic is reviewed
in A.J. L
, S. C
, A.T. D
, M.P.A. F
, A. G & W.
Z
, Dynamics of dissipative -state systems, Review of Modern Physics 59, pp. – ,
. Cited on page .
780 For example, superpositions were observed in Josephson junctions by R.F. V & R.A. W , Macroscopic quantum tunnelling in mm Nb Josephson junctions, Physical Review Letters 47, pp. – , , Cited on page .
781 S. H
, Entanglement, decoherence and the quantum-classical transition, Physics
Today 51, pp. – , July . An experiment putting atom at two places at once, distant
about nm, was published by C. M
& al., Science 272, p. , . Cited on page
.
782 M.R. A
& al., Observations of interference between two Bose condensates, Science
275, pp. – , January . See also the www.aip.org/physnews/special.htm website.
Cited on page .
783 e most famous reference on the wave function collapse is chapter IV of the book by K
G
, Quantum Mechanics, Benjamin, New York, , It is the favorite reference
by Victor Weisskopf, and cited by him on every occasion he talks about the topic. Cited on
page .
784 An experimental measurement of superpositions of le and right owing currents with
electrons was J.E. M , T.P. O
, L. L
, L. T , C.H.
W
& S. L
, Josephson persistent-current qubit, Science 285, pp. – , . In the
year , superpositions of µA clockwise and anticlockwise have been detected; for more
details, see J.R. F
& al., Quantum superposition of distinct macroscopic states,
Nature 406, p. , . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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785 Collapse times have been measured ... Cited on page .
786 A clear discussion can be found in S. H
& J.-M. R
, Quantum computing:
dream or nightmare?, Physics Today 49, pp. – , , as well as the comments in Physics
Today 49, pp. – , . Cited on page .
787 S. K
& E.P. S
, e problem of hidden variables in quantum mechanics, 17,
pp. – , . Cited on page .
788 J.F. C
, M.A. H
, A. S
& R.A. H , Physical Review Letters 23,
p. , . e more general and original result is found in J.S. B , Physics 1, p. ,
, Cited on page .
789 B
W &N G
, eds., e Many–Worlds Interpretation of Quantum
Mechanics, Princeton University Press, . is interpretation talks about entities which
cannot be observed, namely the many worlds, and o en assumes that the wave function of
the universe exists. Both habits are beliefs and in contrast with facts. Cited on page .
790 ‘On the other had I think I can safely say that nobody understands quantum mechanics."
From R
P. F
, e Character of Physical Law, MIT Press, Cambridge, ,
p. . He repeatedly made this statement, e.g. in the introduction of his otherwise excellent
QED: the Strange eory of Light and Matter, Penguin Books, .. Cited on page .
791 M. T
, e importance of quantum decoherence in brain processes, Physical Re-
view D 61, pp. – , , or also http://www.arxiv.org/abs/quant-ph/
. Cited
on page .
792 Connections between quantum theory and information theory can be found in Cited on page .
793 J.A. W
, pp. – , in Batelle Recontres: 1967 Lectures in Mathematics and Phys-
ics, C. D W & J.A. W
, editors, W.A. Benjamin, . For a pedagogical ex-
planation, see J W. N
, From Newton’s laws to the Wheeler-DeWitt equation,
http://www.arxiv.org/abs/physics/
or European Journal of Physics 19, pp. – ,
. Cited on page .
794 E. F tions.
, Quantum theory of radiation, Review of Modern Physics 4, p. ,
. No cita-
795 G. H
, Causality problems for Fermi’s two-atom system, Physical Review Letters
72, pp. – , , No citations.
796 e beautiful analogy between optics and two-dimensional electron systems, in GaAs at he-
lium temperature, has been studied in detail in C.W.J. B
& H. H
,
Solid State Physics, 44, pp. – , , No citations.
797 e use of radioactivity for breeding is described by ... Cited on page .
798 S.M. B
, Real engines of creation, Nature 386, pp. – , . Cited on page .
799 Early results and ideas on molecular motors are summarised by B
G L,
Measured steps advance the understanding of molecular motors, Physics Today pp. – ,
April . Newer results are described in R.D. A
, Making molecules into motors,
Scienti c American pp. – , July . Cited on page .
800 e motorized screw used by viruses was described by A.A. S the bacteriophage phi DNA packaging motor, Nature 408, pp. page .
& al., Structure of – , . Cited on
801 R. B
& P. H
. Cited on page .
, Brownsche Motoren, Physikalische Blätter 51, pp. – ,
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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802 e table and the evolutionary tree are taken from J.O. M I
, M. M
,
M.E. W
& R. P
, Bacteria and Archaea: Molecular techniques reveal as-
tonishing diversity, Biodiversity 3, pp. – , . e tree might still change a little in the
coming years. Cited on page .
803 is is taken from the delightful children text H J. P Ravensburger Buchverlag , . Cited on page .
, Spiel das Wissen scha ,
804 e discovery of a speci c taste for fat was published by F. L
, P. P
-
D
, B. P
, I. N , M. F
, J.P. M
& P. B
,
CD involvement in orosensory detection of dietary lipids, spontaneous fat preference,
and digestive secretions, Journal of Clinical Investigation 115, pp. – , . Cited on
page .
805 Linda Buck and Richard Axel received the
Nobel Prize for medicine and physiology
for their unravelling of the working of the sense of smell. Cited on page .
806 To learn to enjoy life to the maximum there is no standard way. A good foundation can be
found in those books which teach the ability to those which have lost it. Cited on page .
e best experts are those who help others to overcome traumas. P
A. L
&
AF
, Waking the Tiger – Healing Trauma – e Innate Capacity to Transform
Overwhelming Experiences, North Atlantic Books, . G
G
, How to Become
the Parent You Never Had - a Treatment for Extremes of Fear, Anger and Guilt, Real Options
Press, .
A good complement to these texts is the systemic approach, as presented by B
H
, Zweierlei Glück, Carl Auer Verlag, . Some of his books are also available
in English. e author explains how to achieve a life of contentness, living with the highest
possible responsibility of one’s actions, by reducing entaglements with one’s past and with
other people. He presents a simple and e cient technique for realising this disentanglement.
e next step, namely full mastery in the enjoyment of life, can be found in any book
written by somebody who has achieved mastery in any one topic. e topic itself is not
important, only the passion is.
A.
G
, Le dialogue pédagogique avec l’élève, Centurion, , A. de la
Garanderie/ Pour une pédagogie de l’intelligence, Centurion, , A. de la Garanderie/
Réussir ça s’apprend, Bayard, . De la Garanderie explains how the results of teaching
and learning depend in particular on the importance of evocation, imagination and motiv-
ation.
P , Phaedrus, Athens,
.
F
D , La cause des enfants, La ont, , and her other books. Dolto
( – ), a child psychiatrist, is one of the world experts on the growth of the child; her
main theme was that growth is only possible by giving the highest possible responsibility to
every child during its development.
In the domain of art, many had the passion to achieve full pleasure. A good piece of
music, a beautiful painting, an expressive statue or a good movie can show it. On a smaller
scale, the art to typeset beautiful books, so di erent from what many computer programs
do by default, the best introduction is by Jan Tschichold ( – ), the undisputed master
of the eld. Among the many books he designed are the beautiful Penguin books of the late
s; he also was a type designer, e.g. of the Sabon typeface. A beautiful summary of his
views is the short but condensed text J T
, Ausgewählte Aufsätze über Fragen
der Gestalt des Buches und der Typographie, Birkhäuser Verlag, Basel, . An extensive and
beautiful textbook on the topic is H P
W
&F
F
,
Lesetypographie, Verlag Hermann Schmidt, Mainz, . See also R
B
,
e Elements of Typographic Style, Hartley & Marks, .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Many scientists passionately enjoyed their occupation. Any biography of Charles Darwin will purvey his fascination for biology, of Friedrich Bessel for astronomy, of Albert Einstein for physics and of Linus Pauling for chemistry.
807 e group of John Wearden in Manchester has shown by experiments with persons that the
accuracy of a few per cent is possible for any action with a duration between a tenth of a
second and a minute. See J. M C
, When a second lasts forever, New Scientist pp. –
, November . Cited on page .
808 E.J. Z pp. – ,
, e macroscopic nature of space-time, American Journal of Physics 30, . Cited on page .
809 e potential of single-ion clocks is explained in ... Cited on page .
810 P.D. P , e smallest clock, European Journal of Physics 14, pp. – , page .
. Cited on
811 D.J M
& al., Biochemical basis for the biological clock, Biochemistry 41, pp. –
, . Cited on page .
812 e chemical clocks in our body are described in J
D. P
, e Living Clock,
Oxford University Press, , or in A. A
& F. H
, Cycles of Nature: An
Introduction to Biological Rhythms, National Science Teachers Association, . See also
the http://www.msi.umn.edu/~halberg/introd/ website. Cited on page .
813 An introduction to the sense of time as result of clocks in the brain is found in R.B. I
& R. S
, e neural representation of time, Current Opinion in Neurobiology 14,
pp. – , . e interval timer is explain in simple words in K. W
, Times in
our lives, Scienti c American pp. – , September . e MRI research used is S.M.
R , A.R. M
& D.L. H
, e evolution of brain activation during tem-
poral processing, Nature Neuroscience 4, pp. – , . Cited on page .
814 See for example ... Cited on page .
815 A pretty example of a quantum mechanical system showing exponential behaviour at all
times is given by H. N
, M. N
& S. P
, Exponential behaviour
of a quantum system in a macroscopic medium, Physical Review Letters 73, pp. – ,
. Cited on page .
816 See the delightful book about the topic by P
F
&S
P
, La
regola d’oro di Fermi, Bibliopolis, . An experiment observing deviations at short times is
S.R. W
, C.F. B
, M.C. F
, K.W. M
, P.R. M
,
Q. N , B. S
& M.G. R
, Nature 387, p. , . Cited on page .
817 See ... Cited on page .
818 H. K pp. – ,
& S. K
, Unique morphology of the human eye, Nature 387,
. Cited on page .
819 J
W. P
, Body pleasure and the origins of violence, e Futurist Bethseda,
, also available at http://www.violence.de/prescott/bullettin/article.html. Cited on page
.
820 See for example P. P
, Relativity, gold, closed-shell interactions, and CsAu.NH , An-
gewandte Chemie, International Edition 41, pp. – , , or L.J. N
, Why is
mercury liquid? Or, why do relativistic e ects not get into chemistry textbooks?, Journal of
Chemical Education 68, pp. – , . Cited on page .
821 e switchable mirror e ect was discovered by J.N. H
, R. G
, J.H.
R
, R.J. W
, J.P. D
, D.G. G
& N.J. K
, Yt-
trium and lanthanum hydride lms with switchable optical properties, Nature 380, pp. –
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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, . A good introduction is R. G
, Schaltbare Spiegel aus Metallhydriden,
Physikalische Blätter 53, pp. – , . Cited on page .
822 e present attempts to build cheap THz wave sensors – which allow to see through clothes – is described by D. C , Brainstorming their way to an imaging revolution, Science 297, pp. – , . Cited on page .
823 V.F. W 187, p. ,
, Of atoms, mountains and stars: a study in qualitative physics, Science . Cited on page .
824 K. A
, Y. L , T. H , W. Z
, W.P. C , T. K
R.J. F , Adhesive force of a single gecko foot-hair, Nature 405, pp.
on page .
, R. F –,
& . Cited
825 A.B. K , A. M
& T. S , Getting a grip on spider attachment: an AFM
approach to microstructure adhesion in arthropods, Smart Materials and Structures 13,
pp. – , . Cited on page .
826 H. I
, T. S
, K.-I. T
& H. H
, E ect of the magnetic eld on
the melting transition of H O and D O measured by a high resolution and supersensitive
di erential scanning calorimeter, Journal of Applied Physics 96, pp. – , . Cited
on page .
827 R. H
& al., presented these results at the International Conference on New Dia-
mond Science in Tsukuba (Japan) . See gures at http://www.carnegieinstitution.org/
diamond- may /. Cited on page .
828 On the internet, the ‘spherical’ periodic table is credited to Timothy Stowe; but there is no reference for that claim, except an obscure calendar from a small chemical company. e table (containing a few errors) is from the http://chemlab.pc.maricopa.edu/periodic/stowetable. html website. Cited on page .
829 For good gures of atomic orbitals, take any modern chemistry text. See for example the text by ... Cited on page .
830 For experimentally determined pictures of the orbitals of dangling bonds, see for example
F. G
& al., Subatomic features on the silicon ( )-( x ) surface observed by atomic
force microscopy, Science 289, pp. – , . Cited on page .
831 A.P. F , T. A , R. B
, V.B. E
, N.B. K
, M. K
,
L. S
, M. T
& G.E. V
, An intrinsic velocity-independent criterion
for super uid turbulence, Nature 424, pp. – , . Cited on page .
832 J.L. C –K
, N. G
, P. G
–M
& P.A. S
, Con-
ductance in disordered nanowires: foward and backscattering, Physical Review B 53,
p.
, . Cited on page .
833 e rst experimental evidence is by V.J. G
& B. S , Resonance tunnelling in the
fractional quantum Hall regime: measurement of fractional charge, Science 267, pp. –
, . e rst unambiguous measurements are by R. P
& al., Direct
observation of a fractional charge, Nature 389, pp. – , , and L. S
,
D.C. G
, Y. J & B. E
, Observation of the e, fractionally charged Laugh-
lin quasiparticle/ Physical Review Letters 79, pp. – , . or http://www.arxiv.org/
abs/cond-mat/
. Cited on page .
834 e original prediction, which earned him a Nobel Prize for physics, is in R.B. L
,
Anomalous quantum Hall e ect: an incompressible quantum uid with fractionally charged
excitations, Physical Review Letters 50, pp. – , . Cited on page .
835 e fractional quantum Hall e ect was discovered by D.C. T , H.L. S
& A.C.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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G
, Two-dimensional magnetotransport in the extreme quantum limit, Physical Re-
view Letters 48, pp. – , . Cited on page .
836 e discovery of the original quantum Hall e ect, which earned von Klitzing a Nobel Prize,
was published as K.
K
, G. D
& M. P
, New method for high-
accuracy determination of the ne-structure constant based on quantised Hall resistance,
Physical Review Letters 45, pp. – , . Cited on page .
837 [ ] e rst paper was J. B
, L.N. C
& J.R. S
eory of Superconductivity, Physical Review 106, pp. – ,
theory, is then given in the masterly paper J. B
, L.N. C
, eory of Superconductivity, Physical Review 108, pp. –
.
, Microscopic
. e full, so-called BCS
& J.R. S
-
, . Cited on page
838 e story is found on many places in the internet. Cited on page .
839 H.B.G. C
, On the attraction between two perfectly conducting bodies, Proc. Kon.
Ned. Acad. Wetensch. 51, pp. – , . No citations.
840 H.B.G. C
& D. P
, e in uence of retardation on the London-Van der
Waals forces, Physical Review 73, pp. – , . No citations.
841 S.L. B
, Aantrekkende kracht tussen schepen, Nederlands tidschri voor
natuuurkunde 67, pp. – , Augustus . See also his paper S.L. B
,A
maritime analogy of the Casimir e ect, American Journal of Physics 64, pp. – , .
Cited on page .
842 A. L
& B. D
, An acoustic Casimir e ect, Physics Letters A 248, pp. –
, . See also A. L
, A demonstration apparatus for an acoustic analog of the
Casimir e ect, American Journal of Physics 67, pp. – , . Cited on page .
843 B.W. D
,I.I. A
& E.M. L
, Direct measurement of molecu-
lar attraction between solids separated by a narrow gap, Quaterly Review of the Chemical
Society (London) 10, pp. – , . Cited on page .
844 M.J. S
, Nature 180, p. , . M.J. S
, Measurements of attractive
forces between at plates, Physica (Utrecht) 24, pp. – , . Cited on page .
845 S.K. L
, Demonstration of the Casimir force in the . to µm range, Physical
Review Letters 78, pp. – , . U. M
& A. R , Precision measurement of the
Casimir force from . to . µm, Physical Review Letters 81, pp. – , . Cited on
page .
846 A. L
, Das Vakuum kommt zu Krä en, Physik in unserer Zeit 36, pp. – , .
Cited on page .
847 T.H. B
, Van der Waals forces and zero-point energy for dielectric and permeable ma-
terials, Physical Review A 9, pp. – , . is was taken up again in O. K
,
I. K , A. M & M. R
, Repulsive Casimir forces, Physical Review Letters 89,
p.
, . However, none of these e ects has been veri ed yet. Cited on page .
848 e e ect was discovered by K. S
, On propagation of light in the vacuum
between plates, Physics Letters B 236, pp. – , . See also G. B
, Faster-than-
c light between parallel mirrors: the Scharnhorst e ect rederived, Physics Letters B 237,
pp. – , , and P.W. M
& K. S
, Impossibility of measuring faster-
than-c signalling by the Scharnhorst e ect, Physics Letters B 248, pp. – , . e
latter corrects an error in the rst paper. Cited on page .
849 D.L. B
& al., Positron production in multiphoton light-by-light scattering, Phys-
ical Review Letters 79, pp. – , . A simple summary is given in B
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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S
, Gamma rays create matter just by plowing into laser light, Physics
Today 51, pp. – , February . Cited on page .
850 M. K
& R. G
, e ‘friction’ of the vacuum, and other uctuation-
induced forces, Reviews of Modern Physics 71, pp. – , . Cited on page .
851 G. G & L. S
, Will small particles exhibit Brownian motion in the
quantum vacuum?, http://www.arxiv.org/abs/quant/phys/
Cited on page .
852 P.A.M. D , ..., . is article transcribes a speech by Paul Dirac just before his death. Disturbed by the in nities of quantum electrodynamics, he rejects as wrong the theory which he himself found, even though it correctly describes all experiments. e humility of a great man who is so dismissive of his main and great achievement was impressive when hearing him, and still is impressive when reading the speech. Cited on page .
853 See for example the overview by M. B & T. J
, Umweltfreundliche Lichtquellen,
Physik Journal 2, pp. – , . Cited on page .
854 J. K
, Les lasers femtosecondes : applications atmosphériques, La Recherche
pp. RE – , February / See also the http://www.teramobile.org website. Cited on
page .
855 Misleading statements are given in the introduction and in the conclusion of the review by H. P , Interference between independent photons, Review of Modern Physics, volume , pages - ; however, in the bulk of the article the author in practice retracts the statement, e.g. on page . Cited on pages and .
856 L. C , G . T
, G. K
, I. S
& B. F
of atoms using thermal neutrons, Physical Review Letters 89, p.
.
, Holographic imaging , . Cited on page
857 H. E
& B. K
, Naturwissenscha en 23, p. , , H. E , Annalen der
Physik 26, p. , , W. H
& H. E , Folgerung aus der Diracschen e-
orie des Electrons, Zeitschri für Physik 98, pp. – , . Cited on page .
858 T
S
& al., Physical Review Letters October . Cited on page .
859 M. R
& al., Observation of an electric octupole transition in a single ion, Physical
Review Letters 78, pp. – , . A lifetime of days was determined. Cited on
page .
860 K. A , J.J. C
, R.R. D
& M.S. F , Microlaser: a laser with one atom in
an optical resonator, Physical Review Letters 73, pp. – , December . Cited on
page .
861 A. T
, H. Y
, Y. H
, N. N
& H. M
, ree-
dimensional optical memory using a human ngernail, Optics Express 13, pp. – ,
. ey used a Ti:sapphire oscillator and a Ti:sapphire ampli er. Cited on page .
862 See the website by Tom Hales on http://www.math.lsa.umich.edu/~hales/countdown. An
earlier book on the issue is the delightful text by M L
, Kugelpackungen von
Kepler bis heute, Vieweg Verlag, . Cited on page .
863 For a short overview, also covering binary systems, see C. B
, H.-H.
G
& P. L
, Entropische Krä e - warum sich repulsiv wechselwirkende
Teilchen anziehen können, Physikalische Blätter 55, pp. – , . Cited on page .
864 H –W
F , Direct observation of three-body interactions in adsorbed layers:
Re on W( ), Physical Review Letters 52, pp. – , . Cited on page .
865 See for example the textbook by A. P
& J. V
Cambridge University Press, . Cited on page .
, Physics of Crystal Growth,
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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866 Y. F
, Faszination der Schneekristalle - wie ihre bezaubernden Formen entstehen,
Chemie in unserer Zeit 31, pp. – , . His http://www.lowtem.hokudai.ac.jp/~frkw/
index_e.html website gives more information. Cited on page .
867 P.-G. G .
, So matter, Reviews of Modern Physics 63, p. , . Cited on page
868 e uid helium based quantum gyroscope was rst built by the Czech group ... and the Berkeley group, Nature . April . Cited on page .
869 K. S
& al., Nature 404, p. ,
page .
. For the optical analog, see Ref. . Cited on
870 J.P. B
& al., Production of hollow atoms by the excitation of highly charged ions in
interaction with a metallic surface, Physical Review Letters 65, pp. – , . See also C.
R
, Hollow atoms, ..., and ...Physikalische Blätter Oktober . Cited on page .
871 e present knowledge on how humans move is summarised by ... Cited on page .
872 See e.g. D.A. B
, Electron acceleration in the aurora, Contemporary Physics 35,
pp. – , . Cited on page .
873 e discharges above clouds are described e.g. by ... Cited on page .
874 See e.g. L.O’C. D , Acceleration of cosmic rays, Contemporary Physics 35, pp. – , . Cited on page .
875 About Coulomb explosions, see ... Cited on page .
876 See D K
, New mathematical structures in renormalizable quantum eld the-
ories, http://www.arxiv.org/abs/hep-th/
or Annals of Physics 303, pp. – , ,
and the erratum ibid., 305, p. , . Cited on page .
877 S. F , C. A
D , T.W. H
&M
W , Atomic interferometer
with amplitude gratings of light and its applications to atom based tests of the equivalence
principle, Physical Review Letters 93, p.
, . Cited on page .
878 R. C
, A.W. O
& S.A. W
, Observation of gravitationally in-
duced quantum mechanics, Physical Review Letters 34, pp. – , . is experiment
is usually called the COW experiment. Cited on page .
879 G.C. G
, A. R
& T. W , Uni ed dynamics for microscopic and macro-
scopic systems, Physical Review D 34, pp. – , . Cited on page .
880 C .J. B
& C. L
, Atominterferometrie und Gravitation, Physikalische
Blätter 52, pp. – , . See also A. P
, K.Y. C
& S. C , Observation
of gravitational acceleration by dropping atoms, Nature 400, pp. – , August .
Cited on page .
881 V.V. N
& al., Quantum states of neutrons in the Earth’s gravitational eld,
Nature 415, p. , January . Cited on page .
882 J.D. B
, Black holes and entropy, Physical Review D 7, pp. – , .
Cited on page .
883 R. B
, e holographic principle, Review of Modern Physics 74, pp. – , ,
also available as http://www.arxiv.org/abs/hep-th/
. e paper is an excellent review;
however, it has some argumentation errors, as explained on page . Cited on page .
884 is is clearly argued by S. C
, Black hole entropy from horizon conformal eld the-
ory, Nuclear Physics B Proceedings Supplement 88, pp. – , . Cited on page .
885 S.A. F
, Nonuniqueness of canonical eld quantization in Riemannian space-time,
Physical Review D 7, pp. – , , P. Davies/ Scalar particle production in Schwar-
zchild and Rindler metrics, Journal of Physics A: General Physics 8, pp. – , , W.G.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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U
, Notes on black hole evaporation, Physical Review D 14, pp. – ,
on page .
. Cited
886 About the possibility to measure Fulling–Davies–Unruh radiation, see for example the pa-
per by H. R , On the estimates to measure Hawking e ect und Unruh e ect in the labor-
atory, International Journal of Modern Physics D3 p. , . http://www.arxiv.org/abs/
gr-qc/
or the paper P. C & T. T
, Testing Unruh radiation with ultra-
intense lasers, Physical Review Letters 83, pp. – , . Cited on page .
887 W.G. U
& R.M. W , Acceleration radiation and the generalised second law of
thermodynamics, Physical Review D 25, pp. – , . Cited on page .
888 R.M. W , e thermodynamics of black holes, Living Reviews of Relativity , www-livingreviews.org/lrr- - . Cited on page .
889 For a delightful popular account from a black hole expert, see I N
, Black Holes
and the Universe, Cambridge University Press . Cited on pages and .
890 e original paper is W.G. U
, Experimental black hole evaporation?, Physical Review
Letters 46, pp. – , . A good explanation with good literature overview is the one
by M V , Acoustical black holes: horizons, ergospheres and Hawking radiation,
http://www.arxiv.org/abs/gr-qc/
. Cited on page .
891 G.W. G
& S.W. H
, Action integrals and partition functions in quantum
gravity, Physical Review D 15, pp. – , . No citations.
892 Optical black holes ... Cited on page .
893 T. D
& R. R
, Quantum electrodynamical e ects in Kerr–Newman geomet-
ries, Physical Review Letters 35, pp. – , . Cited on page .
894 ese were the Vela satellites; their existence and results were announced o cially only in , even though they were working already for many years. Cited on page .
895 e best general introduction into the topic of gamma ray bursts is S. K , J. G
& D. H
, Kosmische Gammastrahlenausbrüche – Beobachtungen und Modelle,
Teil I und II, Sterne und Weltraum March and April . Cited on page .
896 When the gamma ray burst encounters the matter around the black hole, it is broadened.
e larger the amount of matter, the broader the pulse is. See G. P
,R. R
& S.-S. X , e dyadosphere of black holes and gamma-ray bursts, Astronomy and As-
trophysics 338, pp. L –L , , R. R
, J.D. S
, J.R. W
& S.-S.
X , On the pair electromagnetic pulse of a black hole with electromagnetic structure, As-
tronomy and Astrophysics 350, pp. – , , R. R
, J.D. S
, J.R.
W
& S.-S. X , On the pair electromagnetic pulse from an electromagnetic black
hole surrounded by a baryonic remnant, Astronomy and Astrophysics 359, pp. – ,
, and C.L. B
, R. R
& S.-S. X , e elementary spike produced by
a pure e+e− pair-electromagnetic pulse from a black hole: the PEM pulse, Astronomy and
Astrophysics 368, pp. – , . For a very personal account by Ru ni on his involve-
ment in gamma ray bursts, see his paper Black hole formation and gamma ray bursts, http://
www.arxiv.org/abs/astro-ph/
. Cited on page .
897 About the membrane despription of black holes, see ... Cited on page .
898 http://www.arxiv.org/abs/astro-ph/
, http://www.arxiv.org/abs/astro-ph/
,
D.W. F & al., Early optical emission from the γ-ray burst of October , Nature
422, pp. – , . Cited on page .
899 Negative heat capacity has also been found in atom clusters and in nuclei. See e.g. M.
S
& al., Negative heat capacity for a cluster of sodium atoms, Physical Review
Letters 86, pp. – , . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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900 H.-P. N
, Quasinormal modes: the characteristic ‘sound’ of black holes and neutron
stars, Classical and Quantum Gravity 16, pp. R –R , . Cited on page .
901 He wrote a series of papers on the topic; a beautiful summary is D N. P , How fast
does a black hole radiate information?, International Journal of Modern Physics 3, pp. –
, , based on his earlier papers, such as Information in black hole radiation, Physical
Review Letters 71, pp. – , . See also his electronic preprint at http://www.arxiv.
org/abs/hep-th/
. Cited on page .
902 See D N. P , Average entropy of a subsystem, Physical Review Letters 71, pp. –
, . e entropy formula of this paper, used above, was proven by S.K. F
& S.
K
, Proof of Page’s conjecture on the average entropy of a subsystem, Physical Review
Letters 72, pp. – , . Cited on page .
903 e author speculates that this version of the coincidences could be original; he has not found it in the literature. Cited on page .
904 S
W
, Gravitation and Cosmology, Wiley,
and also page . Cited on page .
. See equation . . on page
905 It could be that knot theory provides a relation between local knot invariant and a global one. Cited on page .
906 R. L
& R.C. M , Gravity’s rainbow: limits for the applicability of the equi-
valence principle, Physical Review D 51, pp. – , , http://www.arxiv.org/abs/
hep-th/
. Cited on page .
907 M.Y . K
& V.V. F
, Scattering of scalar particles by a black hole, http://
www.arxiv.org/abs/gr-qc/
. See also M.Y . K
& V.V. F
, Re ec-
tion on event horizon and escape of particles from con nement inside black holes, http://
www.arxiv.org/abs/gr-qc/
. Cited on page .
908 E. J , Why do we observe a classical space-time?, Physics Letters A 116, pp. – , . Cited on page .
909 is point was made repeatedly by S
W
, namely in Derivation of gauge
invariance and the equivalence principle from Lorentz invariance of the S-matrix , Phys-
ics Letters 9, pp. – , , in Photons and gravitons in S-matrix theory: derivation of
charge conservation and equality of gravitational and inertial mass, Physical Review B 135,
pp. – , , and in Photons and gravitons in perturbation theory: derivation of
Maxwell’s and Einstein’s equations, Physical Review B 138, pp. – , . Cited on
page .
910 L.H. F & T.A. R
, Quantum eld theory constrains traversable wormhole geo-
metries, http://www.arxiv.org/abs/gr-qc/
or Physical Review D 53, pp. – ,
. Cited on page .
911 M.J. P
& L.H. F , e unphysical nature of ‘warp drive’, Classical and
Quantum Gravity 14, pp. – , . Cited on page .
912 B.S. K , M. R
& R.M. W , Quantum eld theory on spacetimes with a
compactly generated Cauchy horizon, http://www.arxiv.org/abs/gr-qc/
or Commu-
nications in Mathematical Physics 183 pp. – , . Cited on page .
913 An overview on squeezed light is given in ... No citations.
914 An introduction into the history of the concept of precision is given by N.W. W , e values of precision, Princeton University Press, . No citations.
915 E.H. K
, Zur Quantenmechanik einfacher Bewegungstypen, Zeitschri für Physik
44, pp. – , . e priority of the result is o en wrongly attributed to the much
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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later paper by H.P. R
, e uncertainty principle, Physical Review 34, pp. –
, . e complete result is by E
S
, Zum Heisenbergschem Un-
schärfeprinzip, Sitzungsberichte der Preussischen Akademie der Wissenscha en, Physikalisch-
Mathematische Klasse Berlin, pp. – , . No citations.
916 W. H
, Über den anschaulichen Inhalt der quantentheoretischen Kinematik
und Mechanik, Zeitschri für Physik 43, pp. – , . No citations.
917 M.G. R
, Uncertainty principle for joint measurement of noncommuting variables,
American Journal of Physics 62, pp. – , . No citations.
918 e question about the origin of blue colour in owers has been discussed already in the
beginning of the twentieth century. A short summary is given by K. Y
& al., Cause
of blue petal colour, Nature 373, p. , January . No citations.
919 e main reference on nondemolition measurements is the text by C.M. C
& P.D.
D
, Quantum limits on bosonic communication rates, Reviews of Modern Physics
66, pp. – , . No citations.
920 M P. S
, And Yet it Moves: Strange Systems and Subtle Questions in Physics,
Cambridge University Press, . It is a treasure chest for anybody interested in the details
of modern physics.
e rst of the two other books by M
P. S
, More than one Mystery,
Springer , talks about the Aharonov-Bohm e ect, about the EPR paradox and about the
Hanbury–Brown–Twiss experiments; his book Waves and Grains: Re ections on Light and
Learning, Princeton University Press, , completes his exposition of interesting thoughts
and insights about physics. No citations.
921 N. A
& D. M
, Solid State Physics, Saunders, , is probably the best
as well as the most original and entertaining introduction to the topic available in English
language. No citations.
922 e famous quantum theory book is P.A.M. D , Quantum Mechanics, th revised edition, Oxford University Press, . No citations.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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VIII
INSIDE THE NUCLEUS
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
N physics was born in
in France, but is now a small activity.
ot many researchers are working on the topic. It produced
ot more than one daughter, experimental high energy physics, which was born
around . But since , also these activities are in strong decline. Despite the short-
Ref. 923 ness, the family history is impressive; the two elds uncovered why stars shine, how
powerful bombs work, how cosmic evolution produced the atoms we are made of and
how medical doctors can dramatically improve their healing rate.
“ ” Nuclear physics is just low-density astrophysics.
A
–
Arguably, the most spectacular tool that physical research has produced in the twenti-
eth century was magnetic resonance imaging, or MRI for short. is technique allows to
image human bodies with a high resolution and with (almost) no damage, in strong con-
trast to X-ray imaging. ough the machines are still expensive – costing
euro
and more – there is hope that they will become cheaper in the future. Such a machine
consists essentially of a large magnetic coil, a radio transmitter and a computer. Some
results of putting part of a person into the coil are shown in Figure .
In these machines, a radio transmitter emits radio waves that are absorbed because
hydrogen nuclei are small spinning magnets. e magnets can be parallel or antiparallel
to the magnetic eld produced by the coil. e transition energy E can be absorbed from
a radio wave whose frequency ω is tuned to the magnetic eld B. e energy absorbed
by a single hydrogen nucleus is given by
E = ħω = ħγB
(614)
e material constant γ π has a value of . MHz T for hydrogen nuclei; it results from the non-vanishing spin of the proton. is is a quantum e ect, as stressed by the appearance of the quantum of action ħ. Using some cleverly applied magnetic elds, typically with a strength between . and . T, the machines are able to measure the absorption for each volume element separately. Interestingly, the precise absorption level depends on the chemical compound the nucleus is built into. us the absorption value will depend on
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 344 Sagittal images of the head and the spine – used with permission from Joseph P. Hornak, e Basics of MRI, http://www.cis.rit.edu/htbooks/mri, Copyright 2003
Ref. 924
the chemical environment. When the intensity of the absorption is plotted as grey scale, an image is formed that retraces the di erent chemical composition. Two examples are shown in Figure . Using additional tricks, modern machines can picture blood ow in the heart or air ow in lungs; they can even make lms of the heart beat. Other techniques show how the location of sugar metabolism in the brain depends on what you are thinking about.* In fact, also what you are thinking about all the time has been imaged: the rst image of people making love has been taken by Willibrord Weijmar Schultz and his group in . It is shown in Figure .
Each magnetic resonance image thus proves that atoms have spinning nuclei. Like for any other object, nuclei have size, colour, composition and interactions that ask to be explored.
T
Page 761
e magnetic resonance signal shows that hydrogen nuclei are quite sensitive to magnetic elds. e -factor of protons, de ned using the magnetic moment µ, their mass and charge as = µ m eħ, is about . . Using expression ( ) that relates the -factor and the radius of a composite object, we deduce that the radius of the proton is about . fm; this value is con rmed by experiment. Protons are thus much smaller than hydrogen atoms, the smallest of atoms, whose radius is about pm. In turn, the proton is the smallest of all nuclei; the largest nuclei have radii times the proton value.
* e website http://www.cis.rit.edu/htbooks/mri by Joseph P. Hornak gives an excellent introduction to magnetic resonance imaging, both in English and Russian, including the physical basis, the working of the machines, and numerous beautiful pictures. e method of studying nuclei by putting them at the same time into magnetic and radio elds is also called nuclear magnetic resonance.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
e small size of nuclei is no news. It is known since the beginning of the twentieth century. e story starts on the rst of March in , when Henri Becquerel* discovered a puzzling phenomenon: minerals of uranium potassium sulphate blacken photographic plates. Becquerel had heard that the material is strongly uorescent; he conjectured that uorescence might have some connection to the X-rays discovered by Conrad Röngten the year before. His conjecture was wrong; nevertheless it led him to an important new discovery. Investigating the reason for the e ect of uranium on photographic plates, Becquerel found that these minerals emit an undiscovered type of radiation, different from anything known at that time; in addition, the radiation is emitted by any substance containing uranium. In , Bémont named the property of these minerals radioactivity.
Radioactive rays are also emitted from many elements other than uranium. e radiation can be ‘seen’: it can be detected by the tiny
ashes of light that are emitted when the rays hit a scintillation screen. e light ashes are tiny even at a distance of several metre from the source; thus the rays must be emitted from point-like sources. Radioactivity has to be emitted from single atoms. us radioactivity con rmed unambiguously that atoms do exist. In fact, radioactivity even allows to count them, as we will nd out shortly.
e intensity of radioactivity cannot be in uenced by magnetic or electric elds; it does not depend on temperature or light irradiation. In short, radioactivity does not depend on electromagnetism and is not related to it. Also the high energy of the emitted radi- Henri Becquerel ation cannot be explained by electromagnetic e ects. Radioactivity must thus be due to another, new type of force. In fact, it took years and a dozen of Nobel Prizes to fully understand the details. It turns out that several types of radioactivity exist; the types behave di erently when they y through a magnetic eld or when they encounter matter. ey are listed in Table . All have been studied in great detail, with the aim to understand the nature of the emitted entity and its interaction with matter.
In , radioactivity inspired the year old physicist Ernest Rutherford,**who had won the Nobel Prize just the year before, to another of his brilliant experiments. He asked his collaborator Hans Geiger to take an emitter of alpha radiation – a type of radioactivity which Rutherford had identi ed and named years earlier – and to point the radiation at a thin metal foil. e quest was to nd out where the alpha rays would end up. e
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* Henri Becquerel (b. 1852 Paris, d. 1908 Le Croisic), important French physicist; his primary topic was the study of radioactivity. He was the thesis adviser of Marie Curie, the wife of Pierre Curie, and was central to bringing her to fame. e SI unit for radioactivity is named a er him. For his discovery of radioactivity he received the 1903 Nobel Prize for physics; he shared it with the Curies. ** Ernest Rutherford (1871–1937), important New Zealand physicist. He emigrated to Britain and became professor at the University of Manchester. He coined the terms alpha particle, beta particle, proton and neutron. A gi ed experimentalist, he discovered that radioactivity transmutes the elements, explained the nature of alpha rays, discovered the nucleus, measured its size and performed the rst nuclear reactions. Ironically, in 1908 he received the Nobel price for chemistry, much to the amusement of himself and of the world-wide physics community; this was necessary as it was impossible to give enough physics prizes to the numerous discoverers of the time. He founded a successful research school of nuclear physics and many famous physicists spent some time at his institute. Ever an experimentalist, Rutherford deeply disliked quantum theory, even though it was and is the only possible explanation for his discoveries.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 345 The origin of human life (© Willibrord Weijmar Schultz)
Challenge 1385 n
research group followed the path of the particles by using scintillation screens; later on
they used an invention by Charles Wilson: the cloud chamber. A cloud chamber, like its
successor, the bubble chamber, produces white traces along the path of charged particles;
the mechanism is the same as the one than leads to the white lines in the sky when an
aeroplane ies by.
e radiation detectors gave a strange result: most alpha
particles pass through the metal foil undisturbed, whereas
a few are re ected. In addition, those few which are re ec-
ted are not re ected by the surface, but in the inside of
the foil. (Can you imagine how they showed this?) Ruther-
ford deduced from this scattering experiment that rst of all,
atoms are mainly transparent. Only transparency explains
why most alpha particles pass the foil without disturbance,
even though it was over atoms thick. But some particles
were scattered by large angles or even re ected. Rutherford
showed that the re ections must be due to a single scattering
point. By counting the particles that were re ected (about
in
for his . µm gold foil), Rutherford was also able Marie Curie
to deduce the size of the re ecting entity and to estimate its
mass. He found that it contains almost all of the mass of the atom in a diameter of around
fm. He thus named it the nucleus. Using the knowledge that atoms contain electrons,
Rutherford then deduced from this experiment that atoms consist of an electron cloud
that determines the size of atoms – of the order of . nm – and of a tiny but heavy nucleus
at the centre. If an atom had the size of a basketball, its nucleus would have the size of a
dust particle, yet contain . % of the basketball’s mass. Atoms resemble thus candy oss
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TA B L E 66 The main types of radioactivity and rays emitted by matter
T
P -E
R
D-
U
S
α rays
helium
3 to MeV nuclei
U, U, Pu, a few cm in when any
thickness
Pu, Am air
eaten, material, measurement
inhaled, e.g. paper
touched
β rays 0 to MeV
electrons and
antineutrinos
C, K, H, Tc
< mm in metal
light years
serious none
metals none
cancer treatment
research
β+ rays
positrons and
neutrinos
K, C, C, N, less than β O
light years
medium any
tomography
material
none none
research
γ rays
high
Ag
energy
photons
several m in high air
thick lead preservation of herbs, disinfection
n reactions neutrons Cf, Po-Li
...
c. MeV
(α,n), Cl-Be
(γ,n)
high
. m of ...
para n
n emission neutrons He, N
...
typ. MeV
high
. m of ...
para n
p emission protons Be, Re typ. MeV
like α rays small solids
spontaneous nuclei
ssion typ. MeV
Cm, Rf
like α rays small
solids
detection of new elements
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 1386 e
around a heavy dust particle. Even though the candy oss – the electron cloud – around the nucleus is extremely thin and light, it is strong enough to avoid that two atoms interpenetrate; thus it keeps the neighbouring nuclei at constant distance. For the tiny and massive alpha however, particles the candy oss is essentially empty space, so that they simply y through the electron clouds until they exit on the other side or hit a nucleus.
e density of the nucleus is impressive: about . ë kg m . At that density, the mass of the Earth would t in a sphere of m radius and a grain of sand would have a mass larger than the largest existing oil tanker. (True?) Now we know that oil tankers are complex structures. What then is the structure of a nucleus?
I now know how an atom looks like!
“ ” Ernest Rutherford
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
N
Page 760 Challenge 1387 n
e magnetic resonance images also show that nuclei are composed. Images can be taken also using heavier nuclei instead of hydrogen, such as certain uorine or oxygen nuclei.
e -factors of these nuclei also depart from the value characteristic of point particles; the more massive they are, the bigger the departure. Such objects have a nite size; indeed, the size of nuclei can be measured directly and con rm the values predicted by the factor. Both the values of the -factor and the non-vanishing sizes show that nuclei are composed.
Interestingly, the idea that nuclei are composed is older than the concept of nucleus itself. Already in , a er the rst mass measurements of atoms by John Dalton and others, researchers noted that the mass of the various chemical elements seem to be almost perfect multiples of the weight of the hydrogen atom. William Prout then formulated the hypothesis that all elements are composed of hydrogen. When the nucleus was discovered, knowing that it contains almost all mass of the atom, it was therefore rst thought that all nuclei are made of hydrogen nuclei. Being at the origin of the list of constituents, the hydrogen nucleus was named proton, from the greek term for ‘ rst’ and reminding the name of Prout at the same time. Protons carry a positive unit of electric charge, just the opposite of that of electrons, but are almost times as heavy.
However, the charge and the mass numbers of the other nuclei do not match. On average, a nucleus that has n times the charge of a proton, has a mass that is about . n times than of the proton. Additional experiments then con rmed an idea formulated by Werner Heisenberg: all nuclei heavier than hydrogen nuclei are made of positively charged protons and neutral neutrons. Neutrons are particles a tiny bit more massive than protons (the di erence is less than a part in ), but without any electrical charge. Since the mass is almost the same, the mass of nuclei – and thus that of atoms – is still an (almost perfect) integer multiple of the proton mass. But since neutrons are neutral, the mass and the charge number of nuclei di er. Being neutral, neutrons do not leave tracks in clouds chambers and are more di cult to detect. For this reason, they were discovered much later than other subatomic particles.
Today it is possible to keep single neutrons suspended between suitably shaped coils, with the aid of te on ‘windows’. Such traps were proposed in by Wolfgang Paul. ey work because neutrons, though they have no charge, do have a small magnetic moment. (By the way, this implies that neutrons are composed of charged particles.) With a suitable arrangement of magnetic elds, neutrons can be kept in place, in other words, they can be levitated. Obviously, a trap only makes sense if the trapped particle can be observed. In case of neutrons, this is achieved by the radio waves absorbed when the magnetic moment switches direction with respect to an applied magnetic eld. e result of these experiments is simple: the lifetime of free neutrons is around ( ) s. Nevertheless, inside most nuclei we are made of, neutrons do not decay, as the result does not lead to a state of lower energy. (Why not?)
Magnetic resonance images also show that some elements have di erent types of atoms. ese elements have atoms that with the same number of protons, but with different numbers of neutrons. One says that these elements have several isotopes.* is
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* e name is derived from the Greek words for ‘same’ and ‘spot’, as the atoms are on the same spot in the periodic table of the elements.
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Half-life
> 10+15 s 10+10 s 10+7 s 10+5 s 10+4 s 10+3 s 10+2 s 10+1 s 10+0 s
unknown
10-1 s 10-2 s 10-3 s 10-4 s 10-5 s 10-6 s 10-7 s 10-15 s < 10-15 s
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Decay type
stable el. capture (beta+) beta - emission alpha emission proton emission neutron emission spontaneous fission unknown
F I G U R E 346 All known nuclides with their lifetimes and main decay modes (data from http://www.nndc.bnl.gov/nudat2)
also explains why some elements radiate with a mixture of di erent decay times. ough chemically they are (almost) indistinguishable, isotopes can di er strongly in their nuclear properties. Some elements, such as tin, caesium, or polonium, have over thirty isotopes each. Together, the about known elements have over nuclides.*
e motion of protons and neutrons inside nuclei allows to understand the spin and the magnetic moment of nuclei. Since nuclei are so extremely dense despite containing numerous positively charged protons, there must be a force that keeps everything together against the electrostatic repulsion. We saw that the force is not in uenced by electromagnetic or gravitational elds; it must be something di erent. e force must be short range; otherwise nuclei would not decay by emitting high energy alpha rays. e new force is called the strong nuclear interaction. We shall study it in detail shortly.
* Nuclides is the standard expression for a nucleus with a given number of neutrons and protons.
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metal wire (e.g. paper clip)
thin aluminium foils
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F I G U R E 347 An electroscope (or electrometer) (© Harald Chmela) and its charged (left) and uncharged state (right)
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Challenge 1388 e
In everyday life, nuclei are mostly found inside atoms. But in some situations, they move all by themselves. e rst to discover an example was Rutherford, who had shown that alpha particles are helium nuclei. Like all nuclei, alpha particles are small, so that they are quite useful as projectiles.
en, in , Viktor Heß* made a completely unexpected discovery. Heß was intrigued by electroscopes (also called electrometers). ese are the simplest possible detectors of electric charge.
ey mainly consist of two hanging, thin metal foils, such as two strips of aluminium foil taken from a chocolate bar. When the electroscope is charged, the strips repel each other and move apart, as shown in Figure . (You can build one easily yourself by covering an empty glass with some transparent cellophane foil and suspending a paper clip and the aluminium strips from the foil.) An electroscope thus measures electrical charge. Like many before him, Heß noted that even for a completely isolated electroscope, the charge disappears a er a while. He asked: why? By careful study he elim- Viktor Heß inated one explanation a er the other, he and others were le with only one possibility: that the discharge could be due to charged rays, such as those of the recently discovered radioactivity. He thus prepared a sensitive electrometer and took it with him on a balloon ight.
As expected, the balloon ight showed that the discharge e ect diminished with height, due to the larger distance from the radioactive substances on the Earth’s surface.
* Viktor Franz Heß, (1883–1964), Austrian nuclear physicist, received the Nobel Prize for physics in 1936 for his discovery of cosmic radiation. Heß was one of the pioneers of research into radioactivity. Heß’ discovery also explained why the atmosphere is always somewhat charged, a result important for the formation and behaviour of clouds. Twenty years a er the discovery of cosmic radiation, in 1932 Carl Anderson discovered the rst antiparticle, the positron, in cosmic radiation; in 1937 Seth Neddermeyer and Carl Anderson discovered the muon; in 1947 a team led by Cecil Powell discovered the pion; in 1951, the Λ and the kaon K are discovered. All discoveries used cosmic rays and most of these discoveries led to Nobel Prizes.
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TA B L E 67 The main types of cosmic radiation
P
E
O
D
S
At high altitude, the primary particles:
Protons (90 %)
to eV stars, supernovae, ex- scintillator tragalactic, unknown
in mines
Alpha rays (9 %) ...
...
...
...
Other nuclei, such to eV stars, novae
...
...
as iron (1 %)
Neutrinos
MeV, GeV
Sun, stars
chlorine,
none
gallium, water
Electrons (0.1 %)
to
eV supernova remnants
Gammas ( − )
eV to TeV stars, pulsars, galactic, semiconductor in mines
extragalactic
detectors
At sea level, secondary particles are produced in the atmosphere:
Muons
GeV, ms
Oxygen and other nuclei
protons hit atmosphere, dri chamber produce pions which decay into muons
Positrons
Neutrons
...
Pions
...
In addition, there are slowed down primary beam particles.
m of water or . m of soil
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But above about m of height, the discharge e ect increased again, and the higher he ew, the stronger it became. Risking his health and life, he continued upwards to more
than m; there the discharge e ect was several times stronger than on the surface of the Earth. is result is exactly what is expected from a radiation coming from outer space and absorbed by the atmosphere. In one of his most important ights, performed during an (almost total) solar eclipse, Heß showed that most of the ‘height radiation’ did not come from the Sun, but from further away. He – and Millikan – thus called the radiation cosmic rays. During the last few centuries, many people have drunk from a glass and eaten chocolate; but only Heß combined these activities with such careful observation and deduction that he earned a Nobel Prize.*
Today, the most impressive detectors for cosmic rays are Geiger–Müller counters and spark chambers. Both share the same idea; a high voltage is applied between two metal parts kept in a thin and suitably chosen gas (a wire and a cylindrical mesh for the GeigerMüller counter, two plates or wire meshes in the spark chambers). When a high energy
* In fact, Hess gold foils in his electrometer.
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Ref. 944 Page 938
ionizing particle crosses the counter, a spark is generated, which can either be observed
through the generated spark (as you can do yourself in the entrance hall of the CERN
main building), or detected by the sudden current ow. Historically, the current was rst
ampli ed and sent to a loudspeaker, so that the particles can be heard by a ‘click’ noise.
With a Geiger counter, one cannot see atoms or particles, but one can hear them. Finally,
ionized atoms could be counted. Finding the right gas mixture is tricky; it is the reason
that the counter has a double name. One needs a gas that extinguishes the spark a er a
while, to make the detector ready for the next particle. Müller was Geiger’s assistant; he
made the best counters by adding the right mixture of alcohol to the gas in the chamber.
Nasty rumours maintained that this was discovered when another assistant tried, without
success, to build counters while Müller was absent. When Müller, supposedly a heavy
drinker, came back, everything worked again. However, the story is apocryphal. Today,
Geiger–Müller counters are used around the world to detect radioactivity; the smallest
t in mobile phones and inside wrist watches.
e particle energy in cosmic rays spans a large range between eV and at least
eV; the latter is the same energy as a tennis ball a er serve. Understanding the origin
of cosmic rays is a science by its own. Some are galactic in origin, some are extragalactic.
For most energies, supernova remnants – pulsars and the like – seem the best candidates.
However, the source of the highest energy particles is still unknown.
In other words, cosmic rays are probably the only
type of radiation discovered without the help of shad-
ows. But shadows have been found later on. In a
beautiful experiment performed in , the shadow
thrown by the Moon on high energy cosmic rays
(about TeV) was studied. When the position of the
shadow is compared with the actual position of the
Moon, a shi is found. Due to the magnetic eld of
the Earth, the cosmic ray Moon shadow would be shif-
ted westwards for protons and eastwards for antipro-
Figure to be included
tons. e data are consistent with a ratio of antiprotons
between % and %. By studying the shadow, the
experiment thus showed that high energy cosmic rays F I G U RE 348 A Geiger–Müller
are mainly positively charged and thus consist mainly counter
of matter, and only in small part, if at all, of antimatter.
Detailed observations showed that cosmic rays arrive on the surface of the Earth as a
mixture of many types of particles, as shown in Table . ey arrive from outside the
atmosphere as a mixture of which the largest fraction are protons, alpha particles, iron
and other nuclei. Nuclei can thus travel alone over large distances. e number of charged
cosmic rays depends on their energy. At the lowest energies, charged cosmic rays hit the
human body many times a second. e measurements also show that the rays arrive in
irregular groups, called showers. e neutrino ux is many orders of magnitude higher,
but does not have any e ect on human bodies.
e distribution of the incoming direction of cosmic rays shows that many rays must
be extragalactic in origin. e typical nuclei of cosmic radiation are ejected from stars
and accelerated by supernova explosions. When they arrive on Earth, they interact with
the atmosphere before they reach the surface of the Earth. e detailed acceleration mech-
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F I G U R E 349 An aurora borealis produced by charged particles in the night sky
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anisms are still a topic of research. Cosmic rays have several e ects on everyday life. rough the charges they produce in
the atmosphere, they are probably responsible for the non-straight propagation of lightning. Cosmic rays are also important in the creation of rain drops and ice particles inside clouds, and thus indirectly in the charging of the clouds. Cosmic rays, together with ambient radioactivity, also start the Kelvin generator.
If the Moon would not exist, we would die from cosmic rays. e Moon helps to give the Earth a high magnetic eld via a dynamo e ect, which then diverts most rays towards the magnetic poles. Also the upper atmosphere helps animal life to survive, by shielding life from the harmful e ects of cosmic rays. Indeed, aeroplane pilots and airline employees have a strong radiation exposure that is not favourable to their health. Cosmic rays are one of several reasons that long space travel, such as a trip to mars, is not an option for humans. When cosmonauts get too much radiation exposure, the body weakens and eventually they die. Space heroes, including those of science ction, would not survive much longer than two or three years.
Cosmic rays also produce beautifully coloured ashes inside the eyes of cosmonauts; they regularly enjoy these events in their trips. But cosmic rays are not only dangerous and beautiful. ey are also useful. If cosmic rays would not exist at all, we would not exist either. Cosmic rays are responsible for mutations of life forms and thus are one of the causes of biological evolution. Today, this e ect is even used arti cially; putting cells into a radioactive environment yields new strains. Breeders regularly derive new mutants in this way.
Cosmic rays cannot be seen directly, but their cousins, the ‘solar’ rays, can. is is most spectacular when they arrive in high numbers. In such cases, the particles are inevitably deviated to the poles by the magnetic eld of the Earth and form a so-called aurora borealis (at the North Pole) or an aurora australis (at the South pole). ese slowly mov-
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F I G U R E 350 An aurora australis on Earth seen from space (in the X-ray domain) and one on Saturn
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Page 602
ing and variously coloured curtains of light belong to the most spectacular e ects in the night sky. Visible light and X-rays are emitted at altitudes between and km. Seen from space, the aurora curtains typically form a circle with a few thousand kilometres diameter around the magnetic poles.*
Cosmic rays are mainly free nuclei. With time, researchers found that nuclei appear without electron clouds also in other situations. In fact, the vast majority of nuclei in the universe have no electron clouds at all: in the inside of stars no nucleus is surrounded by bound electrons; similarly, a large part of intergalactic matter is made of protons. It is known today that most of the matter in the universe is found as protons or alpha particles inside stars and as thin gas between the galaxies. In other words, in contrast to what the Greeks said, matter is not usually made of atoms; it is mostly made of nuclei. Our everyday environment is an exception when seen on cosmic scales. In nature, atoms are rare.
By the way, nuclei are in no way forced to move; nuclei can also be stored with almost no motion. ere are methods – now commonly used in research groups – to superpose electric and magnetic elds in such a way that a single nucleus can be kept oating in mid-air; we discussed this possibility in the section on levitation earlier on.
N
Not all nuclei are stable over time. e rst measurement that provided a hint was the way radioactivity changes with time. e number N of atoms decreases with time. More precisely, radioactivity follows an exponential decay:
N(t) = N( )e−t τ
(615)
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e parameter τ, the so-called life time, depends on the type of nucleus emitting the rays. It can vary from much less than a microsecond to millions of millions of years. e expression has been checked for as long as multiples of the duration τ; its validity and precision is well-established by experiments. Radioactivity is the decay of unstable nuclei. Formula ( ) is an approximation for large numbers of atoms, as it assumes that N(t) is a continuous variable. Despite this approximation, deriving this expression from quantum theory is not a simple exercise, as we saw in the section on atomic physics. ough the
* In the solar system, aurorae due to core magnetic elds have been observed on Jupiter, Saturn, Uranus,
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quantum Zeno e ect can appear for small times t, for the case of radioactivity it has not been observed so far.
Most of all, the expression ( ) allows to count the number of atoms in a given mass of material. Imagine to have measured the mass of radioactive material at the beginning of your experiment; you have chosen an element that has a lifetime of about a day. en you put the material inside a scintillation box. A er a few weeks the number of ashes has become so low that you can count them; using the formula you can then determine how many atoms have been in the mass to begin with. Radioactivity thus allows us to determine the number of atoms, and thus their size, in addition to the size of nuclei.
e decay ( ) and the release of energy is typical of metastable systems. In , Rutherford and Soddy discovered what the state of lower energy is for alpha and beta emitters. In these cases, radioactivity changes the emitting atom; it is a spontaneous transmutation of the atom. An atom emitting alpha or beta rays changes its chemical nature. Radioactivity con rms what statistical mechanics of gases had concluded long time before: atoms have a structure that can change. In alpha decay, the radiating nucleus emits a (doubly charged) helium nucleus. e kinetic energy is typically a handful of MeV. A er the emission, the nucleus has changed to one situated two places earlier in the periodic system of the elements.
In beta decay, a neutron transforms itself into a proton, emitting an electron and an antineutrino. Also beta decay changes the chemical nature of the atom, but to the place following the original atom in the periodic table of the elements. A variation is the beta+ decay, in which a proton changes into a neutron and emits a neutrino and a positron. We will study these important decay processes below.
In gamma decay, the nucleus changes from an excited to a lower energy state by emitting a high energy photon. In this case, the chemical nature is not changed. Typical energies are in the MeV range. Due to the high energy, such rays ionize the material they encounter; since they are not charged, they are not well absorbed by matter and penetrate deep into materials. Gamma radiation is thus by far the most dangerous type of (outside) radioactivity.
By the way, in every human body about nine thousand radioactive decays take place every second, mainly . kBq ( . mSv a) from K and kBq from C ( . mSv a). Why is this not dangerous?
All radioactivity is accompanied by emission of energy. e energy emitted by an atom trough radioactive decay or reactions is regularly a million time large than that emitted by a chemical process. at is the reason for the danger of nuclear weapons. More than a decay, a radioactive process is thus an explosion.
What distinguishes those atoms that decay from those which do not? An exponential decay law implies that the probability of decay is independent of the age of the atom. Age or time plays no role. We also know from thermodynamics, that all atoms are exactly identical. So how is the decaying atom singled out? It took around years to discover that decays are triggered by the statistical uctuations of the vacuum, as described by quantum theory. Indeed, radioactivity is one of the clearest observations that classical physics is not su cient to describe nature. Radioactivity, like all decays, is a pure quantum e ect. Only a nite quantum of action makes it possible that a system remains unchanged
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Neptune, Earth, Io and Ganymede. Aurorae due to other mechanisms have been seen on Venus and Mars.
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Challenge 1390 ny Page 919
Ref. 926
until it suddenly decays. Indeed, in George Gamow explained alpha decay with the tunnelling e ect. e tunnelling e ect explains the relation between the lifetime and the range of the rays, as well as the measured variation of lifetimes – between ns and years – as the consequence of the varying potentials to be overcome.
By the way, massless particles cannot decay. ere is a simple reason for it: massless particles do not experience time, as their paths are null. A particle that does not experience time cannot have a half-life. (Can you nd another argument?)
As a result of the chemical e ects of radioactivity, the composition ratio of certain elements in minerals allows to determine the age of the mineral. Using radioactive decay to deduce the age of a sample is called radiometric dating. With this technique, geologists determined the age of mountains, the age of sediments and the age of the continents.
ey determined the time that continents moved apart, the time that mountains formed when the continents collided and the time when igneous rocks were formed. e times found in this way are consistent with the relative time scale that geologists had de ned independently for centuries before the technique appeared. With the appearance of radiometric dating, all fell into place. Equally successful was the radiocarbon method; with it, historians determined the age of civilizations and the age of human artefacts.* Many false beliefs were shattered. In some communities the shock is still not over, even though over hundred years have passed since these results became known.
With the advent of radiometric dating, for the rst time it became possible to reliably date the age of rocks, to compare it with the age of meteorites and, when space travel became fashionable, with the age of the Moon. e result of the eld of radiometric dating was beyond all estimates and expectations: the oldest rocks and the oldest meteorites studied independently using di erent dating methods, are ( ) million years old. But if the Earth is that old, why did the Earth not cool down in its core in the meantime?
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Challenge 1391 ny
Ref. 925 Page 920
e lava seas and streams found in and around volcanoes are the origin of the images that many cultures ascribe to hell: re and su ering. Because of the high temperature of lava, hell is inevitably depicted as a hot place. A striking example is the volcano Erta Ale, shown in Figure . But why is lava still hot, a er million years?
A straightforward calculation shows that if the Earth had been a hot sphere in the beginning, it should have cooled down and solidi ed already long time ago. e Earth should be a solid object, like the moon: the Earth should not contain any lava and hell would not be hot.
e solution to the riddle is provided by radioactivity: the centre of the Earth contains an oven fuelled by radioactive potassium K, radioactive uranium U and U and radioactive thorium . e radioactivity of these elements, and to minor degree a few others, keeps the centre of the Earth glowing. More precise investigations, taking into account the decay times and material concentrations, show that this mechanism indeed explains the internal heat of the Earth. (By the way, the decay of potassium is the origin for the % of argon found in the Earth’s atmosphere.)
* In 1960, the developer of the radiocarbon dating technique, Willard Libby, received the Nobel Prize for chemistry.
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F I G U R E 351 The lava sea in the volcano Erta Ale in Ethiopia (© Marco Fulle)
is brings up a challenge: why is the radioactivity of lava and of the Earth in general Challenge 1392 n not dangerous to humans?
N Nuclei are highly unstable when they contain more than about nucleons. Higher mass values inevitably decay into smaller fragments. But when the mass is above nucleons, nuclear composites are stable again: such systems are then called neutron stars. is is the most extreme example of pure nuclear matter found in nature. Neutron stars are le overs of (type II) supernova explosions. ey do not run any fusion reactions any more, as other stars do; in rst approximation they are simply a large nucleus.
Neutron stars are made of degenerate matter. eir density of kg m is a few times that of a nucleus, as gravity compresses the star. is density value means that tea spoon of such a star has a mass of several million tons. Neutron stars are about km in diameter. ey are never much smaller, as such stars are unstable. ey are never much larger, because more massive neutron stars turn into black holes. N In everyday life, the colour of objects is determined by the wavelength of light that is least absorbed, or if they shine, by the wavelength that is emitted. Also nuclei can absorb photons of suitably tuned energies and get into an excited state. In this case, the photon energy is converted into a higher energy of one or several of the nucleons whirling around inside the nucleus. Many radioactive nuclei also emit high energy photons, which then are called gamma rays, in the range of keV (or . fJ) to about MeV (or . pJ). e process is similar to the emission of light by electrons in atoms. From the energy, the
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F I G U R E 352 Various nuclear shapes – fixed (left) and oscillating (right), shown realistically as clouds (above) and simplified as geometric shapes (below)
Ref. 939 Ref. 940
number and the lifetime of the excited states – they range from ps to d – researchers can deduce how the nucleons move inside the nucleus.
e photon energies de ne the ‘colour’ of the nucleus. It can be used, like all colours, to distinguish nuclei from each other and to study their motion. in particular, the colour of the γ-rays emitted by excited nuclei can be used to determine the chemical composition of a piece of matter. Some of these transition lines are so narrow that they can been used to study the change due to the chemical environment of the nucleus, to measure their motion or to detect the gravitational Doppler e ect.
e study of γ-rays also allows to determine the shape of nuclei. Many nuclei are spherical; but many are prolate or oblate ellipsoids. Ellipsoids are favoured if the reduction in average electrostatic repulsion is larger than the increase in surface energy. All nuclei – except the lightest ones such as helium, lithium and beryllium – have a constant mass density at their centre, given by about . fermions per fm , and a skin thickness of about . fm, where their density decreases. Nuclei are thus small clouds, as shown in Figure .
We know that molecules can be of extremely involved shape. In contrast, nuclei are mostly spheres, ellipsoids or small variations of these. e reason is the short range, or better, the fast spatial decay of nuclear interactions. To get interesting shapes like in molecules, one needs, apart from nearest neighbour interactions, also next neighbour interactions and next next neighbour interactions. e strong nuclear interaction is too short ranged to make this possible. Or does it? It might be that future studies will discover that some nuclei are of more unusual shape, such as smoothed pyramids. Some predictions have been made in this direction; however, the experiments have not been performed yet.
e shape of nuclei does not have to be xed; nuclei can also oscillate in shape. Such oscillations have been studied in great detail. e two simplest cases, the quadrupole and octupole oscillations, are shown in Figure . Obviously, nuclei can also rotate. Rapidly spinning nuclei, with a spin of up to ħ and more, exist. ey usually slow down step by step, emitting a photon and reducing their angular momentum at each step. Recently it was discovered that nuclei can also have bulges that rotate around a xed core, a bit like tides rotate around the Earth.
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Nuclei are small because the nuclear interactions are short-ranged. Due to this short range, nuclear interactions play a role only in types of motion: scattering, bound motion, decay and a combination of these three called nuclear reactions. e history of nuclear physics showed that the whole range of observed phenomena can be reduced to these four fundamental processes. In each motion type, the main interest is what happens at the start and at the end; the intermediate situations are less interesting. Nuclear interactions thus lack the complex types of motion which characterize everyday life. at is the reason for the shortness of this chapter.
Scattering is performed in all accelerator experiments. Such experiments repeat for nuclei what we do when we look at an object. Seeing is a scattering process, as seeing is the detection of scattered light. Scattering of X-rays was used to see atoms for the rst time; scattering of high energy alpha particles was used to discover and study the nucleus, and later the scattering of electrons with even higher energy was used to discover and study the components of the proton.
Bound motion is the motion of protons and neutrons inside nuclei or the motion of quarks inside hadrons. Bound motion determines shape and shape changes of compounds.
Decay is obviously the basis of radioactivity. Decay can be due to the electromagnetic, the strong or the weak nuclear interaction. Decay allows to study the conserved quantities of nuclear interactions.
Nuclear reactions are combinations of scattering, decay and possibly bound motion. Nuclear reactions are for nuclei what the touching of objects is in everyday life. Touching an object we can take it apart, break it, solder two objects together, throw it away, and much more. e same can be done with nuclei. In particular, nuclear reactions are responsible for the burning of the Sun and the other stars; they also tell the history of the nuclei inside our bodies.
Quantum theory showed that all four types of motion can be described in the same way. Each type of motion is due to the exchange of virtual particles. For example, scattering due to charge repulsion is due to exchange of virtual photons, the bound motion inside nuclei due to the strong nuclear interaction is due to exchange of virtual gluons, beta decay is due to the exchange of virtual W bosons, and neutrino reactions are due to the exchange of virtual Z bosons. e rest of this chapter explains these mechanisms in more details.
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e rst man who thought to have made transuranic elements, the Italian genius Enrico Fermi, received the Nobel Prize for the discovery. Shortly a erwards, Otto Hahn and his collaborators Lise Meitner and Fritz Strassman showed that Fermi was wrong, and that his prize was based on a mistake. Fermi was allowed to keep his prize, the Nobel committee gave Hahn the Nobel Prize as well, and to make the matter unclear to everybody and to women physicists in particular, the prize was not given to Lise Meitner. (A er her death, a new element was named a er her.)
When protons or neutrons were shot into nuclei, they usually remained stuck inside them, and usually lead to the transformation of an element into a heavier one. A er hav-
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•
ing done this with all elements, Fermi used uranium; he found that bombarding it with neutrons, a new element appeared, and concluded that he had created a transuranic element. Alas, Hahn and his collaborators found that the element formed was well-known: it was barium, a nucleus with less than half the mass of uranium. Instead of remaining stuck as in the previous elements, the neutrons had split the uranium nucleus. Hahn, Meitner and Strassmann had observed reactions such as:
U + n Ba + Kr + n + MeV .
(616)
Meitner called the splitting process nuclear ssion. A large amount of energy is liberated in ssion. In addition, several neutrons are emitted; they can thus start a chain reaction. Later, and (of course) against the will of the team, the discovery would be used to make nuclear bombs.
Reactions and decays are transformations. In each transformation, already the Greek taught us to search, rst of all, for conserved quantities. Besides the well-known cases of energy, momentum, electric charge and angular momentum conservation, the results of nuclear physics lead to several new conserved quantities. e behaviour is quite constrained. Quantum eld theory implies that particles and antiparticles (commonly denoted by a bar) must behave in compatible ways. Both experiment and quantum eld theory show for example that every reaction of the type A + B C + D implies that the reactions A + C B + D or C + D A + B or, if energy is su cient, A C + D + B, are also possible. Particles thus behave like conserved mathematical entities.
Experiments show that antineutrinos di er from neutrinos. In fact, all reactions conrm that the so-called lepton number is conserved in nature. e lepton number L is zero for nucleons or quarks, is for the electron and the neutrino, and is − for the positron and the antineutrino.
In addition, all reactions conserve the so-called baryon number. e baryon number B is for protons and neutrons (and / for quarks), and − for antiprotons and antineutrons (and thus − for antiquarks). So far, no process with baryon number violation has ever been observed. Baryon conservation is one reason for the danger of radioactivity, ssion and fusion.
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B
Uranium ssion is triggered by a neutron, liberates energy and produces several additional neutrons. It can trigger a chain reaction which can lead to an explosion or a controlled generation of heat. Once upon a time, in the middle of the twentieth century, these processes were studied by quite a number of researchers. Most of them were interested in making weapons or in using nuclear energy, despite the high toll these activities place on the economy, on human health and on the environment.
Most stories around this topic are absurd. e rst nuclear weapons were built during the second world war with the smartest physicists that could be found. Everything was ready, including the most complex physical models, factories and an organization of incredible size. ere was just one little problem: there was no uranium of su cient quality.
e mighty United States thus had to go around the world to shop for good uranium. ey found it in the Belgian colony of Congo, in central Africa. In short, without the sup-
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Ref. 943
port of Belgium, which sold the Congolese uranium to the USA, there would have been
no nuclear bomb, no early war end and no superpower status.
Congo paid a high price for this important status. It was ruled by a long chain of milit-
ary dictators up to this day. But the highest price was paid by the countries that actually
built nuclear weapons. Some went bankrupt, others remained underdeveloped, still other
countries have amassed huge debts and have a large underprivileged population. ere is
no exception. e price of nuclear weapons has also been that some regions of our planet
became uninhabitable, such as numerous islands, deserts and marine environments. But
it could have been worse. When the most violent physicist ever, Edward Teller, made his
rst calculations about the hydrogen bomb, he predicted that the bomb would set the
atmosphere into re. Nobel Prize winner Hans Bethe corrected the mistake and showed
that nothing of this sort would happen. Nevertheless, the military preferred to explode
the hydrogen bomb in the Bikini atoll, the most distant place from their homeland they
could nd. Today it is even dangerous simply to y over that island. It was them noticed
that nuclear test explosions increased ambient radioactivity in the atmosphere all over
the world. Of the produced radioactive elements, H is absorbed by humans in drinking
water, C and Sr through food, and Cs in both ways. In the meantime, all countries
have agreed to perform their nuclear tests underground.
But even peaceful nuclear reactors are dangerous. e reason was discovered in
by Frédéric Joliot and his wife Irène, the daughter of Pierre and Marie Curie: arti cial
radioactivity. e Joliot–Curies discovered that materials irradiated by alpha rays become
radioactive in turn. ey found that alpha rays transformed aluminium into radioactive
phosphorous:
Al + α P .
(617)
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In fact, almost all materials become radioactive when irradiated with alpha particles, neutrons or gamma rays. As a result, radioactivity itself can only be contained with di culty. A er a time which depends on the material and the radiation, the box that contains radioactive material has itself become radioactive.
e dangers of natural and arti cial radioactivity are the reason for the high costs of nuclear reactors. A er about thirty years of operation, reactors have to be dismantled.
e radioactive pieces have to be stored in specially chosen, inaccessible places, and at the same time the workers’ health must not be put in danger. e world over, many dismantlings are now imminent. e companies performing the job sell the service at high price. All operate in a region not far from the border to criminal activity, and since radioactivity cannot be detected by the human senses, many crossed it. In fact, an important nuclear reactor is (usually) not dangerous to humans: the Sun.
TS
Nuclear physics is the most violent part of physics. But despite this bad image, nuclear physics has something to o er which is deeply fascinating: the understanding of the Sun, the stars and the early universe.
e Sun emits YW of light. Where does this energy come from? If it came by burning coal, the Sun would stop burning a er a few thousands of years. When radioactivity was discovered, researchers tested the possibility that this process was at the heart of the
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TA B L E 68 Some radioactivity measurements
M
A
B/
air sea water human body cow milk pure U metal highly radioactive α emitters radiocarbon: C (β emitter) highly radioactive β and γ emitters main nuclear fallout: Cs, Sr (α emitter) polonium, one of the most radioactive materials (α)
c. − c. max. c.
ë
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Sun’s shining. However, even though radioactivity can produce more energy than chem-
ical burning, the composition of the Sun – mostly hydrogen and helium – makes this
impossible. In fact, the study of nuclei showed that the Sun burns by hydrogen fusion.
Fusion is the composition of a large nucleus from smaller ones. In the Sun, the fusion
reaction
H He + e+ + ν + . pJ
(618)
is the result of a continuous cycle of three separate nuclear reactions:
H+ H H+ H He + He
H + e+ + ν(a weak nuclear reaction) He + γ(a strong nuclear reaction) He + H + γ .
(619)
Ref. 941
In total, four protons are thus fused to one helium nucleus; if we include the electrons, four hydrogen atoms are fused to one helium atom with the emission of neutrinos and light with a total energy of . pJ ( . MeV). Most of the energy is emitted as light; around % is carried away by neutrinos. e rst of the three reaction of equation is due to the weak nuclear interaction; this avoids that it happens too rapidly and ensures that the Sun will shine still for some time. Indeed, in the Sun, with a luminosity of YW, there are thus about fusions per second. is allows to deduce that the Sun will last another handful of Ga (Gigayears) before it runs out of fuel.
e fusion reaction ( ) takes place in the centre of the Sun. e energy carried away by the photons arrives at the Sun’s surface about two hundred thousand years later; this delay is due to the repeated scattering of the photon by the constituents inside the Sun. A er two-hundred thousand years, the photons take another . minutes to reach the Earth and to sustain the life of all plants and animals.
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TA B L E 69 Human exposure to radioactivity and the corresponding doses
E
D
Daily human exposure:
Average exposure to cosmic radiation in Europe, at sea levelc. . mSv a ( . mSv a) ( km)
Average (and maximum) exposure to soil radiation, without . mSv a ( mSv a)
radon
Average (and maximum) inhalation of radon
mSv a ( mSv a)
Average exposure due to internal radionuclides
. mSv a
natural content of K in human muscles
− Gy and
Bq
natural content of Ra in human bones
ë − Gy and
Bq
natural content of C in humans
− Gy
Total average (and maximum) human exposure
mSv a ( mSv a)
Common situations:
Dental X-ray Lung X-ray Short one hour ight (see http://www.gsf.de/epcard) Transatlantic ight Maximum allowed dose at work
c. mSv equivalent dose c. . mSv equivalent dose c. µSv c. . mSv
mSv a
Deadly exposures: Ionization Dose
Equivalent dose
. C kg can be deadly Gy= J kg is deadly in 1 to 3
days more than Sv a leads to death
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C
It is still not clear whether the radiation of the Sun is constant over long time scales. ere is an 11 year periodicity, the famous solar cycle, but the long term trend is still unknown. Precise measurements cover only the years from 1978 onwards, which makes only about 3 cycles. A possible variation of the solar constant might have important consequences for climate research; however, the issue is still open.
Ref. 937
**
Not all γ-rays are due to radioactivity. In the year 2000, an Italian group discovered that thunderstorms also emit γ-rays, of energies up to MeV. e mechanisms are still being investigated.
**
Chain reactions are quite common in nature. Fire is a chemical chain reaction, as are exploding reworks. In both cases, material needs heat to burn; this heat is supplied by a
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•
neighbouring region that is already burning.
**
Radioactivity can be extremely dangerous to humans. e best example is plutonium. Only µg of this alpha emitter inside the human body are su cient to cause lung cancer.
**
Lead is slightly radioactive, because it contains the Pb isotope, a beta emitter. is lead isotope is produced by the uranium and thorium contained in the rock from where the lead is extracted. For sensitive experiments, such as for neutrino experiments, one needs radioactivity shields. e best shield material is lead, but obviously it has to be low radioactivity lead. Since the isotope Pb has a half-life of 22 years, one way to do it is to use old lead. In a precision neutrino experiment in the Gran Sasso in Italy, the research team uses lead mined during Roman times in order to reduce spurious signals.
**
Not all reactors are human made. Natural reactors have been predicted in 1956 by Paul Kuroda. In 1972 the rst example was found. In Oklo, in the African country of Gabon, there is a now famous geological formation where uranium is so common that two thousand million years ago a natural nuclear reactor has formed spontaneously – albeit a small one, with an estimated power generation of kW. It has been burning for over 150 000 years, during the time when the uranium 235 percentage was 3% or more, as required for chain reaction. (Nowadays, the uranium 235 content on Earth is 0.7%.) e water of a nearby river was periodically heated to steam during an estimated 30 minutes; then the reactor cooled down again for an estimated 2.5 hours, since water is necessary to moderate the neutrons and sustain the chain reaction. e system has been studied in great detail, from its geological history up to the statements it makes about the constancy of the ‘laws’ of nature. e studies showed that 2000 million years ago the mechanisms were the same as those used today.
Challenge 1393 ny
**
High energy radiation is dangerous to humans. In the 1950s, when nuclear tests were still made above ground by the large armies in the world, the generals overruled the orders of the medical doctors. ey positioned many soldiers nearby to watch the explosion, and worse, even ordered them to walk to the explosion site as soon as possible a er the explosion. One does not need to comment on the orders of these generals. Several of these unlucky soldiers made a strange observation: during the ash of the explosion, they were able to see the bones in their own hand and arms. How can this be?
**
e SI units for radioactivity are now common; in the old days, Sv was called rem or ‘Röntgen equivalent man’; e SI unit for dose, Gy = J kg, replaces what used to be called rd or Rad. e SI unit for exposition, C kg, replaces the older unit ‘Röntgen’, for which the relation is R = . ë − C kg.
**
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Ref. 938
Nuclear bombs are terrible weapons. To experience their violence but also the criminal actions of many military people during the tests, have a look at the pictures of explosions. In the 1950 and 60s, nuclear tests were performed by generals who refused to listen to doctors and scientists. Generals ordered to explode these weapons in the air, making the complete atmosphere of the world radioactive, hurting all mankind in doing so; worse, they even obliged soldiers to visit the radioactive explosion site a few minutes a er the explosion, thus doing their best to let their own soldiers die from cancer and leukaemia.
Ref. 926 Challenge 1394 n
**
e technique of radiometric dating has deeply impacted astronomy, geology, evolutionary biology, archaeology and history. (And it has reduced the number of violent believers.) Half-lives can usually be measured to within one or two percent of accuracy, and they are known both experimentally and theoretically not to change over geological time scales. As a result, radiometric dating methods can be be surprisingly precise. But how does one measure half-lives of thousands of millions of years to high precision?
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e beta decay of the radioactive carbon isotope C has a decay time of a. is isotope is continually created in the atmosphere through the in uence of cosmic rays.
is happens through the reaction N + n p + C. As a result, the concentration of radiocarbon in air is relatively constant over time. Inside living plants, the metabolism thus (unknowingly) maintains the same concentration. In dead plants, the decay sets in.
e decay time of a few thousand years is particularly useful to date historic material. e method, called radiocarbon dating, has been used to determine the age of mummies, the age of prehistoric tools and the age of religious relics. e original version of the technique measured the radiocarbon content through its radioactive decay and the scintillations it produced. A quality jump was achieved when accelerator mass spectroscopy became commonplace. It was not necessary any more to wait for decays: it is now possible to determine the C content directly. As a result, only a tiny amount of carbon, as low as
. mg, is necessary for a precise dating. is technique showed that numerous religious relics are forgeries, such as a cloth in Turin, and several of their wardens turned out to be crooks.
**
Researchers have developed an additional method to date stones using radioactivity. Whenever an alpha ray is emitted, the emitting atom gets a recoil. If the atom is part of a crystal, the crystal is damaged by the recoil. e damage can be seen under the microscope. By counting the damaged regions it is possible to date the time at which rocks have been crystallized. In this way it has been possible to determine when material from volcanic eruptions has become rock.
**
Several methods to date wine are used, and more are in development. A few are given in Table 70.
**
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•
TA B L E 70 Natural isotopes used in radiometric dating
I
D
H-
M
E
Sm
Nd
Rb
Sr
Rh
Os
Lu
Hf
Pb
U
Pb
K
Ar
U
Pb
Be
B
Fe
...
Cl
Ar
U
Ra
C
N
Cs
Pb H
Ga . Ga Ga Ga Ga . Ga . Ga
. Ga . Ma . Ma . Ma
ka , ka
a a
a .a
samarium–neodynium method
rubidium–strontium method
rhenium–osmium method
lutetium–hafnium method
thorium–lead method, lead–lead method
uranium–lead method, lead–lead method
potassium–argon method, argon–argon method
uranium–lead method, lead–lead method
cosmogenic radiometric dating
supernova debris dating
cosmogenic radiometric dating
uranium–thorium method
rocks, lunar soil, meteorites rocks, lunar soil, meteorites rocks, lunar soil, meteorites rocks, lunar soil, meteorites rocks, lunar soil, meteorites rocks, lunar soil, meteorites rocks, lunar soil, meteorites
rocks, lunar soil, meteorites ice cores
deep sea crust ice cores
corals, stalactites, bones, teeth
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radiocarbon method gamma-ray counting
gamma-ray counting gamma-ray counting
wood, clothing, bones, organic material, wine
dating food and wine a er Chernobyl nuclear accident
dating wine
dating wine
Selected radioactive decay times can be changed by external in uence. Electron capture, as observed in beryllium-7, is one of the rare examples were the decay time can change, by up to 1.5%, depending on the chemical environment. e decay time for the same isotope has also been found to change by a fraction of a percent under pressures of GPa. On the other hand, these e ects are predicted (and measured) to be negligible for nuclei of larger mass.
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.
Lernen ist Vorfreude auf sich selbst.*
“ ” Peter Sloterdijk
Since protons are positively charged, inside nuclei they must be bound by a force strong enough to keep them together against their electromagnetic repulsion. is is the strong nuclear interaction. Most of all, the strong interaction tells a good story about the stu we are made of.
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Page 916 Ref. 927
Don’t the stars shine beautifully? I am the only person in the world who knows why they do.
“ Frits Houtermans ” All stars shine because of fusion. When two light nuclei are fused to a heavier one, some
energy is set free, as the average nucleon is bound more strongly. is energy gain is
possible until the nuclei of iron Fe are made. For nuclei beyond this nucleus, the binding
energies per nucleon then decrease again; thus fusion is not energetically possible.**
e di erent stars observed in the sky*** can be distinguished by the type of fusion
nuclear reaction that dominates. Most stars, in particular young or light stars run hy-
drogen fusion. In fact, there are at least two main types of hydrogen fusion: the direct
hydrogen-hydrogen (p-p) cycle and the CNO cycle(s).
e hydrogen cycle described above is the main energy source of the Sun. e simple
description does not fully purvey the fascination of the process. On average, protons in
the Sun’s centre move with km s. Only if they hit each other precisely head-on can
a nuclear reaction occur; in all other cases, the electrostatic repulsion between the pro-
tons keeps them apart. For an average proton, a head-on collision happens once every
thousand million years. Nevertheless, there are so many proton collisions in the Sun that
every second four million tons of hydrogen are burned to helium.
Fortunately for us, the photons generated in the Sun’s centre are ‘slowed’ down by the
outer parts of the Sun. In this process, gamma photons are progressively converted to
visible photons. As a result, the sunlight of today was in fact generated at the time of the
Neandertalers: a typical estimate is about
years ago. In other words, the e ective
speed of light right at the centre of the Sun is estimated to be around km year.
If a star has heavier elements inside it, the hydrogen fusion uses these elements as
* ‘Learning is anticipated joy about yourself.’ ** us ssion becomes interesting as energy source for heavy nuclei. *** For the stars above you, see the http://me.in-berlin-de/~jd/himmel/himmel.00.11.html website.
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F I G U R E 353 Photographs of the Sun at wavelengths of 30.4 nm (in the extreme ultraviolet, left) and around 677 nm (visible light, right, at a different date), by the SOHO mission (ESA and NASA)
catalysts. is happens through the so-called CNO cycle, which runs as
C+ H N
C+ H N+ H
O N+ H
N+γ C + e+ + ν N+γ O+γ N + e+ + ν C + He
(620)
Challenge 1395 n
e end result of the cycle is the same as that of the hydrogen cycle, both in nuclei and in energy. e CNO cycle is faster than hydrogen fusion, but requires higher temperatures, as the protons must overcome a higher energy barrier before reacting with carbon or nitrogen than when they react with another proton. (Why?) Due to the comparatively low temperature of a few tens of million kelvin inside the Sun, the CNO cycle is less important than the hydrogen cycle. ( is is also the case for the other CNO cycles that exist.) ese studies also explain why the Sun does not collapse. e Sun is a ball of hot gas, and the high temperature of its constituents prevents their concentration into a small volume. For some stars, the radiation pressure of the emitted photons prevents collapse; for others it is the Pauli pressure; for the Sun, like for the majority of stars, it is the usual thermal motion of the gas.
e nuclear reaction rates at the interior of a star are extremely sensitive to temperature. e carbon cycle reaction rate is proportional to between T for hot massive O stars and T for stars like the Sun. In red giants and supergiants, the triple alpha reaction rate is proportional to T ; these strong dependencies imply that stars shine with constancy over medium times, since any change in temperature would be damped by a very e cient feedback mechanism. (Of course, there are exceptions: variable stars get brighter
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and darker with periods of a few days; and the Sun shows small oscillations in the minute range.)
How can the Sun’s surface have a temperature of K, whereas the corona around it, the thin gas emanating from the Sun, reaches two million Kelvin? In the latter part of the twentieth century it was shown, using satellites, that the magnetic eld of the Sun is the cause; through the violent ows in the Sun’s matter, magnetic energy is transferred to the corona in those places were ux tubes form knots, above the bright spots in the le of Figure or above the dark spots in the right photograph. As a result, the particles of the corona are accelerated and heat the whole corona.
When the Sun erupts, as shown in the lower le corner in Figure , matter is ejected far into space. When this matter reaches the Earth,* a er being diluted by the journey, it a ects the environment. Solar storms can deplete the higher atmosphere and can thus possibly trigger usual Earth storms. Other e ects of the Sun are the formation of auroras and the loss of orientation of birds during their migration; this happens during exceptionally strong solar storms, as the magnetic eld of the Earth is disturbed in these situations.
e most famous e ect of a solar storm was the loss of electricity in large parts of Canada in March of . e ow of charged solar particles triggered large induced currents in the power lines, blew fuses and destroyed parts of the network, shutting down the power system. Millions of Canadians had no electricity, and in the most remote places it took two weeks to restore the electricity supply. Due to the coldness of the winter and a train accident resulting from the power loss, over people died. In the meantime the network has been redesigned to withstand such events.
e proton cycle and the CNO cycles are not the only options. Heavier and older stars than the Sun can also shine through other fusion reactions. In particular, when hydrogen is consumed, such stars run helium burning:
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He C .
(621)
Page 689
is fusion reaction is of low probability, since it depends on three particles being at the same point in space at the same time. In addition, small amounts of carbon disappear rapidly via the reaction α + C O. Nevertheless, since Be is unstable, the reaction with alpha particles is the only way for the universe to produce carbon. All these negative odds are countered only by one feature carbon has an excited state at . MeV, which is
. MeV above the sum of the alpha particle masses; the excited state resonantly enhances the low probability of the three particle reaction. Only in this way the universe is able to produce the atoms necessary for pigs, apes and people. e prediction of this resonance by Fred Hoyle is one of the few predictions in physics that used the simple experimental observation that humans exist. e story has lead to an huge out ow of metaphysical speculations, most of which are unworthy of being even mentioned.
W
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Across the world, for over years, a large number of physicists and engineers have tried to build fusion reactors. Fusion reactors try to copy the mechanism of energy release used
* It might even be that the planets a ect the solar wind; the issue is not settled and is still under study.
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F I G U R E 354 A simplified drawing of the Joint European Torus in operation at Culham, showing the large toroidal chamber and the magnets for the plasma confinement (© EFDA-JET)
Ref. 928
by the Sun. e rst machine that realized macroscopic energy production was the Joint European Torus* (JET for short) located in Culham in the United Kingdom.
e idea of JET is to produce an extremely hot plasma that is as dense as possible. At high enough temperature and density, fusion takes place; the energy is released as a particle ux that is transformed (like in a ssion reactor) into heat and then into electricity. To achieve ignition, JET used the fusion between deuterium and tritium, because this reaction has the largest cross section and energy gain:
D + T He + n + . MeV .
(622)
Because tritium is radioactive, most research experiments are performed with the much less e cient deuterium–deuterium reactions, which have a lower cross section and a lower energy gain:
D + D T + H + MeV D + D He + n + . MeV .
(623)
Fusion takes place when deuterium and tritium (or deuterium) collide at high energy. e high energy is necessary to overcome the electrostatic repulsion of the nuclei. In
other words, the material has to be hot. To release energy from deuterium and tritium,
* See www.jet.edfa.org.
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Ref. 929
one therefore rst needs energy to heat it up. is is akin to the ignition of wood: in order to use wood as a fuel, one rst has to heat it with a match.
Following the so-called Lawson criterium, published in by the English engineer John Lawson, (but already known to Russian researchers) a fusion reaction releases energy only if the triple product of density n, reaction (or containment) time τ and temperature T exceeds a certain value. Nowadays this criterium is written as
nτT ë s K m .
(624)
In order to realize the Lawson criterium, JET uses temperatures of to MK, particle densities of to ë m− , and con nement times of s. e temperature is much higher than the MK at the centre of the Sun, because the densities and the con nement times are lower for JET.
Matter at these temperatures is in form of plasma: nuclei and electrons are completely separated. Obviously, it is impossible to pour a plasma at MK into a container: the walls would instantaneously evaporate. e only option is to make the plasma oat in a vacuum, and to avoid that the plasma touches the container wall. e main challenge of fusion research in the past has been to nd a way to keep a hot gas mixture of deuterium and tritium suspended in a chamber so that the gas never touches the chamber walls. e best way is to suspend the gas using a magnetic eld. is works because in the fusion plasma, charges are separated, so that they react to magnetic elds. e most successful geometric arrangement was invented by the famous Russian physicists Igor Tamm and Andrei Sakharov: the tokamak. Of the numerous tokamaks around the world, JET is the largest and most successful. Its concrete realization is shown in Figure . JET manages to keep the plasma from touching the walls for about a second; then the situation becomes unstable: the plasma touches the wall and is absorbed there. A er such a disruption, the cycle consisting of gas injection, plasma heating and fusion has to be restarted. As mentioned, JET has already achieved ignition, that is the state were more energy is released than is added for plasma heating. However, so far, no sustained commercial energy production is planned or possible, because JET has no attached electrical power generator.
e successor project, ITER, an international tokamak built with European, Japanese, US-American and Russian funding, aims to pave the way for commercial energy generation. Its linear reactor size will be twice that of JET; more importantly, ITER plans to achieve s containment time. ITER will use superconducting magnets, so that it will have extremely cold matter at K only a few metres from extremely hot matter at MK. In other words, ITER will be a high point of engineering. e facility will be located in Cadarache in France and is planned to start operation in the year .
Like many large projects, fusion started with a dream: scientists spread the idea that fusion energy is safe, clean and inexhaustible. ese three statements are still found on every fusion website across the world. In particular, it is stated that fusion reactors are not dangerous, produce much lower radioactive contamination than ssion reactors, and use water as basic fuel. ‘Solar fusion energy would be as clean, safe and limitless as the Sun.’ In reality, the only reason that we do not feel the radioactivity of the Sun is that we are far away from it. Fusion reactors, like the Sun, are highly radioactive. e management of radioactive fusion reactors is much more complex than the management of radioactive
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ssion reactors. Fusion fuels are almost inexhaustible: deuterium is extracted from water and the tri-
tium – a short-lived radioactive element not found in nature in large quantities – is produced from lithium. e lithium must be enriched, but since material is not radioactive, this is not problematic. However, the production of tritium from lithium is a dirty process that produces large amounts of radioactivity. Fusion energy is thus inexhaustible, but not safe and clean.
In short, of all technical projects ever started by mankind, fusion is by far the most challenging and ambitious. Whether fusion will ever be successful – or whether it ever should be successful – is another issue.
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Page 462
People consist of electrons and various nuclei. Electrons, hydrogen and helium nuclei are formed during the big bang. All other nuclei are formed in stars. Young stars run hydrogen burning or helium burning; heavier and older stars run neon-burning or even silicon-burning. ese latter processes require high temperatures and pressures, which are found only in stars with a mass at least eight times that of the Sun. However, all fusion processes are limited by photodissociation and will not lead to nuclei heavier than Fe.
Heavier nuclei can only be made by neutron capture. ere are two main processes; the s-process (for ‘slow’) runs inside stars, and gradually builds up heavy elements until the most heavy, lead, from neutron ying around. e rapid r-process occurs in stellar explosions. Many stars die this violent death. Such an explosion has two main e ects: on one hand it distributes most of the matter of the star, such carbon, nitrogen or oxygen, into space in the form of neutral atoms. On the other hand, new elements are synthesized during the explosion. e abundances of the elements in the solar system can be precisely measured. ese several hundred data points correspond exactly with what is expected from the material ejected by a (type II) supernova explosion. In other words, the solar system formed from the remnants of a supernova, as did, somewhat later, life on Earth.* We all are recycled stardust.
T
Ref. 930
Both radioactivity and medical images show that nuclei are composed. But quantum theory makes an additional prediction: protons and neutrons themselves must be composed.
ere are two reasons: nucleons have a nite size and their magnetic moments do not match the value predicted for point particles. e prediction of components inside the protons was con rmed in the late s when Kendall, Friedman and Taylor shot high energy electrons into hydrogen atoms. ey found what that a proton contains three constituents with spin / , which they called called partons. e experiment was able to ‘see’ the constituents through large angle scattering of electrons, in the same way that we see objects through large angle scattering of photons. ese constituents correspond in number and properties to the so-called quarks predicted in by Murray Gell-Mann** and,
* By chance, the composition ratios between carbon, nitrogen and oxygen inside the Sun are the same as inside the human body. ** Murray Gell-Mann (b. 1929 New York, d. ) received the Nobel Prize for physics in 1969. He is the originator of the term ‘quark’. ( e term has two origins: o cially, it is said to be taken from Finnegans Wake, a
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figure to be inserted
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F I G U R E 355 A selection of mesons and baryons and their classification as bound states of quarks
Ref. 931
independently, by George Zweig. It turns out that the interaction keeping the protons together in a nucleus, which was
rst described by Yukawa Hideki,* is only a shadow of the interaction that keeps quarks together in a proton. Both are called by the same name. e two cases correspond somewhat to the two cases of electromagnetism found in atomic matter. Neon atoms show the cases most clearly: the strongest aspect of electromagnetism is responsible for the attraction of the electrons to the neon nuclei and its feebler ‘shadow’ is responsible for the attraction of neon atoms in liquid neon and for processes like evaporation. Both attractions are electromagnetic, but the strengths di er markedly. Similarly, the strongest aspect of the strong interaction leads to the formation of the proton and the neutron; the feeble aspect leads to the formation of nuclei and to alpha decay. Obviously, most can be learned by studying the strongest aspect.
B
,
Physicists are simple people. To understand the constituents of matter, and of nuclei in particular, they had no better idea than to take all particles they could get hold of and to smash them into each other. Many played this game for several decades.**
Imagine that you want to study how cars are built just by crashing them into each other. Before you get a list of all components, you must perform and study a non-negligible number of crashes. Most give the same result, and if you are looking for a particular part, you
Challenge 1396 n
novel by James Joyce; in reality, he took it from a German and Yiddish term meaning ‘lean so cheese’ and used guratively in those langauges to mean ‘silly idea’.)
Gell-Mann is the central gure of particle physics; he introduced the concept of strangeness, the renormalization group, the V-A interaction, the conserved vector current, the partially conserved axial current, the eightfold way, the quark model and quantum chromodynamics.
Gell-Mann is also known for his constant battle with Richard Feynman about who deserves to be called the most arrogant physicist of their university. * Yukawa Hideki (1907–1981), important Japanese physicist specialized in nuclear and particle physics. He founded the journal Progress of eoretical Physics and together with his class mate Tomonaga he was an example to many scientists in Japan. He received the 1949 Nobel Prize for physics for this theory of mesons. ** In fact, quantum theory forbids any other method. Can you explain why?
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•
might have to wait for a long time. If the part is tightly attached to others, the crashes have
to be especially energetic. Since quantum theory adds the possibility of transformations,
reactions and excited states, the required diligence and patience is even greater than for
car crashes. erefore, for many decades, researchers collected an ever increasing num-
ber of debris. e list was overwhelming. en came the quark model, which explained
the whole mess as a consequence of only a few types of bound constituents.
Other physicists then added a few details and as a result, the whole list of debris could
be ordered in tables such as the ones given in Figure
ese tables were the beginning
of the end of high energy physics. When the proton scattering experiments found that
protons are made of three constituents, the quark model became accepted all over the
world.
e proton and the neutron are seen as combinations of two quarks, called up (u) and
down (d). Later, other particles lead to the addition of four additional types of quarks.
eir names are somewhat confusing: they are called strange (s), charm (c), bottom (b) –
also called ‘beauty’ in the old days – and top (t) – called ‘truth’ in the past.
All quarks have spin one half; their electric charges are multiples of of the electron
charge. In addition, quarks carry a strong charge, which in modern terminology is called
colour. In contrast to electromagnetism, which has only positive, negative, and neutral
charges, the strong interaction has red, blue, green charges on one side, and anti-red,
anti-blue and anti-green on the other. e neutral state is called ‘white’. All baryons and
mesons are white, in the same way that all atoms are neutral.
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– CS – details to be added – CS –
T
,
Frank Wilczek mentions that one of the main results of QCD, the theory of strong interRef. 932 actions, is to explain mass relations such as
mproton e−k α mPlanck and k = π , αunif =
.
(625)
Page 941 Ref. 933
Here, the value of the coupling constant αunif is taken at the unifying energy, a factor of below the Planck energy. (See the section of uni cation below.) In other words, a
general understanding of masses of bound states of the strong interaction, such as the
proton, requires almost purely a knowledge of the uni cation energy and the coupling
constant at that energy. e approximate value αunif = low energy value, using experimental data.
is an extrapolation from the
e proportionality factor in expression ( ) is still missing. Indeed, it is not easy to
calculate. Many calculations are now done on computers. e most promising calcula-
tion simplify space-time to a lattice and then reduce QCD to lattice QCD. Using the most
powerful computers available, these calculations have given predictions of the mass of
the proton and other baryons within a few per cent.
But the mass is not the only property of the proton. Being a cloud of quarks and gluons,
it also has a shape. Surprisingly, it took a long time before people started to become inter-
ested in this aspect. e proton is made of two u quarks and one d quark. It thus resembles
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figure to be inserted
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F I G U R E 356 The spectrum of the excited states of proton and neutron
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Ref. 934 Ref. 935
a ionized H+ molecule, where one electron forms a cloud around two protons. Obviously, the H+ molecule is elongated. Is that also the case for the proton? First results from seem to point into this direction.
e shape of a molecule will depend on whether other molecules surround it. Recent research showed that both the size and the shape of the proton in nuclei is slightly variable; both seem to depend on the nucleus in which the proton is built-in.
Apart from shapes, molecules also have a colour. e colour of a molecule, like that of any object, is due to the energy absorbed when it is irradiated. For example, the H+ molecule can absorb certain light frequencies by changing to an excited state. Protons and neutrons can also be excited; in fact, their excited states have been studied in detail; a summary is shown in Figure . It turns out that all these excitations can be explained as excited quarks states. For several excitations, the masses (or colours) have been calculated by lattice QCD to within %. e quark model and QCD thus structure and explain a large part of the baryon spectrum.
Obviously, in our everyday environment the energies necessary to excite nucleons do not appear – in fact, they do not even appear inside the Sun – and these excited states can be neglected. ey only appear in particle accelerators. In a way, we can say that in our corner of the universe energies are to low to show the colour of protons.
E
How can we pretend that quarks exist, even though they are never found alone? ere are a number of arguments in favour.
— e quark model explains the non-vanishing magnetic moment of the neutron and explains the magnetic moments µ of the baryons. By describing the proton as a uud state and the neutron a udd state with no orbital angular momentum, we get
µu = ( µp + µn)µ
µd = ( µn + µp)µ
where µ = ħ Mc (626)
is means that mu = md = MeV,a bit more than a third of the nucleon, whose mass is c. MeV. If we assume that the quark magnetic moment is proportional to
their charge, we predict a ratio of the magnetic moments of the proton and the neut-
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g
g
g
g
g
g
g
F I G U R E 357 The essence of the QCD Lagrangian
q g
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Ref. 945
ron of µp µn = − . ; this prediction di ers from measurements only by 3 %. Using the values for the magnetic moment of the quarks, magnetic moment values of over half a dozen of other baryons can be predicted. e results typically deviate from measurements by around 10 %; the sign is always correctly calculated. — e quark model describes the quantum numbers of mesons and baryons. All texts on the quark model are full of diagrams such as those shown in Figure 355. ese generalizations of the periodic table of the elements were lled in during the twentieth century; they allow a complete classi cation of all mesons and baryons as bound states of quarks. — e quark model also explains the mass spectrum of baryons and mesons. e best predictions are made by lattice calculations. A er one year of computer time, researchers were able to reproduce the masses of proton and neutron to within a few per cent. Even if one sets the u and d quark masses to zero, the resulting proton and neutron mass di er from experimental values only by 10 %.
TL
All motion due to the strong interaction can be described by the three fundamental processes shown in Figure . Two gluons can scatter, a gluon can emit another, and a quark can emit or absorb a gluon. In electrodynamics, only the last diagram is possible, in the strong interaction, the rst two appear as well. Among others, the rst two diagrams are responsible for the con nement of quarks.
Let us have now a look at the o cial Lagrangian of the strong interaction, and show that it is just a complicated rewriting of Figure . e Lagrangian density of quantum chromodynamics, abbreviated QCD, is
Q C D = − Fµ(aν)F(a)µν − c where
mq
ψ
k q
ψ
qk
+
i
ħ c
c
ψ
k q
γ
µ
(D
µ
)k
l
ψ
l q
q
q
Fµ(aν) = ∂µ Aaν − ∂ν Aaµ + s fabc Abµ Acν
(Dµ)kl = δkl ∂µ − i s
λ
a k,
l
Aaµ
.
a
(627)
We remember from the section on the principle of least action that Lagrangians are always
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Page 197
sums of scalar products; this is clearly seen in the formula. Furthermore, the index a =
. . . numbers the eight types of gluons, k = , , numbers the three colours and q =
, . . . numbers the six quark avours. e elds Aaµ(x) are the eight gluon elds: they
are the coiled lines in Figure . e eld ψqk(x) is the eld of the quark of avour q and colour k: it is the straight line in the gure. e quark elds are -component Dirac
spinors.
e rst term of the Lagrangian ( ) represents the kinetic energy of the radiation
(gluons), the second term the kinetic energy of the matter particles (the quarks) and the
third term the interaction between the two. e third term in the Lagrangian, the inter-
action term, thus corresponds to the third diagram in Figure .
Gluons are massless; therefore no gluon mass term appears in the Lagrangian. In op-
position to electromagnetism, where the gauge group U( ) is abelian, the gauge group
SU( ) of the strong interactions is non-abelian. As a consequence, the colour eld itself
is charged, i.e. carries colour: the index a appears on A and F. As a result, gluons can
interact with each other, in contrast to photons, which pass each other undisturbed. e
rst two diagrams of Figure are thus re ected in the somewhat complicated de ni-
tion of the function Fµ(aν). In contrast to electrodynamics, the de nition has an extra term that is quadratic in the elds A; it is described by the interaction strength s and by the
so-called structure constants fabc. ese are the structure constants of the SU( ) algebra. e interaction between the quarks and the gluons, the third term of the Lagrangian,
is
described
by
the
matrices
λ
a k
,
l
;
they
are
a
fundamental,
-dimensional representation
of the generators of the SU( ) algebra.*
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* In their simplest form, the matrices γµ can be written as
γ=
I −I
and γn = −σ i σ i
for n = , ,
(628)
Page 1201 where the σ i are the Pauli spin matrices. e matrices λa , a = .. , and the structure constants fabc obey the relations
[λa, λb] = i fabc λc λa, λb = δabI + dabc λc
(629)
where I is the unit matrix. e structure constants fabc, which are odd under permutation of any pair of indices, and dabc, which are even, are
abc
fabc
abc dabc
abc dabc
−
−
−
−( )
(630)
−
−( )
−( )
−
−( )
−
All other elements vanish. A fundamental 3-dimensional representation of the generators λa is given for
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•
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We see that only quarks and gluons appear in the Lagrangian of QCD, because only quarks and gluons interact via the strong force. is can be also expressed by saying that only quarks and gluons carry colour; colour is the source of the strong force in the same way that electric charge is the source of the electromagnetic eld. In the same way as electric charge, colour charge is conserved in all interactions. Electric charge comes in two types, positive and negative; in contrast, colour comes in three types, called red, green and blue. e neutral state, with no colour charge, is called white. Protons and neutrons, but also electrons or neutrinos, are thus white in this sense.
Like in all of quantum eld theory, also in the case of QCD the mathematical form of the Lagrangian is uniquely de ned by requiring renormalizability, Lorentz invariance and gauge invariance (SU( ) in this case) and by specifying the di erent particles types ( quarks in this case) with their masses and the coupling constant.
e Lagrangian is thus almost xed by construction. We say ‘almost’, because it contains a few parameters that remain unexplained:
— e number and the masses of the quarks are not explained by QCD.
— e coupling constant s of the strong interaction is unexplained. O en also the equivalent quantity αs = s π is used to describe the coupling. Like for the case of the electroweak interactions, αs and thus s depend on the energy Q of the experiment.
is energy dependence is indeed observed in experiments; it is described by the renor-
malization procedure:
αs(Q ) =
π
−
nf
ln
Q Λ
+ ...
(632)
where n f is the number of quarks with mass less than the energy scale Q and lies between 3 and 6. e strong coupling is thus completely described by the energy parameter Λ . GeV c . If αs is known for one energy, it is known for all of them. Presently, the experimental value is αs(Q = GeV) = . . . Expression (632) also illustrates asymptotic freedom: αs vanishes for high energies. In other words, at
high energies quarks are freed from the strong interaction.*
At low energies, the coupling increases, and leads to quark con nement.** is be-
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example by the set of the Gell-Mann matrices
λ=
λ=
−i i
λ=
−
−i
λ=
λ=
λ=
i
λ=
−i λ =
i
. −
(631)
ere are eight matrices, one for each gluon type, with elements, corresponding to the three colours of the strong interactions. * Asymptotic freedom was rst discovered by Gerard ‘t Hoo ; since he had the Nobel Prize already, it the 2004 Prize was then given to the next people who found it: David Gross, David Politzer and Frank Wilczek. ** Only at energies much larger than Λ can a perturbation expansion be applied.
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haviour is in contrast to the electroweak interactions, where the coupling increases with energy. We thus nd that one parameter describing the strong coupling, Λ, remains unexplained and must be introduced into the Lagrangian from the beginning. — e properties of space-time, its Lorentz invariance, its continuity and the number of its dimensions are obviously all unexplained and assumed from the outset.
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T
Ref. 946 Page 941
Ref. 947
e size of quarks, like that of all elementary particles, is predicted to be zero by quantum eld theory. So far, no experiment has found an e ect due to a nite quark size. Measurements show that quarks are surely smaller than − m. No de nite size predictions have been made; quarks might have a size of the order of the grand uni cation scale, i.e. − m; however, so far this is pure speculation.
We noted in several places that a compound is always less massive than its components. But when the mass values for quarks are looked up in most tables, the masses of u and d quarks are only of the order of a few MeV c , whereas the proton’s mass is MeV c . What is the story here?
e de nition of the masses for quarks is more involved than for other particles. Quarks are never found as free particles, but only in bound states. Quarks behave almost like free particles at high energies; this property is called asymptotic freedom. e mass of such a free quark is called current quark mass; for the light quarks it is only a few MeV c .
At low energy, for example inside a proton, quarks are not free, but must carry along a large amount of energy due to the con nement process. As a result, bound quarks have a much larger constituent quark mass, which takes into account this con nement energy. To give an idea of the values, take a proton; the indeterminacy relation for a particle inside a sphere of radius . fm gives a momentum indeterminacy of around MeV c. In three dimensions this gives an energy of times that value, or a mass of about MeV c .
ree con ned quarks are thus heavier than a proton, whose mass is MeV c ; we can thus still say that a compound proton is less massive than its constituents. In short, the mass of the proton and the neutron is (almost exclusively) the kinetic energy of the quarks inside them, as their own rest mass is negligible. As Frank Wilczek says, some people put on weight even though they never eat anything heavy.
To complicate the picture, the distinction of the two mass types makes no sense for the top quark; this quark decays so rapidly that the con nement process has no time to set in. As a result, the top mass is again a mass of the type we are used to.
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C
e description of the proton mass using con ned quarks should not hide the fact that the complete explanation of quark con nement, the lack of single quarks in nature, is the biggest challenge of theoretical high energy physics.
e Lagrangian of QCD di ers from that of electromagnetism in a central aspect. So far, bound states cannot be deduced with a simple approximation method. In particular, the force dependence between two coloured particles, which does not decrease with increasing distance, but levels o at a constant value, does not follow directly from the Lagrangian. e constant value, which then leads to con nement, has been reproduced only in involved computer calculations.
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In fact, the challenge is so tough that the brightest minds have been unable to solve it, so far. In a sense, it can be seen as the biggest challenge of all of physics, as its solution probably requires the uni cation of all interactions and most probably the uni cation with gravity. We have to leave this issue for later in our adventure.
C
e computer calculations necessary to extract particle data from the Lagrangian of quantum chromodynamics are among the most complex calculations ever performed.
ey beat weather forecasts, uid simulations and the like by orders of magnitude. Nobody knows whether this is necessary: the race for a simple approximation method for nding solutions is still open.
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Ref. 948
**
Even though gluons are massless, like photons and gravitons, there is no colour radiation. Gluons carry colour and couple to themselves; as a result, free gluons were predicted to directly decay into quark–antiquark pairs. is decay has indeed been observed in experiments at particle accelerators.
Something similar to colour radiation, but still stranger might have been found in 1997. First results seem to con rm the prediction of glueballs from numerical calculations.
Ref. 948
**
e latest fashion in high energy physics is the search for hybrid mesons, particles made of gluons and quarks. is fashion is not over yet; the coming years should settle whether the candidates known so far really are hybrids.
**
Do particles made of ve quarks, so-called pentaquarks, exist? So far, they seem to exist only in a few laboratories in Japan, whereas other laboratories across the world fail to see them. e issue is still open.
**
Whenever we look at a periodic table of the elements, we look at a manifestation of the strong interaction. e Lagrangian of the strong interaction describes the origin and properties of the presently known 115 elements.
Nevertheless a central aspect of nuclei is determined together with the electromagnetic interaction. Why are there around one hundred di erent elements? Because the electromagnetic coupling constant α is . ( ). Indeed, if the charge of a nucleus was much higher than around 130, the electric eld around nuclei would lead to spontaneous electron–positron pair generation; the electron would fall into the nucleus and transform one proton into a neutron, thus inhibiting a larger proton number.
**
To know more about radioactivity, its e ects, its dangers and what a government can do about it, see the English and German language site of the Federal O ce for Radiation
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Protection at http://www.bfs.de.
**
From the years 1990 onwards, it has o en been claimed that extremely poor countries Challenge 1397 n are building nuclear weapons. Why is this not possible?
**
In the 1960s and 70s, it was discovered that the Sun pulsates with a frequency of 5 minutes. e e ect is small, only 3 kilometres out of 1.4 million; still it is measurable. In the mean-
time, helioseismologists have discovered numerous additional oscillations of the Sun, and in 1993, even on other stars. Such oscillations allow to study what is happening inside stars, even separately in each of the layers they consist of.
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**
Historically, nuclear reactions also provided the rst test of the relation E = γmc . is was achieved in 1932 by Cockcro and Walton. ey showed that by shooting protons into lithium one gets the reaction
Li + H Be He + He + MeV .
(633)
e measured energy on the right is exactly the value that is derived from the di erences in total mass of the nuclei on both sides.
**
Some stars shine like a police siren: their luminosity increases and decreases regularly. Such stars, called Cepheids, are important because their period depends on their average (absolute) brightness. Measuring their period and their brightness on Earth thus allows astronomers to determine their distance.
.
No interaction is as weird as the weak interaction. First of all, the corresponding ‘weak radiation’ consists of massive particles; there are two types, the neutral Z boson with a mass of . GeV – that is the mass of a silver atom – and the electrically charged W boson with a mass of . GeV. e masses are so large that free radiation exists only for an extremely short time, about . ys; then the particles decay. e large mass is the reason that the interaction is extremely short range and weak; any exchange of virtual particles scales with the negative exponential of the intermediate particle’s mass.
e existence of a massive intermediate vector boson was already deduced in the s; but theoretical physicists did not accept the idea until the Dutch physicist Gerard ’t Hoo proved that it was possible to have such a mass without having problems in the rest of the theory. For this proof he later received the Nobel price of physics. Experimentally,
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Ref. 952
Ref. 953 Ref. 958
the Z boson was found found as a virtual particle in and as a real particle in , both times at CERN in Geneva. e last experiment was a year-long e ort by thousands of people working together.
A central e ect of the weak interaction is its ability to transform quarks. It is this property that is responsible for beta decay, where a d quark in a neutron is changed into a u quark, or for a crucial step in the Sun, where the opposite happens.
e next weird characteristic of the weak interaction is the nonconservation of parity under spatial inversion. e weak interaction distinguishes between mirror systems, in contrast to everyday life, gravitation, electromagnetism, and the strong interactions. Parity non-conservation had been predicted by by Lee Tsung-Dao and Yang Chen Ning, and was con rmed a few months later, earning them a Nobel Prize. ( ey had predicted the e ect in order to explain the ability of K mesons to decay either into or into pions.)
e most beautiful consequence of parity non-conservation property is its in uence on the colour of certain atoms. is prediction was made in by Bouchiat and Bouchiat.
e weak interaction is triggered by the weak charge of electrons and nuclei. erefore, electrons in atoms do not exchange only virtual photons with the nucleus, but also virtual Z particles. e chance for this latter process is extremely small, around − times smaller than virtual photon exchange. But since the weak interaction is not parity conserving, this process allows electron transitions which are impossible by purely electromagnetic effects. In , measurements con rmed that certain optical transitions of caesium atoms that are impossible via the electromagnetic interaction, are allowed when the weak interaction is taken into account. Several groups have improved these results and have been able to con rm the prediction of the weak interaction, including the charge of the nucleus, to within a few per cent.
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– CS – e section on weak interactions will be inserted here – CS –
C
e weak interaction is responsible for the burning of hydrogen to helium. Without helium, there would be no path to make still heavier elements. us we owe our own existence to the weak interaction.
** e weak interaction is required to have an excess of matter over antimatter. Without the parity breaking of the weak interactions, there would be no matter at all in the universe.
** rough the emitted neutrinos, the weak interaction helps to get the energy out of a supernova. If that were not the case, black holes would form, heavier elements – of which we are made – would not have been spread out into space, and we would not exist.
** Ref. 951 e paper by Peter Higgs on the boson named a er him is only 79 lines long, and has
only ve equations.
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**
e weak interaction is not parity invariant. In other words, when two electrons collide, the fraction of the collisions that happens through the weak interaction should behave di erently than a mirror experiment. In 2004, polarized beams of electrons – either le handed or right-handed – were shot at a matter target and the re ected electrons were counted. e di erence was 175 parts per billion – small, but measurable. e experiment also con rmed the predicted weak charge of -0.046 of the electron.
**
e weak interaction is also responsible for the heat produced inside the Earth. is heat keeps the magma liquid. As a result, the weak interaction, despite its weakness, is responsible for all earthquakes, tsunamis and volcanic eruptions.
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**
Beta decay, due to the weak interaction, separates electrons and protons. Only in 2005 people have managed to propose practical ways to use this e ect to build long-life batteries that could be used in satellites. Future will tell whether the method will be successful.
M, H
In the years
and
Di cile est satiram non scribere.
Juvenal*
“ ” an intense marketing campaign was carried out across the
United States of America by numerous particle physicists. ey sought funding for the
‘superconducting supercollider’, a particle accelerator with a circumference of km. is
should have been the largest human machine ever built, with a planned cost of more than
ten thousand million dollars, aiming at nding the Higgs boson before the Europeans
would do so, at a fraction of the cost. e central argument brought forward was the
following: since the Higgs boson was the basis of mass, it was central to science to know
about it. Apart from the discussion on the relevance of the argument, the worst is that it
is wrong.
We have even seen that % of the mass of protons, and thus of the universe, is due
to con nement; it appears even if the quarks are approximated as massless. e Higgs
boson is not responsible for the origin of mass itself; it just might shed some light on the
issue. e whole campaign was a classic case of disinformation and many people involved
have shown their lack of honesty. In the end, the project was stopped, mainly for nancial
reasons. But the disinformation campaign had deep consequences. US physicist lost their
credibility. Even in Europe the budget cuts became so severe that the competing project
in Geneva, though over ten times cheaper and nanced by thirty countries instead of
only one, was almost stopped as well. (Despite this hick-up, the project is now under way,
scheduled for completion in / .)
* ‘It is hard not to be satirical.’ 1, 30
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•
N
e weak interaction, with its breaking of parity and the elusive neutrino, exerts a deep Ref. 949 fascination on all those who have explored it.
Ref. 950 Page 910
**
Every second around neutrinos y through our body. ey have ve sources:
— Solar neutrinos arrive on Earth at ë m s, with an energy from 0 to . MeV; they are due to the p-p reaction in the sun; a tiny is due to the B reaction and has energies up to MeV.
— Atmospheric neutrinos are products of cosmic rays hitting the atmosphere, consist of
2/3 of muon neutrinos and one third of electron neutrinos, and have energies mainly between MeV and GeV. — Earth neutrinos from the radioactivity that keeps the Earth warm form a ux of
ë m s. — Fossil neutrinos from the big bang, with a temperature of . K are found in the uni-
verse with a density of cm− , corresponding to a ux of m s. — Man-made neutrinos are produced in nuclear reactors (at MeV) and in as neutrino
beams in accelerators, using pion and kaon decay. A standard nuclear plant produces ë neutrinos per second. Neutrino beams are produced, for example, at the CERN
in Geneva. ey are routinely sent km across the Earth to central Italy, where they
are detected.
ey are mainly created in the atmosphere by cosmic radiation, but also coming directly from the background radiation and from the centre of the Sun. Nevertheless,
during our whole life – around 3 thousand million seconds – we have only a 10 % chance that one of them interacts with one of the ë atoms of our body. e reason is that the weak interaction is felt only over distances less than − m, about 1/100th
of the diameter of a proton. e weak interaction is indeed weak.
**
e weak interaction is so weak that a neutrino–antineutrino annihilation – which is only possible by producing a massive intermediate Z boson – has never been observed up to this day.
**
Only one type of particles interacts (almost) only weakly: neutrinos. Neutrinos carry no electric charge, no colour charge and almost no gravitational charge (mass). To get an impression of the weakness of the weak interaction, it is usually said that the probability of a neutrino to be absorbed by a lead screen of the thickness of one light-year is less than 50 %. e universe is thus essentially empty for neutrinos. Is there room for bound states of neutrinos circling masses? How large would such a bound state be? Can we imagine bound states, which would be called neutrinium, of neutrinos and antineutrinos circling each other? e answer depends on the mass of the neutrino. Bound states of massless
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Challenge 1398 ny
particles do not exist. ey could and would decay into two free massless particles.* Since neutrinos are massive, a neutrino–antineutrino bound state is possible in prin-
ciple. How large would it be? Does it have excited states? Can they ever be detected? ese issues are still open.
Ref. 954
**
Do ruminating cows move their jaws equally o en in clockwise and anticlockwise direction? In 1927, the theoretical physicists Pascual Jordan and Ralph de Laer Kronig published a study showing that in Denmark the two directions are almost equally distributed.
e rumination direction of cows is thus not related to the weak interaction.
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**
Page 916 Challenge 1399 e
e weak interaction plays an important part in daily life. First of all, the Sun is shining. e fusion of two protons to deuterium, the rst reaction of the hydrogen cycle, implies that one proton changes into a neutron. is transmutation and the normal beta decay have the same rst-order Feynman diagram. e weak interaction is thus essential for the burning of the Sun. e weakness of the process is one of the guarantees that the Sun will continue burning for quite some time.
**
Of course, the weak interaction is responsible for radioactive beta decay, and thus for part of the radiation background that leads to mutations and thus to biological evolution.
**
What would happen if the Sun suddenly stopped shining? Obviously, temperatures would fall by several tens of degrees within a few hours. It would rain, and then all water would freeze. A er four or ve days, all animal life would stop. A er a few weeks, the oceans would freeze; a er a few months, air would liquefy.
**
Not everything about the Sun is known. For example, the neutrino ux from the Sun oscillates with a period of 28.4 days. at is the same period with which the magnetic
eld of the Sun oscillates. e connections are still being studied.
**
e energy carried away by neutrinos is important in supernovas; if neutrinos would not carry it away, supernovas would collapse instead of explode. at would have prevented the distribution of heavier elements into space, and thus our own existence.
**
Even earlier on in the history of the universe, the weak interaction is important, as it prevents the symmetry between matter and antimatter, which is required to have an excess of one over the other in the universe.
* In particular, this is valid for photons bound by gravitation; this state is not possible.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•
**
Due to the large toll it placed on society, research in nuclear physics, like poliomyelitis, has almost disappeared from the planet. Like poliomyelitis, nuclear research is kept alive only in a few highly guarded laboratories around the world, mostly by questionable gures, in order to build dangerous weapons. Only a small number of experiments carried on by a few researchers are able to avoid this involvement and continue to advance the topic.
Ref. 955
**
Interesting aspects of nuclear physics appear when powerful lasers are used. In 1999, a British team led by Ken Ledingham observed laser induced uranium ssion in U nuclei. In the meantime, this has even be achieved with table-top lasers. e latest feat, in 2003, was the transmutation of I to I with a laser. is was achieved by focussing a J laser pulse onto a gold foil; the ensuing plasma accelerates electrons to relativistic speed, which hit the gold and produce high energy γ rays that can be used for the transmutation.
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.
–
– CS – the section will appear soon – CS –
C
e standard model clearly distinguishes elementary from composed particles. It provides the full list of properties that characterizes a particle and thus any object in nature: charge, spin, isospin, parity, charge parity, strangeness, charm, topness, beauty, lepton number, baryon number and mass. e standard model also describes interactions as exchange of virtual radiation particles. It describes the types of radiation that are found in nature at experimentally accessible energy. In short, the standard model realizes the dream of the ancient Greeks, plus a bit more: we have the bricks that compose all of matter and radiation, and in addition we know precisely how they move and interact.
But we also know what we still do not know:
— we do not know the origin of the coupling constants; — we do not know the origin of the symmetry groups; — we do not know the details of con nement; — we do not know whether the particle concept survives at high energy; — we do not know what happens in curved space-time.
To study these issues, the simplest way is to explore nature at particle energies that are as high as possible. ere are two methods: building large experiments or making some calculations. Both are important.
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–
.
–
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 956
Materie ist geronnenes Licht.*
“ ” Albertus Magnus
Is there a common origin of the three particle interactions? We have seen in the preceding sections that the Lagrangians of the electromagnetic, the weak and the strong nuclear interactions are determined almost uniquely by two types of requirements: to possess a certain symmetry and to possess mathematical consistency. e search for uni cation of the interactions thus requires the identi cation of th uni ed symmetry of nature. In recent decades, several candidate symmetries have fuelled the hope to achieve this program: grand uni cation, supersymmetry, conformal invariance and coupling constant duality.
e rst of them is conceptually the simplest. At energies below GeV there are no contradictions between the Lagrangian of the standard model and observation. e Lagrangian looks like a low energy approximation. It should thus be possible (attention, this a belief) to nd a unifying symmetry that contains the symmetries of the electroweak and strong interactions as subgroups and thus as di erent aspects of a single, uni ed interaction; we can then examine the physical properties that follow and compare them with observation. is approach, called grand uni cation, attempts the uni ed description of all types of matter. All known elementary particles are seen as elds which appear in a Lagrangian determined by a single symmetry group. Like for each gauge theory described so far, also the grand uni ed Lagrangian is mainly determined by the symmetry group, the representation assignments for each particle, and the corresponding coupling constant. A general search for the symmetry group starts with all those (semisimple) Lie groups which contain U( ) SU( ) SU( ). e smallest groups with these properties are SU( ), SO( ) and E( ); they are de ned in Appendix D. For each of these candidate groups, the experimental consequences of the model must be studied and compared with experiment.
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E
Grand uni cation makes several clear experimental predictions. Any grand uni ed model predicts relations between the quantum numbers of all ele-
mentary particles – quarks and leptons. As a result, grand uni cation explains why the electron charge is exactly the opposite of the proton charge.
Grand uni cation predicts a value for the weak mixing angle θW that is not determined by the standard model. e predicted value,
is close to the measured value of
sin θW,th = .
(634)
sin θW,ex = . ( ) .
(635)
* ‘Matter is coagulated light.’ Albertus Magnus (b. c. 1192 Lauingen, d. 1280 Cologne), the most important thinker of his time.
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•.
–
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 959
All grand uni ed models predict the existence of magnetic monopoles, as was shown by Gerard ’t Hoo . However, despite extensive searches, no such particles have been found yet. Monopoles are important even if there is only one of them in the whole universe: the existence of a single monopole implies that electric charge is quantized. Grand uni cation thus explains why electric charge appears in multiples of a smallest unit.
Grand uni cation predicts the existence of heavy intermediate vector bosons, called X bosons. Interactions involving these bosons do not conserve baryon or lepton number, but only the di erence B − L between baryon and lepton number. To be consistent with experiment, the X bosons must have a mass of the order of GeV.
Most spectacularly, the X bosons grand uni cation implies that the proton decays. is prediction was rst made by Pati and Salam in . If protons decay, means that neither coal nor diamond* – nor any other material – is for ever. Depending on the precise symmetry group, grand uni cation predicts that protons decay into pions, electrons, kaons or other particles. Obviously, we know ‘in our bones’ that the proton lifetime is rather high, otherwise we would die of leukaemia; in other words, the low level of cancer already implies that the lifetime of the proton is larger than a. Detailed calculations for the proton lifetime τp using SU( ) yield the expression
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τp
MX αG(MX) Mp
a
(636)
Ref. 960
where the uncertainty is due to the uncertainty of the mass MX of the gauge bosons involved and to the exact decay mechanism. Several large experiments aim to measure this
lifetime. So far, the result is simple but clear. Not a single proton decay has ever been
observed. e experiments can be summed up by
τ(p e+ π ) ë a τ(p K+ ν¯) . ë a τ(n e+ π−) ë a τ(n K ν¯) . ë a
(637)
ese values are higher than the prediction by SU( ). To settle the issue de nitively, one last prediction of grand uni cation remains to be checked: the uni cation of the coupling constants.
T
Ref. 961
e estimates of the grand uni cation energy are near the Planck energy, the energy at which gravitation starts to play a role even between elementary particles. As grand uni cation does not take gravity into account, for a long time there was a doubt whether something was lacking in the approach. is doubt changed into certainty when the precision measurements of the coupling constants became available. is happened in , when
* As is well known, diamond is not stable, but metastable; thus diamonds are not for ever, but coal might be, if protons do not decay.
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–
F I G U R E 358 The behaviour of the three coupling constants with energy for the standard model (left) and for the minimal supersymmetric model (right) (© Dmitri Kazakov)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
these measurements were shown as Figure . It turned out that the SU( ) prediction of the way the constants evolve with energy imply that the three constants do not meet at the grand uni cation energy. Simple grand uni cation by SU( ) is thus de nitively ruled out.
is state of a airs is changed if supersymmetry is taken into account. Supersymmetry is the low-energy e ect of gravitation in the particle world. Supersymmetry predicts new particles that change the curves at intermediate energies, so that they all meet at a grand uni cation energy of about GeV. e inclusion of supersymmetry also puts the proton lifetime prediction back to a value higher (but not by much) than the present experimental bound and predicts the correct value of the mixing angle. With supersymmetry, we can thus retain all advantages of grand uni cation (charge quantization, fewer parameters) without being in contradiction with experiments. e predicted particles, not yet found, are in a region accessible to the LHC collider presently being built at CERN in Geneva. We will explore supersymmetry later on.
Eventually, some decay and particle data will become available. Even though these experimental results will require time and e ort, a little bit of thinking shows that they probably will be only partially useful. Grand uni cation started out with the idea to unify the description of matter. But this ambitious goal cannot been achieved in this way. Grand uni cation does eliminate a certain number of parameters from the Lagrangians of QCD and QFD; on the other hand, some parameters remain, even if supersymmetry is added. Most of all, the symmetry group must be put in from the beginning, as grand uni cation cannot deduce it from a general principle.
If we look at the open points of the standard model, grand uni cation reduces their number. However, grand uni cation only shi s the open questions of high energy physics to the next level, while keeping them unanswered. Grand uni cation remains a low energy e ective theory. Grand uni cation does not tell us what elementary particles are; the name ‘grand uni cation’ is ridiculous. In fact, the story of grand uni cation is a rst hint that looking at higher energies using only low-energy concepts is not the way to solve the mystery of motion. We de nitively need to continue our adventure.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
–
•.
B
923 A excellent technical introduction to nuclear physics is B
P ,K
R,
C
S
&F
Z
, Teilchen und Kerne, Springer, th edition,
. It is also available in English translation.
One of the best introductions into particle physics is K G
&V
F.
W
, Concepts of Particle Physics, Clarendon Press, Oxford, . Victor Weisskopf
was one of the heroes of the eld, both in theoretical research and in the management of
CERN, the particle research institution. Cited on page .
924 W.C.M. W
S
& al., Magnetic resonance imaging of male and female
genitals during coitus and female sexual arousal, British Medical Journal 319, pp. – ,
December , , available online as http://www.bmj.com/cgi/content/full/ / / .
Cited on page .
925 A good overview is given by A.N. H
, Radioactivity, the discovery of time and the
earliest history of the Earth, Contemporary Physics 38, pp. – , . Cited on page .
926 An excellent summary on radiometric dating is by R. W , Radiometric dating – a christian perpective, http://www.asa .org/ASA/resources/Wiens.html. e absurd title is due to the habit in many religious circles to put into question radiometric dating results. Apart from the extremely few religious statements in the review, the content is well explained. Cited on pages and .
927 e slowness of the speed of light inside stars is due to the frequent scattering of photons
by the star matter. e most common estimate for the Sun is an escape time of
to
million years, but estimates between
years and million years can be found in the
literature. Cited on page .
928 See the freely downloadable book by J W
, e Science of JET - e Achievements
of the Scientists and Engineers Who Worked on the Joint European Torus 1973-1999, JET
Joint Undertaking, , available at http://www.jet.edfa.org/documents/wesson/wesson.
html. Cited on page .
929 J.D. L
, Some criteria for a power producing thermonuclear reactor, Proceedings of
the Physical Society, London B 70, pp. – , . e paper had been kept secret for two
years. Cited on page .
930 Kendall, Friedman and Taylor received the Nobel Prize for Physics for a series of ex-
periments they conducted in the years to . e story is told in the three Nobel
lectures R.E. T
, Deep inelastic scattering: the early years, Review of Modern Phys-
ics 63, pp. – , , H.W. K
, Deep inelastic scattering: Experiments on the
proton and the observation of scaling, Review of Modern Physics 63, pp. – , , and
J.I. F
, Deep inelastic scattering: Comparisons with the quark model, Review of
Modern Physics 63, pp. – , . Cited on page .
931 G. Z , An SU model for strong interaction symmetry and its breaking II, CERN Report No. TH. , February , . Cited on page .
932 F. W
, Getting its from bits, Nature 397, pp. – , . Cited on page .
933 For an overview of lattice QCD calculations, see ... Cited on page .
934 S. S
& al., Polarization transfer in the He (e,e’p) H reaction up to Q = . (GeV/
c) , Physical Review Letters 91, p.
, . Cited on page .
935 e excited states of the proton and the neutron can be found in on the particle data group website at http://pdg.web.cern.ch. Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Dvipsbugw
936 An older but fascinating summary of solar physics is R. K
, Hundert Milliarden
Sonnen, Piper, . which is available also in English translation. No citations.
937 M. B
, S. C
, M. G , G. G
& A. P
, Gamma-
ray bursts of atmospheric origin in the MeV energy range, Geophysical Research Letters 27,
p. , June . Cited on page .
938 A book with nuclear explosion photographs is M . Cited on page .
L , 100 Suns, Jonathan Cape,
939 J. D
, A. G , N. S
& M. M
, Nuclear tetrahedral symmetry:
possibly present throughout the periodic table, Physical Review Letters 88, p.
,
June . Cited on page .
940 A good introduction is R. C
& B. W
pp. – , July . Cited on page .
, A new spin on nuclei, Physics World
941 M. N
& V.F. W
, Why does the sun shine?, American Journal of Phys-
ics 46, pp. – , . Cited on page .
942 R.A. A
, H. B
& G. G
, e Origin of Chemical Elements, Physical Re-
view 73, p. - , . No citations.
943 J H
, e End of Science – Facing the Limits of Knowledge in the Twilight of the
Scienti c Age, Broadway Books, , chapter , note . Cited on page .
944 M. C
, T.C. W
, X. S
& M. U
, Antimatter and the moon,
Nature 367, p. , . M. A
& al., Cosmic ray shadow by the moon observed
with the Tibet air shower array, Proceedings of the rd International Cosmic Ray Confer-
ence, Calgary 4, pp. – , . M. U
& al., Nuclear Physics, Proceedings Supple-
ment 14B, pp. – , . Cited on page .
945 See also C. B
& al., Light hadron spectrum with Kogut–Susskind quarks, Nuclear
Physics, Proceedings Supplement 73, p. , , and references therein. Cited on page .
946 F. A & al., Measurement of dijet angular distributions by the collider detector at Fermilab, Physical Review Letters 77, pp. – , . Cited on page .
947 e approximation of QCD with zero mass quarks is described by F. W from bits, Nature 397, pp. – , . Cited on page .
, Getting its
948 F. C , Glueballs and hybrids: new states of matter, Contemporary Physics 38, pp. – , . Cited on page .
949 K. G
& H.V. K
, Die schwache Wechselwirkung in Kern-, Teilchen- und As-
trophysik, Teubner Verlag, Stuttgart, . Also available in English and in several other
languages. Cited on page .
950 D. T
, Particle physics from the Earth and from teh sky: Part II, Europhysics News
35, no. , . Cited on page .
951 P.W. H , Broken symmetries, massless particles and gauge elds, Physics Letters 12, pp. – , . He then expanded the story in P.W. H , Spontaneous symmetry breakdown without massless bosons, Physical Review 145, p. - , . Higgs gives most credit to Anderson, instead of to himself; he also mentions Brout and Englert, Guralnik, Hagen, Kibble and ‘t Hoo . Cited on page .
952 M.A. B
& C.C. B
, Weak neutral currents in atomic physics, Physics
Letters B 48, pp. – , . U. A
, A. B , L.S. D
, P. L
,
A.K. M , W.J. M
, A. S
& H.H. W
, Comprehensive analysis
of data pertaining to the weak neutral current and the intermediate-vector-boson masses,
Physical Review D 36, pp. – , . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
953 M.C. N
, B.P. M
& C.E. W
, Precision measurement of parity
nonconservation in atomic cesium: a low-energy test of electroweak theory, Physical Review
Letters 61, pp. – , . See also D.M. M
& al., High-precision measurement
of parity nonconserving optical rotation in atomic lead, Physical Review Letters 71, pp. –
, . Cited on page .
954 Rumination is studied in P. J Cited on page .
& R. L K
, in Nature 120, p. , .
955 K.W.D. L
& al., Photonuclear physics when a multiterawatt laser pulse inter-
acts with solid targets, Physical Review Letters 84, pp. – , . K.W.D. L
& al., Laser-driven photo-transmutation of Iodine- – a long lived nuclear waste product,
Journal of Physics D: Applied Physics 36, pp. L –L , . R.P. S
, K.W.D. L -
& P. M K
, Nuclear physics with ultra-intense lasers – present status and fu-
ture prospects, Recent Research Developments in Nuclear Physics 1, pp. – , . Cited
on page .
956 For the bibliographic details of the latest print version of the Review of Particle Physics, see Appendix C. e online version can be found at http://pdg.web.cern.ch/pdg. e present status on grand uni cation can also be found in the respective section of the overview. Cited on page .
957 J. T T
V , ed., CP violation in Particle Physics and Astrophysics, Proc. Conf.
Chateau de Bois, France, May , Editions Frontières, . No citations.
958 S.C. B
& C.E. W
, Measurement of the S – S transition polarizability
in atomic cesium and an improved test of the standard model, Physical Review Letters 82,
pp. – , . e group has also measured the spatial distribution of the weak
charge, the so-called the anapole moment; see C.S. W & al., Measurement of parity
nonconservation and an anapole moment in cesium, Science 275, pp. – , . Cited
on page .
959 H. J & M. L
, Search for magnetic monopoles trapped in matter, Physical Review
Letters 75, pp. – , . Cited on page .
960 On proton decay rates, see the data of the particle data group, at http://pdg.web.cern.ch. Cited on page .
961 U. A
, W. B & H. F
, Comparison of grand uni ed theories with
elektroweak and strong coupling constants measured at LEP, Physics Letters 260, pp. –
, . is widely cited paper is the standard reference for this issue. Cited on page .
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– CS – this chapter will be made available in the future – CS – Dvipsbugw
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
ADVANC ED QUANTUM THEORY (NOT
YET AVAI L ABLE)
IX
C
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
C
X
QUANTUM PHYSICS
IN A NUTSHELL
Q
’
–
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C to classical physics, quantum theory is remarkably more omplex. e basic idea however, is simple: in nature there is a minimum hange, or a minimum action, or again, a minimum angular momentum ħ . e minimum action leads to all the strange observations made in the microscopic domain, such as wave behaviour of matter, tunnelling, indeterminacy relations, randomness in measurements, quantization of angular momentum, pair creation, decay, indistinguishability and particle reactions. e mathematics is o en disturbingly involved. Was this part of the walk worth the e ort? It was. e accuracy is excellent and the results profound. We give an overview of both and then turn to the list of questions that are still le open.
A
Quantum theory improved the accuracy of predictions from the few – if any – digits common in classical mechanics to the full number of digits – sometimes fourteen – that can be measured today. e limited precision is usually not given by the inaccuracy of theory, it is given by the measurement accuracy. In other words, the agreement is only limited by the amount of money the experimenter is willing to spend. Table shows this in more detail.
TA B L E 71 Some comparisons between classical physics, quantum theory and experiment
O
C -P
M
-C
-
-
a
b
-
Simple motion of bodies
Indeterminacy
Wavelength of matter none beams
Tunnelling rate in alpha decay
Compton wavelength
none
∆x∆p ħ λp = πħ
τ = ...
λc = h mec
( − )ħ
k
( − )ħ
k
( − )τ
.M
( − )λ
k
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
O
C -P
M
-C
b
-
-
a
-
Pair creation rate
Radiative decay time in hy- none drogen
Smallest action and angular momentum
Casimir e ect
Colours of objects
Lamb shi
none
Rydberg constant
none
Stefan–Boltzmann con- none stant
Wien displacement con- none stant
Refractive index of ...
none
Photon-photon scattering
Particle and interaction properties
Electron gyromagnetic ra- or tio
Z boson mass
none
proton mass
none
reaction rate
Composite matter properties
Atom lifetime
µs
Molecular size
none
Von Klitzing constant
AC Josephson constant
Heat capacity of metals at K
Water density
none
Minimum electr. conductivity
Proton lifetime
µs
... τn
...
M
...
k
ħ
(
− )ħ
k
ma A = (π ħc) ( r ) ( − ) ma
k
∆λ = . ( ) MHz ( − ) ∆λ
k
R = mecα h σ=π k ħ c
( − )R
k
( ë − )σ
k
b = λmaxT
( − )b
k
...
...
...
...
...
M
.
()
mZ = mW ( + sin θW ) ( %) mp
...
. ()
(
− ) mZ
mp = . yg
...
M
M M
from QED h e =µ c α eh
JK
... G= e ħ
a
k
within −
k
( − )h e M
( −) eh M
< − JK
k
kg m
k
G( − )
k
a
M
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Page 959 Page 1154
Challenge 1400 n
a. All these predictions are calculated from the quantities of Table , and no other input. eir most precise experimental values are given in Appendix B.
b. Sometimes the cost for the calculation of the prediction is higher than that of its measurement. (Can you spot the examples?) e sum of the two is given.
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Page 959
We notice that the predicted values are not noticeably di erent from the measured ones. If we remember that classical physics does not allow to calculate any of the predicted values we get an idea of the progress quantum physics has allowed. But despite this impressive agreement, there still are unexplained observations. In fact, these unexplained observations provide the input for the calculations just cited; we list them in detail below, in Table .
In summary, in the microscopic domain we are le with the impression that quantum theory is in perfect correspondence with nature; despite prospects of fame and riches, despite the largest number of researchers ever, no contradiction with observation has been found yet.
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P
Deorum o ensae diis curae.
“ ” Voltaire, Traité sur la tolérance.
All of quantum theory can be resumed in two sentences.
In nature, actions smaller than ħ = . ë − Js are not observed.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
All intrinsic properties in nature – with the exception of mass – such as electric charge, spin, parities, etc., appear as integer numbers; in composed systems they either add or multiply.
e second statement in fact results from the rst. e existence of a smallest action in nature directly leads to the main lesson we learned about motion in the second part of our adventure:
If it moves, it is made of particles.
Page 1174 Challenge 1401 e
is statement applies to everything, thus to all objects and to all images, i.e. to matter and to radiation. Moving stu is made of quanta. Stones, water waves, light, sound waves, earthquakes, gelatine and everything else we can interact with is made of particles. We started the second part of our mountain ascent with the title question: what is matter and what are interactions? Now we know: they are composites of elementary particles.
To be clear, an elementary particle is a countable entity, smaller than its own Compton wavelength, described by energy, momentum, and the following complete list of intrinsic properties: mass, spin, electric charge, parity, charge parity, colour, isospin, strangeness, charm, topness, beauty, lepton number, baryon number and R-parity. Experiments so far failed to detect a non-vanishing size for any elementary particle.
Moving entities are made of particles. To see how deep this result is, you can apply it to all those moving entities for which it is usually forgotten, such as ghosts, spirits, angels, nymphs, daemons, devils, gods, goddesses and souls. You can check yourself what happens when their particle nature is taken into account.
From the existence of a minimum action, quantum theory deduces all its statements about particle motion. We go through the main ones.
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ere is no rest for microscopic particles. All objects obey the indeterminacy principle, which states that the indeterminacies in position x and momentum p follow
∆x∆p ħ with ħ = . ë − Js
(638)
and making rest an impossibility. e state of particles is de ned by the same observables as in classical physics, with the di erence that observables do not commute. Classical physics appears in the limit that the Planck constant ħ can e ectively be set to zero.
Quantum theory introduces a probabilistic element into motion. It results from the minimum action value through the interactions with the baths in the environment of any system.
Large number of identical particles with the same momentum behave like waves. e so-called de Broglie wavelength λ is given by the momentum p of a single particle through
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
λ
=
h p
=
πħ p
(639)
both in the case of matter and of radiation. is relation is the origin of the wave behaviour of light. e light particles are called photons; their observation is now standard practice. All waves interfere, refract and di ract. is applies to electrons, atoms, photons and molecules. All waves being made of particles, all waves can be seen, touched and moved. Light for example, can be ‘seen’ in photon-photon scattering, can be ‘touched’ using the Compton e ect and it can be ‘moved’ by gravitational bending. Matter particles, such as molecules or atoms, can be seen, e.g. in electron microscopes, as well as touched and moved, e.g. with atomic force microscopes. e interference and di raction of wave particles is observed daily in the electron microscope.
Particles cannot be enclosed. Even though matter is impenetrable, quantum theory shows that tight boxes or insurmountable obstacles do not exist. Waiting long enough always allows to overcome boundaries, since there is a nite probability to overcome any obstacle. is process is called tunnelling when seen from the spatial point of view and is called decay when seen from the temporal point of view. Tunnelling explains the working of television tubes as well as radioactive decay.
Particles are described by an angular momentum called spin, specifying their behaviour under rotations. Bosons have integer spin, fermions have half integer spin. An even number of bound fermions or any number of bound bosons yield a composite boson; an odd number of bound fermions or an in nite number of interacting bosons yield a lowenergy fermion. Solids are impenetrable because of the fermion character of its electrons in the atoms.
Identical particles are indistinguishable. Radiation is made of bosons, matter of fermions. Under exchange, fermions commute at space-like separations, whereas bosons anticommute. All other properties of quantum particles are the same as for classical particles, namely countability, interaction, mass, charge, angular momentum, energy, momentum, position, as well as impenetrability for matter and penetrability for radiation.
In collisions, particles interact locally, through the exchange of other particles. When matter particles collide, they interact through the exchange of virtual bosons, i.e. o -shell bosons. Motion change is thus due to particle exchange. Exchange bosons of even spin
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
’
–
Ref. 962
mediate only attractive interactions. Exchange bosons of odd spin mediate repulsive interactions as well.
Quantum theory de nes elementary particles as particles smaller than e properties of collisions imply the existence of antiparticles, as regularly observed
in experiments. Elementary fermions, in contrast to many elementary bosons, di er from their antiparticles; they can be created and annihilated only in pairs. Apart from neutrinos, elementary fermions have non-vanishing mass and move slower than light.
Images, made of radiation, are described by the same properties as matter. Images can only be localized with a precision of the wavelength λ of the radiation producing it.
e appearance of Planck’s constant ħ implies that length scales exist in nature. Quantum theory introduces a fundamental jitter in every example of motion. us the in nitely small is eliminated. In this way, lower limits to structural dimensions and to many other measurable quantities appear. In particular, quantum theory shows that it is impossible that on the electrons in an atom small creatures live in the same way that humans live on the Earth circling the Sun. Quantum theory shows the impossibility of Lilliput.
Clocks and metre bars have nite precision, due to the existence of a smallest action and due to their interactions with baths. On the other hand, all measurement apparatuses must contain baths, since otherwise they would not be able to record results.
Quantum physics leaves no room for cold fusion, astrology, teleportation, telekinesis, supernatural phenomena, multiple universes, or faster than light phenomena – the EPR paradox notwithstanding.
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Page 713
Quantum eld theory is that part of quantum theory that includes the process of transformation of particles into each other. e possibility of transformation results from the existence of a minimum action. Transformations have several important consequences.
Quantum electrodynamics is the quantum eld description of electromagnetism. Like all the other interactions, its Lagrangian is determined by the gauge group, the requirements of space-time (Poincaré) symmetry, permutation symmetry and renormalizability.
e latter requirement follows from the continuity of space-time. rough the e ects of virtual particles, QED describes decay, pair creation, vacuum energy, Unruh radiation for accelerating observers, the Casimir e ect, i.e. the attraction of neutral conducting bodies, and the limit for the localization of particles. In fact, an object of mass m can be localized only within intervals of the Compton wavelength
λC =
h mc
=
πħ mc
,
(640)
where c is the speed of light. At the latest at these distances we must abandon the classical description and use quantum eld theory. Quantum eld theory introduces corrections to classical electrodynamics; among others, the nonlinearities thus appearing produce small departures from the superposition principle for electromagnetic elds, resulting in photon-photon scattering.
Composite matter is separable because of the nite interaction energies of the con-
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Page 1174 Page 1174
stituents. Atoms are made of a nucleus made of quarks, and of electrons. ey provide an e ective minimal length scale to all everyday matter.
Elementary particles have the same properties as either objects or images, except divisibility. e elementary fermions (objects) are: the six leptons electron, muon, tau, each with its corresponding neutrino, and the six quarks. e elementary bosons (images) are the photon, the eight gluons and the two weak interaction bosons.
Quantum chromodynamics, the eld theory of the strong interactions, explains the masses of mesons and baryons through its descriptions as bound quark states. At fundamental scales, the strong interaction is mediated by the elementary gluons. At femtometer scales, the strong interaction e ectively acts through the exchange of spin pions, and is thus strongly attractive.
e theory of electroweak interactions describes the uni cation of electromagnetism and weak interactions through the Higgs mechanism and the mixing matrix.
Objects are composed of particles. Quantum eld theory provides a complete list of the intrinsic properties which make up what is called an ‘object’ in everyday life, namely the same which characterize particles. All other properties of objects, such as shape, temperature, (everyday) colour, elasticity, density, magnetism, etc., are merely combinations of the properties from the particle properties. In particular, quantum theory speci es an object, like every system, as a part of nature interacting weakly and incoherently with its environment.
Since quantum theory explains the origin of material properties, it also explains the origin of the properties of life. Quantum theory, especially the study of the electroweak and the strong forces, has allowed to give a common basis of concepts and descriptions to materials science, nuclear physics, chemistry, biology, medicine and to most of astronomy.
For example, the same concepts allow to answer questions such as why water is liquid at room temperature, why copper is red, why the rainbow is coloured, why the Sun and the stars continue to shine, why there are about elements, where a tree takes the material to make its wood and why we are able to move our right hand at our own will.
Matter objects are permanent because, in contrast to radiation, matter particles can only disappear when their antiparticles are present. It turns out that in our environment antimatter is almost completely absent, except for the cases of radioactivity and cosmic rays, where it appears in tiny amounts.
e particle description of nature, e.g. particle number conservation, follows from the possibility to describe interactions perturbatively. is is possible only at low and medium energies. At extremely high energies the situation changes and non-perturbative e ects come into play.
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I
?
Studying nature is like experiencing magic. Nature o en looks di erent from what it is. During magic we are fooled – but only if we forget our own limitations. Once we start to see ourselves as part of the game, we start to understand the tricks. at is the fun of it.
e same happens in physics.
**
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’
–
e world looks irreversible, even though it isn’t. We never remember the future. We are fooled because we are macroscopic.
** e world looks decoherent, even though it isn’t. We are fooled again because we are macroscopic.
** ere are no clocks possible in nature. We are fooled because we are surrounded by a huge number of particles.
** Motion seems to disappear, even though it is eternal. We are fooled again, because our senses cannot experience the microscopic domain.
** e world seems dependent on the choice of the frame of reference, even though it is not. We are fooled because we are used to live on the surface of the Earth.
** Objects seem distinguishable, even though the statistical properties of their components show that they are not. We are fooled because we live at low energies.
** Matter looks continuous, even though it isn’t. We are fooled because of the limitations of our senses.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
In short, our human condition permanently fools us. e answer to the title question is a rmative: quantum theory is magic. at is its main attraction.
T
Another summary of our walk so far is given by the ultimate product warning, which according to certain well-informed lawyers should be printed on every cans of beans and Ref. 963 on every product package. It shows in detail how deeply our human condition fools us.
Warning: care should be taken when looking at this product:
It emits heat radiation.
Bright light has the e ect to compress this product.
Warning: care should be taken when touching this product:
Part of it could heat up while another part cools down, causing severe burns.
Warning: care should be taken when handling this product:
is product consists of at least .
% empty space.
is product contains particles moving with speeds higher than one million kilometres per hour.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Every kilogram of this product contains the same amount of energy as liberated by about one hundred nuclear bombs.* In case this product is brought in contact with antimatter, a catastrophic explosion will occur. In case this product is rotated, it will emit gravitational radiation.
Warning: care should be taken when transporting this product:
e force needed depends on its velocity, as does its weight. is product will emit additional radiation when accelerated. is product attracts, with a force that increases with decreasing distance, every other object around, including its purchaser’s kids.
Warning: care should be taken when storing this product:
It is impossible to keep this product in a speci c place and at rest at the same time. Except when stored underground at a depth of several kilometres, over time cosmic radiation will render this product radioactive.
is product may disintegrate in the next years. It could cool down and li itself into the air. Parts of this product are hidden in other dimensions.
is product warps space and time in its vicinity, including the storage container. Even if stored in a closed container, this product is in uenced and in uences all other objects in the universe, including your parents in law.
is product can disappear from its present location and reappear at any random place in the universe, including your neighbour’s garage.
Warning: care should be taken when travelling away from this product:
It will arrive at the expiration date before the purchaser does so.
Warning: care should be taken when using this product:
Any use whatsoever will increase the entropy of the universe. e constituents of this product are exactly the same as those of any other object in the
universe, including those of rotten sh. e use could be disturbed by the (possibly) forthcoming collapse of the universe.
e impression of a certain paranoid side to physics is purely coincidental.
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Ref. 964
* A standard nuclear warhead has an explosive yield of about 0.2 megatons (implied is the standard explosive trinitrotoluene or TNT), about thirteen times the yield of the Hiroshima bomb, which was kilotonne. A megatonne is de ned as Pcal= . PJ, even though TNT delivers about 5 % slightly less energy that this value. In other words, a megaton is the energy content of about g of matter. at is less than a handful for most
solids or liquids.
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T
We can summarize quantum physics with a simple statement: quantum physics is the description of matter and radiation without the concept of in nitely small. Matter and radiation are described by nite quantities. We had already eliminated the in nitely large in our exploration of relativity. On the other hand, some types of in nities remain. We had to retain the in nitely small in the description of space or time, and in topics related to them, such as renormalization. We did not manage to eliminate all in nities yet. We are thus not yet at the end of our quest. Surprisingly, we shall soon nd out that a completely
nite description of all of nature is equally impossible. To nd out more, we focus on the path that remains to be followed.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
W
?
e material gathered in this second part of our mountain ascent, together with the earlier summary of general relativity, allows us to describe all observed phenomena connected to motion. erefore, we are also able to provide a complete list of the unexplained properties of nature. Whenever we ask ‘why?’ about an observation and continue doing so a er each answer, we arrive at one of the points listed in Table .
TA B L E 72 Everything quantum field theory and general relativity do not explain; in other words, a list of the only experimental data and criteria available for tests of the unified description of motion
O
P
Local quantities, from quantum theory
αem αw αs mq ml mW θW β ,β ,β
the low energy value of the electromagnetic coupling constant the low energy value of the weak coupling constant the low energy value of the strong coupling constant the values of the quark masses the values of lepton masses (or , if neutrinos have masses) the values of the independent mass of the W vector boson the value of the Weinberg angle three mixing angles (or , if neutrinos have masses)
θCP
the value of the CP parameter
θst
the value of the strong topological angle
the number of particle generations
+
the number of space and time dimensions
. nJ m
the value of the observed vacuum energy density or cosmological constant
Global quantities, from general relativity
. ( ) ë m (?) the distance of the horizon, i.e. the ‘size’ of the universe (if it makes sense)
(?)
the number of baryons in the universe, i.e. the average matter density in the
universe (if it makes sense)
(?)
the initial conditions for more than particle elds in the universe, includ-
ing those at the origin of galaxies or stars (if or as long as they make sense)
Local structures, from quantum theory
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O
P
S(n)
the origin of particle identity, i.e. of permutation symmetry
Ren. group
the renormalization properties, i.e. the existence of point particles
SO( , )
the origin of Lorentz (or Poincaré) symmetry
(i.e. of spin, position, energy, momentum)
C
the origin of the algebra of observables
Gauge group
the origin of gauge symmetry
(and thus of charge, strangeness, beauty, etc.)
in particular, for the standard model:
U( )
the origin of the electromagnetic gauge group (i.e. of the quantization of elec-
tric charge, as well as the vanishing of magnetic charge)
SU( )
the origin of weak interaction gauge group
SU( )
the origin of strong interaction gauge group
Global structures, from general relativity maybe R S (?) the unknown topology of the universe (if it makes sense)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
e table has several notable aspects.* First of all, neither quantum mechanics nor general relativity explain any property unexplained in the other eld. e two theories do not help each other; the unexplained parts of both elds simply add up. Secondly, both in quantum theory and in general relativity, motion still remains the change of position with time. In short, in the rst two parts of this walk we did not achieve our goal: we still do not understand motion. Our basic questions remain: What is time and space? What is mass? What is charge and what are the other properties of objects? What are elds? Why are all the electrons the same?
We also note that Table lists extremely di erent concepts. at means that at this point of our walk there is a lot we do not understand. Finding the answers will not be easy, but will require e ort.
On the other hand, the list of unexplained properties of nature is also short. e description of nature our adventure has produced so far is concise and precise. No discrepancies from experiments are known. In other words, we have a good description of motion in practice. Going further is unnecessary if we only want to improve measurement precision. Simplifying the above list is mainly important from the conceptual point of view. For this reason, the study of physics at university o en stops at this point. However, even though we have no known discrepancies with experiments, we are not at the top of Motion Mountain, as Table shows.
An even more suggestive summary of the progress and open issues of physics is shown in Figure . From one corner of a cube, representing Galilean physics, three edges – la-
* Every now and then, researchers provide other lists of open questions. However, they all fall into the list above. e elucidation of dark matter and of dark energy, the details of the big bang, the modi cations of general relativity by quantum theory, the mass of neutrinos, the quest for unknown elementary particles such as the in aton eld, magnetic monopoles or others, the functioning of cosmic high-energy particle accelerators, the stability or decay of protons, the origins of the heavy chemical elements, other interactions between matter and radiation or the possibility of higher spatial dimensions are questions that all fall into the table above.
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Classical gravitation 1685
Quantum gravity ca 1990
Classical mechanics 1650
G
ħ, e, k Quantum theory 1925
c
General relativity 1915
Theory of motion 2020?
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Special relativity 1905
Quantum field theory 1950
F I G U R E 359 A simplified history of the description of motion in physics, by giving the limits to motion included in each description
belled G, c and ħ, e, k – lead to classical gravity, special relativity and quantum theory. Each constant implies a limit to motion; in the corresponding theory, this limit is taken into account. From these rst level theories, corresponding parallel edges lead to general relativity, quantum eld theory and quantum gravity, which take into account two of the limits.* From the second level theories, all edges lead to the last missing corner; that is the theory of motion. It takes onto account all limits found so far. Only this theory is a full or uni ed description of motion. e important point is that we already know all limits to motion. To arrive at the last point, no new experiments are necessary. No new knowledge is required. We only have to advance in the right direction, with careful thinking. Reaching the nal theory of motion is the topic of the third part of our adventure.
Finally, we note from Table that all progress we can expect about the foundations of motion will take place in two speci c elds: cosmology and high energy physics.
H
M
M
Nowadays it is deemed chic to pretend that the adventure is over at the stage we have just reached.** e reasoning is as follows. If we change the values of the unexplained con-
Challenge 1402 ny Ref. 965
* Of course, Figure 359 gives a simpli ed view of the history of physics. A more precise diagram would use three di erent arrows for ħ, e and k, making the gure a ve-dimensional cube. However, not all of its corners would have dedicated theories (can you con rm this?), and moreover, the diagram would be much less appealing. ** Actually this attitude is not new. Only the arguments have changed. Maybe the greatest physicist ever, James Clerk Maxwell, already fought against this attitude over a hundred years ago: ‘ e opinion seems to
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stants from Table only ever so slightly, nature would look completely di erent from Ref. 966 what it does. e consequences have been studied in great detail; Table gives an over-
view of the results.
TA B L E 73 A selection of the consequences of changing the properties of nature
O
C
R
Local quantities, from quantum theory
αem
smaller:
larger:
+ %:
+ %:
αw
− %:
very weak:
+ %:
GF me Gme :
much larger:
αs
− %:
− %:
+ . %:
much larger:
n-p mass di er-larger: ence
smaller:
ml changes: e-p mass ratio
generations
smaller than me:
much di erent: much smaller: -:
:
only short lived, smaller and hotter stars; no Sun darker Sun, animals die of electromagnetic radiation, too much proton decay, no planets, no stellar explosions, no star formation, no galaxy formation quarks decay into leptons proton-proton repulsion makes nuclei impossible carbon nucleus unstable no hydrogen, no p-p cycle in stars, no C-N-O cycle no protons from quarks either no or only helium in the universe
no stellar explosions, faster stellar burning no deuteron, stars much less bright no C resonance, no life diproton stable, faster star burning carbon unstable, heavy nuclei unstable, widespread leukaemia neutron decays in proton inside nuclei; no elements
free neutron not unstable, all protons into neutrons during big bang; no elements protons would capture electrons, no hydrogen atoms, star life much shorter
no molecules
no solids only helium in nature no asymptotic freedom and con nement
Global quantities, from general relativity
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have got abroad that, in a few years, all great physical constants will have been approximately estimated, and that the only occupation which will be le to men of science will be to carry these measurements to another place of decimals. ... e history of science shows that even during that phase of her progress in which she devotes herself to improving the accuracy of the numerical measurement of quantities with which she has long been familiar, she is preparing the materials for the subjugation of new regions, which would have remained unknown if she had been contented with the rough methods of her early pioneers.’
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
O
C
R
horizon size
much smaller:
baryon number very di erent:
much higher:
Initial condition changes:
Moon mass
smaller:
Moon mass
larger:
Sun’s mass Sun’s mass Jupiter mass
smaller: larger: smaller:
Jupiter mass
larger:
Oort cloud ob-smaller: ject number
galaxy centre dis-smaller: tance
initial cosmicspeed + . %:
− . %:
vacuum energy change by
density
−:
+ dimensions di erent:
no people no smoothness no solar system
small Earth magnetic eld; too much cosmic radiation; widespread child skin cancer
large Earth magnetic eld; too little cosmic radiation; no evolution into humans
too cold for the evolution of life
Sun too short lived for the evolution of life
too many comet impacts on Earth; extinction of animal life
too little comet impacts on Earth; no Moon; no dinosaur extinction
no comets; no irregular asteroids; no Moon; still dinosaurs
irregular planet motion; supernova dangers
times faster universe expansion
universe recollapses a er
years
no atness
no atoms, no planetary systems
Local structures, from quantum theory
permutation symmetry
none:
Lorentz symmetry none:
U( )
di erent:
SU( )
di erent:
SU( )
di erent:
no matter
no communication possible no Huygens principle, no way to see anything no radioactivity, no Sun, no life no stable quarks and nuclei
Global structures, from general relativity
topology
other:
unknown; possibly correlated gamma ray bursts or star images at the antipodes
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Ref. 967 Challenge 1403 r
Some even speculate that the table can be condensed into a single sentence: if any parameter in nature is changed, the universe would either have too many or too few black holes. However, the proof of this condensed summary is not complete yet.
Table , on the e ects of changing nature, is overwhelming. Obviously, even the tiniest changes in the properties of nature are incompatible with our existence. What does this mean? Answering this question too rapidly is dangerous. Many fall into a common trap, namely to refuse admitting that the unexplained numbers and other properties need
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Page 683 Page 689
Challenge 1404 n
to be explained, i.e. deduced from more general principles. It is easier to throw in some irrational belief. e three most fashionable beliefs are that the universe is created or designed, that the universe is designed for people, or that the values are random, as our universe happens to be one of many others.
All these beliefs have in common that they have no factual basis, that they discourage further search and that they sell many books. Physicists call the issue of the rst belief ne tuning, and usually, but not always, steer clear from the logical errors contained in the so common belief in ‘creation’ discussed earlier on. However, many physicists subscribe to the second belief, namely that the universe is designed for people, calling it the anthropic principle, even though we saw that it is indistinguishable both from the simian principle or from the simple request that statements be based on observations. In , this belief has even become fashionable among older string theorists. e third belief, namely multiple universes, is a minority view, but also sells well.
Stopping our mountain ascent with a belief at the present point is not di erent from doing so directly at the beginning. Doing so used to be the case in societies which lacked the passion for rational investigation, and still is the case in circles which discourage the use of reason among their members. Looking for beliefs instead of looking for answers means to give up the ascent of Motion Mountain while pretending to have reached the top.
at is a pity. In our adventure, accepting the powerful message of Table is one of the most awe-inspiring, touching and motivating moments. ere is only one possible implication based on facts: the evidence implies that we are only a tiny part of the universe, but linked with all other aspects of it. Due to our small size and to all the connections with our environment, any imagined tiny change would make us disappear, like a water droplet is swept away by large wave. Our walk has repeatedly reminded us of this smallness and dependence, and overwhelmingly does so again at this point.
Having faced this powerful experience, everybody has to make up his own mind on whether to proceed with the adventure or not. Of course, there is no obligation to do so.
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W
?
e shortness of the list of unexplained aspects of nature means that no additional experimental data are available as check of the nal description of nature. Everything we need to arrive at the nal description of motion will probably be deduced from the experimental data given in this list, and from nothing else. In other words, future experiments will not help us – except if they change something in the list, as supersymmetry might do with the gauge groups or astronomical experiments with the topology issue.
is lack of new experimental data means that to continue the walk is a conceptual adventure only. We have to walk into storms raging near the top of Motion Mountain, keeping our eyes open, without any other guidance except our reason: this is not an adventure of action, but an adventure of the mind. And it is an incredible one, as we shall soon nd out. To provide a feeling of what awaits us, we rephrase the remaining issues in six simple challenges.
What determines colours? In other words, what relations of nature x the famous ne structure constant? Like the hero of Douglas Adams’ books, physicists know the answer to the greatest of questions: it is . . But they do not know the question.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
What xes the contents of a teapot? It is given by its size to the third power. But why
are there only three dimensions? Why is the tea content limited in this way?
Was Democritus right? Our adventure has con rmed his statement up to this point;
nature is indeed well described by the concepts of particles and of vacuum. At large scales,
relativity has added a horizon, and at small scales, quantum theory added vacuum energy
and pair creation. Nevertheless, both theories assume the existence of particles and the
existence of space-time, and neither predicts them. Even worse, both theories completely
fail to predict the existence of any of the properties either of space-time – such as its
dimensionality – or of particles – such as their masses and other quantum numbers. A
lot is missing.
Was Democritus wrong? It is o en said that the standard model has only about twenty
unknown parameters; this common mistake negates about initial conditions! To get
an idea of the problem, we simply estimate the number N of possible states of all particles
in the universe by
N=nvd p f
(641)
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where n is the number of particles, v is the number of variables (position, momentum, spin), d is the number of di erent values each of them can take (limited by the maximum of decimal digits), p is the number of visible space-time points (about ) and f is a factor expressing how many of all these initial conditions are actually independent of each other. We thus have the following number of possibilities
N= ë ë ë ëf=
ëf
(642)
Page 815 Page 157 Challenge 1405 e
from which the actual initial conditions have to be explained. ere is a small problem that we know nothing whatsoever about f . Its value could be , if all data were interdependent, or , if none were. Worse, above we noted that initial conditions cannot be de ned for the universe at all; thus f should be unde ned and not be a number at all! Whatever the case, we need to understand how all the visible particles get their states assigned from this range of options.
Were our e orts up to this point in vain? Quite at the beginning of our walk we noted that in classical physics, space and time are de ned using matter, whereas matter is de ned using space-time. Hundred years of general relativity and of quantum theory, including dozens of geniuses, have not solved this oldest paradox of all. e issue is still open at this point of our walk, as you might want to check by yourself.
e answers to these six questions de ne the top of Motion Mountain. Answering them means to know everything about motion. In summary, our quest for the unravelling of the essence of motion gets really interesting only from this point onwards!
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Ref. 968
“ at is why Leucippus and Democritus, who say that the atoms move always in the void and the unlimited, must say what movement is, and in what their natural motion consists. ” Aristotle, Treaty of the Heaven
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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B
962 An informative account of the world of psychokinesis and the paranormal is given by the
famous professional magician J
R , Flim- am!, Prometheus Books, Bu alo ,
as well as in several of his other books. See also the http://www.randi.org website. Cited on
page .
963 is way to look at things goes back to the text by S
H
&E
S
,
A call for more scienti c truth in product warning labels, Journal of Irreproducible Results
36, nr. , . Cited on page .
964 J. M , e yields of the Hiroshima and Nagasaki nuclear explosions, Technical Report LA- , Los Alamos National Laboratory, September . Cited on page .
965 J
C
M
, Scienti c Papers, 2, p. , October . Cited on page .
966 A good introduction is C.J. H
, Why the universe is just so, Reviews of Modern Physics
72, pp. – , . Most of the material of Table is from the mighty book by J D.
B
&F
J. T
, e Anthropic Cosmological Principle, Oxford University
Press, . Discarding unrealistic options is also an interesting pastime. See for example
the reasons why life can only be carbon-based, as explained in the essay by I. A
,e
one and only, in his book e Tragedy of the Moon, Doubleday, Garden City, New York, .
Cited on page .
967 L. S
, e fate of black hole singularities and the parameters of the standard models
of particle physics and cosmology, http://www.arxiv.org/abs/gr-qc/
. Cited on page
.
968 A
, Treaty of the heaven, III, II, b . See J -P D
présocratiques, Folio Essais, Gallimard, p. , . Cited on page .
, Les écoles
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
I
BACTERIA, FLIES AND KNOTS
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“La première et la plus belle qualité de la nature est le mouvement qui l’agite sans cesse ; mais ce mouvement n’est qu’une suite perpétuelle de crimes ; ce n’est que par des crimes qu’elle le conserve. Donatien de Sade, Justine, ou les malheurs de la vertu.*
” entities, in particular jelly sh or amoebas, open up a fresh vision of the
orld of motion, if we allow to be led by the curiosity to study them in detail.
We have missed many delightful insights by leaving them aside. In particular,
wobbly entities yield surprising connections between shape change and motion which will be of great use in the last part of our mountain ascent. Instead of continuing to look at the smaller and smaller, we now take a look back, towards everyday motion and its mathematical description.
To enjoy this intermezzo, we change a dear habit. So far, we always described any general example of motion as composed of the motion of point particles. is worked well in classical physics, in general relativity and in quantum theory; we based the approach on the silent assumption that during motion, each point of a complex system can be followed separately. We will soon discover that this assumption is not realized at smallest scales. erefore the most useful description of motion of extended bodies uses methods that do not require that body parts be followed one by one. We explore this issue in this intermezzo; doing so is a lot of fun in its own right.
If we imagine particles as extended entities – as we soon will have to – a particle moving through space is similar to a dolphin swimming through water or to a bee ying through air. Let us explore how these animals do this.
B
When a butter y passes by, as can happen to anybody ascending a mountain as long as owers are present, we can stop a moment to appreciate a simple fact: a butter y ies, and
it is rather small. If you leave some cut fruit in the kitchen until it is rotten, we nd the even smaller fruit ies. If you have ever tried to build small model aeroplanes, or if you even only compare them to paper aeroplanes (probably the smallest man-made ying
* ‘ e primary and most beautiful of nature’s qualities is motion, which agitates her at all times; but this motion is simply a perpetual consequence of crimes; she conserves it by means of crimes only.’ Donatien Alphonse François de Sade (1740–1814) is the French writer from whom the term ‘sadism’ was deduced.
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F I G U R E 360 A flying fruit fly, tethered to a string
F I G U R E 361 Vortices around a butterfly wing (© Robert Srygley/Adrian Thomas)
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Ref. 970
thing you ever saw) you start to get a feeling for how well evolution optimized insects. Compared to paper planes, insects also have engines, apping wings, sensors, navigation systems, gyroscopic stabilizers, landing gear and of course all the features due to life, reproduction and metabolism, built into an incredibly small volume. Evolution really is an excellent engineering team. e most incredible yers, such as the common house y (Musca domestica), can change ying direction in only ms, using the stabilizers that nature has built by reshaping the original second pair of wings. Human engineers are getting more and more interested in the technical solutions evolution has chosen and are trying to achieve the same miniaturization. e topic is extremely vast, so that we will pick out only a few examples.
How does an insect such as a fruit y (Drosophila melanogaster) y? e li m generated by a xed wing follows the relation
m = f Av ρ
(643)
Ref. 969 Challenge 1406 e
Challenge 1407 n
where A is the surface of the wing, v is the speed of the wing in the uid of density ρ. e factor f is a pure number, usually with a value between . and . , that depends on
the angle of the wing and its shape; here we use the average value . . For a Boeing , the surface is m , the top speed is m s; at an altitude of km the density of air is only a quarter of that on the ground, thus only . kg m . We deduce (correctly) that a Boeing has a mass of about ton. For bumblebees with a speed of m s and a wing surface of cm , we get a li ed mass of about mg, much less than the weight of the bee, namely about g. In other words, a bee, like any other insect, cannot y if it keeps its wings xed. It could not y with xed wings even if it had propellers! erefore, all insects must move their wings, in contrast to aeroplanes, not only to advance or to gain height, but also to simply remain airborne. Aeroplanes generate enough li with xed wings. Indeed, if you look at ying animals, you note that the larger they are, the less they need to move their wings.
Can you deduce from equation ( ) that birds or insects can y but people cannot? e formula also (partly) explains why human powered aeroplanes must be so large.*
* e rest of the explanation requires some aerodynamics, which we will not study here. Aerodynamics
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Ref. 971 Ref. 972
But how do insects, small birds, ying sh or bats have to move their wings? is is a
tricky question. In fact, the answer is just being uncovered by modern research. e main
point is that insect wings move in a way to produce eddies at the front edge which in turn
thrust the insect upwards. e aerodynamic studies of butter ies – shown in Figure –
and the studies of enlarged insect models moving in oil instead of in air are exploring the
way insects make use of vortices. Researchers try to understand how vortices allow con-
trolled ight at small dimensions. At the same time, more and more mechanical birds and
model ‘aeroplanes’ that use apping wings for their propulsion are being built around the
world. e eld is literally in full swing.* e aim is to reduce the size of ying machines.
However, none of the human-built systems is yet small enough that it actually requires
wing motion to y, as is the case for insects.
Formula ( ) also shows what is necessary for take-o and landing. e li of wings
decreases for smaller speeds. us both animals and aeroplanes increase their wing sur-
face in these occasions. But even strongly apping enlarged wings o en are not su cient
at take-o . Many ying animals, such as swallows, therefore avoid landing completely.
For ying animals which do take o from the ground, nature most commonly makes
them hit the wings against each other, over their back, so that when the wings separate
again, the vacuum between them provides the rst li . is method is used by insects
and many birds, such as pheasants. As every hunter knows, pheasants make a loud ‘clap’
when they take o .
Both wing use and wing construction thus depend
on size. ere are four types of wings in nature. First
of all, all large ying objects, such aeroplanes and
large birds, y using xed wings, except during take-
o and landing. Second, common size birds use ap-
ping wings. ese rst two types of wings have a thick-
ness of about to % of the wing depth. At smaller dimensions, a third wing type appears, as seen in
Figure to be inserted
dragon ies and other insects. At these scales, at Reyn-
olds numbers of around
and below, thin mem- F I G U R E 362 Two large wing types
brane wings are the most e cient. e Reynolds num-
ber measures the ratio between inertial and viscous e ects in a uid. It is de ned as
R
=
lvρ η
(644)
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where l is a typical length of the system, v the speed, ρ the density and η the dynamic viscosity of the uid.** A Reynolds number much larger than one is typical for rapid air
shows that the power consumption, and thus the resistance of a wing with given mass and given cruise speed, is inversely proportional to the square of the wingspan. Large wingspans with long slender wings are thus of advantage in (subsonic) ying, especially when energy needs to be conserved. * e website http://www.aniprop.de presents a typical research approach and the sites http://ovirc.free.fr and http://www.ornithopter.org give introductions into the way to build such systems for hobbyists. ** e viscosity is the resistance to ow a uid poses. It is de ned by the force F necessary to move a layer of surface A with respect to a second, parallel one at distance d; in short, the (coe cient of) dynamic viscosity is de ned as η = d F A v. e unit is kg s m or Pa s or N s m , once also called P or 10 poise. In other
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Ref. 973
ow and fast moving water. In fact, the Reynolds numbers speci es what is meant by a
‘rapid’ or ‘ uid’ ow on one hand, and a ‘slow’ or ‘viscous’ ow on the other. e rst three
wing types are all for rapid ows.
e fourth type of wings is found at the smallest possible dimensions, for insects smal-
ler than one millimetre; their wings are not membranes at all. Typical are the cases of
thrips and of parasitic wasps, which can be as small as . mm. All these small insects
have wings which consist of a central stalk surrounded by hair. In fact, Figure shows
that some species of thrips have wings which look like miniature toilet brushes.
At even smaller dimensions, corresponding to
Reynolds number below , nature does not use
wings any more, though it still makes use of air trans-
port. In principle, at the smallest Reynolds numbers
gravity plays no role any more, and the process of y-
ing merges with that of swimming. However, air cur-
rents are too strong compared with the speeds that
such a tiny system could realize. No active naviga-
tion is then possible any more. At these small dimen-
sions, which are important for the transport through air of spores and pollen, nature uses the air currents for passive transport, making use of special, but xed
F I G U R E 363 The wings of a few types of insects smaller than mm (thrips,
Encarsia, Anagrus, Dicomorpha)
shapes.
(HortNET )
We can summarize that active ying is only pos-
sible through shape change. Only two types of shape changes are possible: that of pro-
pellers (or turbines) and that of wings.* Engineers are studying with intensity how these
shape changes have to take place in order to make ying most e ective. Interestingly, the
same challenge is posed by swimming.
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S
Swimming is a fascinating phenomenon. e Greeks argued that the ability of sh to swim is a proof that water is made of atoms. If atoms would not exist, a sh could not advance through it. Indeed, swimming is an activity that shows that matter cannot be continuous. Studying swimming can thus be quite enlightening. But how exactly do sh swim?
Whenever dolphins, jelly sh, submarines or humans swim, they take water with their ns, body, propellers, hands or feet and push it backwards. Due to momentum conserva-
Challenge 1408 ny
words, given a horizontal tube, the viscosity determines how strong the pump needs to be to pump the uid
through the tube at a given speed. e viscosity of air °C is . − kg s m or µPa s and increases
with temperature. In contrast, the viscosity of liquids decreases with temperature. (Why?) e viscosity of
water at °C is . mPa s, at °C it is . mPa s (or cP), and at °C is . mPa s. Hydrogen has a viscosity
smaller than µPa s, whereas honey has Pa s and pitch MPa s.
Physicists also use a quantity ν called the kinematic viscosity. It is de ned with the help of the mass density
of the uid as ν = η ρ and is measured in m s, once called stokes. e kinematic viscosity of water at
°C is mm s (or cSt). One of the smallest values is that of acetone, with . mm s; a larger one is
glycerine, with mm s. Gases range between mm s and mm s.
* e book by J B
, Insects in Flight, 1992. is a wonderful introduction into the biomech-
anics of insects, combining interesting science and beautiful photographs.
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Ref. 976
tion they then move forward.* In short, people swim in the same way that reworks or rockets y: by throwing matter behind them. Does all swimming work in this way? In particular, do small organisms advancing through the molecules of a liquid use the same method? No.
It turns out that small organisms such as bacteria do not have the capacity to propel or accelerate water against their surroundings. From far away, the swimming of microorganisms thus resembles the motion of particles through vacuum. Like microorganisms, also particles have nothing to throw behind them. Indeed, the water remains attached around a microorganism without ever moving away from it. Physically speaking, in these cases of swimming the kinetic energy of the water is negligible. In order to swim, unicellular beings thus need to use other e ects. In fact, their only possibility is to change their body shape in controlled ways.
Let us go back to everyday scale for a moment. Swimming scallops, molluscs up to a few cm in size, can be used to clarify the di erence between macroscopic and microscopic swimming. Scallops have a double shell connected by a hinge that they can open and close. If they close it rapidly, water is expelled and the mollusc is accelerated; the scallop then can glide for a while through the water. en the scallop opens the shell again, this time slowly, and repeats the feat. When swimming, the larger F I G U RE 364 A swimming scallop (here scallops look like clockwork false teeth. If we re- from the genus Chlamys) (© Dave duce the size of the scallop by a thousand times to Colwell) the size of single cells we get a simple result: such a tiny scallop cannot swim.
e origin of the lack of scalability of swimming methods is the changing ratio between inertial and dissipative e ects at di erent scales. is ratio is measured by the Reynolds number. For the scallop the Reynolds number is about , which shows that when it swims, inertial e ects are much more important than dissipative, viscous e ects. For a bacterium the Reynolds number is much smaller than , so that inertial e ects e ectively play no role. ere is no way to accelerate water away from a bacterial-sized scallop, and thus no way to glide. In fact one can even show the stronger result that no cell-sized being can move if the shape change is the same in the two halves of the motion (opening and closing). Such a shape change would simply make it move back and forward. us there is no way to move at cell dimensions with a method the scallop uses on centimetre scale; in fact the so-called scallop theorem states that no microscopic system can swim if it uses movable parts with only one degree of freedom.
Microorganisms thus need to use a more evolved, two-dimensional motion of their shape to be able to swim. Indeed, biologists found that all microorganisms use one of the following three swimming styles:
— Microorganisms of compact shape of diameter between µm and about mm, use
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Page 73 * Fish could use propellers, as the arguments against wheels we collected at the beginning of our walk do not apply for swimming. But propellers with blood supply would be a weak point in the construction, and thus in the defence of a sh.
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figure to be added
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F I G U R E 365 Ciliated and flagellate motion
Ref. 977
Ref. 980 Ref. 984 Ref. 981 Page 73
cilia. Cilia are hundreds of little hairs on the surface of the organism. e organisms move the cilia in waves wandering around their surface, and these surface waves make the body advance through the uid. All children watch with wonder Paramecium, the unicellular animal they nd under the microscope when they explore the water in which some grass has been le for a few hours. Paramecium, which is between µm and µm in size, as well as many plankton species* use cilia for its motion. e cilia and their motion are clearly visible in the microscope. A similar swimming method is even used by some large animals; you might have seen similar waves on the borders of certain ink sh; even the motion of the manta (partially) belongs into this class. Ciliate motion is an e cient way to change the shape of a body making use of two dimensions and thus avoiding the scallop theorem. — Sperm and eukaryote microorganisms whose sizes are in the range between µm and
µm swim using an (eukaryote) agellum.** Flagella, Latin for ‘small whips’, work like exible oars. Even though their motion sometimes appears to be just an oscillation, agella get a kick only during one half of their motion, e.g. at every swing to the le . Flagella are indeed used by the cells like miniature oars. Some cells even twist their agellum in a similar way that people rotate an arm. Some microorganisms, such as Chlamydomonas, even have two agella which move in the same way as people move their legs when they perform the breast stroke. Most cells can also change the sense in which the agellum is kicked, thus allowing them to move either forward or backward. rough their twisted oar motion, bacterial agella avoid retracing the same path when going back and forward. As a result, the bacteria avoid the scallop theorem and manage to swim despite their small dimensions. e exible oar motion they use is an example of a non-adiabatic mechanism; an important fraction of the energy is dissipated. — e smallest swimming organisms, bacteria with sizes between . µm and µm, swim using bacterial agella. ese agella, also called prokaryote agella, are di erent from the ones just mentioned. Bacterial agella move like turning corkscrews. ey are used by the famous Escherichia coli bacterium and by all bacteria of the genus Salmonella.
is type of motion is one of the prominent exceptions to the non-existence of wheels in nature; we mentioned it in the beginning of our walk. Corkscrew motion is an example of an adiabatic mechanism.
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Ref. 978
* See the http://www.liv.ac.uk/ciliate/ website for an overview. ** e largest sperm, of cm length, are produced by the . mm sized Drosophila bifurca y, a relative of the famous Drosophila melanogaster. Even when thinking about the theory of motion, it is impossible to avoid thinking about sex.
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F I G U R E 366 A well-known ability of cats
Ref. 979
A Coli bacterium typically has a handful of agella, each about nm thick and of corkscrew shape, with up to six turns; the turns have a ‘wavelength’ of . µm. Each
agellum is turned by a sophisticated rotation motor built into the cell, which the cell can control both in rotation direction and in angular velocity. For Coli bacteria, the range is between 0 and about Hz.
A turning agellum does not propel a bacterium like a propeller; as mentioned, the velocities involved are much too small, the Reynolds number being only about − . At these dimensions and velocities, the e ect is better described by a corkscrew turning in honey or in cork: a turning corkscrew produces a motion against the material around it, in the direction of the corkscrew axis. e agellum moves the bacterium in the same way that a corkscrew moves the turning hand with respect to the cork.
Ref. 982
Note that still smaller bacteria do not swim at all. Each bacterium faces a minimum swimming speed requirement: is must outpace di usion in the liquid it lives in. Slow swimming capability makes no sense; numerous microorganisms therefore do not manage or do not try to swim at all. Some microorganisms are specialized to move along liquid–air interfaces. Others attach themselves to solid bodies they nd in the liquid.
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Page 820 Challenge 1409 ny
Some of them are able to move along these solids. e amoeba is an example for a microorganism moving in this way. Also the smallest active motion mechanisms known, namely the motion of molecules in muscles and in cell membranes, work this way.
Let us summarize these observations in a di erent way. All known active motion, or self-propulsion, takes place in uids – be it air or liquids. All active motion requires shape change. In order that shape change leads to motion, the environment, e.g. the water, must itself consist of moving components always pushing onto the swimming entity. e motion of the swimming entity can then be deduced from the particular shape change it performs. To test your intuition, you may try the following puzzle: is microscopic swimming possible in two spatial dimensions? In four?
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In the last decades, the theory of shape change has changed from a fashionable piece of research to a topic whose results are both appealing and useful. We have seen that shape change of a body in a uid can lead to translation. But shape change can also lead to a rotation of the body. In particular, the theory of shape change is useful in explaining how falling cats manage to fall on their feet. Cats are not born with this ability; they have to learn it. But the feat remains fascinating. e great British physicist Michael Berry understood that this ability of cats can be described by an angular phase in a suitably de ned shape space.
– CS – to be inserted – CS –
Page 91 In fact, cats con rm in three dimensions what we already knew for two dimensions: a deformable body can change its own orientation in space without outside help. But shape change bears more surprises.
T
A text should be like a lady’s dress; long enough to cover the subject, yet short enough to keep it
“interesting. ” Continuing the theme of motion of wobbly entities, a famous example cannot be avoided.
In , the mathematician Stephen Smale proved that a sphere can be turned inside out. e discovery brought him the Fields medal in , the highest prize for discoveries in
mathematics. Mathematicians call his discovery the eversion of the sphere. To understand the result, we need to describe more clearly the rules of mathematical
eversion. First of all, it is assumed that the sphere is made of a thin membrane which has the ability to stretch and bend without limits. Secondly, the membrane is assumed to be able to intersect itself. Of course, such a ghostly material does not exist in everyday life; but in mathematics, it can be imagined. A third rule requires that the moves must be performed in such a way that the membrane is not punctured, ripped nor creased; in short, everything must happen smoothly (or di erentiably, as mathematicians like to say).
Even though Smale proved that eversion is possible, the rst way to actually perform it was discovered by the blind topologist Bernard Morin in , based on ideas of Arnold
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F I G U R E 367 A way to turn a sphere inside out, with intermediate steps ordered clockwise (© John Sullivan)
Shapiro. A er him, several additional methods have been discovered. Several computer videos of sphere eversions are now available.* e most famous ones
are Outside in, which shows an eversion due to William P. urston, and e Optiverse, which shows the most e cient method known so far, discovered by a team led by John Sullivan and shown in Figure .
Why is sphere eversion of interest to physicists? If elementary particles were extended and at the same time were of spherical shape, eversion might be a symmetry of particles. To make you think, we mention the e ects of eversion on the whole surrounding space, not only on the sphere itself. e nal e ect of eversion is the transformation
(x, y, z)
(x, y, −z) R r
(645)
where R is the radius of the sphere and r is the length of the coordinate vector (x, y, z),
* Summaries of the videos can be seen at the http://www.geom.umn.edu/docs/outreach/oi website, which also has a good pedagogical introduction. Another simple eversion and explanation is given by Erik de Neve on the http://www.xs4all.nl/~alife/sphere1.htm website. It is even possible to run the lm so ware at home; see the http://www.cslub.uwaterloo.ca/~mjmcgu /eversion website. Figure 367 is from the http://new.math. uiuc.edu/optiverse website.
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F I G U R E 368 The knot diagrams for the simplest prime knots (© Robert Scharein)
thus r = x + y + z . Due to the minus sign in the z-coordinate, eversion is thus different from inversion, but not by too much. As we will nd out shortly, a transformation Page 1051 similar to eversion, space-time duality, is a fundamental symmetry of nature.
K,
“Don’t touch this, or I shall tie your ngers into knots! (Surprisingly e cient child education ” technique.)
Knots and their generalization are central to the study of wobbly entity motion. A (mathematical) knot is a closed piece of rubber string, i.e. a string whose ends have been glued together, which cannot be deformed into a circle or a simple loop. e simple loop is also called the trivial knot. If knots are ordered by their crossing numbers, as shown in Figure , the trivial knot ( ) is followed by the trefoil knot ( ) and by the gure-eight knot ( ). e gure only shows prime knots, i.e., knots that cannot be decomposed into two knots that are connected by two parallel strands. In addition, the gure only shows one of two possible mirror images.
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right-hand crossing +1
left-hand crossing -1
Redemeister move I
Redemeister move II
Redemeister move III
a nugatory crossing
F I G U R E 369 Crossing types in knots
the flype F I G U R E 370 The Reidemeister moves and the flype
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F I G U R E 371 The diagrams for the simplest links with two and three components (© Robert Scharein)
Ref. 990
Knots are of importance in the context of this intermezzo as they visualize the limitations of the motion of wobbly entities. In addition, we will nd other reasons to study knots later on. In this section, we just have a bit of fun.*
How do we describe such a knot through the telephone? Mathematicians have spent a lot of time to gure out smart ways to achieve it. e simplest way is to atten the knot onto a plane and to list the position and the type (below or above) of the crossings.
Mathematicians are studying the simplest way to describe knots by the telephone. e task is not completely nished, but the end is in sight. Of course, the at diagrams can be characterized by the minimal number of crossings. e knots in Figure are ordered in this way. ere is knot with zero, with three and with four crossings (not counting mirror knots); there are knots with ve and with six crossings, knots with seven,
knots with eight, with nine, with ten, with eleven, with twelve,
* Pretty pictures and other information about knots can be found on the KnotPlot site, i.e. at the http://www. cs.ubc.ca/nest/imager/contributions/scharein/KnotPlot.html site.
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Ref. 990 Ref. 992
with thirteen,
with fourteen,
with een and
knots with sixteen
crossings.
Mathematicians do not talk about ‘telephone messages’, they talk about invariants,
i.e. about quantities that do not depend on the precise shape of the knot. At present,
the best description of knots is a polynomial invariant based on a discovery by Vaughan
Jones in . However, though the polynomial allows to uniquely describe most simple
knots, it fails to do so for more complex ones. But the Jones polynomial nally allowed
to prove that a diagram which is alternating and eliminates nugatory crossings (i.e. if it
is ‘reduced’) is indeed one which has minimal number of crossings. e polynomial also
allows to show that any two reduced alternating diagrams are related by a sequence of
ypes.
Together with the search for invariants, the tabulation of knots is a modern mathemat-
ical sport. In , Schubert proved that every knot can be decomposed in a unique way
as sum of prime knots. Knots thus behave similarly to integers.
e mirror image of a knot usually, but not always, is di erent from the original. If
you want a challenge, try to show that the trefoil knot, the knot with three crossings, is
di erent from its mirror image. e rst proof was by Max Dehn in .
Antiknots do not exist. An antiknot would be a knot on a rope that cancels out the
corresponding knot when the two are made to meet along the rope. It is easy to prove
that this is impossible. We take an in nite sequence of knots and antiknots on a string,
K − K + K − K + K − K.... On one hand, we could make them disappear in this way
K − K + K − K + K − K... = (K − K)+(K − K)+(K − K)... = . On the other hand, we could
do the same thing using K −K +K −K +K −K... = K(−K +K)+(−K +K)+(−K +K)... = K.
e only knot K with an antiknot is thus the unknot K = .*
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– CS – Several topics to be included – CS –
Challenge 1411 e
Since knots are stable in time, a knotted line in three dimensions is equivalent to a knotted surface in space-time. When thinking in higher dimensions, we need to be careful. Every knot (or knotted line) can be untied in four or more dimensions; however, there is no surface embedded in four dimensions which has as t = slice a knot, and as t = slice the circle. Such a surface embedding needs at least ve dimensions.
In higher dimensions, knots are possible only n-spheres are tied instead of circles; for example, as just said, -spheres can be tied into knots in dimensions, -spheres in dimensions and so forth.
Mathematicians also study more elaborate structures. Links are the generalization of knots to several closed strands. Braids are the generalization of links to open strands. Braids are especially interesting, as they form a group; can you state what the group operation is?
K Knots do not play a role only in shoe laces and in sailing boats.
* is proof does not work when performed with numbers; we would be able to deduce = by setting K=1. Challenge 1410 n Why is this proof valid with knots but not with numbers?
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F I G U R E 372 A hagfish tied into a knot
Figure to be included F I G U R E 373 How to simulate order for long ropes
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Ref. 994
**
Proteins, the molecules that make up many cell structures, are chains of aminoacids. It seems that very few proteins are knotted, and that most of these form trefoil knots. However, a gure-eight knotted protein has been discovered in 2000 by William Taylor.
**
Knots form also in other polymers. ey seem to play a role in the formation of radicals in carbohydrates. Research on knots in polymers is presently in full swing.
Challenge 1412 r
**
is is the simplest unsolved knot problem: Imagine an ideally wobbly rope, that is, a rope that has the same radius everywhere, but whose curvature can be changed as one prefers. Tie a trefoil knot into the rope. By how much do the ends of the rope get nearer? In 2006, there are only numerical estimates for the answer: about 10.1 radiuses. ere is no formula yielding the number 10.1. Alternatively, solve the following problem: what is the rope length of a closed trefoil knot? Also in this case, only numerical values are known – about 16.33 radiuses – but no exact formula. e same is valid for any other knot, of course.
Ref. 991
**
A famous type of eel, the knot sh Myxine glutinosa, also called hag sh or slime eel, is able to make a knot in his body and move this knot from head to tail. It uses this motion to cover its body with a slime that prevents predators from grabbing it; it also uses this motion to escape the grip of predators, to get rid of the slime a er the danger is over, and to push against a prey it is biting in order to extract a piece of meat. All studied knot sh form only le handed trefoil knots, by the way; this is another example of chirality in nature.
**
One of the most incredible discoveries of recent years is related to knots in DNA molecules. e DNA molecules inside cell nuclei can be hundreds of millions of base pairs long; they
regularly need to be packed and unpacked. When this is done, o en the same happens as when a long piece of rope or a long cable is taken out of a closet.
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It is well known that you can roll up a rope and put it into a closet in such a way that it looks orderly stored, but when it is pulled out at one end, a large number of knots is suddenly found. Figure 373 shows how to achieve this.
To make a long story short, this also happens to nature when it unpacks DNA in cell nuclei. Life requires that DNA molecules move inside the cell nucleus without hindrance. So what does nature do? Nature takes a simpler approach: when there are unwanted crossings, it cuts the DNA, moves it over and puts the ends together again. In cell nuclei, there are special enzymes, the so-called topoisomerases, which perform this process. e details of this fascinating process are still object of modern research.
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Ref. 995
**
e great mathematician Carl-Friedrich Gauß was the rst person to ask what would happen when an electrical current I ows along a wire A linked with a wire B. He discovered a beautiful result by calculating the e ect of the magnetic eld of one wire onto the other. Gauss found the expression
∫ ∫ ∫ πI
dxA ë BB =
A
π
dxAë
A
dxB
B
(xA − xB) xA − xB
=
n
,
(646)
where the integrals are performed along the wires. Gauss found that the number n does not depend on the precise shape of the wires, but only on the way they are linked. Deforming the wires does not change it. Mathematicians call such a number a topological invariant. In short, Gauss discovered a physical method to calculate a mathematical invariant for links; the research race to do the same for other invariants, also for knots and braids, is still going on today.
In the 1980s, Edward Witten was able to generalize this approach to include the nuclear interactions, and to de ne more elaborate knot invariants, a discovery that brought him the Fields medal.
**
Knots are also of importance at Planck scales, the smallest dimensions possible in nature. We will soon explore how knots and the structure of elementary particles are related.
Challenge 1413 r
**
Knots appear rarely in nature. For example, tree roots do not seem to grow many knots during the lifetime of a plant. How do plants avoid this? In other words, why are there no knotted bananas in nature?
Challenge 1414 ny
**
If we move along the knot and count the crossings where we stay above and subtract the number of crossings where we pass below, we get a number called the writhe of the knot. It is not an invariant, but usually a tool in building them. e writhe is not necessarily invariant under one of the three Reidemeister moves. Can you see which one? However, the writhe is invariant under ypes.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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,
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Page 597 Page 183 Challenge 1415 ny
Clouds are another important class of extended entities. e lack of a de nite boundary makes them even more fascinating than amoebas, bacteria or falling cats. We can observe the varieties of clouds from an aeroplane. We also have encountered clouds as the basic structure determining the size of atoms. Comparing these two and other types of clouds teaches us several interesting things about nature.
Galaxies are clouds of stars; stars are clouds of plasma; the atmosphere is a gas cloud. Obviously, the common cumulus or cumulonimbus in the sky are vapour and water droplet clouds. Clouds of all types can be described by a shape and a size, even though in theory they have no bound. An e ective shape can be de ned by that region in which the cloud density is only, say, % of the maximum density; slightly di erent procedures can also be used. All clouds are described by probability densities of the components making up the cloud. All clouds show conservation of their number of constituents.
Whenever we see a cloud, we can ask why it does not collapse. Every cloud is an aggregate. All aggregates are kept from collapse in only three ways: through rotation, through pressure or through the Pauli principle, i.e. the quantum of action. Galaxies are kept from collapsing by rotation. Most stars and the atmosphere are kept from collapsing by gas pressure. Neutron stars, the Earth, atomic nuclei, protons or the electron clouds of atoms are kept apart by the quantum of action.
A rain cloud can contain several thousand tons of water; can you explain what keeps it a oat, and what else keeps it from continuously di using into a thinner and thinner structure?
Two rain clouds can merge. So can two atomic electron clouds. But only atomic clouds are able to cross each other. We remember that a normal atom can be inside a Rydberg atom and leave it again without change. Rain clouds, stars, galaxies or other macroscopic clouds cannot cross each other. When their paths cross, they can only merge or be ripped into pieces. Due to this lack of crossing ability, it is in fact easier to count atomic clouds than macroscopic clouds. In the macroscopic case, there is no real way to de ne a ‘single’ cloud in an accurate way. If we aim for full precision, we are unable to claim that there is more than one rain cloud, as there is no clear-cut boundary between them. Electronic clouds are di erent. True, in a piece of solid matter we can argue that there is only a single electronic cloud throughout the object; however, when the object is divided, the cloud is divided in a way that makes the original atomic clouds reappear. We thus can speak of ‘single’ electronic clouds.
Let us explore the limits of the topic. In our de nition of the term ‘cloud’ we assumed that space and time are continuous. We also assumed that the cloud constituents were localized entities. is does not have to be the case.
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A-
S
Ref. 974
Fluid dynamics is a topic with many interesting aspects. A beautiful result from the s is that a linear, deformable vortex in a rotating liquid is (almost) described by the onedimensional Schrödinger equation. A simple physical system of this type is the vortex that can be observed in any emptying bath tub: it is extended in one dimension, and it wriggles around. As we will see shortly, this wriggling motion is described by a Schrödinger-like
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:
,
wv
e n
F I G U R E 374 The mutually perpendicular tangent e, normal n, torsion w and velocity v of a vortex in a rotating fluid
Dvipsbugw
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
equation. Any deformable linear vortex, as illustrated in Figure , is described by a continuous
set of position vectors r(t, s) that depend on time t and on a single parameter s. e parameter s speci es the relative position along the vortex. At each point on the vortex, there is a unit tangent vector e(t, s), a unit normal curvature vector n(t, s) and a unit torsion vector w(t, s). e three vectors, shown in Figure , are de ned as usual as
e
=
∂r ∂s
,
κn =
∂e ∂s
,
τw
=
− ∂(e ∂s
n)
,
(647)
where κ speci es the value of the curvature and τ speci es the value of the torsion. In general, both numbers depend on time and on the position along the line.
We assume that the rotating environment induces a local velocity v for the vortex that is proportional to the curvature κ, perpendicular to the tangent vector e and perpendicular to the normal curvature vector n:
v = ηκ(e n) ,
(648)
Ref. 974 Ref. 975
where η is the so-called coe cient of local self-induction that describes the coupling between the liquid and the vortex motion.
Any vortex described by the evolution equation ( ) obeys the one-dimensional Schrödinger equation. To repeat his argument, we assume that the lament is deformed only slightly from the straight con guration. (Technically, we are thus in the linear regime.) For such a lament, directed along the x-axis, we can write
r = (x, y(x, t), z(x, t)) .
(649)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 375 Motion of a vortex: the fundamental helical solution and a moving helical ‘wave packet’
Slight deformations imply ∂s ∂x and therefore
e
=
(
,
∂y ∂x
,
∂z ∂x
)
(, , ),
κn
(
,
∂y ∂x
,
∂z ∂x
)
,
and
v=(
,
∂y ∂t
,
∂z ∂t
)
.
We can thus rewrite equation ( ) as
(650)
(
,
∂y ∂t
,
∂z ∂t
)
=
η
(
,
−
∂z ∂x
,
∂y ∂x
)
.
(651)
is equation is well known; if we drop the rst coordinate and introduce complex numbers by setting Φ = y + iz, we can rewrite it as
∂Φ ∂t
=
iη
∂Φ ∂x
.
(652)
is is the one-dimensional Schrödinger equation for the evolution of a free wave function! e complex function Φ speci es the transverse deformation of the vortex. In other words, we can say that the Schrödinger equation in one dimension describes the evolution of the deformation for an almost linear vortex surrounded by a rotating liquid. We note that there is no constant ħ in the equation, as we are exploring a classical system.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
:
,
Schrödinger’s equation is linear in Φ. erefore the fundamental solution is
Φ(x, y, z, t) = a ei(τx−ωt) with ω = ητ and κ = aτ .,
(653)
Challenge 1416 ny
e amplitude a and the wavelength or pitch b = τ can be freely chosen, as long as the approximation of small deviation is ful lled; this condition translates as a  b.* In the present interpretation, the fundamental solution corresponds to a vortex line that is deformed into a helix, as shown in Figure . e angular speed ω is the rotation speed around the axis of the helix.
A helix moves along the axis with a speed given by
vhelix along axis = ητ .
(654)
In other words, for extended entities following evolution equation ( ), rotation and translation are coupled.** e momentum p can be de ned using ∂Φ ∂x, leading to
Dvipsbugw
p=τ= b .
(655)
Momentum is thus inversely proportional to the helix wavelength or pitch, as expected. e energy E is de ned using ∂Φ ∂t, leading to
E = ητ
=
η b
.
(656)
Energy and momentum are connected by
E=
p µ
where
µ= η .
(657)
Page 984 Challenge 1418 ny
In other words, a vortex with a coe cient η – describing the coupling between environment and vortex – is thus described by a number µ that behaves like an e ective mass. We can also de ne the (real) quantity Φ = a; it describes the amplitude of the deformation.
In the Schrödinger equation ( ), the second derivative implies that the deformation ‘wave packet’ has tendency to spread out over space. Can you con rm that the wavelength– frequency relation for a vortex wave group leads to something like the indeterminacy relation (however, without a ħ appearing explicitly)?
In summary, the complex amplitude Φ for a linear vortex in a rotating liquid behaves like the one-dimensional wave function of a non-relativistic free particle. In addition, we found a suggestion for the reason why complex numbers appear in the Schrödinger equation of quantum theory : they could be due to the intrinsic rotation of an underlying substrate. We will see later on whether this is correct.
Challenge 1417 ny
* e curvature is given by κ = a b , the torsion by τ = b. Instead of a  b one can thus also write κ  τ. ** A wave packet moves along the axis with a speed given by vpacket = ητ , where τ is the torsion of the helix of central wavelength.
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:
,
F
-
Ref. 996 Page 383 Page 858 Page 1072
So far, we have looked at the motion of wobbly entities in continuous space-time. But that is an unnecessary restriction. Looking at space-time itself in this way is also interesting. e most intriguing approach was published in by Ted Jacobson. He explored what happens if space-time, instead of assumed to be continuous, is assumed to be the statistical average of numerous components moving in a disordered fashion.
e standard description of general relativity describes space-time as an entity similar to a exible mattress. Jacobson studied what happens if the mattress is assumed to be made of a liquid. A liquid is a collection of (unde ned) components moving randomly and described by a temperature varying from place to place. He thus explored what happens if space-time is made of uctuating entities.
Jacobson started from the Fulling–Davies–Unruh e ect and assumed that the local temperature is given by the same multiple of the local gravitational acceleration. He also used the proportionality – correct on horizons – between area and entropy. Since the energy owing through a horizon can be called heat, one can thus translate the expression δQ = TδS into the expression δE = aδA(c G), which describes the behaviour of spacetime at horizons. As we have seen, this expression is fully equivalent to general relativity.
In other words, imagining space-time as a liquid is a powerful analogy that allows to deduce general relativity. Does this mean that space-time actually is similar to a liquid? So far, the analogy is not su cient to answer the question. In fact, just to confuse the reader a bit more, there is an old argument for the opposite statement.
Dvipsbugw
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
S
-
e main reason to try to model empty space as a solid is a famous property of the motion of dislocations. To understand it, a few concepts need to be introduced. Dislocations are one-dimensional construction faults in crystals, as shown in Figure . A general dislocation is a mixture of the two pure dislocation types: edge dislocations and screw dislocations. Both are shown in Figure . If one studies how the involved atoms can rearrange themselves, one nds that edge dislocations can only move perpendicularly to the added plane. In contrast, screw dislocations can move in all directions.* An important case of general, mixed dislocations, i.e. of mixtures of edge and screw dislocations, are closed dislocation rings. On such a dislocation ring, the degree of mixture changes continuously from place to place.
A dislocation is described by its strength and by its e ective size; they are shown, respectively, in red and blue in Figure . e strength of a dislocation is measured by the so-called Burgers vector; it measures the mis ts of the crystal around the dislocation. More precisely, the Burgers vector speci es by how much a section of perfect crystal needs to be displaced, a er it has been cut open, to produce the dislocation. Obviously, the strength of a dislocation is quantized in multiples of a minimal Burgers vector. In fact, dislocations with large Burgers vectors can be seen as composed of dislocations of minimal Burgers vector.
e size or width of a dislocation is measured by an e ective width w. Also the width
* See the http://uet.edu.pk/dmems/edge_dislocation.htm, http://uet.edu.pk/dmems/screw_dislocation.htm and http://uet.edu.pk/dmems/mixed_dislocation.htm web pages for seeing a moving dislocation.
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:
,
effective size Burgers vector
F I G U R E 376 The two pure dislocation types: edge and screw dislocations
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
is a multiple of the lattice vector. e width measures the size of the deformed region of the crystal around the dislocation. Obviously, the size of the dislocation depends on the elastic properties of the crystal, can take continuous values and is direction-dependent.
Dvipsbugw
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
:
,
Ref. 997
e width is thus related to the energy content of a dislocation.
A general dislocation can move, though only in directions which are both perpendic-
ular to its own orientation and to its Burgers vector. Let us study this motion in more
detail. We call c the speed of sound in a pure (cubic) crystal. As Frenkel and Kontorowa
found in it turns out that when a screw dislocation moves with velocity v, its width
w changes as
w= w
.
−v c
(658)
In addition, the energy of the moving dislocation obeys
Dvipsbugw
E= E
.
−v c
(659)
Page 1010 Challenge 1419 n
A screw dislocation thus cannot move faster than the speed of sound in a crystal and its width shows a speed-dependent contraction. (Edge dislocations have similar, but more complex behaviour.) e motion of screw dislocations in solids is described by the same e ects and formulae that describe the motion of bodies in special relativity; the speed of sound is the limit speed for dislocations in the same way that the speed of light is the limit speed for objects.
Does this mean that elementary particles are dislocations of space or even of spacetime, maybe even dislocation rings? e speculation is appealing, even though it supposes that space-time is a solid, and thus contradicts the model of space or space-time as a uid. Worse, we will soon encounter good reasons to reject modelling space-time as a lattice; maybe you can nd a few ones already by yourself. Still, expressions ( ) and ( ) for dislocations continue to fascinate. For the time being, we do not study them further.
S
Ref. 998 Challenge 1420 ny
ere is an additional reason to see space as a liquid. It is possible to swim through empty space. is discovery was published in by Jack Wisdom. He found that cyclic changes in the shape of a body can lead to net translation, a rotation of the body, or both.
Swimming in space-time does not happen at high Reynolds numbers. at would imply that a system would be able to throw empty space behind it, and to propel itself forward as a result. No such e ects have ever been found. However, Jack Wisdom found a way to swim that corresponds to low Reynolds numbers, where swimming results of simple shape change.
ere is a simple system that shows the main idea. We know from Galilean physics that on a frictionless surface it is impossible to move, but that it is possible to turn oneself.
is is true only for a at surface. On a curved surface, one can use the ability to turn and translate it into motion.
Take to massive discs that lie on the surface of a frictionless, spherical planet, as shown in Figure . Consider the following four steps: e disc separation φ is increased by the angle ∆φ, then the discs are rotated oppositely about their centres by the angle ∆θ, their separation is decreased by −∆φ, and they are rotated back by −∆θ. Due to to the conservation of angular momentum, the two-disc system changes its longitude ∆ψ as
Dvipsbugw
:
,
z
φ ψφ x
y θ
Equator
θ
F I G U R E 377 Swimming on a curved surface using two discs
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 378 A large raindrop falling downwards
F I G U R E 379 Is this possible?
∆ψ = γ ∆θ∆φ ,
(660)
where γ is the angular radius of the discs. is cycle can be repeated over and over. e cycle it allows a body on the surface of the Earth, to swim along the surface. However, for a body of metre size, the motion for each swimming cycle is only around − m.
Wisdom showed that the mechanism also works in curved space-time. e mechanism thus allows a falling body to swim away from the path of free fall. Unfortunately, the achievable distances for everyday objects are negligible. Nevertheless, the e ect exists.
At this point, we are thoroughly confused. Space-time seems to be solid and liquid at the same time. Despite this contrast, the situation gives the impression that extended, wobbly and uctuating entities might lead us towards a better understanding of the structure of space and time. at exploration is le for the third and last part of our adventure.
C
“Any pair of shoes proves that we live on the inside of a sphere. eir soles are worn out at the ends, and hardly at all in between. ” Anonymous
e topic of wobbly entities is full of fascinating details. Here are a few.
**
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:
,
Challenge 1421 n Ref. 999
What is the shape of raindrops? Try to picture it. However, use your reason, not your
prejudice! By the way, it turns out that there is a maximum size for raindrops, with a value of about mm. e shape of such a large raindrop is shown in Figure 378. Can you
imagine where the limit comes from? For comparison, the drops in clouds, fog or mist are in the range of 1 to µm, with
a peak at 10 to µm. In those cases when all droplets are of similar size one and when
light is scattered only once by the droplets, one can observe coronae, glories or fogbows.
** Challenge 1422 n What is the entity shown in Figure 379 – a knot, a braid or a link?
** Challenge 1423 d Can you nd a way to classify tie knots?
** Challenge 1424 n Are you able to nd a way to classify the way shoe laces can be threaded?
Dvipsbugw
O
Challenge 1425 d
We have studied one example of motion of extended bodies already earlier on: solitons. We can thus sum up the possible motions of extended entities in four key themes. We rst studied solitons and interpenetration, then knots and their rearrangement, continued with duality and eversion and nally explored clouds and extension. e sum of it all seems to be half liquid and half solid.
e motion of wobbly bodies probably is the most neglected topic in all textbooks on motion. Research is progressing at full speed; it is expected that many beautiful analogies will be discovered in the near future. For example, in this intermezzo we have not described any good analogy for the motion of light; similarly, including quantum theory into the description of wobbly bodies’ motion remains a fascinating issue for anybody aiming to publish in a new eld.
e ideas introduced in this intermezzo were su cient to prepare us for the third part of our ascent of Motion Mountain. We can now tackle the nal part of our adventure.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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:
,
B
969 H T mith Uitgevers, page .
, De wetten van de vliegkunst – over stijgen, dalen, vliegen en zweven, Ara. is clear and interesting text is also available in English. Cited on
970 In , the smallest human-made ying object was the helicopter built by a group of the
Institut für Mikrotechnik in Mainz, in Germany. A picture is available at their web page,
to be found at http://www.imm-mainz.de/English/billboard/f_hubi.html. e helicopter is
mm long, weighs mg and ies (though not freely) using two built-in electric mo-
tors driving two rotors, running at between
and
revolutions per minute. See
also the helicopter from Stanford University at http://www-rpl.stanford.edu/RPL/htmls/
mesoscopic/mesicopter/mesicopter.html, with an explanation of its battery problems. Cited
on page .
971 e most recent computational models of li still describe only two-dimensional wing motion, e.g. Z.J. W , Two dimensional mechanism for insect hovering, Physical Review Letters 85 pp. – , . A rst example of a mechanical bird has been constructed by Wolfgang Send; it can be studied on the http://www.aniprop.de website. See also W. S , Physik des Fliegens, Physikalische Blätter 57, pp. – , June . Cited on page .
972 R.B. S
& A.L.R. T
, Unconventional li -generating mechanisms in free-
ying butter ies, Nature 420, pp. – , . Cited on page .
973 e simulation of insect ight using enlarged wing models apping in oil instead of air is described for example in http://www.dickinson.caltech.edu/research_robo y.html. e higher viscosity of oil allows to achieve the same Reynolds number with larger sizes and lower frequencies than in air. Cited on page .
974 G
K. B
, An Introduction to Fluid Mechanics, Cambridge University
Press, , and H. H
, A soliton on a vortex lament, Journal of Fluid Mech-
anics 51, pp. – , . A summary is found in H. Z , On the motion of slender
vortex laments, Physics of Fluids 9, p. - , . Cited on pages and .
975 V.P. D
, Helical waves on a vortex lament, American Journal of Physics 73,
pp. – , , and V.P. D
, Mechanical analogy for the wave-particle: helix
on a vortex lament, http://www.arxiv.org/abs/quant-ph/
. Cited on page .
976 E. P
, Life at low Reynolds number, American Journal of Physics 45, p. ,
on page .
. Cited
977 Most bacteria are attened, ellipsoidal sacks kept in shape by the membrane enclosing the cytoplasma. But there are exceptions; in salt water, quadratic and triangular bacteria have been found. More is told in the corresponding section in the interesting book by B D , Power Unseen – How Microbes Rule the World, W.H. Freeman, New York, . Cited on page .
978 S. P
, G. S
& T.A. M
. Cited on page .
, How long is a giant sperm?, Nature 375, p. ,
979 For an overview of the construction and the motion of coli bacteria, see H.C. B , Motile behavior of bacteria, Physics Today 53, pp. – , January . Cited on page .
980 M. K
, A. S
& S. N
, Swimming of microorganisms viewed
from string and membrane theories, Modern Journal of Physics Letters A 9, pp. – ,
. Also available as http://www.arxiv.org/abs/hep-th/
. Cited on page .
981 ey are also called prokaryote agella. See for example S.C. S
& S. K ,
e bacterial agellar motor, Annual Review of Biophysics and Biomolecular Structure 23,
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Dvipsbugw
Dvipsbugw
pp. – , , or S.R. C
& M. K -I
, e bacterial agellar motor,
International Review of Cytology 147, pp. – , . See also the information on the
topic that can be found on the website http://www.id.ucsb.edu: /fscf/library/origins/
graphics-captions/ agellum.html. Cited on page .
982 is is from the book by D
D
Library, . Cited on page .
, Life at a Small Scale, Scienti c American
983 A. S
& F. W
, Gauge kinematics of deformable bodies, American Journal
of Physics 57, pp. – , , A. S
& F. W
, Geometry of self-propulsion
at low Reynolds number, Journal of Fluid Mechanics 198, pp. – , , A. S
& F. W
, E ciencies of self-propulsion at low Reynolds number, Journal of Fluid
Mechanics 198, pp. – , . See also R. M
, Gauge theory of the falling
cat, Field Institute Communications 1, pp. – , . No citations.
984 W. N
& U. R
, Die Orientierung freibeweglicher Organismen zum Licht,
dargestellt am Beispiel des Flagellaten Chlamydomonas reinhardtii, Naturwissenscha en 81,
pp. – , . Cited on page .
985 A lot here is from L
H. K
, Knots and Physics, World Scienti c, second
edition, , which gives a good, simple, and visual introduction to the mathematics of
knots and its main applications to physics. No citations.
986 S. S
, A classi cation of immersions of the two-sphere, Transactions of the American
Mathematical Society 90, pp. – , . No citations.
987 G.K. F
& B. M , Arnold Shapiro’s Eversion of the Sphere, Mathematical Intel-
ligencer pp. – , . See also Ref. . No citations.
988 S
L ,D
M
&T
M
the Ideas Behind Outside In, Peters, . No citations.
, Making Waves – a Guide to
989 B. M
& J.-P. P , Le retournement de la sphere, Pour la Science 15, pp. – , .
See also the clear article by A. P
, Turning a surface inside out, Scienti c American
pp. – , May . No citations.
990 A good introduction to knots is the paper by J. H , M. T W , e rst , , knots, e Mathematical Intelligencer 20, pp. – , on pages and .
& J. . Cited
991 A
S
, Nœuds – histoire d’une théorie mathématique, Editions du Seuil, .
D. J
, Le poisson noué, Pour la science, dossier hors série, pp. – , April . Cited
on page .
992 I. S
, Game, Set and Math, Penguin Books, , pp. – . Cited on page .
993 For some modern knot research, see P. H
, R.B. K
& A. S
, Quantiz-
ation of energy and writhe in self-repelling knots, New Journal of Physics 4, pp. . – . ,
. No citations.
994 W.R. T
, A deeply knotted protein structure and how it might fold, Nature 406,
pp. – , . Cited on page .
995 A.C. H
, Knots and physics: Old wine in new bottles, American Journal of Physics
66, pp. – , . Cited on page .
996 T. J
, ermodynamics of spacetime: the Einstein equation of state, Physical Re-
view Letters 75, pp. – , , or http://www.arxiv.org/abs/gr-qc/
. Cited on
page .
997 J. F
& T. K
, Über die eorie der plastischen Verformung, Physikalis-
che Zeitschri der Sowietunion 13, pp. – , . F.C. F
, On the equations of mo-
tion of crystal dislocations, Proceedings of the Physical Society A 62, pp. – , , J.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Dvipsbugw
Dvipsbugw
:
,
E
, Uniformly moving dislocations, Proceedings of the Physical Society A 62, pp. –
, . See also G. L
& H. D
, Zeitschri für Physik 126, p. , . A
general introduction can be found in A. S
& P. S
, Kinks in dislocations
lines and their e ects in internal friction in crystals, Physical Acoustics 3A, W.P. M ,
ed., Academic Press, . See also the textbooks by F
R.N. N
, eory of
Crystal Dislocations, Oxford University Press, , or J.P. H
& J. L
, eory of
Dislocations, McGraw Hills Book Company, . Cited on page .
998 J. W
, Swimming in spacetime: motion by cyclic changes in body shape, Science 299,
pp. – , st of March, . Cited on page .
999 H.R. P
& J.D. K , Microphysics of Clouds and Precipitation, Reidel, ,
pp. – . Falling drops are attened and look like a pill, due to the interplay between
surface tension and air ow. Cited on page .
1000 A manual for the ways to draw manifolds is the unique text by G
F
,e
Topological Picturebook, Springer Verlag, . It also contains a chapter on sphere eversion.
Cited on page .
1001 M.Y . K
, Some problems of topology change description in the theory of
space-time, http://www.arxiv.org/abs/gr-qc/
preprint. No citations.
1002 A. U
, Teleparallel space-time with torsion yields geometrization of electro-
dynamcis with quantised charges, http://www.arxiv.org/abs/gr-qc/
preprint. No
citations.
Dvipsbugw
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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ird Part
M
W
M
:
W A S ,T
P
?
Where through the combination of quantum mechanics and general relativity, the top of Motion Mountain is reached and it is discovered that vacuum is indistinguishable from matter, that space, time and mass are easily confused, that there is no di erence between the very large and the very small, and that a complete description of motion is possible. (Well, wait a few more years for the last line.)
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
C
XI
GENERAL RELATIVIT Y VERSUS
QUANTUM MECHANICS
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Man muß die Denkgewohnheiten durch Denknotwendigkeiten ersetzen.*
“ Albert Einstein ” two stories told in the two parts of the path we have followed up to now, namely
hat on general relativity and that on quantum eld theory, are both beautiful and
Thoroughly successful. Both are con rmed by experiments. We have reached a con-
siderable height in our mountain ascent. e precision we achieved in the description of nature is impressive, and we are now able to describe all known examples of motion. So far we have encountered no exceptions.
However, the most important aspects of any type of motion, the masses of the particles involved and the strength of their coupling, are still unexplained. Furthermore, the origin of the number of particles in the universe, their initial conditions and the dimensionality of space-time remain hidden from us. Obviously, our adventure is not yet complete.
is last part of our hike will be the most demanding. In the ascent of any high mountain, the head gets dizzy because of the lack of oxygen. e nite energy at our disposal requires that we leave behind all unnecessary baggage and everything that slows us down. In order to determine what is unnecessary, we need to focus on what we want to achieve. Our aim is the precise description of motion. But even though the general relativity and quantum theory are extremely precise, we carry are a burden: the two theories and their concepts contradict each other. To pinpoint this useless baggage, we rst list these contradictions.
T
Ref. 1003 Page 460
In classical physics and in general relativity, the vacuum, or empty space-time, is a region with no mass, no energy and no momentum. If matter or gravitational elds are present, space-time is curved. e best way to measure the mass or energy content of space-time is to measure the average curvature of the universe. Cosmology tells us how we can do this; measurements yield an average energy density of the ‘vacuum’ of
E V nJ m .
(661)
Ref. 1004 However, quantum eld theory tells a di erent story. Vacuum is a region with zero-point uctuations. e energy content of vacuum is the sum of the zero-point energies of all
* ‘One needs to replace habits of thought by necessities of thought.’
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
the elds it contains. Indeed, the Casimir e ect ‘proves’ the reality of these zero-point Page 858 energies. eir energy density is given, within one order of magnitude, by
∫ E
V
=
πh c
νmax
ν
dν
=
π c
h
νmax
.
(662)
e approximation is valid for the case in which the cut-o frequency νmax is much larger than the rest mass m of the particles corresponding to the eld under consideration. Particle physicists argue that the cut-o energy has to be at least the energy of grand
uni cation, about GeV= . MJ. at would give a vacuum energy density of
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E V
Jm ,
(663)
Ref. 1005
Page 655 Page 828
which is about
times higher than the experimental limit deduced from spatial
curvature using general relativity estimates. In other words, something is slightly wrong
here.
General relativity and quantum theory contradict each other in other ways. Gravity is
curved space-time. Extensive research has shown that quantum eld theory, the descrip-
tion of electrodynamics and of nuclear forces, fails for situations with strongly curved
space-times. In these cases the concept of ‘particle’ is not uniquely de ned; quantum eld
theory cannot be extended to include gravity consistently and thus to include general re-
lativity. Without the concept of the particle as a countable entity, the ability to perform per-
turbation calculations is also lost; and these are the only calculations possible in quantum
eld theory. In short, quantum theory only works because it assumes that gravity does
not exist! Indeed, the gravitational constant does not appear in any consistent quantum
eld theory.
On the other hand, general relativity neglects the commutation rules between physical
quantities discovered in experiments on a microscopic scale. General relativity assumes
that the position and the momentum of material objects can be given the meaning that
they have in classical physics. It thus ignores Planck’s constant ħ and only works by neg-
lecting quantum theory.
Measurements also lead to problems. In general relativity, as in classical physics, it is
assumed that in nite precision of measurement is possible, e.g. by using ner and ner
ruler marks. In contrast, in quantum mechanics the precision of measurement is limited.
e indeterminacy principle gives the limits that result from the mass M of the apparatus.
Time shows the contradictions most clearly. Relativity explains that time is what is
read from clocks. Quantum theory says that precise clocks do not exist, especially if the
coupling with gravitation is included. What does waiting minutes mean, if the clock
goes into a quantum mechanical superposition as a result of its coupling to space-time
geometry?
In addition, quantum theory associates mass with an inverse length via the Compton
wavelength; general relativity associates mass with length via the Schwarzschild radius.
Similarly, general relativity shows that space and time cannot be distinguished,
whereas quantum theory says that matter does make a distinction. Quantum theory is a
theory of – admittedly weirdly constructed – local observables. General relativity doesn’t
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Page 494 Ref. 1006, Ref. 1007 Ref. 1008, Ref. 1009
have any local observables, as Einstein’s hole argument shows. Most dramatically, the contradiction is shown by the failure of general relativity to de-
scribe the pair creation of particles with spin / , a typical and essential quantum process. John Wheeler and others have shown that, in such a case, the topology of space necessarily has to change; in general relativity, however, the topology of space is xed. In short, quantum theory says that matter is made of fermions, while general relativity cannot incorporate fermions.
To sum up, general relativity and quantum theory clash. As long as an existing description of nature contains contradictions, it cannot lead to a uni ed description, to useful explanations, or even to a correct description. In order to proceed, let us take the shortest and fastest path: let us investigate the contradictions in more detail.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
.
?
Ref. 1010 Ref. 1011
ere is a simple way to state the origin of all contradictions between general relativity and quantum mechanics.* Both theories describe motion with objects made up of particles and with space-time made up of events. Let us see how these two concepts are de ned.
A particle – and in general any object – is de ned as a conserved entity to which a position can be ascribed and which can move. ( e etymology of the term ‘object’ is connected to the latter fact.) In other words, a particle is a small entity with conserved mass, charge etc., which can vary its position with time.
In every physics text time is de ned with the help of moving objects, usually called ‘clocks’, or with the help of moving particles, such as those emitted by light sources. Similarly, the length is de ned in terms of objects, either with an old-fashioned ruler or with the help of the motion of light, which in turn is motion of particles.
Modern physics has further sharpened the de nitions of particle and space-time. Quantum mechanics assumes that space-time is given (it is included as a symmetry of the Hamiltonian), and studies the properties and the motion of particles, both for matter and for radiation. General relativity, and especially cosmology, takes the opposite approach: it assumes that the properties of matter and radiation are given, e.g. via their equations of state, and describes in detail the space-time that follows from them, in particular its curvature.
However, one fact remains unchanged throughout all these advances in physics: the two concepts of particles and of space-time are each de ned with the help of the other. To avoid the contradiction between quantum mechanics and general relativity and to eliminate their incompleteness requires the elimination of this circular de nition. As argued in the following, this necessitates a radical change in our description of nature, and in particular of the continuity of space-time.
For a long time, the contradictions between the two descriptions of nature were avoided by keeping them separate. One o en hears the statement that quantum mech-
* e main results of this section are standard knowledge among specialists of uni cation; there are given here in simple arguments. For another way to derive the results, see the summary section on limit statements in nature, on page 1068.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 380 ‘Tekenen’ by Maurits Escher, 1948 – a metaphor for the way in which ‘particles’ and ‘space-time’ are usually defined: each with the help of the other (© M.C. Escher Heirs)
Ref. 1012, Ref. 1013 Ref. 1014, Ref. 1015
anics is valid at small dimensions and general relativity is valid at large dimensions, but this arti cial separation is not justi ed; worse, it prevents the solution of the problem. e situation resembles the well-known drawing (Figure ) by Maurits Escher ( – ) where two hands, each holding a pencil, seem to be drawing each other. If one hand is taken as a symbol of space-time and the other as a symbol of particles, with the act of drawing taken as a symbol of the act of de ning, the picture gives a description of standard twentieth century physics. e apparent contradiction is solved by recognizing that the two concepts (the two hands) result from a third, hidden concept from which the other two originate. In the picture, this third entity is the hand of the painter.
In the case of space-time and matter, the search for the underlying common concept is presently making renewed progress. e required conceptual changes are so dramatic that they should be of interest to anybody who has an interest in physics. e most effective way to deduce the new concepts is to focus in detail on that domain where the contradiction between the two standard theories becomes most dramatic and where both theories are necessary at the same time. at domain is given by a well-known argument.
P
Both general relativity and quantum mechanics are successful theories for the description of nature. Each provides a criterion for determining when classical Galilean physics is no longer applicable. (In the following, we use the terms ‘vacuum’ and ‘empty space-time’ interchangeably.)
General relativity shows that it is necessary to take into account the curvature of spacetime whenever we approach an object of mass m to within a distance of the order of the
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TA B L E 74 The size, Schwarzschild radius and Compton wavelength of some objects appearing in nature. The lengths between quotes make no physical sense, as explained in the text.
O
S :M
mS
-R
.
d rS
d
rS
galaxy
Zm ë kg
Tm
neutron star km . ë kg . km
.
Sun
. Gm . ë kg . km
.ë
Earth
Mm . ë kg . mm
.ë
human
.m
kg
. ym
.ë
molecule
nm . zg
‘ . ë − m’ . ë
atom ( C) . nm yg
‘ . ë − m’ . ë
proton p
fm . yg
‘ . ë − m’ . ë
pion π
fm . yg
‘ . ë − m’ . ë
up-quark u < . fm . yg
‘ . ë − m’ < . ë
electron e < am . ë − kg ‘ . ë − m’ . ë
neutrino νe < am < . ë − kg ‘< . ë − m’ n.a.
C
R
-
d λC
λC
−m ‘ . ë − m’ . ë ‘ . ë − m’ . ë ‘ . ë − m’ . ë ‘ . ë − m’ . ë
.ë − m .ë .ë − m .ë .ë − m . .ë − m . .ë − m <. .ë − m <. ë −
.ë −m < . ë −
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Schwarzschild radius rS, given by
rS = Gm c .
(664)
Ref. 1006, Ref. 1016 Ref. 1017
e gravitational constant G and the speed of light c act as conversion constants. Indeed, as the Schwarzschild radius of an object is approached, the di erence between general relativity and the classical r description of gravity becomes larger and larger. For example, the barely measurable gravitational de ection of light by the Sun is due to approaching it to within . ë times its Schwarzschild radius. Usually however, we are forced to stay away from objects at a distance that is an even larger multiple of the Schwarzschild radius, as shown in Table . For this reason, general relativity is unnecessary in everyday life. (An object smaller than its own Schwarzschild radius is called a black hole. According to general relativity, no signals from inside the Schwarzschild radius can reach the outside world; hence the name ‘black hole’.)
Similarly, quantum mechanics shows that Galilean physics must be abandoned and quantum e ects must be taken into account whenever an object is approached to within distances of the order of the (reduced) Compton wavelength λC, given by
λC
=
ħ mc
.
(665)
In this case, Planck’s constant h and the speed of light c act as conversion factors to transform the mass m into a length scale. Of course, this length only plays a role if the object itself is smaller than its own Compton wavelength. At these dimensions we get relativistic
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quantum e ects, such as particle–antiparticle creation or annihilation. Table shows that the approach distance is near or smaller than the Compton wavelength only in the microscopic world, so that such e ects are not observed in everyday life. We do not therefore need quantum eld theory to describe common observations.
e combined concepts of quantum eld theory and general relativity are required in situations in which both conditions are satis ed simultaneously. e necessary approach distance for such situations is calculated by setting rS = λC (the factor is introduced for simplicity). We nd that this is the case when lengths or times are (of the order of)
lPl = ħG c tPl = ħG c
= . ë − m, the Planck length, = . ë − s , the Planck time.
(666)
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Challenge 1426 n Ref. 1018
Whenever we approach objects at these scales, both general relativity and quantum mechanics play a role; at these scales e ects of quantum gravity appear. Because the values of the Planck dimensions are extremely small, this level of sophistication is unnecessary in everyday life, in astronomy and even in particle physics.
However, to answer the questions posted at the beginning of the book – why do we live in three dimensions and why is the proton . times heavier than the electron? – we require a precise and complete description of nature. e contradictions between quantum mechanics and general relativity appear to make the search for answers impossible. However, while the uni ed theory describing quantum gravity is not yet complete, we can already get a few glimpses at its implications from its present stage of development.
Note that the Planck scales specify one of only two domains of nature where quantum mechanics and general relativity apply at the same time. (What is the other?) As Planck scales are the easier of the two to study, they provide the best starting point for the following discussion. When Max Planck discovered the existence of Planck scales or Planck units, he was interested in them mainly as natural units of measurement, and that is what he called them. However, their importance in nature is much more widespread, as we will shall see in the new section. We will discover that they determine what is commonly called quantum geometry.
F
Ref. 1019, Ref. 1020
Time is composed of time atoms ... which in fact are indivisible.
“ Moses Maimonides, th century. ” e appearance of the quantum of action in the description of motion leads to quantum
limits to all measurements. ese limits have important consequences at Planck dimensions. Measurement limits appear most clearly when we investigate the properties of clocks and metre rules. Is it possible to construct a clock that is able to measure time intervals shorter than the Planck time? Surprisingly, the answer is no, even though the time–energy indeterminacy relation ∆E∆t ħ seems to indicate that by making ∆E arbitrary large, we can make ∆t arbitrary small.
Every clock is a device with some moving parts. Parts can be mechanical wheels, particles of matter in motion, changing electrodynamic elds, i.e. photons, or decaying
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Ref. 1021, Ref. 1022 Ref. 1023
radioactive particles. For each moving component of a clock, such as the two hands, the indeterminacy principle applies. As discussed most clearly by Michael Raymer, the indeterminacy relation for two non-commuting variables describes two di erent, but related situations: it makes a statement about standard deviations of separate measurements on many identical systems; and it describes the measurement precision for a joint measurement on a single system. roughout this article, only the second situation is considered.
For any clock to work, we need to know both the time and the energy of each hand. Otherwise it would not be a recording device. Put more generally, a clock must be a classical system. We need the combined knowledge of the non-commuting variables for each moving component of the clock. Let us focus on the component with the largest time indeterminacy ∆t. It is evident that the smallest time interval δt that can be measured by a clock is always larger than the quantum limit, i.e. larger than the time indeterminacy ∆t for the most ‘uncertain’ component. us we have
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δt
∆t
ħ ∆E
,
(667)
where ∆E is the energy indeterminacy of the moving component, and this energy indeterminacy ∆E must be smaller than the total energy E = mc of the component itself.* Furthermore, a clock provides information and thus signals have to be able to leave it. To make this possible, the clock must not be a black hole and its mass m must therefore be smaller than the Schwarzschild mass for its size, i.e. m c l G, where l is the size of the clock (neglecting factors of order unity). Finally, for a sensible measurement of the time interval δt, the size l of the clock must be smaller than c δt itself, because otherwise di erent parts of the clock could not work together to produce the same time display.** If we combine all these conditions, we get
δt
ħG c δt
(668)
or
δt
ħG c
= tPl .
(669)
Ref. 1027
In summary, from three simple properties of any clock, namely that there is only a single clock, that we can read its dial and that it gives sensible read-outs, we get the general conclusion that clocks cannot measure time intervals shorter than the Planck time. Note that this argument is independent of the nature of the clock mechanism. Whether the clock is powered by gravitational, electrical, plain mechanical or even nuclear means, the limit still applies.***
e same result can also be found in other ways. For example, any clock small enough to measure small time intervals necessarily has a certain energy indeterminacy due to
* Physically, this condition means being sure that there is only one clock; the case ∆E E would mean that it is impossible to distinguish between a single clock and a clock–anticlock pair created from the vacuum, or a component plus two such pairs, etc. ** It is amusing to explore how a clock larger than c δt would stop working, as a result of the loss of rigidity
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Page 410 Ref. 1016
the indeterminacy relation. At the same time, on the basis of general relativity, any energy
density induces a deformation of space-time and signals from the deformed region arrive
with a certain delay due to that deformation. e energy indeterminacy of the source
leads to an indeterminacy in the deformation and thus in the delay. e expression from
general relativity for the deformation of the time part of the line element due to a mass
m is δt = mG l c . From the mass–energy relation, an energy spread ∆E produces an
indeterminacy ∆t in the delay
∆t =
∆E G lc
.
(670)
is indeterminacy determines the precision of the clock. Furthermore, the energy indeterminacy of the clock is xed by the indeterminacy relation for time and energy ∆E ħ ∆t, in turn xed by the precision of the clock. Combining all this, we again
nd the relation δt tPl for the minimum measurable time. We are forced to conclude that in nature there is a minimum time interval. In other words, at Planck scales the term ‘instant of time’ has no theoretical or experimental basis. It therefore makes no sense to use the term.
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F
Ref. 1028
In a similar way, we can deduce that it is impossible to make a metre rule or any other
length measuring device that is able to measure lengths shorter than the Planck length.
Obviously, we can already deduce this from lPl = c tPl, but a separate proof is also possible. e straightforward way to measure the distance between two points is to put an object
at rest at each position. In other words, joint measurements of position and momentum
are necessary for every length measurement. Now, the minimal length δl that can be
measured must be larger than the position indeterminacy of the two objects. From the
indeterminacy principle it is known that each object’s position cannot be determined with
a precision ∆l better than that given by the indeterminacy relation ∆l ∆p = ħ, where ∆p
is the momentum indeterminacy. e requirement that there is only one object at each
end, i.e. avoiding pair production from the vacuum, means that ∆p < mc; together, these
requirements give
δl
∆l
ħ mc
.
(671)
Furthermore, the measurement cannot be performed if signals cannot leave the objects; thus they may not be black holes. erefore their masses must be small enough for their Schwarzschild radius rS = Gm c to be smaller than the distance δl separating them.
Challenge 1427 n
Ref. 10245 Ref. 1026
in its components. *** Note that gravitation is essential here. e present argument di ers from the well-known study on the limitations of clocks due to their mass and their measuring time which was published by Salecker and Wigner and summarized in pedagogical form by Zimmerman. Here, both quantum mechanics and gravity are included, and therefore a di erent, lower and much more fundamental limit is found. Note also that the discovery of black hole radiation does not change the argument; black hole radiation notwithstanding, measurement devices cannot exist inside black holes.
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Again omitting the factor of , we get
δl
ħG c
= lPl .
(672)
Another way to deduce this limit reverses the roles of general relativity and quantum the-
ory. To measure the distance between two objects, we have to localize the rst object with
respect to the other within a certain interval ∆x. e corresponding energy indetermin-
acy obeys ∆E = c(c m + (∆p) ) cħ ∆x. However, general relativity shows that a
Ref. 1006, Ref. 1016 small volume lled with energy changes the curvature of space-time, and thus changes
the metric of the surrounding space. For the resulting distance change ∆l, compared to
Ref. 1028, Ref. 1029, empty space, we nd the expression ∆l G∆E c . In short, if we localize the rst particle
Ref. 1030 in space with a precision ∆x, the distance to a second particle is known only with preci-
Ref. 1031, Ref. 1032, sion ∆l. e minimum length δl that can be measured is obviously larger than either of
Ref. 1033 these quantities; inserting the expression for ∆E, we nd again that the minimum meas-
urable length δl is given by the Planck length.
We note that, as the Planck length is the shortest possible length, it follows that there
can be no observations of quantum mechanical e ects for situations in which the corres-
ponding de Broglie or Compton wavelength is smaller than the Planck length. In proton–
proton collisions we observe both pair production and interference e ects. In contrast,
the Planck limit implies that in everyday, macroscopic situations, such as car–car colli-
sions, we cannot observe embryo–antiembryo pair production and quantum interference
e ects.
In summary, from two simple properties common to all length measuring devices,
namely that they can be counted and that they can be read out, we arrive at the conclusion
that lengths smaller than the Planck length cannot be found in measurements. Whatever
method is used, be it a metre rule or time-of- ight measurement, we cannot overcome
this fundamental limit. It follows that the concept of a ‘point in space’ has no experimental
basis. In the same way, the term ‘event’, being a combination of a ‘point in space’ and an
‘instant of time’, also loses its meaning for the description of nature.
A simple way to deduce the minimum length using the limit statements which struc-
ture this ascent is the following. General relativity is based on a maximum force in nature,
Page 349 or alternatively, on a minimum mass change per time; its value is given by dm dt = c G.
Quantum theory is based on a minimum action in nature, given by L = ħ . Since a dis-
tance d can be expressed like
d
=
L dm dt
,
(673)
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Ref. 1034
one sees directly that a minimum action and a maximum mass rate imply a minimum distance. In other words, quantum theory and general relativity, when put together, imply a minimum distance.
ese results are o en expressed by the so-called generalized indeterminacy principle
∆p∆x
ħ
+
f
G c
(∆p)
(674)
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or
∆p∆x
ħ
+
f
lPl ħ
(∆p)
,
(675)
Challenge 1428 e
Ref. 1034 Ref. 1035 Ref. 1036 Ref. 1037, Ref. 1038, Ref. 1039
Ref. 1040
where f is a numerical factor of order unity. A similar expression holds for the time–
energy indeterminacy relation. e rst term on the right hand side is the usual quantum
mechanical indeterminacy. e second term, negligible for everyday life energies, plays a
role only near Planck energies and is due to the changes in space-time induced by gravity
at these high energies. You should be able to show that the generalized principle ( )
automatically implies that ∆x can never be smaller than f lPl. e generalized indeterminacy principle is derived in exactly the same way in which
Heisenberg derived the original indeterminacy principle ∆p∆x ħ , namely by study-
ing the de ection of light by an object under a microscope. A careful re-evaluation of the
process, this time including gravity, yields equation ( ). For this reason, all approaches
that try to unify quantum mechanics and gravity must yield this relation; indeed, it
appears in the theory of canonical quantum gravity, in superstring theory and in the
quantum group approach.
We remember that quantum mechanics starts when we realize that the classical
concept of action makes no sense below the value of ħ ; similarly, uni ed theories start
when we realize that the classical concepts of time and length make no sense below Planck
values. However, the usual description of space-time does contain such small values; the
usual description involves the existence of intervals smaller than the smallest measurable
one. erefore, the continuum description of space-time has to be abandoned in favour of
a more appropriate description.
e new indeterminacy relation appearing at Planck scales shows that continuity can-
not be a good description of space-time. Inserting c∆p ∆E ħ ∆t into equation ( ),
we get
∆x∆t ħG c = tPllPl ,
(676)
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which of course has no counterpart in standard quantum mechanics. It shows that spacetime events do not exist. A nal way to convince oneself that points have no meaning is that a point is an entity with vanishing volume; however, the minimum volume possible in nature is the Planck volume VPl = lPl.
While space-time points are idealizations of events, this idealization is incorrect. e use of the concept of ‘point’ is similar to the use of the concept of ‘aether’ a century ago: it is impossible to detect and it is only useful for describing observations until a way to describe nature without it has been found. Like ‘aether´, also ‘point´ leads reason astray.
In other words, the Planck units do not only provide natural units, they also provide — within a factor of order one — the limit values of space and time intervals.
F
-
e consequences of the Planck limits for measurements of time and space can be taken much further. It is commonplace to say that given any two points in space or any two instants of time, there is always a third in between. Physicists sloppily call this property continuity, while mathematicians call it denseness. However, at Planck dimensions this
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Ref. 1041 Ref. 1028, Ref. 1042
property cannot exist, since intervals smaller than the Planck time can never be found. us points and instants are not dense, and between two points there is not always a third. is means that space and time are not continuous. Of course, at large scales they are –
approximately – continuous, in the same way that a piece of rubber or a liquid seems continuous at everyday dimensions, even though it is not at a small scale.
All paradoxes resulting from the in nite divisibility of space and time, such as Zeno’s argument on the impossibility to distinguish motion from rest, or the Banach–Tarski paradox, are now avoided. We can dismiss the paradoxes straight away because of their incorrect premises concerning the nature of space and time.
But let us go on. Special relativity, quantum mechanics and general relativity all rely on the idea that time can be de ned for all points of a given reference frame. However, two clocks a distance l apart cannot be synchronized with arbitrary precision. Since the distance between two clocks cannot be measured with an error smaller than the Planck length lPl, and transmission of signals is necessary for synchronization, it is not possible to synchronize two clocks with a better precision than the time lPl c = tPl, the Planck time. Because it is impossible to synchronize clocks precisely, a single time coordinate for a whole reference frame is only an approximation, and this idea cannot be maintained in a precise description of nature.
Moreover, since the time di erence between events can only be measured within a Planck time, for two events distant in time by this order of magnitude, it is not possible to say with complete certainty which of the two precedes the other! is is an important result. If events cannot be ordered, the concept of time, which was introduced into physics to describe sequences, cannot be de ned at all at Planck scales. In other words, a er dropping the idea of a common time coordinate for a complete frame of reference, we are forced to drop the idea of time at a single ‘point’ as well. erefore, the concept of ‘proper time’ loses its meaning at Planck scales.
It is straightforward to use the same arguments to show that length measurements do not allow us to speak of continuous space, but only of approximately continuous space. As a result of the lack of measurement precision at Planck scales, the concepts of spatial order, translation invariance, isotropy of vacuum and global coordinate systems have no experimental basis.
But there is more to come. e very existence of a minimum length contradicts special relativity theory, in which it is shown that lengths undergo Lorentz contraction when the frame of reference is changed. A minimum length thus cannot exist in special relativity. But we just deduced that there must be such a minimum distance in nature. ere is only one conclusion: special relativity cannot be correct at smallest distances. us, space-time
is neither Lorentz invariant nor di eomorphism invariant nor dilatation invariant at Planck dimensions. All symmetries that are at the basis of special and general relativity are thus only approximately valid at Planck scales.
As a result of the imprecision of measurement, most familiar concepts used to describe spatial relations become useless. For example, the concept of metric loses its usefulness at Planck scales. Since distances cannot be measured with precision, the metric cannot be determined. We deduce that it is impossible to say precisely whether space is at or curved. In other words, the impossibility of measuring lengths exactly is equivalent to uctuations of the curvature, and thus equivalent to uctuations of gravity.
In addition, even the number of spatial dimensions makes no sense at Planck scales.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 1043
Let us remind ourselves how to determine this number experimentally. One possible way is to determine how many points we can choose in space such that all the distances between them are equal. If we can nd at most n such points, the space has n − dimensions. We can see that if reliable length measurement at Planck scale is not possible, there is no way to determine reliably the number of dimensions of a space with this method.
Another way to check for three spatial dimensions is to make a knot in a shoe string and glue the ends together: since it stays knotted we know that space has three dimensions, because there is a mathematical theorem that in spaces with greater or fewer than three dimensions, knots do not exist. Again, at Planck dimensions the errors in measurement do not allow to say whether a string is knotted or not, because measurement limits at crossings make it impossible to say which strand lies above the other; in short, at Planck scales we cannot check whether space has three dimensions or not.
ere are many other methods for determining the dimensionality of space.* All these methods start from the de nition of the concept of dimensionality, which is based on a precise de nition of the concept of neighbourhood. However, at Planck scales, as just mentioned, length measurements do not allow us to say whether a given point is inside or outside a given volume. In short, whatever method we use, the lack of reliable length measurements means that at Planck scales, the dimensionality of physical space is not de ned. It should therefore not come as a surprise that when we approach these scales, we may get a scale-dependent answer for the number of dimensions, that may be di erent from three.
e reason for the problems with space-time become most evident when we remember Euclid’s well-known de nition: ‘A point is that which has no part.’ As Euclid clearly understood, a physical point, and here the stress is on physical, cannot be de ned without some measurement method. A physical point is an idealization of position, and as such includes measurement right from the start. In mathematics, however, Euclid’s de nition is rejected; mathematical points do not need a metric for their de nition. Mathematical points are elements of a set, usually called a space. In mathematics, a measurable or metric space is a set of points equipped a erwards with a measure or a metric. Mathematical points do not need a metric for their de nition; they are basic entities. In contrast to the mathematical situation, the case of physical space-time, the concepts of measure and of metric are more fundamental than that of a point. e di culties distinguishing physical and mathematical space and points arise from the failure to distinguish a mathematical metric from a physical length measurement.**
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* For example, we can determine the dimension using only the topological properties of space. If we draw a so-called covering of a topological space with open sets, there are always points that are elements of several sets of the covering. Let us call p the maximal number of sets of which a point can be an element in a given covering. is number can be determined for all possible coverings. e minimum value of p, minus one, gives the dimension of the space.
In fact, if physical space is not a manifold, the various methods may give di erent answers for the dimensionality. Indeed, for linear spaces without norm, a unique number of dimensions cannot be de ned. e value then depends on the speci c de nition used and is called e.g. fractal dimension, Lyapunov dimension, etc. ** Where does the incorrect idea of continuous space-time have its roots? In everyday life, as well as in physics, space-time is introduced to describe observations. Space-time is a book-keeping device. Its properties are extracted from the properties of observables. Since observables can be added and multiplied, we extrapolate that they can take continuous values. is extrapolation implies that length and time intervals can take continuous values, and, in particular, arbitrary small values. From this result we get the possibil-
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Ref. 1044
Perhaps the most beautiful way to make this point is the Banach–Tarski theorem, which clearly shows the limits of the concept of volume. e theorem states that a sphere made up of mathematical points can be cut into ve pieces in such a way that the pieces can be put together to form two spheres, each of the same volume as the original one. However, the necessary cuts are ‘in nitely’ curved and detailed: they are wildly disconnected. For physical matter such as gold, unfortunately – or fortunately – the existence of a minimum length, namely the atomic distance, makes it impossible to perform such a cut. For vacuum, the puzzle reappears: for example, the energy of zero-point uctuations is given by the density times the volume; following the Banach–Tarski theorem, the zero point energy content of a single sphere should be equal to the zero point energy of two similar spheres each of the same volume as the original one. e paradox is solved by the Planck length, because it also provides a fundamental length scale for vacuum, thus making in nitely complex cuts impossible. erefore, the concept of volume is only well de ned at Planck scales if a minimum length is introduced.
To sum up, physical space-time cannot be a set of mathematical points. But the surprises are not nished. At Planck dimensions, since both temporal and spatial order break down, there is no way to say if the distance between two space-time regions that are close enough together is space-like or time-like. Measurement limits make it impossible to distinguish the two cases. At Planck scales, time and space cannot be distinguished from each other.
In addition, it is impossible to state that the topology of space-time is xed, as general relativity implies. e topology changes – mentioned above – required for particle reactions do become possible. In this way another of the contradictions between general relativity and quantum theory is resolved.
In summary, space-time at Planck scales is not continuous, not ordered, not endowed with a metric, not four-dimensional and not made up of points. If we compare this with the de nition of the term manifold,* not one of its de ning properties is ful lled. We arrive at the conclusion that the concept of a space-time manifold has no backing at Planck scales. is is a strong result. Even though both general relativity and quantum mechanics
use continuous space-time, the combination of both theories does not.
“ ere is nothing in the world but matter in motion, and matter in motion cannot move otherwise than in space and time. ”Lenin
F
To complete this review of the situation, if space and time are not continuous, no quantities de ned as derivatives with respect to space or time are precisely de ned. Velocity, acceleration, momentum, energy, etc., are only well-de ned under the assumption of continuous space and time. at important tool, the evolution equation, is based on derivatives and thus can no longer be used. erefore the Schrödinger or the Dirac equation
ity of de ning points and sets of points. A special eld of mathematics, topology, shows how to start from a set of points and construct, with the help of neighbourhood relations and separation properties, rst a topological space. en, with the help of a metric, a metric space can be built. With the appropriate compactness and connectedness relations, a manifold, characterized by its dimension, metric and topology, can be constructed. * A manifold is what locally looks like an Euclidean space. e exact de nition can be found in Appendix D.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
lose their basis. Concepts such as ‘derivative’, ‘divergence-free’, ‘source free’ etc., lose their meaning at Planck scales.
In fact, all physical observables are de ned using length and time measurements. A list of physical units shows that each is a product of powers of length, time (and mass) units. (Even though in the SI system electrical quantities have a separate base quantity, the ampere, the argument still holds; the ampere is itself de ned in terms of a force, which is measured using the three base units length, time and mass.) Since time and length are not continuous, observables themselves are not de ned, because their value is not xed.
is means that at Planck scales, observables cannot be described by real numbers. In addition, if time and space are not continuous, the usual expression for an observable eld A, namely A(t, x), does not make sense: we have to nd a more appropriate description. Physical elds cannot exist at Planck scales.
e consequences for quantum mechanics are severe. It makes no sense to de ne multiplication of observables by real numbers, thus by a continuous range of values, but only by a discrete set of numbers. Among other implications, this means that observables do not form a linear algebra. We recognize that, because of measurement errors, we cannot prove that observables do form such an algebra. is means that observables are not described by operators at Planck scales. And, because quantum mechanics is based on the superposition principle, without it, everything comes crumbling down. In particular, the most important observables are the gauge potentials. Since they do not now form an algebra, gauge symmetry is not valid at Planck scales. Even innocuous looking expressions such as [xi , x j] = for xi x j, which are at the root of quantum eld theory, become meaningless at Planck scales. Since at those scales also the superposition principle cannot be backed up by experiment, even the famous Wheeler–DeWitt equation, o en assumed to describe quantum gravity, cannot be valid.
Similarly, permutation symmetry is based on the premise that we can distinguish two points by their coordinates, and then exchange particles between those two locations. As we have just seen, this is not possible if the distance between the two particles is small; we conclude that permutation symmetry has no experimental basis at Planck scales.
Even discrete symmetries, like charge conjugation, space inversion and time reversal cannot be correct in this domain, because there is no way to verify them exactly by measurement. CPT symmetry is not valid at Planck scales.
Finally we note that all types of scaling relations do not work at smallest scales. As a
result, renormalization symmetry is also destroyed at Planck scales. All these results are consistent: if there are no symmetries at Planck scales, there are
also no observables, since physical observables are representations of symmetry groups. In fact, the limits on time and length measurements imply that the concept of measurement has no signi cance at Planck scales.
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C
-
?–A
Ref. 1045 Ref. 1046 Ref. 1047 Ref. 1049
Discretization of space-time has been studied already since s. Recently, the idea that space-time could be described as a lattice has also been explored most notably by David Finkelstein and by Gerard ’t Hoo . e idea of space-time as a lattice is based on the idea that, if a minimum distance exists, then all distances are a multiple of this minimum. It is generally agreed that, in order to get an isotropic and homogeneous situation for large,
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Ref. 1048 Ref. 1050
everyday scales, the structure of space-time cannot be periodic, but must be random. In addition, any xed structure of space-time violates the result that there are no lengths smaller than the Planck length: as a result of the Lorentz contraction, any moving observer would nd lattice distances smaller than the Planck value. Worse still, the lattice idea con icts with general relativity, in particular with the di eomorphism invariance of vacuum. Finally, where would a particle be during the jump from one lattice point to the next? us, in summary, space-time cannot be a lattice. A minimum distance does exist in nature; however, the hope that all other distances are simple multiples of the smallest distance is not correct. We will discover more evidence for this later on.
If space-time is not a set of points or events, it must be something else. ree hints already appear at this stage. e rst step necessary to improve the description of motion is the recognition that abandoning ‘points’ means abandoning the local description of nature. Both quantum mechanics and general relativity assume that the phrase ‘observable at a point’ has a precise meaning. Because it is impossible to describe space as a manifold, this expression is no longer useful. e uni cation of general relativity and quantum physics forces the adoption of a non-local description of nature at Planck scales.
e existence of a minimum length implies that there is no way to physically distinguish locations that are even closer together. We are tempted to conclude therefore that no pair of locations can be distinguished, even if they are one metre apart, since on any path joining two points, no two locations that are close together can be distinguished.
is situation is similar to the question about the size of a cloud or of an atom. If we measure water density or electron density, we nd non-vanishing values at any distance from the centre of the cloud or the atom; however, an e ective size can still be de ned, because it is very unlikely that the e ects of the presence of a cloud or of an atom can be seen at distances much larger than this e ective size. Similarly, we can guess that two points in space-time at a macroscopic distance from each other can be distinguished because the probability that they will be confused drops rapidly with increasing distance. In short, we are thus led to a probabilistic description of space-time. Space-time becomes a macroscopic observable, a statistical or thermodynamic limit of some microscopic entities.
We note that a uctuating structure for space-time would also avoid the problems of xed structures with Lorentz invariance. is property is of course compatible with a statistical description. In summary, the experimental observations of special relativity, i.e. Lorentz invariance, isotropy and homogeneity, together with that of a minimum distance, point towards a uctuating description of space-time. Research e orts in quantum gravity, superstring theory and quantum groups have con rmed independently of each other that a probabilistic and non-local description of space-time at Planck dimensions, resolves the contradictions between general relativity and quantum theory. is is our
rst result on quantum geometry. To clarify the issue, we have to turn to the concept of the particle.
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In every example of motion, some object is involved. One of the important discoveries of the natural sciences was that all objects are composed of small constituents, called elementary particles. Quantum theory shows that all composite, non-elementary objects
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Ref. 1051
have a nite, non-vanishing size. is property allows us to determine whether a particle is elementary or not. If it behaves like a point particle, it is elementary. At present, only the leptons (electron, muon, tau and the neutrinos), the quarks and the radiation quanta of the electromagnetic, weak and strong nuclear interactions (the photon, the W and Z bosons, the gluons) have been found to be elementary. A few more elementary particles are predicted by various re nements of the standard model. Protons, atoms, molecules, cheese, people, galaxies etc., are all composite, as shown in Table . Elementary particles are characterized by their vanishing size, their spin and their mass.
Even though the de nition of ‘elementary particle’ as point particle is all we need in the following argument, it is not complete, because it seems to leave open the possibility that future experiments could show that electrons or quarks are not elementary. is is not so! In fact, any particle smaller than its own Compton wavelength is elementary. If it were composite, there would be a lighter component inside it and this lighter particle would have a larger Compton wavelength than the composite particle. is is impossible, since the size of a composite particle must be larger than the Compton wavelength of its components. ( e possibility that all components are heavier than the composite, which would avoid this argument, does not lead to satisfying physical properties; for example, it leads to intrinsically unstable components.)
e size of an object, such as those given in Table , is de ned as the length at which di erences from point-like behaviour are observed. is is the way in which, using alpha particle scattering, the radius of the atomic nucleus was determined for the rst time in Rutherford’s experiment. In other words, the size d of an object is determined by measuring how it scatters a beam of probe particles. In daily life as well, when we look at objects, we make use of scattered photons. In general, in order to make use of scattering, the effective wavelength λ = ħ mv of the probe must be smaller than the object size d to be determined. We thus need d λ = ħ (mv) ħ (mc). In addition, in order to make a scattering experiment possible, the object must not be a black hole, since, if it were, it would simply swallow the approaching particle. is means that its mass m must be smaller than that of a black hole of the same size; in other words, from equation ( ) we must have m < dc G. Combining this with the previous condition we get
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d
ħG c
= lPl .
(677)
In other words, there is no way to observe that an object is smaller than the Planck length. ere is thus no way in principle to deduce from observations that a particle is point-like. In
fact, it makes no sense to use the term ‘point particle’ at all! Of course, there is a relation between the existence of a minimum length for empty space and a minimum length for objects. If the term ‘point of space’ is meaningless, then the term ‘point particle’ is also meaningless. As in the case of time, the lower limit on length results from the combination of quantum mechanics and general relativity.*
e size d of any elementary particle must by de nition be smaller than its own Compton wavelength ħ (mc). Moreover, the size of a particle is always larger than the
* Obviously, the minimum size of a particle has nothing to do with the impossibility, in quantum theory, of localizing a particle to within less than its Compton wavelength.
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Planck length: d lPl. Combining these two requirements and eliminating the size d we get the condition for the mass m of any elementary particle, namely
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
m
<
ħ c lPl
=
ħc G
= mPl =
.
ë
− kg = . ë
GeV c .
(678)
Ref. 1052
Ref. 1053 Ref. 1054
e limit mPl, the so-called Planck mass, corresponds roughly to the mass of a human embryo that is ten days old, or equivalently, to that of a small ea. In short, the mass of any elementary particle must be smaller than the Planck mass. is fact was already noted as ‘well-known’ by Andrei Sakharov* in ; he explains that these hypothetical particles are sometimes called ‘maximons’. And indeed, the known elementary particles all have masses well below the Planck mass. (In fact, the question why their masses are so incredibly much smaller than the Planck mass is one of the most important questions of high-energy physics. We will come back to it.)
ere are many other ways to arrive at the mass limit for particles. For example, in order to measure mass by scattering – and that is the only way for very small objects – the Compton wavelength of the scatterer must be larger than the Schwarzschild radius; otherwise the probe will be swallowed. Inserting the de nition of the two quantities and neglecting the factor , we get again the limit m < mPl. (In fact it is a general property of descriptions of nature that a minimum spacetime interval leads to an upper limit for elementary particle masses.)
e importance of the Planck mass will become clear shortly. Another property connected with the size of a particle is its electric dipole moment. It describes the deviation of its charge distribu- Andrei Sakharov tion from spherical. Some predictions from the standard model of elementary particles give as upper limit for the electron dipole moment de a value of
de e
<
−
m,
(679)
Ref. 1055
where e is the charge of the electron. is value is ten thousand times smaller than the Planck length lPl e. Since the Planck length is the smallest possible length, we seem to have a potential contradiction here. However, a more recent prediction from the standard
model is more careful and only states
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de e
<
ë
− m,
(680)
which is not in contradiction with a minimal length in nature. e issue is still not settled. We will see below that the experimental limit is expected to allow to test these predictions
* Andrei Dmitrievich Sakharov, famous Soviet nuclear physicist (1921–1989). One of the keenest thinkers in physics, Sakharov, among others, invented the Tokamak, directed the construction of nuclear bombs, and explained the matter-antimatter asymmetry of nature. Like many others, he later campaigned against nuclear weapons, a cause for which he was put into jail and exile, together with his wife, Yelena Bonner. He received the Nobel Peace Prize in 1975.
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in the foreseeable future. Planck scales have other strange consequences. In quantum eld theory, the di erence
between a virtual particle and a real particle is that a real particle is ‘on shell’, obeying E = m c + p c , whereas a virtual particle is ‘o shell’, obeying E m c + p c . Because of the fundamental limits of measurement precision, at Planck scales we cannot
determine whether a particle is real or virtual. However, that is not all. Antimatter can be described as matter moving backwards
in time. Since the di erence between backwards and forwards cannot be determined at Planck scales, matter and antimatter cannot be distinguished at Planck scales.
Particles are also characterized by their spin. Spin describes two properties of a particle: its behaviour under rotations (and if the particle is charged, its behaviour in magnetic
elds) and its behaviour under particle exchange. e wave function of particles with spin remains invariant under a rotation of π, whereas that of particles with spin / changes sign. Similarly, the combined wave function of two particles with spin does not change sign under exchange of particles, whereas for two particles with spin / it does.
We see directly that both transformations are impossible to study at Planck scales. Given the limit on position measurements, the position of a rotation axis cannot be well de ned, and rotations become impossible to distinguish from translations. Similarly, position imprecision makes impossible the determination of precise separate positions for exchange experiments. In short, spin cannot be de ned at Planck scales, and fermions cannot
be distinguished from bosons, or, phrased di erently, matter cannot be distinguished from radiation at Planck scales. We can thus easily see that supersymmetry, a unifying symmetry between bosons and fermions, somehow becomes natural at Planck dimensions.
But let us now move to the main property of elementary particles.
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F e Planck mass divided by the Planck volume, i.e. the Planck density, is given by
ρPl
=
c G
ħ
=
.
ë
kg m
(681)
Challenge 1429 e
and is a useful concept in the following. If we want to measure the (gravitational) mass
M enclosed in a sphere of size R and thus (roughly) of volume R , one way to do this is
to put a test particle in orbit around it at that same distance R. Universal gravitation then
gives for the mass M the expression M = Rv G, where v is the speed of the orbiting test
particle. From v < c, we thus deduce that M < c R G; since the minimum value for R is
the Planck distance, we get (neglecting again factors of order unity) a limit for the mass
density ρ, namely
ρ < ρPl .
(682)
In other words, the Planck density is the maximum possible value for mass density. Unsurprisingly, a volume of Planck dimensions cannot contain a mass larger than the Planck mass.
Interesting things happen when we start to determine the error ∆M of a mass measurement in a Planck volume. Let us return to the mass measurement by an orbiting probe.
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M
R
R
F I G U R E 381 A Gedanken experiment showing that at Planck scales, matter and vacuum cannot be distinguished
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From the relation GM = rv we deduce by di erentiation that G∆M = v ∆r + vr∆v vr∆v = GM∆v v. For the error ∆v in the velocity measurement we have the indeterminacy relation ∆v ħ (m∆r) + ħ (MR) ħ (MR). Inserting this in the previous inequality, and forgetting again the factor of , we nd that the mass measurement error ∆M of a mass M enclosed in a volume of size R is subject to the condition
∆M
ħ cR
.
(683)
Note that for everyday situations, this error is extremely small, and other errors, such as the technical limits of the balance, are much larger.
To check this result, we can explore another situation. We even use relativistic expressions, in order to show that the result does not depend on the details of the situation or the approximations. Imagine having a mass M in a box of size R and weighing the box with a scale. (It is assumed that either the box is massless or that its mass is subtracted by the scale.) e mass error is given by ∆M = ∆E c , where ∆E is due to the indeterminacy in the kinetic energy of the mass inside the box. Using the expression E = m c + p c , we get that ∆M ∆p c, which again reduces to equation ( ). Now that we are sure of the result, let us continue.
From equation ( ) we deduce that for a box of Planck dimensions, the mass measurement error is given by the Planck mass. But from above we also know that the mass that can be put inside such a box must not be larger than the Planck mass. erefore, for a box of Planck dimensions, the mass measurement error is larger than (or at best equal to) the mass contained in it: ∆M MPl. In other words, if we build a balance with two boxes of Planck size, one empty and the other full, as shown in Figure , nature cannot decide which way the balance should hang! Note that even a repeated or a continuous measurement will not resolve the situation: the balance will only randomly change inclination, staying horizontal on average.
e argument can be rephrased as follows. e largest mass that we can put in a box
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of size R is a black hole with a Schwarzschild radius of the same value; the smallest mass
present in such a box – corresponding to what we call vacuum – is due to the indeterm-
inacy relation and is given by the mass with a Compton wavelength that matches the size
of the box. In other words, inside any box of size R we have a mass m, the limits of which
are given by:
(full
box)
cR G
m
ħ cR
(empty box)
.
(684)
We see directly that for sizes R of the order of the Planck scale, the two limits coincide; in other words, we cannot distinguish a full box from an empty box in that case.
To be sure of this strange result, we check whether it also occurs if, instead of measuring the gravitational mass, as we have just done, we measure the inertial mass. e inertial mass for a small object is determined by touching it, i.e. physically speaking, by performing a scattering experiment. To determine the inertial mass inside a region of size R, a probe must have a wavelength smaller than R, and thus a correspondingly high energy. A high energy means that the probe also attracts the particle through gravity. (We thus
nd the intermediate result that at Planck scales, inertial and gravitational mass cannot be distinguished. Even the balance experiment shown in Figure illustrates this: at Planck scales, the two types of mass are always inextricably linked.) Now, in any scattering experiment, e.g. in a Compton-type experiment, the mass measurement is performed by measuring the wavelength change δλ of the probe before and a er the scattering experiment. e mass indeterminacy is given by
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∆M M
=
∆δλ δλ
.
(685)
In order to determine the mass in a Planck volume, the probe has to have a wavelength of the Planck length. But we know from above that there always is a minimum wavelength indeterminacy, given by the Planck length lPl. In other words, for a Planck volume the mass error is always as large as the Planck mass itself: ∆M MPl. Again, this limit is a direct consequence of the limit on length and space measurements.
is result has an astonishing consequence. In these examples, the measurement error is independent of the mass of the scatterer, i.e. independent of whether or not we start with a situation in which there is a particle in the original volume. We thus nd that in a volume of Planck size, it is impossible to say whether or not there is something there when we probe it with a beam!
In short, all arguments lead to the same conclusion: vacuum, i.e. empty space-time, cannot be distinguished from matter at Planck scales. Another, o en used way to express this state of a airs is to say that when a particle of Planck energy travels through space it will be scattered by the uctuations of space-time itself, thus making it impossible to say whether it was scattered by empty space-time or by matter. ese surprising results rely on a simple fact: whatever de nition of mass we use, it is always measured via combined length and time measurements. ( is is even the case for normal weighing scales: mass is measured by the displacement of some part of the machine.) e error in these measurements makes it impossible to distinguish vacuum from matter.
We can put this result in another way. If on one hand, we measure the mass of a piece
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Ref. 1056 Page 858
of vacuum of size R, the result is always at least ħ cR; there is no possible way to nd a perfect vacuum in an experiment. On the other hand, if we measure the mass of a particle, we nd that the result is size dependent; at Planck dimensions it approaches the Planck mass for every type of particle, be it matter or radiation.
If we use another image, when two particles approach each other to a separation of the order of the Planck length, the indeterminacy in the length measurements makes it impossible to say whether there is something or nothing between the two objects. In short, matter and vacuum are interchangeable at Planck dimensions. is is an important result: since both mass and empty space-time cannot be di erentiated, we have con rmed that they are made of the same ‘fabric’. is approach, already suggested above, is now commonplace in all attempts to nd a uni ed description of nature.
is approach is corroborated by the attempts to apply quantum mechanics in highly curved space-time, where a clear distinction between vacuum and particles is impossible.
is has already been shown by Fulling–Davies–Unruh radiation. Any accelerated observer and any observer in a gravitational eld detects particles hitting him, even if he is in vacuum. e e ect shows that for curved space-time the idea of vacuum as a particlefree space does not work. Since at Planck scales it is impossible to say whether space is
at or not, it again follows that it is impossible to say whether it contains particles or not.
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e strange results at Planck scales imply many other consequences.
**
Observes are made of matter. Observer are thus biased, because they take a speci c standpoint. But at Planck scale, vacuum, radiation and matter cannot me distinguished. Two conclusions result: rst, only at those scales would a description be free of any bias in favour of matter; but secondly, observers do not exist at all at Planck energy.
Challenge 1430 n Challenge 1431 n
**
e Planck energy is rather large. Imagine that we want to impart this amount of energy to protons using a particle accelerator. How large would that accelerator have to be? In contrast, in everyday life, the Planck energy is rather small. Measured in litres of gasoline, how much fuel does it correspond to?
**
e usual concepts of matter and of radiation are not applicable at Planck dimensions. Usually, it is assumed that matter and radiation are made up of interacting elementary particles. e concept of an elementary particle is one of an entity that is countable, pointlike, real and not virtual, that has a de nite mass and a de nite spin, that is distinct from its antiparticle, and, most of all, that is distinct from vacuum, which is assumed to have zero mass. All these properties are found to be incorrect at Planck scales. At Planck dimensions, it does not make sense to use the concepts of ‘mass’, ‘vacuum’, ‘elementary particle’, ‘radiation’ and ‘matter’.
**
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Do the large mass measurement errors make it possible to claim that mass can be negative Challenge 1432 n at Planck energy?
**
We now have a new answer to the old question: why is there anything rather than nothing? Well, we now see that at Planck scales there is no di erence between anything and nothing. In addition, we now can honestly say about ourselves that we are made of nothing.
**
If vacuum and matter or radiation cannot be distinguished, then it is incorrect to claim that the universe appeared from nothing. e impossibility of making this distinction thus shows that naive creation is a logical impossibility. Creation is not a description of reality. e term ‘creation’ turns out to be a result of lack of imagination.
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Challenge 1433 r Ref. 1058
**
Special relativity implies that no length or energy can be invariant. Since we have come to the conclusion that the Planck energy and the Planck length are invariant, there must be deviations from Lorentz invariance at high energy. Can you imagine what the e ects would be? In what experiment could they be measured? If you nd an answer, publish it; you might get known. First attempts are appearing in the research papers. We return to the issue in the third part, with some interesting insights.
Ref. 1057 Ref. 1038
Challenge 1434 ny
**
Quantum mechanics alone gives, via the Heisenberg indeterminacy relation, a lower limit on the spread of measurements, but strangely enough not on their precision, i.e. not on the number of signi cant digits. Jauch gives the example that atomic lattice constants are known much more precisely than the position indeterminacy of single atoms inside the crystal.
It is sometimes claimed that measurement indeterminacies smaller than the Planck values are possible for large enough numbers of particles. Can you show why this is incorrect, at least for space and time?
**
Of course, the idea that vacuum is not empty is not new. More than two thousand years ago, Aristotle argued for a lled vacuum, even though he used incorrect arguments, as seen from today’s perspective. In the fourteenth century the discussion on whether empty space was composed of indivisible entities was rather common, but died down again later.
Challenge 1435 n
**
A Planck energy particle falling in a gravitational eld would gain energy. However, this is impossible, as the Planck energy is the highest energy in nature. What does this imply for this situation?
**
One way to generalize the results presented here is to assume that, at Planck energy, nature Ref. 1027 is event symmetric, i.e. nature is symmetric under exchange of any two events. is idea,
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developed by Phil Gibbs, provides an additional formulation of the strange behaviour of nature at extreme scales.
Page 489
**
Because there is a minimum length in nature, so-called naked singularities do not exist. e issue, so hotly debated in the twentieth century, becomes uninteresting, thus ending
decades of speculation.
Page 870 Ref. 1059
**
Since mass density and thus energy density are limited, we know that the number of degrees of freedom of any object of nite volume is nite. e entropy of black holes has shown us already that entropy values are always nite. is implies that perfect baths do not exist. Baths play an important role in thermodynamics (which is thus found to be only an approximation) and also in recording and measuring devices: when a device measures, it switches from a neutral state to a state in which it shows the result of the measurement. In order to avoid the device returning to the neutral state, it must be coupled to a bath. Without a bath, a reliable measuring device cannot be made. In short, perfect clocks and length measuring devices do not exist because nature puts a limit on their storage ability.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Page 37
**
If vacuum and matter cannot be distinguished, we cannot distinguish between objects and their environment. However, this was one the starting points of our journey. Some interesting adventures thus still await us!
Page 676
**
We have seen earlier that characterizing nature as made up of particles and vacuum creates problems when interactions are included, since on one hand interactions are the difference between the parts and the whole, while on the other hand, according to quantum theory, interactions are exchanges of particles. is apparent contradiction can be used to show either that vacuum and particles are not the only components of nature, or that something is counted twice. However, since matter and space-time are both made of the same ‘stu ,’ the contradiction is resolved.
** Challenge 1436 d Is there a smallest possible momentum? And a smallest momentum error?
**
ere is a maximum acceleration in nature. Can you deduce the value of this so-called Challenge 1437 n Planck acceleration? Does it require quantum theory?
**
Given that time becomes an approximation at Planck scales, can we still say whether nature is deterministic? Let us go back to the beginning. We can de ne time, because in nature change is not random, but gradual. What is the situation now that we know that time is only approximate? Is non-gradual change possible? Is energy conserved? In other words, are surprises possible?
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Challenge 1438 n
To say that time is not de ned at Planck scales and that therefore determinism is an unde nable concept is correct, but not a satisfying answer. What happens at daily life scales? e rst answer is that at our everyday scales, the probability of surprises is so small that the world indeed is e ectively deterministic. e second answer is that nature is not deterministic, but that the di erence is not measurable, since every measurement and observation, by de nition, implies a deterministic world. e lack of surprises would be due to the limitations of our human nature, and more precisely to the limitations of our senses and brain. e third answer is that the lack of surprises is only apparent, and that we have not yet experienced them yet.
Can you imagine any other possibility? To be honest, it is not possible to answer at this point. But we need to keep the alternatives in mind. We have to continue searching, but with every step we take, we have to consider carefully what we are doing.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
**
If matter and vacuum cannot be distinguished, matter and vacuum each has the properties of the other. For example, since space-time is an extended entity, matter and radiation are also extended entities. Furthermore, as space-time is an entity that reaches the borders of the system under scrutiny, particles must also do so. is is the rst hint at the extension of matter; in the following, we will examine this argument in more detail.
Ref. 1060
**
Vacuum has zero mass density at large scales, but Planck mass density at Planck scales. Cosmological measurements show that the cosmos is at or almost at at large scales, i.e. its energy density is quite low. In contrast, quantum eld theory maintains that vacuum has a high energy density (or mass density) at small scales. Since mass is scale dependent, both viewpoints are right, providing a hint to the solution of what is usually called the cosmological constant problem. e contradiction is only apparent; more about this issue later on.
** Challenge 1439 n When can matter and vacuum be distinguished? At what energy?
**
If matter and vacuum cannot be distinguished, there is a lack of information, which in turn produces an intrinsic basic entropy associated with any part of the universe. We will come back to this topic shortly, in the discussion of the entropy of black holes.
**
Can we distinguish between liquids and gases by looking at a single atom? No, only by looking at many. In the same way, we cannot distinguish between matter and vacuum by looking at one point, but only by looking at many. We must always average. However, even averaging is not completely successful. Distinguishing matter from vacuum is like distinguishing clouds from the clear sky; like clouds, matter also has no de ned boundary.
** In our exploration we have found that there is no argument which shows that space and
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time are either continuous or made up of points. Indeed, in contrast, we have found that
the combination of relativity and quantum theory makes this impossible. In order to pro-
ceed in our ascent of Motion Mountain, we need to leave behind us the usual concept of space-time. At Planck dimensions, the concept of ‘space-time point’ or ‘mass point’ is not applicable in the description of nature.
F
A minimum length, or equivalently,* a minimum action, both imply that there is a maximum curvature for space-time. Curvature can be measured in several ways; for example, surface curvature is an inverse area. A minimum length thus implies a maximum curvature. Within a factor of order one, we nd
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K
<
c Għ
=
.
ë
m− .
(686)
Challenge 1440 n
as limit for surface curvature K in nature. In other words, the universe never has been a point, never had zero age, never had in nite density and never had in nite curvature. It is not di cult to get a similar limit for temperature or any other physical quantity.
In short, since events do not exist, also the big bang cannot have been an event. ere never was an initial singularity or a beginning of the universe.
T
Page 1068
Ref. 1012 Ref. 1013
In this rapid journey, we have destroyed all the experimental pillars of quantum theory: the superposition principle, space-time symmetry, gauge symmetry, renormalization symmetry and permutation symmetry. We also have destroyed the foundations of general relativity, namely the existence of the space-time manifold, the eld concept, the particle concept and the concept of mass. We have even seen that matter and space-time cannot be distinguished.
All these conclusions can be drawn in a simpler manner, by using the minimum action of quantum theory and the maximum force of general relativity. All the mentioned results above are con rmed. It seems that we have lost every concept used for the description of motion, and thus made its description impossible. We naturally ask whether we can save the situation.
First of all, since matter is not distinguishable from vacuum, and since this is true for all types of particles, be they matter or radiation, we have an argument which demonstrates that the quest for uni cation in the description of elementary particles is correct and necessary.
Moreover, since the concepts ‘mass’, ‘time’ and ‘space’ cannot be distinguished from each other, we also know that a new, single entity is necessary to de ne both particles and space-time. To nd out more about this new entity, three approaches are being pursued at the beginning of the twenty- rst century. e rst, quantum gravity, especially the approach using the loop representation and Ashtekar’s new variables, starts by generalizing space-time symmetry. e second, string theory, starts by generalizing gauge symmetries
* e big bang section was added in summer 2002.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 1014
and interactions, while the third, the algebraic quantum group approach, looks for generalized permutation symmetries. We will describe these approaches in more detail later on.
Before we go on however, we should check with experiments what we have deduced so far.
S
Ref. 1061
At present, there is a race going on both in experimental and in theoretical physics: which will be the rst experiment that will detect quantum gravity e ects, i.e. e ects sensitive to the Planck energy?*
One might think that the uctuations of space and time might make images from far away galaxies unsharp or destroy the phase relation between the photons. However, this e ect has been shown to be unmeasurable in all possible cases.
A better candidate is measurement of the speed of light at di erent frequencies in far away light ashes. ere are natural ashes, called gamma ray bursts, which have an extremely broad spectrum, from GeV down to visible light of about eV. ese ashes o en originate at cosmological distances d. From the di erence in arrival time ∆t for two frequencies we can de ne a characteristic energy by setting
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Echar
=
ħ(ω − ω c∆t
)d
.
(687)
Ref. 1062 Ref. 1063, Ref. 1064 Ref. 1065, Ref. 1064
Ref. 1066
is energy value is ë GeV for the best measurement to date. is value is not far from the Planck energy; in fact, it is even closer when the missing factors of order unity are included. It is expected that the Planck scale will be reached in a few years, so that tests will become possible on whether the quantum nature of space-time in uences the dispersion of light signals. Planck scale e ects should produce a minimum dispersion, di erent from zero. Detecting it would con rm that Lorentz symmetry is not valid at Planck scales.
Another candidate experiment is the direct detection of distance uctuations between bodies. Gravitational wave detectors are sensitive to extremely small noise signals in length measurements. ere should be a noise signal due to the distance uctuations induced near Planck energies. e length indeterminacy with which a length l can be measured is predicted to be
δl lPl ll
(688)
Page 1036 Ref. 1067
e expression is deduced simply by combining the measurement limit of a ruler in quantum theory with the requirement that the ruler cannot be a black hole. We will discuss this result in more detail in the next section. e sensitivity to noise of the detectors might reach the required level in the early twenty- rst century. e noise induced by quantum gravity e ects is also predicted to lead to detectable quantum decoherence and vacuum uctuations.
* As more candidates appear, they will be added to this section.
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Ref. 1064 Ref. 1068
Ref. 1069 Ref. 1067
A third candidate for measurable quantum gravity is the detection of the loss of CPT symmetry at high energies. Especially in the case of the decay of certain elementary particles, such as neutral kaons, the precision of experimental measurement is approaching the detection of Planck scale e ects.
A fourth candidate is the possibility that quantum gravity e ects may change the threshold energy at which certain particle reactions become possible. It may be that extremely high energy photons or cosmic rays will make it possible to prove that Lorentz invariance is indeed broken near the Planck scale.
A h candidate is the possibility that the phase of light that travels over long distances gets washed out. However, the rst tests show that this is not the case; light form extremely distant galaxies still interferes. e precise prediction of the phase washing e ect is still in discussion; most probably the e ect is too small to be measured.
In the domain of atomic physics, it has also been predicted that quantum gravity e ects will induce a gravitational Stark e ect and a gravitational Lamb shi in atomic transitions. Either e ect could be measurable.
A few candidates for quantum gravity e ects have also been predicted by the author. To get an overview, we summarize and clarify the results found so far.* Special relativity starts with the discovery that observable speeds are limited by the speed of light c. Quantum theory starts with the result that observable actions are limited by ħ . Gravitation shows that for every system with length L and mass M, the observable ratio L M is limited by the constant G c . Combining these results, we have deduced that all physical observables are bound, namely by what are usually called the Planck values, though modi ed by a factor of square root of (or several of them) to compensate for the numerical factors omitted from the previous sentence.
We need to replace ħ by ħ and G by G in all the de ning expressions for Planck quantities, in order to nd the corresponding measurement limits. In particular, the limit for lengths and times is times the Planck value and the limit for energy is the Planck value divided by .** Interestingly, the existence of bounds on all observables makes it possible to deduce several experimentally testable predictions for the uni cation of quantum theory and general relativity. ese predictions do not depend on the detailed
nal theory. However, rst we need to correct the argument that we have just presented. e argu-
ment is only half the story, because we have cheated. e (corrected) Planck values do not seem to be the actual limits to measurements. e actual measurement limits are even stricter still.
First of all, for any measurement, we need certain fundamental conditions to be realized. Take the length measurement of an object. We need to be able to distinguish between matter and radiation, since the object to be measured is made up of matter, and since radiation is the measurement tool that is used to read distances from the ruler. For a measurement process, we need an interaction, which implies the use of radiation. Note that even the use of particle scattering to determine lengths does not invalidate this general requirement.
* is subsection, in contrast to the ones so far, is speculative; it was added in February 2001. ** e entropy of a black hole is thus given by the ratio between its horizon and half the minimum area. Of course, a detailed investigation also shows that the Planck mass (divided by ) is the limit for elementary particles from below and for black holes from above. For everyday systems, there is no limit.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Page 1104 Ref. 1070
Ref. 1071
In addition, for the measurement of wavelengths we need to distinguish between matter and radiation, because matter is necessary to compare two wavelengths. In fact, all length measurements require the distinction between matter and radiation.* However, this distinction is impossible at the energy of grand uni cation, when the electroweak and the strong nuclear interactions are uni ed. At and above this energy, particles of matter and particles of radiation transform into each other; in practice they cannot be distinguished from each other.
If all matter and radiation particles were the same or mixtures of each other, mass could not be de ned. Similarly, spin, charge or any other quantum numbers could be de ned. To sum up, no measurement can be performed at energies equal to or greater than the GUT uni cation energy.
In other words, the particle concept (and thus the matter concept) does not run into trouble at the Planck scale, it has already done so at the uni cation scale. Only below the uni cation scale do our standard particle and space-time concepts apply. Only below the
uni cation scale can particles and vacuum be e ectively distinguished. As a result, the smallest length in nature is times the Planck length reduced by
the ratio between the maximal energy EPl and the uni cation energy EGUT. Present estimates give EGUT = GeV, implying that the smallest accessible length Lmin is
Lmin =
lPl
EPl EGUT
−m
lPl .
(689)
It is unlikely that measurements at these dimensions will ever be possible. Anyway, the smallest measurable length is signi cantly larger than the Planck scale of nature discussed above. e reason for this is that the Planck scale is that length for which particles and vacuum cannot be distinguished, whereas the minimal measurable length is the distance at which particles of matter and particles of radiation cannot be distinguished. e latter happens at lower energy than the former. We thus have to correct our previous statement to: the minimum measurable length cannot be smaller than Lmin.
e experimentally determined factor of about is one of the great riddles of physics. It is the high-energy equivalent of the quest to understand why the electromagnetic coupling constant is about , or more simply, why all things have the colours they have. Only the nal theory of motion will provide the answer.
In particular, the minimum length puts a bound on the electric dipole moment d of elementary particles, i.e. on any particles without constituents. We get the limit
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d dmin = e Lmin = e − m = . ë − Cm .
(690)
Page 1012 Ref. 1072
We saw that this result is in contradiction with one of the predictions deduced from the standard model, but not with others. More interestingly, the prediction is in the reach of future experiments. is improved limit may be the simplest possible measurement of yet unpredicted quantum gravity e ects. Measuring the dipole moment could thus be a way to determine the uni cation energy (the factor ) independently of high-energy physics experiments and possibly to a higher precision.
* To speak in modern high energy concepts, all measurements require broken supersymmetry.
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Inverse of coupling
1/26
electromagnetism weak interaction
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
strong interaction
105
1010
1015
"1020" Energy
GUT Planck in GeV
energy energy
F I G U R E 382 Coupling constants and their spread as a function of energy
Interestingly, the bound on the measurability of observables also puts a bound on the measurement precision for each observable. is bound is of no importance in everyday life, but it is important at high energy. What is the precision with which a coupling constant can be measured? We can illustrate this by taking the electromagnetic coupling constant as an example. is constant α, also called the ne structure constant, is related to the charge q by
q = πε ħcα .
(691)
Now, any electrical charge itself is de ned and measured by comparing, in an electrical eld, the acceleration to which the charged object is subjected with the acceleration of
some unit charge qunit. In other words, we have
q qunit
=
ma munit aunit
.
(692)
Page 860
erefore any error in mass and acceleration measurements implies errors in measurements of charge and the coupling constant.
We found in the part on quantum theory that the electromagnetic, the weak and the strong interactions are characterized by coupling constants, the inverse of which depend linearly on the logarithm of the energy. It is usually assumed that these three lines meet at the uni cation energy already mentioned. Measurements put the uni cation coupling
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Ref. 1071 Challenge 1441 n
value at about / . We know from the above discussions that the minimum measurement error for any
energy measurement at high energies is given by the ratio between the energy to be measured and the limit energy. Inserting this into the graph of the coupling constants ‘running’ with energy – as physicist like to say – we get the result shown in Figure . e search for the consequences of this fan-out e ect is delightful. One way to put the result is to say that coupling constants are by de nition subject to an error. However, all measurement devices, be they clocks, metre rules, scales or any other device, use electromagnetic effects at energies of around eV plus the electron rest energy. is is about − times the GUT energy. As a consequence, the measurement precision of any observable is limited to about digits. e maximum precision presently achieved is digits, and, for the electromagnetic coupling constant, about digits. It will thus be quite some time before this prediction can be tested.
e fun is thus to nd a system in which the spreading coupling constant value appears more clearly in the measurements. For example, it may be that high-precision measurements of the -factor of elementary particles or of high-energy cosmic ray reactions will show some e ects of the fan-out. e lifetime of elementary particles could also be affected. Can you nd another e ect?
In summary, the experimental detection of quantum gravity e ects should be possible, despite their weakness, at some time during the twenty- rst century. e successful detection of any such e ect will be one of the highlights of physics, as it will challenge the usual description of space and time even more than general relativity did.
We now know that the fundamental entity describing space-time and matter that we are looking for is not point-like. What does it look like? To get to the top of Motion Mountain as rapidly as possible, we will make use of some explosive to blast away a few disturbing obstacles.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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B
1003 S
W
, e cosmological constant problem, Reviews of Modern Physics 61,
pp. – , . Cited on page .
1004 S I,
W , and II,
, e quantum theory of elds, Cambridge University Press, volume . Cited on page .
1005 e di culties are summarised by B.S. W , Quantum eld theory in curved spacetime, Physics Reports 19, pp. – , . Cited on page .
1006 C.W. M pages ,
& K.S. T , and .
, J.A. W
, Gravitation, Freeman, . Cited on
1007 J.A. W
, in Relativity, Groups and Topology, edited by C. DeWitt & B. DeWitt,
Gordon and Breach, . See also J.A. W
, Physics at the Planck length, Interna-
tional Journal of Modern Physics A 8, pp. – , . Cited on page .
1008 J.L. F
& R.D. S
, Spin / from gravity, Physical Review Letters 44,
pp. – , . Cited on page .
1009 A.P. B
, G. B
, G. M
and quantum physics, Nuclear Physics B 446, pp.
hep-th/
. Cited on page .
& A. S
, Topology change
– , , http://www.arxiv.org/abs/
1010 Many themes discussed here resulted from an intense exchange with Luca Bombelli, now
at the Department of Physics and Astronomy of the University of Mississippi. ese pages
are a reworked version of those published in French as C. S
, Le vide di ère-t-il
de la matière? in E. G
& S. D , editeurs, Le vide – Univers du tout et du rien
– Des physiciens et des philosophes s’interrogent, Les Éditions de l’Université de Bruxelles,
. An older English version is also available as C. S
, Does matter di er from
vacuum? http://www.arxiv.org/abs/gr-qc/
. Cited on page .
1011 See for example R
P. F
Lectures on Physics, Addison Wesley,
, R.B. L
& M. S
. Cited on page .
, e Feynman
1012 A. A
, Recent mathematical developments in quantum general relativity, pre-
print found at http://www.arxiv.org/abs/gr-qc/
. C. R
, Ashtekar Formula-
tion of general relativity and loop-space non-perturbative quantum gravity: a report, Clas-
sical and Quantum Gravity 8, pp. – , . Cited on pages and .
1013 M.B. G
, J.H. S
bridge University Press,
Europhysics Letters 2, pp.
& E. W
, Superstring theory, volume and (Cam-
). G. V
, A stringy nature needs just two constants,
– , . Cited on pages and .
1014 S. M , Introduction to Braided Geometry and q-Minkowski Space, preprint down-
loadable at http://www.arxiv.org/abs/hep-th/
. See also S. M , Beyond su-
persymmetry and quantum symmetry (an introduction to braided groups and braided
matrices), in Quantum Groups, Integrable Statistical Models and Knot eory, edited by
M.-L. G & H.J. V , World Scienti c, , pp. – . Cited on pages
and .
1015 C.J. I
, Prima facie questions in quantum gravity, http://www.arxiv.org/abs/gr-qc/
. Cited on page .
1016 S
W
and .
, Gravitation and Cosmology, Wiley, . Cited on pages , ,
1017 e observations of black holes at the centre of galaxies and elsewhere are summarised by
R. B
& N. G
, Revisiting the black hole, Physics Today 52, June .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Cited on page .
1018 M. P
, Über irreversible Strahlungsvorgänge, Sitzungsberichte der Kaiserlichen
Akademie der Wissenscha en zu Berlin, pp. – , . Today it is commonplace to
use Dirac’s ħ instead of Planck’s h, which Planck originally called b. Cited on page .
1019 e argument is given e.g. in E.P. W
, Relativistic invariance and quantum phenom-
ena, Reviews of Modern Physics 29, pp. – , . Cited on page .
1020 e starting point for the following arguments is taken from M. S
, Operative time
de nition and principal indeterminacy, preprint http://www.arxiv.org/abs/gr-qc/
,
and from T. P
, Limitations on the operational de nition of space-time
events and quantum gravity, Classical and Quantum Gravity 4, pp. L –L , ; see
also Padmanabhan’s earlier papers referenced there. Cited on page .
1021 W. H
, Über den anschaulichen Inhalt der quantentheoretischen Kinematik
und Mechanik, Zeitschri für Physik 43, pp. – , . Cited on page .
1022 E.H. K
, Zur Quantenmechanik einfacher Bewegungstypen, Zeitschri für Physik
44, pp. – , . Cited on page .
1023 M.G. R
, Uncertainty principle for joint measurement of noncommuting variables,
American Journal of Physics 62, pp. – , . Cited on page .
1024 H. S
& E.P. W
, Quantum limitations of the measurement of space-time
distances, Physical Review 109, pp. – , . Cited on page .
1025 E.J. Z 30, pp. – ,
, e macroscopic nature of space-time, American Journal of Physics . Cited on page .
1026 S.W. H
, Particle creation by black holes, Communications in Mathematical Phys-
ics 43, pp. – , ; see also S.W. H
, Black hole thermodynamics, Physical
Review D13, pp. – , . Cited on page .
1027 P. G , e small structure of space-time: a bibliographical review, unpublished pre-
print found at http://www.arxiv.org/abs/hep-th/
. Cited on pages and .
1028 C.A. M , Possible connection between gravitation and fundamental length, Physical Review B 135, pp. – , . Cited on pages , , , and .
1029 P.K. T
, Small-scale structure of space-time as the origin of the gravitational
constant, Physical Review D 15, pp. – , . Cited on page .
1030 M.-T. J
& S. R
, Gravitational quantum limit for length measurement, Phys-
ics Letters A 185, pp. – , . Cited on page .
1031 D.V. A 339, pp. – page .
, Quantum measurement, gravitation and locality, Physics Letters B
, , also preprint http://www.arxiv.org/abs/gr-qc/
. Cited on
1032 L. G
, Quantum gravity and minimum length, International Journal of Modern Phys-
ics A 10, pp. – , , also preprint http://www.arxiv.org/abs/gr-qc/
; this pa-
per also includes an extensive bibliography. Cited on page .
1033 G. A
–C
, Limits on the measurability of space-time distances in (the
semiclassical approximation of) quantum gravity, Modern Physics Letters A 9, pp. –
, , also preprint http://www.arxiv.org/abs/gr-qc/
. See also Y.J. N & H.
V D , Limit to space-time measurement, Modern Physics Letters A 9, pp. – ,
. Cited on page .
1034 e generalised indeterminacy relation is an implicit part of Ref. , especially on page , but the issue is explained rather unclearly. Obviously the author found the result too
simple to be mentioned explicitly. Cited on pages and .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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1035 C. R
& L. S
, Discreteness of area and volume in quantum gravity, Nuclear
Physics B 442, pp. – , . R. L , e volume operator in discretized quantum
gravity, preprint http://www.arxiv.org/abs/gr-qc/
. Cited on page .
1036 D. A
, M. C
& G. V
, Superstring collisions at Planckian en-
ergies, Physics Letters B 197, pp. – , . D.J. G
& P.F. M
, e high en-
ergy behavior of string scattering amplitudes, Physics Letters B 197, pp. – , . K.
K
, G. P
& P. P
, Minimum physical length and the generalised
uncertainty principle, Physics Letters B 234, pp. – , . P. A
, Minimum
Distances in Non-Trivial String Target Spaces, Nuclear Physics B 431, pp. – , , also
preprint http://www.arxiv.org/abs/hep-th/
. Cited on page .
1037 M. M
, A generalised uncertainty principle in quantum mechanics, Physics Let-
ters B 304, pp. – , . Cited on page .
1038 A simple approach is S. D
, K. F
& J.E. R
, Space-time
quantization induced by classical gravity, Physics Letters B 331, pp. – , , Cited on
pages and .
1039 A. K
, Uncertainty relation in quantum mechanics with quantum group symmetry,
Journal of Mathematical Physics 35, pp. – , . A. K
, Quantum Groups
and quantum eld theory with Nonzero minimal uncertainties in positions and momenta,
Czechoslovak Journal of Physics 44, pp. – , . Cited on page .
1040 E.J. H
& K. T
. Cited on page .
, Quantized space-time, Physical Review 94, pp. – ,
1041 e impossibility to determine temporal ordering in quantum theory is discussed by J.
O
, B. R
& W.G. U
, Temporal ordering in quantum mechan-
ics, Journal of Physics A 35, pp. – , , or http://www.arxiv.org/abs/quant-ph/
. Cited on page .
1042 A. P & N. R , Quantum Limitations on the measurement of gravitational elds, Physical Review 118, pp. – , . Cited on page .
1043 It is the rst de nition in Euclid’s Elements, c.
. For an English translation see T.
H
, e irteen Books of the Elements, Dover, . Cited on page .
1044 A beautiful description of the Banach–Tarski paradox is the one by I S dox of the spheres, New Scientist, January , pp. – . Cited on page
, Para.
1045 H.S. S
, Quantized space-time, Physical Review 71, pp. – , . H.S. S
,
e electromagnetic eld in quantised of space-time, Physical Review 72, pp. – , .
A. S
, Discrete space-time and integral Lorentz transformations, Physical Review
73, pp. – , . E.L. H , Relativistic theory of discrete momentum space and dis-
crete space-time, Physical Review 100, pp. – , . H.T. F , e quantization
of space-time, Physical Review 74, pp. – , . A. D , Cellular space-time and
quantum eld theory, Il Nuovo Cimento 18, pp. – , . Cited on page .
1046 D. F
, ‘Superconducting’ causal nets, International Journal of eoretical Phys-
ics 27, pp. – , . Cited on page .
1047 N.H. C
, R. F
& T.D. L , Random lattice eld theory: general formu-
lation, Nuclear Physics B 202, pp. – , . G. ’ H
, Quantum eld theory for
elementary particles – is quantum eld theory a theory?, Physics Reports 104, pp. – ,
. Cited on page .
1048 See for example L. B
, J. L , D. M
& R.D. S
, Space-time as
a causal set, Physical Review Letters 59, pp. – , . G. B
& R.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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G
, Structure of random space-time, Physical Review Letters 66, pp. – , .
Cited on page .
1049 For a discussion, see R. S
, Time, Creation and the continuum: theories in antiquity
and the early Middle Ages, Duckworth, London . Cited on page .
1050 For quantum gravity, see for example A. A
, Quantum geometry in action: big
bang and black holes, http://www.arxiv.org/abs/math-ph/
. Cited on page .
1051 e false belief that particles like quarks or electrons are composite is slow to die out. See,
e.g, S. F
, Preon prophecies by the standard Model, http://www.arxiv.org/
abs/hep-ph/
. Preon models gained popularity in the s and s, in particular
through the papers by J.C. P & A. S
, Physical Review D 10,, p. , , H.
H
, Physics Letters B 86,, p. , , M.A. S
, Physics Letters B 86,, p. , ,
and H. F
& G. M
, Physics Letters B 102,, p. , . Cited on
page .
1052 A.D. S
, Vacuum quantum uctuations in curved space and the theory of grav-
itation, Soviet Physics – Doklady, 12, pp. – , . Cited on page .
1053 C. W , Upper limit for the mass of an elementary particle due to discrete time quantum mechanics, Il Nuovo Cimento B 109, pp. – , . Cited on page .
1054 N.F. R
& A. W , Suche nach permanenten elektrischen Dipolmomenten: ein
Test der Zeitumkehrinvarianz, Physikalische Blätter 52, pp. – , . e paper by
E.D. C
, S.B. R , D. D M
& B.C. R
, Improved experimental
limit on the electric dipole moment of the electron, Physical Review A 50, p. , ,
gives an upper experimental limit to dipole moment of the electron of . ë − e m. W.
B
& M. S
, e electric dipole moment of the electron, Reviews of
Modern Physics 63, pp. – , . See also the musings in H D
, Is the
electron a composite particle?, Hyper ne Interactions 81, pp. – , . Cited on page .
1055 K. A
, T. H
& K. K
, Naturalness bounds on dipole moments
from new physics, http://www.arxiv.org/abs/hep-ph/
Cited on page .
1056 W.G. U
, Notes on black hole evaporation, Physical Review D 14, pp. – , .
W.G. U
& R.M. W , What happens when an accelerating observer detects a
Rindler particle, Physical Review D 29, pp. – , . Cited on page .
1057 W. J
, Heisenberg’s uncertainty relation and thermal vibrations in crystals, American
Journal of Physics 61, pp. – , . Cited on page .
1058 One way out is proposed in J. M
& L. S
invariant energy scale, Physical Review Letters 88, p.
a modi cation of the mass energy relation of the kind
, Lorentz invariance with an , April . ey propose
E=
γmc
+
γmc EPl
and
p=
γmv .
+
γmc EPl
(693)
Another, similar approach of recent years, with a di erent proposal, is called doubly special
relativity. A recent summary is G. A
–C
, Doubly-special relativity: rst
results and key open problems, http://www.arxiv.org/abs/gr-qc/
. e paper shows
how conceptual problems hinder the advance of the eld. Cited on page .
1059 H.D. Z , On the interpretation of measurement in quantum theory, Foundations of Physics 1, pp. – , . Cited on page .
1060 S
W
, e cosmological constant problem, Reviews of Modern Physics 61,
pp. – , . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Dvipsbugw
Dvipsbugw
1061 See Y.J. N , W.A. C
& H. D , Probing Planck-scale physics with
extragalactic sources?, Astrophysical Journal 591, pp. L –L , , also downloadable
electronically at http://www.arxiv.org/abs/astro-ph/
; D.H. C
, Planck scale
still safe from stellar images, Classical and Quantum Gravity 20, pp. – , , also
available as electronic preprint at http://www.arxiv.org/abs/astro-ph/
. Negative ex-
perimental results (and not always correct calculations) are found in R. L & L. H -
, e phase coherence of light from extragalactic sources - direct evidence against rst
order Planck scale uctuations in time and space, Astrophysical Journal 585, pp. L –L ,
, and R. R
, M. T
& W. G
, e lack of observational
evidence for the quantum structure of spacetime at Planck scales, Astrophysical Journal
587, pp. L –L , . Cited on page .
1062 B.E. S
, Severe limits on variations of the speed of light with frequency, Physical
Review Letters 82, pp. – , June . Cited on page .
1063 G. A
–C
, J. E , N.E. M
, D.V. N
& S.
S
, Potential sensitivity of gamma-ray-burster observations to wave dispersion in
vacuo, Nature 393, p. , , and preprint http://www.arxiv.org/abs/astro-ph/
.
Cited on page .
1064 G. A
–C
, Phenomenological description of space-time foam, preprint
found at http://www.arxiv.org/abs/gr-qc/
, . e paper includes a clearly writ-
ten overview of present experimental approaches to detecting quantum gravity e ects.
See also his update G. A
–C
, Quantum-gravity phenomenology: status
and prospects, preprint found at http://www.arxiv.org/abs/gr-qc/
, . Cited on
pages and .
1065 G. A
–C
, An interferometric gravitational wave detector as a quantum
gravity apparatus, Nature 398, p. , , and preprint http://www.arxiv.org/abs/gr-qc/
. Cited on page .
1066 F. K
, Gravitation and quantum mechanics of macroscopic objects, Il Nuovo
Cimento A42, pp. – , , Y.J. N & H. D , Limit to space-time measure-
ment, Modern Physics Letters A 9, pp. – , , Y.J. N & H. V D , Modern
Physics Letters A Remarks on gravitational sources, 10, pp. – , . e discussion
is neatly summarised in Y.J. N & H. D , Comment on ‘Uncertainty in measure-
ments of distance’, http://www.arxiv.org/abs/gr-qc/
. See also Y.J. N , Spacetime
foam, preprint found at http://www.arxiv.org/abs/gr-qc/
. Cited on page .
1067 L.J. G pp. – on pages
, Spacetime foam as a quantum thermal bath, Physics Review Letters 80,
, , also downloadable as http://www.arxiv.org/abs/gr-qc/
. Cited
and .
1068 G. A
–C
& T. P , Planck scale-deformation of Lorentz symmetry
as a solution to the UHECR and the TeV γ paradoxes, preprint http://www.arxiv.org/
astro-ph/
, . Cited on page .
1069 Experimental tests were described by R. L & L.W. H
, e phase coher-
ence of light from extragalactic sources: direct evidence against rst-order Planck-scale
uctuations in time and space, Astrophysical Journal 585, pp. L –L , , and R.
R
& al., e lack of observational evidence for the quantum structure of space-
time at Planck scales, Astrophysical Journal 587, pp. L –L , , Discussions on the pre-
dicted phase values are found in http://www.arxiv.org/abs/astro-ph/
and http://
www.arxiv.org/abs/astro-ph/
. Cited on page .
1070 At the uni cation scale, supersymmetry is manifest; all particles can transform into each other. See for example ... Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Dvipsbugw
1071 See the paper by U. A
, W. B & H. F
, Comparison of grand
uni ed theories with elektroweak and strong coupling constants measured at LEP, Physics
Letters 260, pp. – , . is is the standard reference on the topic. Cited on pages
and .
1072 M.V. R
, W.C. G
, J.P. J
& E.N. F
, New limit on the
permanent electric dipole moment of Hg, Physical Review Letters 86, pp. – ,
March , or http://www.arxiv.org/abs/hep-ex/
. eir upper limit on the dipole
moment of mercury atoms is . ë e m. Research for better measurement methods is
progressing; a modern proposal is by S.K. L
, Solid state systems for electron
electric dipole moment and other fundamental measurements, http://www.arxiv.org/abs/
nucl-ex/
. Cited on page .
Dvipsbugw
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
.
–
?
Ref. 1073
Die Grenze ist der eigentlich fruchtbare Ort der Erkenntnis.*
“ Paul Tillich, Auf der Grenze. ” strange question is the topic of the current leg of our mountain ascent. In
he last section we explored nature in the vicinity of Planck dimensions; but
The other limit, namely to study the description of motion at large, cosmological
scales, is equally fascinating. As we proceed, many incredible results will appear, and at the end we will discover a surprising answer to the question in the section title.
is section is not standard textbook material; a large part of it is original** and thus speculative and open to question. Even though this section aims at explaining in simple words the ongoing research in the domains of quantum gravity and superstring theory, be warned. With every sentence of this section you will nd at least one physicist who disagrees!
We have discussed the universe as a whole several times already. In classical physics we enquired about the initial conditions of the universe and whether it is isolated. In the
rst intermezzo we asked whether the universe is a set or a concept and indeed, whether it exists at all. In general relativity we gave the classical de nition of the term, as the sum of all matter and space-time, we studied the expansion of the universe and we asked about its size and topology. In quantum theory we asked whether the universe has a wave function, whether it is born from a quantum uctuation, and whether it allows the number of particles to be de ned.
Here we will settle all these issues by combining general relativity and quantum theory at cosmological scales. at will lead us to some of the strangest results we will encounter in our journey.
Dvipsbugw
C
“Hic sunt leones.***
” Antiquity
e description of motion requires the application of general relativity whenever the scale
d of the situation are of the order of the Schwarzschild radius, i.e. whenever
d rS = Gm c .
(694)
It is straightforward to con rm that, with the usually quoted mass m and size d of Challenge 1442 n everything visible in the universe, this condition is indeed ful lled. We do need general
relativity and thus curved space-time when talking about the whole of nature.
* ‘ e frontier is the really productive place of understanding’. Paul Tillich (1886–1965), German theologian, socialist and philosopher. ** Written between June and December 2000. *** ‘Here are lions.’ Written across unknown and dangerous regions on ancient maps.
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–
Similarly, quantum theory is required for the description of motion of an object
whenever we approach it within a distance d of the order of the Compton wavelength
λC, i.e. whenever
d
λC
=
h mc
.
(695)
Obviously, for the total mass of the universe this condition is not ful lled. However, we are not interested in the motion of the universe itself; we are interested in the motion of its components. In the description of these components, quantum theory is required whenever pair production and annihilation play a role. is is especially the case in the early history of the universe and near the horizon, i.e. for the most distant events that we can observe in space and time. We are thus obliged to include quantum theory in any precise description of the universe.
Since at cosmological scales we need both quantum theory and general relativity, we start our investigation with the study of time, space and mass, by asking at large scales the same questions that we asked above at Planck scales.
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Challenge 1443 n
Is it possible to measure time intervals of any imaginable size? General relativity shows that in nature there is a maximum time interval, with a value of about fourteen thousand million years or Ps, providing an upper limit to the measurement of time. It is called the ‘age’ of the universe and has been deduced from two sets of measurements: the expansion of space-time and the age of matter.
We all know of clocks that have been ticking for a long time: the hydrogen atoms in our body. All hydrogen atoms were formed just a er the big bang. We can almost say that the electrons in these atoms have been orbiting the nuclei since the dawn of time. In fact, inside the protons in these atoms, the quarks have been moving already a few hundred thousand years longer. Anyway, we thus get a common maximum time limit for any clock made of atoms. Even ‘clocks’ made of radiation (can you describe one?) yield a similar maximum time. Also the study of the spatial expansion of the universe leads to the same maximum. No real or imaginable clock or measurement device was ticking before this maximum time and no clock could provide a record of having done so.
In summary, it is not possible to measure time intervals greater than the maximum one, either by using the history of space-time or by using the history of matter or radiation.*
e maximum time is thus rightly called the ‘age’ of the universe. Of course, all this is not new, although looking at the issue in more detail does provide some surprises.
D
?
“One should never trust a woman who tells one her real age. A woman who would tell one that, would tell one anything. ” Oscar Wilde * is conclusion implies that so-called ‘oscillating’ universe models, in which it is claimed that ‘before’ the big bang there were other phenomena, cannot be based on nature or on observations. ey are based on beliefs.
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Page 1167
Asking about the age of the universe may seem a silly question, because we have just discussed it. Furthermore, the value is found in many books and tables, including that of Appendix B, and its precise determination is actually one of the most important quests in modern astrophysics. But is this quest reasonable?
In order to measure the duration of a movement or the age of a system, we need a clock. e clock has to be independent of that movement or system and thus has to be outside the system. However, there are no clocks outside the universe and, inside it, a clock cannot be independent. In fact we have just seen that inside the universe, no clock can run throughout its complete history. Indeed, time can be de ned only once it is possible to distinguish between matter and space-time. Once this distinction can be made, only the two possibilities just discussed remain: we can either talk about the age of space-time, as is done in general relativity, by assuming that matter provides suitable and independent clocks; or we can talk about the age of matter, such as stars or galaxies, by assuming that the extension of space-time or some other matter provides a good clock. Both possibilities are being explored experimentally in modern astrophysics; both give the same result of about fourteen thousand million years that was mentioned previously. However, for the universe as a whole, an age cannot be de ned.
e issue of the starting point of time makes this di culty even more apparent. We may imagine that going back in time leads to only two possibilities: either the starting instant t = is part of time or it is not. (Mathematically, this means that the segment describing time should be either closed or open.) Both cases assume that it is possible to measure arbitrarily small times, but we know from the combination of general relativity and quantum theory that this is not the case. In other words, neither possibility is incorrect: the beginning cannot be part of time, nor can it not be part of it. To this situation there is only one solution: there was no beginning at all.
In other words, the situation is consistently muddled. Neither the age of the universe nor its origin makes sense. What is going wrong? Or, more correctly, how are things going wrong? In other words, what happens if instead of jumping directly to the big bang, we approach it as closely as possible? e best way to clarify the issue is to ask about the measurement error we are making when we say that the universe is fourteen thousand million years old. is turns out to be a fascinating topic.
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H
?
Ref. 1075
No woman should ever be quite accurate about her age. It looks so calculating.
“ Oscar Wilde ” e rst way to measure the age of the universe* is to look at clocks in the usual sense of
the term, namely at clocks made of matter. As explained in the part on quantum theory, Salecker and Wigner showed that a clock built to measure a total time T with a precision
Ref. 1074
* Note that the age t is not the same as the Hubble time T = H . e Hubble time is only a computed quantity and (almost) always larger than the age; the relation between the two depends on the value of the cosmological constant, on the density and on other parameters of the universe. For example, for the standard hot big bang scenario, i.e. for the matter-dominated Einstein–de Sitter model, we have the simple relation T=( )t .
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–
∆t has a minimum mass m given by
m
ħT c (∆t)
.
(696)
Ref. 1076
A simple way to incorporate general relativity into this result was suggested by Ng and
van Dam. Any clock of mass m has a minimum resolution ∆t due to the curvature of
space that it introduces, given by
∆t
Gm c
.
(697)
If m is eliminated, these two results imply that any clock with a precision ∆t can only measure times T up to a certain maximum value, namely
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T
<
(∆t) tPl
,
(698)
Challenge 1444 e Challenge 1445 ny
where tPl = ħG c = . ë − s is the already familiar Planck time. (As usual, we have omitted factors of order one in this and in all the following results of this section.) In other words, the higher the accuracy of a clock, the shorter the time during which the clock works dependably! e precision of a clock is not (only) limited by the budget spent on building it, but by nature itself. Nevertheless, it does not take much to check that for clocks in daily life, this limit is not even remotely approached. For example, you may want to deduce how precisely your own age can be speci ed.
As a consequence of ( ), a clock trying to achieve an accuracy of one Planck time can do so for at most one single Planck time! Simply put, a real clock cannot achieve Planck time accuracy. If we try to go beyond limit ( ), uctuations of space-time hinder the working of the clock and prevent higher precision. With every Planck time that passes, the clock accumulates a measuring error of at least one Planck time. us, the total measurement error is at least as large as the measurement itself. e conclusion is also valid for clocks based on radiation, for example on background radiation.
In short, measuring age with a clock always involves some errors; whenever we try to reduce these errors to Planck level, the clock becomes so imprecise that age measurements become impossible.
D
?
Page 45
“Time is waste of money.
” Oscar Wilde
From the origins of physics onwards, the concept of ‘time’ has designated what is meas-
ured by a clock. Since equation ( ) expresses the non-existence of perfect clocks, it also
implies that time is only an approximate concept, and that perfect time does not exist.
us there is no ‘idea’ of time, in the Platonic sense. In fact, all discussion in the previous
and present sections can be seen as proof that there are no perfect or ‘ideal’ examples of
any classical or everyday concept.
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Challenge 1446 e
Ref. 1073 Page 1004 Challenge 1447 e
Time does not exist. Despite this conclusion, time is obviously a useful concept in everyday life. A simple explanation is provided when we focus on the importance of energy. Any clock, in fact any system of nature, is characterized by a simple number, namely the highest ratio of the kinetic energy to the rest energy of its components. In daily life, this fraction is about eV GeV = − . Such low-energy systems are well suited to building clocks. e more precisely the motion of the main moving part – the pointer of the clock – can be kept constant and can be monitored, the higher the precision of the clock becomes. To achieve the highest possible precision, the highest possible mass of the pointer is required; indeed, both the position and the speed of the pointer must be measured, and the two measurement errors are related by the quantum mechanical indeterminacy relation ∆v ∆x ħ m. High mass implies low intrinsic uctuations. In order to screen the pointer from outside in uences, even more mass is needed. is connection might explain why better clocks are usually more expensive than less accurate ones.
e usually quoted indeterminacy relation is valid only at everyday energies. Increasing the mass does not allow to reach arbitrary small time errors, since general relativity changes the indeterminacy relation to ∆v ∆x ħ m + G(∆v) m c . e additional term on the right hand side, negligible at everyday scales, is proportional to energy. Increasing it by too large an amount limits the achievable precision of the clock. e smallest measurable time interval turns out to be the Planck time.
In summary, time exists as a good approximation only for low-energy systems. Any increase in precision beyond a certain limit will require an increase in the energy of the components; at Planck energy, this energy increase will prevent an increase in precision.
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W
?
Challenge 1448 e
Applying the discussion about the measurement of time to the age of the universe is now
straightforward. Expression ( ) implies that the highest precision possible for a clock
is about − s, or about the time light takes to move across a proton. e nite age of the
universe also yields a maximum relative measurement precision. Expression ( ) can be
written as
∆t T
(
tPl T
)
(699)
which shows that no time interval can be measured with a precision of more than about
decimals.
In order to clarify the issue, we can calculate the error in measurement as a function
of the observation energy Emeas. ere are two limit cases. For small energies, the error is given by quantum e ects as
∆t T Emeas
(700)
and thus decreases with increasing measurement energy. For high energies, however, the error is given by gravitational e ects as
∆t Emeas T EPl
(701)
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–
Relative measurement error 1
quantum error
∆Emin E
total error
quantum gravity error
Energy
Eopt
EPl
F I G U R E 383 Measurement errors as a function of measurement energy
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so that the total result is as shown in Figure . In particular, energies that are too high
do not reduce measurement errors, because any attempt to reduce the measurement error
for the age of the universe below − s would require energies so high that the limits of
space-time would be reached, making the measurement itself impossible.
We reached this conclusion through an argument based on clocks made of particles.
Below we will nd out that even by determining the age of the universe using space-time
expansion leads to the same limit.
Imagine to observe a tree which, as a result of some storm
or strong wind, has fallen towards second tree, touching it at
the very top, as shown in Figure . It is possible to determ-
ine the height of both trees by measuring their separation and
the angles at the base. e error in height will depend on the er-
v
rors in separation and angles. Similarly, the age of the universe
follows from the present distance and speed of objects – such
as galaxies – observed in the night sky. e present distance d corresponds to separation of the trees at ground level and the T
speed v to the angle between the two trees. e Hubble time T
of the universe – which, as has already been mentioned, is usu-
ally assumed to be larger than the age of the universe – then cor-
d
responds to the height at which the two trees meet. is time
since the universe ‘started’, in a naive sense since the galaxies F IGURE 384 Trees and
‘separated’, is then given, within a factor of order one, by
galaxies
T
=
d v
.
(702)
is is in simple words the method used to determine the age of the universe from the expansion of space-time, for galaxies with red-shi s below unity.* Of interest in the fol-
* At higher red-shi s, the speed of light, as well as the details of the expansion, come into play; if we continue
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lowing is the (positive) measurement error ∆T, which becomes
∆T T
=
∆d d
+
∆v v
.
(703)
Exploring this in more detail is worthwhile. For any measurement of T we have to choose the object, i.e. a distance d, as well as an observation time ∆t, or, equivalently, an observation energy ∆E = πħ ∆t. We will now investigate the consequences of these choices for expression ( ), always taking into account both quantum theory and general relativity.
At everyday energies, the result of the determination of the age t is about ë Ga. is value is deduced by measuring red-shi s, i.e. velocities, and dis-
tances, using stars and galaxies in distance ranges from some hundred thousand light years up to a red-shi of about . Measuring red-shi s does not produce large velocity errors. e main source of experimental error is the di culty in determining the distances of galaxies.
What is the smallest possible error in distance? Obviously, equation ( ) implies
Dvipsbugw
∆d lPl Td
(704)
Challenge 1449 e Challenge 1450 e
Ref. 1074
thus giving the same indeterminacy in the age of the universe as found above in the case of material clocks.
We can try to reduce this error in two ways: by choosing objects at either small or large distances. Let us start with the smallest possible distances. In order to get high precision at small distances, we need high observation energies. It is fairly obvious that at observation energies near the Planck value, the value of ∆T T approaches unity. In fact, both terms on the right-hand side of expression ( ) become of order one. At these energies, ∆v approaches c and the maximum value for d approaches the Planck length, for the same reason that at Planck energies the maximum measurable time is the Planck time. In short, at Planck scales it is impossible to say whether the universe is old or young.
Let us continue with the other extreme, namely objects extremely far away, say with a red-shi of z . Relativistic cosmology requires the diagram of Figure to be replaced by the more realistic diagram of Figure . e ‘light onion’ replaces the familiar light cone of special relativity: light converges near the big bang.
In this case the measurement error for the age of the universe also depends on the distance and velocity errors. At the largest possible distances, the signals an object must send out must be of high energy, because the emitted wavelength must be smaller than the universe itself. us, inevitably we reach Planck energies. However, we saw that in such high-energy situations, the emitted radiation, as well as the object itself, are indistinguishable from the space-time background. In other words, the red-shi ed signal we would observe today would have a wavelength as large as the size of the universe, with a correspondingly small frequency.
with the image of inclined trees, we nd that the trees are not straight all the way up to the top and that they grow on a slope, as shown in Figure 385.
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–
Challenge 1451 ny
Another way to describe the situation is the fol- big bang lowing. At Planck energies or near the horizon, the
space
original signal has an error of the same size as the
signal itself. When measured at the present time,
the red-shi ed signal still has an error of the same size as the signal. As a result, the error on the horizon distance becomes as large as the value to be
4ct0/9
light cone: what we
measured. In short, even if space-time expansion and large
time
can see
scales are used, the instant of the so-called begin-
other galaxies
ning of the universe cannot be determined with an
in the night sky
error smaller than the age of the universe itself, a
result we also found at Planck distances. Whenever we aim for perfect precision, we nd that the uni-
t0 our galaxy
verse is
thousand million years old! In other F I G U R E 385 The speed and distance of
words, at both extremal situations it is impossible to remote galaxies
say whether the universe has a non-vanishing age.
We have to conclude that the anthropomorphic concept of ‘age’ does not make any
sense for the universe as a whole. e usual textbook value is useful only for domains in
time, space and energy for which matter and space-time are clearly distinguished, namely
at everyday, human-scale energies; however, this anthropocentric value has no overall
meaning.
By the way, you may like to examine the issue of the fate of the universe using the
same arguments. In the text, however, we continue on the path outlined at the start of
this section; the next topic is the measurement of length.
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Page 460
General relativity shows that in the standard cosmological model, for hyperbolic (open) and parabolic (marginal) evolutions of the universe, the actual size of the universe is in-
nite. It is only the horizon distance, i.e. the distance of objects with in nite red-shi , which is nite. In a hyperbolic or parabolic universe, even though the size is in nite, the most distant visible events (which form the horizon) are at a nite distance.* For elliptical evolution, the total size is nite and depends on the curvature. However, in this case also the present measurement limit yields a minimum size for the universe many times larger than the horizon distance. At least, this is what general relativity says.
On the other hand, quantum eld theory is based on at and in nite space-time. Let us see what happens when the two theories are combined. What can we say about measurements of length in this case? For example, would it be possible to construct and use a metre rule to measure lengths larger than the distance to the horizon? It is true that we would have no time to push it up to there, since in the standard Einstein–de Sitter big
Ref. 1074
* In cosmology, we need to distinguish between the scale factor R, the Hubble radius c H = cR R˙, the horizon distance h and the size d of the universe. e Hubble radius is a computed quantity giving the distance at which objects move away with the speed of light. It is always smaller than the horizon distance, at which in the standard Einstein–de Sitter model, for example, objects move away with twice the speed of light. However, the horizon itself moves away with three times the speed of light.
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Ref. 1074 Ref. 1074
bang model the horizon moves away from us faster than the speed of light. We should have started using the metre rule right at the big bang.
For fun, let us assume that we have actually managed to do this. How far away can we read o distances? In fact, since the universe was smaller in the past and since every observation of the sky is an observation of the past, Figure shows that the maximum spatial distance an object can be seen away from us is only ( )ct . Obviously, for spacetime intervals, the maximum remains ct .
us, in all cases it turns out to be impossible to measure lengths larger than the horizon distance, even though general relativity predicts such distances. is unsurprising result is in obvious agreement with the existence of a limit for measurements of time intervals. e real surprises come now.
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Astronomers and Hollywood movies answer this question in the a rmative. Indeed, the distance to the horizon of the universe is usually included in tables. Cosmological models specify that the scale factor R, which xes the distance to the horizon, grows with time t; for the case of the usually assumed mass-dominated Einstein–de Sitter model, i.e. for a vanishing cosmological constant and at space, we have
R(t) = C t ,
(705)
Challenge 1452 ny
where the numerical constant C relates the commonly accepted horizon distance to the commonly accepted age. Indeed, observation shows that the universe is large and is still getting larger. But let us investigate what happens if we add to this result from general relativity the limitations of quantum theory. Is it really possible to measure the distance to the horizon?
We look rst at the situation at high energies. We saw above that space-time and matter are not distinguishable at Planck scales. erefore, at Planck energies we cannot state whether objects are localized or not. At Planck scales, a basic distinction of our thinking, namely the one between matter and vacuum, becomes obsolete. Equivalently, it is not possible to claim that space-time is extended at Planck scales. Our concept of extension derives from the possibility of measuring distances and time intervals, and from observations such as the ability to align several objects behind one another. Such observations are not possible at Planck scales. In fact, none of the observations in daily life from which we deduce that space is extended are possible at Planck scales. At Planck scales, the basic
distinction between vacuum and matter, namely the opposition between extension and localization, disappears. As a consequence, at Planck energies the size of the universe cannot be measured. It cannot even be called larger than a match box.
At cosmological distances, the situation is even easier. All the arguments given above on the errors in measurement of the age can be repeated for the distance to the horizon. Essentially, at the largest distances and at Planck energies, the measurement errors are of the same magnitude as the measured value. All this happens because length measurements become impossible at nature’s limits. is is corroborated by the lack of any standard with which to compare the size of the universe.
Studying the big bang also produces strange results. At Planck energies, whenever we
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try to determine the size of the big bang, we cannot claim that the universe was smaller than the present size. At Planck energies, there is no way to distinguish length values. Somehow, Planck dimensions and the size of the universe get confused.
ere are also other con rmations. Let us go back to the example above. If we had a metre rule spanning the whole universe, even beyond the horizon, with zero at the place where we live, what measurement error would it produce for the horizon? It does not take long to discover that the expansion of space-time from Planck scales to the present also expands the indeterminacy in the Planck size into one of the order of the distance to the horizon. e error is as large as the measurement result.
Since this also applies when we try to measure the diameter of the universe instead of its radius, it becomes impossible to state whether the antipodes in the sky really are distant from each other!
We can summarize the situation by noting that anything said about the size of the universe is as limited as anything said about its age. e height of the sky depends on the observation energy. At Planck energies, it cannot be distinguished from the Planck length. If we start measuring the sky at standard observation energies, trying to increase the precision of measurement of the distance to the horizon, the measurement error increases beyond all bounds. At Planck energies, the volume of the universe is indistinguishable from the Planck volume!
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e horizon of the universe, essentially the black part of the night sky, is a fascinating entity. Everybody interested in cosmology wants to know what happens there. In newspapers the horizon is sometimes called the boundary of space-time. Some surprising insights, not yet common in newspapers, appear when the approaches of general relativity and quantum mechanics are combined.
We saw above that the errors in measuring the distance of the horizon are substantial. ey imply that we cannot pretend that all points of the sky are equally far away from us. us we cannot say that the sky is a surface. ere is even no way to determine the dimensionality of the horizon or the dimensionality of space-time near the horizon.* Measurements thus do not allow us to determine whether the boundary is a point, a surface, or a line. It may be an arbitrary complex shape, even knotted. In fact, quantum theory tells us that it must be all of these from time to time, in short, that the sky uctuates in height and shape. In short, it is impossible to determine the topology of the sky. But that is nothing new. As is well known, general relativity is unable to describe pair creation particles with spin / . e reason for this is the change in space-time topology required by the process. On the other hand, the universe is full of such processes, implying that it is impossible to de ne a topology for the universe and, in particular, to talk of the topology of the horizon itself. Are you able to nd at least two other arguments to show this? Worse still, quantum theory shows that space-time is not continuous at a horizon, as
Challenge 1454 ny
* In addition, the measurement errors imply that no statement can be made about translational symmetry at cosmological scales. Are you able to con rm this? In addition, at the horizon it is impossible to distinguish between spacelike and timelike distances. Even worse, concepts such as ‘mass’ or ‘momentum’ are muddled at the horizon. is means that, as at Planck energies, we are unable to distinguish between objects and the background, and between state and intrinsic properties. We will come back to this important point shortly.
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Ref. 1073
can easily be deduced by applying the Planck-scale arguments from the previous section. Time and space are not de ned there.
Finally, there is no way to decide whether the various boundary points are di erent from each other. e distance between two points in the night sky is unde ned. In other words, it is unclear what the diameter of the horizon is.
In summary, the horizon has no speci c distance or shape. e horizon, and thus the universe, cannot be shown to be manifolds. is leads to the next question:
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Ref. 1078 Ref. 1073
One o en reads about the quest for the initial conditions of the universe. But before joining this search, we should ask whether and when such initial conditions make any sense. Obviously, our everyday description of motion requires them. Initial conditions describe the state of a system, i.e. all those aspects that di erentiate it from a system with the same intrinsic properties. Initial conditions, like the state of a system, are attributed to a system by an outside observer.
More speci cally, quantum theory tells us that initial conditions or the state of a system can only be de ned by an outside observer with respect to an environment. It is already di cult to be outside the universe. In addition, quite independently of this issue, even inside the universe a state can only be de ned if matter can be distinguished from vacuum. However, this is impossible at Planck energies, near the big bang, or at the horizon. us the universe has no state. No state also means that there is no wave function of the universe.
e limits imposed by the Planck values con rm this conclusion in other ways. First of all, they show that the big bang was not a singularity with in nite curvature, density or temperature, because in nitely large values do not exist in nature. Second, since instants of time do not exist, it is impossible to de ne the state of any system at a given time. ird, as instants of time do not exist, neither do events exist, and thus the big bang was not an event, so that for this more prosaic reason, neither an initial state nor an initial wave function can be ascribed to the universe. (Note that this also means that the universe cannot have been created.)
In short, there are no initial conditions for the universe. Initial conditions make sense only for subsystems and only far away from Planck scales. us, for initial conditions to exist, the system must be far away from the horizon and it must have evolved for some time ‘a er’ the big bang. Only when these two requirements are ful lled can objects move in space. Of course, this is always the case in everyday life.
At this point in our mountain ascent, where time and length are unclearly de ned at cosmological scales, it should come as no surprise that there are similar di culties concerning the concept of mass.
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e number of stars, about
, is included in every book on cosmology, as it is in the
table of Appendix B. A subset of this number can be counted on clear nights. If we ask
the same question about particles instead of stars, the situation is similar. e commonly
quoted number of baryons is , together with
photons. However, this does not
settle the issue. Neither quantum theory nor general relativity alone make predictions
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about the number of particles, either inside or outside the horizon. What happens if we combine them?
In order to de ne the number of particles in a region, quantum theory rst of all requires a vacuum state to be de ned. e number of particles is de ned by comparing the system with the vacuum. If we neglect or omit general relativity by assuming at spacetime, this procedure poses no problem. However, if we include general relativity and thus a curved space-time, especially one with such a strangely behaved horizon as the one we have just found, the answer is simple: there is no vacuum state with which we can compare the universe, for two reasons. First, nobody can explain what an empty universe would look like; second, and more importantly, there is no way to de ne a state of the universe at all. e number of particles in the universe thus becomes unde nable. Only at everyday energies and for nite dimensions are we able to speak of an approximate number of particles.
Comparison between a system and the vacuum is also impossible in the case of the universe for purely practical reasons. e requirement for such a comparison e ectively translates into the requirement that the particle counter be outside the system. (Can you con rm the reason for this connection?) In addition, it is impossible to remove particles from the universe. e impossibility of de ning a vacuum state, and thus the number of particles in the universe, is not surprising. It is an interesting exercise to investigate the measurement errors that appear when we try to determine the number of particles despite this fundamental impossibility.
Can we count the stars? In principle, the same conclusion applies as for particles. However, at everyday energies the stars can be counted classically, i.e. without taking them out of the volume in which they are enclosed. For example, this is possible if the stars are differentiated by mass, colour or any other individual property. Only near Planck energies or near the horizon are these methods inapplicable. In short, the number of stars is only de ned as long as the observation energy is low, i.e. as long as we stay away from Planck energies and from the horizon.
In short, despite what appears to be the case on human scales, there is no de nite number of particles in the universe. e universe cannot be distinguished from vacuum by counting particles. Even though particles are necessary for our own existence and functioning, a complete count of them cannot be made.
is conclusion is so strange that we should not accept it too easily. Let us try another method of determining the content of matter in the universe: instead of counting particles, let us weigh them.
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e average density of the universe, about − kg m , is frequently cited in texts. Is it di erent from a vacuum? Quantum theory shows that, as a result of the indeterminacy relation, even an empty volume of size R has a mass. For a zero-energy photon inside such a vacuum, we have E c = ∆p ħ ∆x, so that in a volume of size R, we have a minimum mass of at least mmin(R) = h cR. For a spherical volume of radius R there is
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
thus a minimal mass density given approximately by
ρmin
mmin(R) R
=
ħ cR
.
(706)
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Page 417
For the universe, if the standard horizon distance R of
million light years is in-
serted, the value becomes about − kg m . is describes the density of the vacuum.
In other words, the universe, with a density of about − kg m , seems to be clearly
di erent from vacuum. But are we sure?
We have just deduced that the radius of the horizon is unde ned: depending on the
observation energy, it can be as small as the Planck length. is implies that the density
of the universe lies somewhere between the lowest possible value, given by the density
of vacuum just mentioned, and the highest possible one, namely the Planck density.* In
short, relation ( ) does not really provide a clear statement.
Another way to measure the mass of the universe would be to apply the original de ni-
tion of mass, as given by Mach and modi ed by special relativity. us, let us try to collide
a standard kilogram with the universe. It is not hard to see that whatever we do, using
either low or high energies for the standard kilogram, the mass of the universe cannot be
constrained by this method. We would need to produce or to measure a velocity change
∆v for the rest of the universe a er the collision. To hit all the mass in the universe at the
same time, we need high energy; but then we are hindered by Planck energy e ects. In
addition, a properly performed collision measurement would require a mass outside the
universe, a rather di cult feat to achieve.
Yet another way to measure the mass would be to determine the gravitational mass
of the universe through straightforward weighing. But the lack of balances outside the
universe makes this an impractical solution, to say the least.
A way out might be to use the most precise de nition of mass provided by general
relativity, the so-called ADM mass. However, for de nition this requires a speci ed beha-
viour at in nity, i.e. a background, which the universe lacks.
We are then le with the other general relativistic method: determining the mass of
the universe by measuring its average curvature. Let us take the de ning expressions for
average curvature κ for a region of size R, namely
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κ = rcurvature =
π
πR R
−S
=
π
πR R
−V
.
(708)
We have to insert the horizon radius R and either its surface area S or its volume V . However, given the error margins on the radius and the volume, especially at Planck Challenge 1460 ny energies, for the radius of curvature we again nd no reliable result.
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* In fact, at everyday energies the density of the universe lies almost exactly between the two values, yielding the strange relation
m R mPl RPl = c G .
(707)
But this is nothing new. e approximate equality can be deduced from equation 16.4.3 (p. 620) of S
W
, Gravitation and Cosmology, Wiley, 1972, namely Gnb mp = t . e relation is required by
several cosmological models.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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An equivalent method starts with the usual expression for the indeterminacy ∆κ in Ref. 1079 the scalar curvature for a region of size R provided by Rosenfeld, namely
∆κ
πlPl . R
(709)
Challenge 1461 ny
However, this expression also shows that the error in the radius of curvature behaves like the error in the distance to the horizon.
In summary, at Planck energies, the average radius of curvature of nature turns out to lie between in nity and the Planck length. is implies that the density of matter lies between the minimum value and the Planck value. ere is thus no method to determine the mass of the universe at Planck energies. (Can you nd one?) e concept of mass cannot be applied to the universe as a whole. us, the universe has no mass.
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We have already seen that at the horizon, space-time translation symmetry breaks down. Let us have a quick look at the other symmetries.
What happens to permutation symmetry? Exchange is an operation on objects in space-time. Exchange thus automatically requires a distinction between matter, space and time. If we cannot distinguish positions, we cannot talk about exchange of particles. However, this is exactly what happens at the horizon. In short, general relativity and quantum theory together make it impossible to de ne permutation symmetry at the horizon.
CPT symmetry su ers the same fate. As a result of measurement errors or of limiting maximum or minimum values, it is impossible to distinguish between the original and the transformed situations. It is therefore impossible to maintain that CPT is a symmetry of nature at horizon scales. In other words, matter and antimatter cannot be distinguished at the horizon.
e same happens with gauge symmetry, as you may wish to check in detail yourself. For its de nition, the concept of gauge eld requires a distinction between time, space and mass; at the horizon this is impossible. We therefore also deduce that at the horizon also concepts such as algebras of observables cannot be used to describe nature. Renormalization also breaks down.
All symmetries of nature break down at the horizon. e complete vocabulary we use when we talk about observations, including terms such as such as magnetic eld, electric
eld, potential, spin, charge, or speed, cannot be used at the horizon. And that is not all.
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It is common to take ‘boundary’ and ‘horizon’ as synonyms in the case of the universe, because they are the same for all practical purposes. To study this concept, knowledge of mathematics does not help us; the properties of mathematical boundaries, e.g. that they themselves have no boundary, are not applicable in the case of nature, since space-time is not continuous. We need other, physical arguments.
e boundary of the universe is obviously supposed to represent the boundary between something and nothing. is gives three possibilities:
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
— ‘Nothing’ could mean ‘no matter’. But we have just seen that this distinction cannot be made at Planck scales. As a consequence, the boundary will either not exist at all or it will encompass the horizon as well as the whole universe.
— ‘Nothing’ could mean ‘no space-time’. We then have to look for those domains where space and time cease to exist. ese occur at Planck scales and at the horizon. Again, the boundary will either not exist or it will encompass the whole universe.
— ‘Nothing’ could mean ‘neither space-time nor matter.’ e only possibility is a boundary that encloses domains beyond the Planck scale and beyond the horizon; but again, such a boundary would also encompass all of nature.
Challenge 1463 ny Ref. 1080
is result is puzzling. When combining quantum theory and relativity, we do not seem to be able to nd a conceptual de nition of the horizon that distinguishes the horizon from what it includes. In fact, if you nd one, publish it! A distinction is possible in general relativity; and equally, a distinction is possible in quantum theory. However, as soon as we combine the two, the boundary becomes indistinguishable from its content.
e interior of the universe cannot be distinguished from its horizon. ere is no boundary. A di culty to distinguish the horizon and its contents is de nitely interesting; it suggests that nature may be symmetric under transformations that exchange interiors and boundaries. is connection is nowadays called holography because it vaguely recalls the working of credit card holograms. It is a busy research eld in present-day high-energy physics. However, for the time being, we shall continue with our original theme, which directly leads us to ask:
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We are used to calling the universe the sum of all matter and all space-time. In other words, we imply that the universe is a set of components, all di erent from each other.
is idea was introduced in three situations: it was assumed that matter consists of particles, that space-time consists of events (or points) and that the set of states consists of di erent initial conditions. However, our discussion so far shows that the universe is not a set of such distinguishable elements. We have encountered several proofs: at the horizon, at the big bang and at Planck scales distinction between events, between particles, between observables and between space-time and matter becomes impossible. In those domains, distinctions of any kind become impossible. We have found that any distinction between two entities, such as between a toothpick and a mountain, is only approximately possible. e approximation is possible because we live at energies much smaller than the Planck energy. Obviously, we are able to distinguish cars from people and from toothpicks; the approximation is so good that we do not notice the error when we perform it. Nevertheless, the discussion of the situation at Planck energies shows that a perfect distinction is impossible in principle. It is impossible to split the universe into separate
entities. Another way to reach this result is the following. Distinguishing between two entit-
ies requires di erent measurement results, such as di erent positions, masses, sizes, etc. Whatever quantity we choose, at Planck energies the distinction becomes impossible. Only at everyday energies is it approximately possible.
In short, since the universe contains no distinguishable entities, the universe is not a
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set. We have already envisaged this possibility in the rst intermezzo; now it is con rmed.
e concept of ‘set’ is already too specialized to describe the universe. e universe must
be described by a mathematical concept that does not contain any set.
is is a powerful result: it means that the universe cannot be described precisely if
any of the concepts used for its description presuppose the use of sets. But all concepts
we have used so far to describe nature, such as space-time, phase space, Hilbert space and
its generalizations, namely Fock space and particle space, are based on sets. ey all must
be abandoned at Planck energies, as well as in any precise description.
Furthermore, many speculations about uni ed descriptions do not satisfy the criterion
that sets must not be included. In particular, all studies of quantum uctuations, mathem-
atical categories, posets, complex mathematical spaces, computer programs, Turing ma-
chines, Gödel’s theorem, creation of any sort, space-time lattices, quantum lattices and
Bohm’s unbroken wholeness fail to satisfy this requirement. In addition, almost none of
the speculations about the origin of the universe can be correct. For example, you may
wish to check the religious explanations you know against this result. In fact, no approach
used by theoretical physics in the year
satis es the requirement that sets must be
abandoned; perhaps a future version of string or M theory will do so. e task is not easy;
do you know of a single concept not based on a set?
Note that this radical conclusion is deduced from only two statements: the necessity
of using quantum theory whenever the dimensions are of the order of the Compton
wavelength, and the necessity to use general relativity whenever the dimensions are of
the order of the Schwarzschild radius. Together, they mean that any precise description
of nature cannot contain sets. We have reached this result a er a long and interesting,
but in a sense unnecessary, digression. e di culties in complying with this result may
explain why the uni cation of the two theories has not so far been successful. Not only
does uni cation require that we stop using space, time and mass for the description of
nature; it also requires that all distinctions, of any kind, should be only approximate. But
all physicists have been educated on the basis of exactly the opposite creed!
Note that, because it is not a set, the universe is not a physical system. Speci cally, it has
no state, no intrinsic properties, no wave function, no initial conditions, no density, no
entropy and no cosmological constant. e universe is thus neither thermodynamically
closed nor open; and it contains no information. All thermodynamic quantities, such as
entropy, temperature and free energy, are de ned using ensembles. Ensembles are limits
of systems which are thermodynamically either open or closed. As the universe is neither
of these two, no thermodynamic quantity can be de ned for it.* All physical properties
are de ned only for parts of nature which are approximated or idealized as sets, and thus
are physical systems.
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* Some people knew this long before physicists; for example, the belief that the universe is or contains in-
formation was ridiculed most thoroughly in the popular science ction parody by D
A
,e
Hitchhiker’s Guide to the Galaxy, 1979, and its sequels.
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“Insofern sich die Sätze der Mathematik auf die Wirklichkeit beziehen, sind sie nicht sicher, und sofern sie sicher sind, beziehen sie sich nicht auf die Wirklichkeit.* ” Albert Einstein
e contradictions between the term ‘universe’ and the concept of ‘set’ lead to numerous fascinating issues. Here are a few.
Challenge 1466 ny
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In mathematics, + = . is statement is an idealization of statements such as ‘two apples plus two apples makes four apples.’ However, we now know that at Planck energies this is not a correct statement about nature. At Planck energies, objects cannot be counted. Are you able to con rm this?
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Ref. 1081
**
In 2002, Seth Lloyd estimated how much information the universe can contain, and how many calculations it has performed since the big bang. is estimate is based on two ideas: that the number of particles in the universe is a well-de ned quantity, and that the universe is a computer, i.e. a physical system. We now know that neither assumption is correct. is example shows the power of the criteria that we deduced for the nal description of motion.
Challenge 1467 ny
**
People take pictures of the cosmic background radiation and its variations. Is it possible that the photographs will show that the spots in one direction of the sky are exactly the same as those in the diametrically opposite direction?
Ref. 1082 Challenge 1468 n
**
In 1714, Leibniz published his Monadologie. In it he explores what he calls a simple substance, which he de ned to be a substance that has no parts. He called it a monade and explores some of its properties. However, due mainly to its incorrect deductions, the term has not been taken over by others. Let us forget the strange deductions and focus only on the de nition: what is the physical concept most related to that of monade?
Challenge 1469 ny
**
We usually speak of the universe, implying that there is only one of them. Yet there is a simple case to be made that ‘universe’ is an observer-dependent concept, since the idea of ‘all’ is observer-dependent. Does this mean that there are many universes?
**
If all particles would be removed – assuming one would know where to put them – there Challenge 1470 ny wouldn’t be much of a universe le . True?
* In so far as mathematical statements describe reality, they are not certain, and as far as they are certain, they are not a description of reality.
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At Planck energies, interactions cannot be de ned. erefore, ‘existence’ cannot be Challenge 1471 ny de ned. In short, at Planck energies we cannot say whether particles exist. True?
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In the year , David Hilbert gave a famous lecture in which he listed of the great challenges facing mathematics in the twentieth century. Most of these problems provided challenges to many mathematicians for decades a erwards. A few are still unsolved, among them the sixth, which challenged mathematicians and physicists to nd an axiomatic treatment of physics.
Since the universe is not even a set, we can deduce that such an axiomatic description of nature is impossible. e reasoning is simple; all mathematical systems, be they algebraic systems, order systems or topological systems, are based on sets. Mathematics does not have axiomatic systems that do not contain sets. e reason for this is simple: any (mathematical) concept contains at least one set. However, nature does not.
e impossibility of an axiomatic system for physics is also con rmed in another way. Physics starts with a circular de nition: space-time is de ned with the help of objects and objects are de ned with the help of space-time. Physics thus has never been modelled on the basis of mathematics. Physicists have always had to live with logical problems.
e situation is similar to a child’s description of the sky as ‘made of air and clouds’. Looking closely, we discover that clouds are made up of water droplets. However, there is air inside clouds, and there is also water vapour elsewhere in the air. When clouds and air are viewed through a microscope, there is no clear boundary between the two. We cannot de ne either of the terms ‘cloud’ and ‘air’ without the other. No axiomatic de nition is possible.
Objects and vacuum also behave in this way. Virtual particles are found in vacuum, and vacuum is found inside objects. At Planck scales there is no clear boundary between the two; we cannot de ne either of them without the other. Despite the lack of a precise de nition and despite the logical problems that ensue, in both cases the description works well at large, everyday scales.
We note that, since the universe is not a set and since it contains no information, the paradox of the physics book containing a full description of nature disappears. Such a book can exist, as it does not contradict any property of the universe. But then a question arises naturally:
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“Drum hab ich mich der Magie ergeben, [...] Daß ich erkenne, was die Welt Im Innersten zusammenhält.* ” Goethe, Faust.
Is the universe really the sum of matter–energy and space-time? Or of particles and vacuum? We have heard this so o en up to now that we may be lulled into forgetting to
* us I have devoted myself to magic, [...] that I understand how the innermost world is held together.
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TA B L E 75 Physical statements about the universe
e universe has no age.
e universe has no beginning.
e universe has no size.
e universe has no volume.
e universe has no shape.
e universe’s particle number is unde ned.
e universe has no mass.
e universe has no energy.
e universe has no density.
e universe contains no matter.
e universe has no cosmological constant.
e universe has no initial conditions.
e universe has no state.
e universe has no wave function.
e universe is not a physical system.
e universe contains no information.
e universe is not isolated.
e universe is not open.
e universe has no boundaries.
e universe does not interact.
e universe cannot be measured.
e universe cannot be said to exist.
e universe cannot be distinguished from e universe cannot be distinguished from a
nothing.
single event.
e universe contains no moments.
e universe is not composite.
e universe is not a set.
e universe is not a concept.
e universe cannot be described.
ere is no plural for ‘universe’.
e universe cannot be distinguished from va- e universe was not created. cuum.
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Page 631 Challenge 1473 r
Page 644
check the statement. To nd the answer, we do not need magic, as Faust thought; we only need to list what we have found so far, especially in this section, in the section on Planck scales, and in the intermezzo on brain and language. Table shows the result.
Not only are we unable to state that the universe is made of space-time and matter; in fact, we are unable to say anything about the universe at all!* It is not even possible to say that it exists, since it is impossible to interact with it. e term ‘universe’ does not allow us to make a single sensible statement. (Can you nd one?) We are only able to say which properties it does not have. We are unable to nd any property the universe does have. us, the universe has no properties! We cannot even say whether the universe is something or nothing. e universe isn’t anything in particular. In other words, the term ‘universe’ is not useful at all for the description of motion.
We can obtain a con rmation of this strange conclusion from the rst intermezzo. ere we found that any concept needs de ned content, de ned limits and a de ned domain of application. In this section, we have found that for the term ‘universe’, not one of these three aspects is de ned; there is thus no such concept. If somebody asks: ‘why does the universe exist?’ the answer is: not only does the use of ‘why’ wrongly suggest that something may exist outside the universe, providing a reason for it and thus contradicting the de nition of the term ‘universe’ itself; most importantly of all, the universe simply does not exist. In summary, any sentence containing the word universe makes no sense.
* ere is also another well-known, non-physical concept about which nothing can be said. Many scholars Challenge 1472 n have explored it in detail. Can you see what it is?
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–
e term ‘universe’ only seems to express something, even if it doesn’t.* e conclusion that the term universe makes no sense may be interesting, even
strangely beautiful; but does it help us to understand motion more precisely? Interestingly so, it does.
A
Challenge 1474 e
e discussion about the term ‘universe’ also shows that the term does not contain any set. In other words, this is the rst term that will help us on the way to a precise description of nature. We will see later on how this happens.
By taking into account the limits on length, time, mass and all other quantities we have encountered, we have deduced a number of almost painful conclusions about nature. However, we also received something in exchange: all the contradictions between general relativity and quantum theory that we mentioned at the beginning of this chapter are now resolved. Although we have had to leave behind us many cherished habits, in exchange we have the promise of a description of nature without contradictions. But we get even more.
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Page 959 Challenge 1475 r
In the chapter Quantum physics in a nutshell we listed all the unexplained properties of nature le open either by general relativity or by quantum theory. e present conclusions provide us with new connections among them. Indeed, many of the cosmological results of this section sound surprisingly familiar; let us compare them systematically with those of the section on Planck scales. Both sections explored topics – some in more detail than others – from the list of unexplained properties of nature. First, Table shows that none of the unexplained properties makes sense at both limits of nature, the small and the large. All open questions are open at both extremes. Second and more importantly, nature behaves in the same way at horizon scales and at Planck scales. In fact, we have not found any di erence between the two cases. (Are you able to discover one?) We are thus led to the hypothesis that nature does not distinguish between the large and the small. Nature seems to be characterized by extremal identity.
I
?
e principle of extremal identity incorporates some rather general points:
— all open questions about nature appear at its two extremes; — a description of nature requires both general relativity and quantum theory; — nature or the universe is not a set; — initial conditions and evolution equations make no sense at nature’s limits; — there is a relation between local and global issues in nature; — the concept of ‘universe’ has no content.
* Of course, the term ‘universe’ still makes sense if it is de ned more restrictively, such as ‘everything interacting with a particular human or animal observer in everyday life’. But such a de nition is not useful for our quest, as it lacks the precision required for any description of motion.
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TA B L E 76 Properties of nature at maximal, everyday and minimal scales
P
A
-A
AP -
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requires quantum theory and relativity intervals can be measured precisely length and time intervals are space-time is not continuous points and events cannot be distinguished space-time is not a manifold space is 3 dimensional space and time are indistinguishable initial conditions make sense space-time uctuates Lorentz and Poincaré symmetry CPT symmetry renormalization permutation symmetry interactions number of particles algebras of observables matter indistinguishable from vacuum boundaries exist nature is a set
true false limited true true true false true false true does not apply does not apply does not apply does not apply do not exist unde ned unde ned true false false
false true unlimited false false false true false true false applies applies applies applies exist de ned de ned false true true
true false limited true true true false true false true does not apply does not apply does not apply does not apply do not exist unde ned unde ned true false false
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Challenge 1476 ny Ref. 1084
Extremal identity thus looks like a good candidate tool for use in the search for a uni ed description of nature. To be a bit more provocative, it may be the only known principle incorporating the idea that the universe is not a set, and thus might be the only candidate tool for use in the quest of uni cation. Extremal identity is beautiful in its simplicity, in its unexpectedness and in the richness of its consequences. You might enjoy exploring it by yourself.
e study of the consequences of extremal identity is currently the focus of much activity in high energy particle physics, although o en under di erent names. e simplest approach to extremal identity – in fact one that is too simple to be correct – is inversion. It looks as if extremal identity implies a connection such as
r
lPl r
or
xµ
lPl xµ xµ xµ
(710)
relating distances r or coordinates xµ with their inverse values using the Planck length lPl. Can this mapping, called inversion, be a symmetry of nature? At every point of space? For
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Challenge 1477 ny Challenge 1478 ny
example, if the horizon distance is inserted, equation ( ) implies that lengths smaller
than lPl
− m never appear in physics. Is this the case? What would inversion
imply for the big bang?
Numerous fascinating questions are contained in the simple hypothesis of extremal
identity. ey lead to two main directions for investigation.
First, we have to search for some stronger arguments for the validity of extremal iden-
tity. We will discover a number of simple arguments, all showing that extremal identity
is indeed a property of nature and producing many beautiful insights.
e other quest then follows. We need to nd the correct version of equation ( ).
at oversimpli ed expression is neither su cient nor correct. It is not su cient because
it does not explain any of the issues le open by general relativity and quantum theory.
It only relates some of them, thus reducing their number, but it does not solve any of
them. You may wish to check this for yourself. In other words, we need to nd the precise
description of quantum geometry and of elementary particles.
However, inversion is also simply wrong. Inversion is not the correct description of
extremal identity because it does not realize a central result discovered above: it does
not connect states and intrinsic properties. Inversion keeps them distinct. is means that
inversion does not take interactions into account. And most open issues at this point of
our mountain ascent are properties of interactions.
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B
1073 Available at http://www.dse.nl/motionmountain/C -QMGR.pdf. ose pages are a re-
worked version of the ones published in French as C. S
, Le vide di ère-t-il de la
matière? in E. G
& S. D , editeurs, Le vide – Univers du tout et du rien – Des
physiciens et des philosophes s’interrogent, Les Éditions de l’Université de Bruxelles, .
See also C. S
, Does matter di er from vacuum? http://www.arxiv.org/abs/gr-qc/
. Cited on pages , , and .
1074 See the lucid discussion by G.F.R. E & T. R Journal of Physics 61, pp. – , . Cited on pages
, Lost horizons, American , , , and .
1075 H. S
& E.P. W
, Quantum limitations of the measurement of space-time
distances, Physical Review 109, pp. – , . See also the popular account of this paper
by E.J. Z
, e macroscopic nature of space-time, American Journal of Physics
30, pp. – , . Cited on page .
1076 F. K
, Gravitation and quantum mechanics of macroscopic objects, Il Nuovo
Cimento A42, pp. – , , Y.J. N & H. V D , Limit to space-time measure-
ment, Modern Physics Letters A 9, pp. – , , Y.J. N & H. V D , Remarks on
gravitational sources, Modern Physics Letters A 10, pp. – , . Newer summaries
are Y.J. N & H.
D , Comment on “Uncertainty in measurements of distance”,
http://www.arxiv.org/abs/gr-qc/
and Y.J. N , Spacetime foam, http://www.arxiv.
org/abs/gr-qc/
. Cited on page .
1077 See, for example, the Hollywood movie Contact, based on the book by C S
,
Contact, . Cited on page .
1078 See, for example, the international bestseller by S. H From the Big Bang to Black Holes, . Cited on page
, A Brief History of Time – .
1079 L. R
, in H.-J. T
, Entstehung, Entwicklung und Perspektiven der Ein-
steinschen Gravitationstheorie, Springer Verlag, . Cited on page .
1080 Holography in high-energy physics is connected with the work by ’t Hoo and by Susskind.
See for example G. ’ H
, Dimensional reduction in quantum gravity, pp. – ,
in A. A , J. E & S. R
-D , Salaamfeest, , or the o en cited paper
by L. S
, e world as a hologram, Journal of Mathematical Physics 36, pp. –
, , or as http://www.arxiv.org/abs/hep-th/
A good modern overview is
R. B
, e holographic principle, Review of Modern Physics 74, pp. – , ,
also available as http://www.arxiv.org/abs/hep-th/
, even though it has some argu-
mentation errors, as explained on page . Cited on page .
1081 S. L
, Computational capacity of the universe, Physical Review Letters 88, p.
,
. Cited on page .
1082 G
W
L
, La Monadologie, . Written in French, it is avail-
able freely at http://www.uqac.uquebec.ca/zone /Classiques_des_sciences_sociales and
in various other languages on other websites. Cited on page .
1083 See, for example, H. W
& P.S. A
, (eds.), Die Hilbertschen Probleme,
Akademische Verlagsgesellscha Geest & Portig, , or B H. Y
, e Honours
Class: Hilbert’s Problems and their Solvers, A.K. Peters, . Cited on page .
1084 Large part of the study of dualities in string and M theory can be seen as investigations into
the detailed consequences of extremal identity. For a review of duality, see ... A classical ver-
sion of duality is discussed by M.C.B. A
, A.L. G
& I.V. V
, Du-
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.
ality between coordinates and the Dirac eld, http://www.arxiv.org/abs/hep-th/
Cited on page .
.
–
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“Sex is the physics urge sublimated.
”Gra to
M you have once met a physicist who has told you, in one of those oments of con dentiality, that studying physics is even more beautiful than aking love. At this statement, many will simply shake their head in disbelief
and strongly disapprove. In this section we shall argue that it is possible to learn so much
about physics while making love that discussions about their relative beauty can be put
aside altogether.
Imagine to be with your partner on a beautiful tropical island, just a er sunset, and to
look together at the evening sky. Imagine as well that you know little of what is taught at
school nowadays, e.g. that your knowledge is that of the late Renaissance, which probably
is a good description of the average modern education level anyway.
Imagine being busy enjoying each other’s company. e most important results of
physics can be deduced from the following experimental facts:*
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Love is communication.
Love is tiring.
Love is an interaction between moving bodies. Love takes time.
Love is attractive.
Love is repulsive.
Love makes noise.
In love, size matters.
Love is for reproduction.
Love can hurt.
Love needs memory.
Love is Greek.
Love uses the sense of sight.
Love is animalic.
Love is motion.
Love is holy.
Love is based on touch.
Love uses motion again.
Love is fun.
Love is private.
Love makes one dream.
Let us see what these observations imply for the description of nature.
**
Love is communication. Communication is possible because nature looks similar from different standpoints and because nature shows no surprises. Without similarity we could not understand each other, and a world of surprises would even make thinking impossible; it would not be possible to form concepts to describe observations. But fortunately, the world is regular; it thus allows to use concepts such as time and space for its description.
Ref. 1085
* Here we deduce physics from love. We could also deduce physics from sexuality. e modern habit of saying ‘sex’ instead of ‘sexuality’ mixes up two completely di erent concepts. In fact, studying the in uences of sex on physics is almost fully a waste of time. We avoid it. Maybe one day we shall understand why there do not seem to be any female crackpots proposing pet physical theories.
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Challenge 1479 n
**
Love is an interaction between moving bodies. Together with the previous result, this implies that we can and need to describe moving bodies with mass, energy and momentum.
at is not a small feat. For example, it implies that the Sun will rise tomorrow if the sea level around the island is the usual one.
**
Love is attractive. When feeling attracted to your partner, you may wonder if this attraction is the same which keeps the Moon going around the Earth. You make a quick calculation and nd that applying the expression for universal gravity
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Epot
=
−GMm r
(711)
to both of you, the involved energy is about as much as the energy added by the leg of a y on the skin. (M and m are your masses, r your distance, and the gravitational constant
has the value G = . ë − m kg s .) In short, your partner teaches you that in nature there are other attractive interactions apart from gravity; the average modern education is incomplete.
Nevertheless, this rst equation is important: it allows to predict the position of the planets, the time of the tides, the time of eclipses, the return of comets, etc., to a high accuracy for thousands of years in advance.
**
Love makes noise. at is no news. However, even a er making love, even when everybody and everything is quiet, in a completely silent environment, we do hear something.
e noises we hear are produced within the ear, partly by the blood owing through the head, partly by the electrical noise generated in the nerves. at is strange. If matter were continuous, there would be no noise even for low signal levels. In fact, all proofs for the discreteness of matter, of electric current, of energy, or of light are based on the increase of
uctuations with the smallness of systems under consideration. e persistence of noise thus makes us suspect that matter is made of smallest entities. Making love con rms this suspicion in several ways.
**
Love is for reproduction. Love is what we owe our life to, as we all are results of reproduction. But the reproduction of a structure is possible only if it can be constructed, in other words if the structure can be built from small standard entities. us we again suspect ourselves to be made of smallest, discrete entities.
Love is also a complicated method of reproduction. Mathematics provides a much simpler one. If matter objects were not made of particles, but were continuous, it would be possible to perform reproduction by cutting and reassembling. A famous mathematical theorem by Banach and Tarski proves that it is possible to take a continuous solid, cut it into ve pieces and rearrange the pieces in such a way that the result are two copies of the same size and volume as the original. In fact, even volume increases can be produced in this way, thus realizing growth without any need for food. Mathematics thus provides
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some interesting methods for growth and reproduction. However, they assume that matter is continuous, without a smallest length scale. e observation that these methods do not work in nature is compatible with the idea that matter is not continuous.
Ref. 1086 Challenge 1480 n
**
Love needs memory. If you would not recognize your partner among all possible ones, your love life would be quite complicated. A memory is a device which, in order to store information, must have small internal uctuations. Obviously, uctuations in systems get smaller as their number of components increase. Since our memory works so well, we can follow that we are made of a large number of small particles.
In summary, love shows that we are made of some kind of lego bricks: depending on the level of magni cation, these bricks are called molecules, atoms, or elementary particles. It is possible to estimate their size using the sea around the tropical island, as well as a bit of oil. Can you imagine how?
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Ref. 1087
**
Love uses the sense of sight. Seeing each other is only possible because we are cold whereas the Sun is hot. If we and our environment all had the same temperature as the Sun, we would not see each other. is can be checked experimentally by looking into a hot oven: Inside a glowing oven lled with glowing objects it is impossible to discern them against the background.
**
Love is motion. Bodies move against each other. Moreover, their speed can be measured. Since measurement is a comparison with a standard, there must be a velocity standard in nature, some special velocity standing out. Such a standard must either be the minimum or the maximum possible value. Now, daily life shows that for velocity a nite minimum value does not exist. We are thus looking for a maximum value. To estimate the value of the maximum, just take your mobile phone and ring home from the island to your family. From the delay in the line and the height of the satellite, you can deduce the telephone speed c and get ë m s.
e existence of a maximum speed c implies that time is di erent for di erent observers. Looking into the details, we nd that this e ect becomes noticeable at energies
Edifferent time mc ,
(712)
where m is the mass of the object. For example, this applies to electrons inside a television tube.
Challenge 1481 n
**
Love is based on touching. When we touch our partner, sometimes we get small shocks. e energies involved are larger than than those of touching y legs. In short, people are
electric. In the dark, we observe that discharges emit light. Light is thus related to electricity. In
addition, touching proves that light is a wave: simply observe the dark lines between two ngers near your eye in front of a bright background. e lines are due to interference
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e ects. Light thus does not move with in nite speed. In fact, it moves with the same speed as that of telephone calls.
Page 210
**
Love is fun. People like to make love in di erent ways, such as in a dark room. But rooms get dark when the light is switched o only because we live in a space of odd dimensions. In even dimensions, a lamp would not turn o directly a er the switch is ipped, but dim only slowly.
Love is also fun because with our legs, arms and bodies we can make knots. Knots are possible only in three dimensions. In short, love is real fun only because we live in 3 dimensions.
**
Love is tiring. e reason is gravity. But what is gravity? A little thinking shows that since there is a maximum speed, gravity is the curvature of space-time. Curved space also means that a horizon can appear, i.e. a largest possible visible distance. From equations (711) and (712), we deduce that this happens when distances are of the order of
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Rhorizon Gm c .
(713)
For example, only due a horizon, albeit one appearing in a di erent way, the night sky is dark.
Ref. 1091
**
Love takes time. It is known that men and women have di erent opinions on durations. It is also known that love happens between your ears. Indeed, biological research has shown that we have a clock inside the brain, due to circulating electrical currents. is clock provides our normal sense of time. Since such a brain clock can be built, there must be a time standard in nature. Again, such a standard must be either a minimum or a maximum time interval. We shall discover it later on.
**
Love is repulsive. And in love, size matters. Both facts turn out to be the two sides of the same coin. Love is based on touch, and touch needs repulsion. Repulsion needs a length scale, but neither gravity nor classical electrodynamics provide one. Classical physics only allows for the measurement of speed. Classical physics cannot explain that the measurement of length, time, or mass is possible.* Classically, matter cannot be hard; it should be possible to compress it. But love shows us that this is not the case. Love shows us that lengths scales do exist in nature and thus that classical physics is not su cient for the description of nature.
** Love can hurt. For example, it can lead to injuries. Atoms can get ripped apart. at hap-
* Note that the classical electron radius is not an exception: it contains the elementary charge e, which contains a length scale, as shown on page 492.
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pens when energies are concentrated on small volumes, such as a few aJ per atom. Investigating such situations more precisely, we nds that strange phenomena appear at distances r if energies exceed the value
E
ħc r
;
(714)
in particular, energy becomes chunky, things become fuzzy, boxes are not tight, and particles get confused. ese are called quantum phenomena. e new constant ħ =
− Js is important: it determines the size of things, because it allows to de ne distance
and time units. In other words, objects tear and break because in nature there is a minimum action, given roughly by ħ.
If even more energy is concentrated in small volumes, such as energies of the order of mc per particle, one even observes transformation of energy into matter, or pair production. From equations (712) and (714), we deduce that this happens at distances of
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rpair production
ħ mc
.
(715)
At such small distances we cannot avoid using the quantum description of nature.
**
Love is not only Greek. e Greeks were the rst to make theories above love, such as Plato in his Phaedrus. But they also described it in another way. Already before Plato, Democritus said that making love is an example of particles moving and interacting in vacuum. If we change ‘vacuum’ to ‘curved 3+1-dimensional space’ and ‘particle’ to ‘quantum particle’, we do indeed make love in the way Democritus described 2500 years ago.
It seems that physics has not made much progress in the mean time. Take the statement made in 1939 by the British astrophysicist Arthur Eddington:
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 1088
I believe there are 15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,914,527,116,709,366,231,425,076,185,631,031,296 protons in the universe and the same number of electrons.
Compare it with the version of 2006:
Baryons in the universe: ; total charge: near zero.
e second is more honest, but which of the two is less sensible? Both sentences show that there are unexplained facts in the Greek description nature, in particular the number of involved particles.
** Love is animalic. We have seen that we can learn a lot about nature from the existence of love. We could be tempted to see this approach of nature as a special case of the so-called
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–
anthropic principle. However, some care is required here. In fact, we could have learned exactly the same if we had taken as starting point the observation that apes or pigs have love. ere is no ‘law’ of nature which distinguishes between them and humans. In fact, there is a simple way to determine whether any ‘anthropic’ statement makes sense: the reasoning must be equally true for humans, apes, and pigs.
A famous anthropic deduction was drawn by the British astrophysicist Fred Hoyle. While studying stars, he predicted a resonance in the carbon-12 nucleus. If it did not exist, he argued, stars could not have produced the carbon which a erwards was spread out by explosions into interstellar space and collected on Earth. Also apes or pigs could reason this way; therefore Hoyle’s statement does make sense.
On the other hand, claiming that the universe is made especially for people is not sensible: using the same arguments, pigs would say it is made for pigs. e existence of either requires all ‘laws’ of nature. In summary, the anthropic principle is true only in so far as its consequences are indistinguishable from the porcine or the simian principle. In short, the animalic side of love puts limits to the philosophy of physics.
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Ref. 1089
**
Love is holy. Following the famous de nition by the theologian Rudolf Otto, holiness results from a mixture of a mysterium tremendum and a mysterium fascinans. Tremendum means that it makes one tremble. Indeed, love produces heat and is a dissipative process. All systems in nature which produce heat have a nite lifetime. at is true for machines, stars, animals, lightning, re, lamps and people. rough heat, love shows us that we are going to die. Physicists call this the second principle of thermodynamics.
But love also fascinates. Everything which fascinates has a story. Indeed, this is a principle of nature: every dissipative structure, every structure which appears or is sustained through the release of energy, tells us that it has a story. Take atoms, for example. All the protons we are made of formed during the big bang. Most hydrogen we are made of is also that old. e other elements were formed in stars and then blown into the sky during nova or supernova explosions. ey then regrouped during planet formation. We truly are made of stardust.
Why do such stories fascinate? If you only think about how you and your partner have met, you will discover that it is through a chain of incredible coincidences. If only one of all these coincidences had not taken place, you and your partner would not be together. And of course, we all owe our existence to such a chain of coincidences, which brought our parents together, our grandparents, and made life appear on Earth.
e realization of the importance of coincidences automatically produces two kinds of questions: why? and what if? Physicists have now produced a list of all the answers to repeated why questions and many are working at the list of what-if questions. e rst list, the why-list of Table 77, gives all facts still unexplained. It can also be called the complete list of all surprises in nature. (Above, it was said that there are no surprises in nature about what happens. However, so far there still are a handful of surprises on how all these things happen.)
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TA B L E 77 Everything quantum field theory and general relativity do not explain; in other words, a list of the only experimental data and criteria available for tests of the unified description of motion
O
P
Local quantities, from quantum theory
αem αw αs mq ml mW θW β ,β ,β
θCP θst 3 . nJ m
+
the low energy value of the electromagnetic coupling constant the low energy value of the weak coupling constant the low energy value of the strong coupling constant the values of the 6 quark masses the values of 3 lepton masses the values of the independent mass of the W vector boson the value of the Weinberg angle three mixing angles the value of the CP parameter the value of the strong topological angle the number of particle generations the value of the observed vacuum energy density or cosmological constant the number of space and time dimensions
Global quantities, from general relativity
. ( ) ë m ? the distance of the horizon, i.e. the ‘size’ of the universe (if it makes sense)
?
the number of baryons in the universe, i.e. the average matter density in the
universe (if it makes sense)
?
the initial conditions for more than particle elds in the universe, includ-
ing those at the origin of galaxies and stars (if they make sense)
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Local structures, from quantum theory
S(n)
the origin of particle identity, i.e. of permutation symmetry
Ren. group
the renormalization properties, i.e. the existence of point particles
SO(3,1)
the origin of Lorentz (or Poincaré) symmetry
(i.e. of spin, position, energy, momentum)
C
the origin of the algebra of observables
Gauge group
the origin of gauge symmetry
(and thus of charge, strangeness, beauty, etc.)
in particular, for the standard model:
U(1)
the origin of the electromagnetic gauge group (i.e. of the quantization of elec-
tric charge, as well as the vanishing of magnetic charge)
SU(2)
the origin of weak interaction gauge group
SU(3)
the origin of strong interaction gauge group
Global structures, from general relativity maybe R S ? the unknown topology of the universe (if it makes sense)
is why-list fascinates through its shortness, which many researchers are still trying
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–
to reduce. But it is equally interesting to study what consequences appear if any of the values from Table 77 were only a tiny bit di erent. It is not a secret that small changes in nature would lead to completely di erent observations, as shown in Table 78.
TA B L E 78 A small selection of the consequences when changing aspects of nature
O
C
R
Moon size
smaller
Moon size
larger
Jupiter
smaller
Jupiter
larger
Oort belt
smaller
Star distance
smaller
Strong coupling smaller constant
small Earth magnetic eld; too much cosmic radiation; widespread child cancers.
large Earth magnetic eld; too little cosmic radiation; no evolution into humans.
too many comet impacts on Earth; extinction of animal life.
too little comet impacts on Earth; no Moon; no dinosaur extinction.
no comets, no irregular asteroids, no Moon; still dinosaurs.
irregular planet motion; supernova dangers.
proton decay; leucemia.
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e large number of coincidences of life force our mind to realize that we are only a tiny part of nature. We are a small droplet shaken around in the ocean of nature. Even the tiniest changes in nature would prevent the existence of humans, apes and pigs. In other words, making love tells us that the universe is much larger than we are and tells us how much we are dependent and connected to the rest of the universe.
**
We said above that love uses motion. It contains a remarkable mystery, worth a second look:
- Motion is the change of position with time of some bodies. - Position is what we measure with a ruler. Time is what we measure with a clock. Both rulers and clocks are bodies. - A body is an entity distinct from its environment by its shape or its mass. Shape is the extension of a body in space (and time). Mass is measured by measuring speed or acceleration, i.e. by measuring space and time.
is means that we de ne space-time with bodies – as done in detail in general relativity – and that we de ne bodies with space-time – as done in detail in quantum theory.
is circular reasoning shows that making love is truly a mystery. e circular reasoning has not yet been eliminated yet; at present, modern theoretical physicists are busy attempting to do so. e most promising approach seems to be M-theory, the modern extension of string theory. But any such attempt has to overcome important di culties which can also be experienced while making love.
**
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M
R
R
F I G U R E 386 A Gedanken experiment showing that at Planck scales, matter and vacuum cannot be distinguished
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 1090
Love is private. But is it? Privacy assumes that a person can separate itself from the rest,
without important interactions, at least for a given time, and come back later. is is
possible if the person puts enough empty space between itself and others. In other words,
privacy is based on the idea that objects can be distinguished from vacuum. Let us check
whether this is always possible.
What is the smallest measurable distance? is question has been almost, but only
almost answered by Max Planck in 1899. e distance δl between two objects of mass m is
surely larger than their position indeterminacy ħ ∆p; and the momentum indeterminacy
must be smaller that the momentum leading to pair production, i.e. ∆p < mc. is means
that
δl
∆l
ħ mc
.
(716)
In addition, the measurements require that signals leave the objects; the two masses must not be black holes. eir masses must be so small that the Schwarzschild radius is smaller than the distance to be measured. is means that rS Gm c < δl or that
δl
ħG c
= lPl =
.
ë
−
m.
(717)
is expression de nes a minimum length in nature, the so-called Planck length. Every other Gedanken experiment leads to this characteristic length as well. In fact, this minimum distance (and the corresponding minimum time interval) provides the measurement standard we were looking for at the beginning of our musings about length and time measurements.
A more detailed discussion shows that the smallest measurable distance is somewhat larger, a multiple of the Planck length, as measurements require the distinction of matter and radiation. is happens at scales about 800 times the Planck length.
In other words, privacy has its limits. In fact, the issue is even more muddled when
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–
we explore the consequences for bodies. A body, also a human one, is something we can touch, throw, hit, carry or weigh. Physicists say that a body is something with energy or momentum. Vacuum has none of it. In addition, vacuum is unbounded, whereas objects are bounded.
What happens if we try to weigh objects at Planck scales? Quantum theory makes a simple prediction. If we put an object of mass M in a box of size R onto a scale – as in Figure 386 – equation (714) implies that there is a minimal mass error ∆M given by
∆M
ħ cR
.
If the box has Planck size, the mass error is the Planck mass
(718)
∆M = MPl = ħc G
µg .
(719)
How large is the mass we can put into a box of Planck size? Obviously it is given by the
maximum possible mass density. To determine it, imagine a planet and put a satellite in
orbit around it, just skimming its surface. e density ρ of the planet with radius r is given
by
ρ
M r
=
v Gr
.
(720)
Using equation (716) we nd that the maximum mass density in nature, within a factor of order one, is the so-called Planck density, given by
ρPl
=
c G
ħ
=
.
ë
kg m .
(721)
Ref. 1090 Challenge 1482 n
erefore the maximum mass that can be contained inside a Planck box is the Planck mass. But that was also the measurement error for that situation. is implies that we cannot say whether the original box we measured was empty or full: vacuum cannot be distinguished from matter at Planck scales. is astonishing result is con rmed by every other Gedanken experiment exploring the issue.
It is straightforward to deduce with similar arguments that objects are not bound in size at Planck scales, i.e. that they are not localized, and that the vacuum is not necessarily extended at those scales. In addition, the concept of particle number cannot be de ned at Planck scales.
So, why is there something instead of nothing? Making love shows that there is no di erence between the two options!
**
Love makes us dream. When we dream, especially at night, we o en look at the sky. How far is it away? How many atoms are enclosed by it? How old is it? ese questions have an answer for small distances and for large distances; but for the whole of the sky or the whole of nature they cannot have one, as there is no way to be outside of the sky in order to measure it. In fact, each of the impossibilities to measure nature at smallest distances
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Ref. 1092
are found again at the largest scales. ere seems to be a fundamental equivalence, or, as physicists say, a duality between the largest and the smallest distances.
e coming years will hopefully show how we can translate these results into an even more precise description of motion and of nature. In particular, this description should allow us to reduce the number of unexplained properties of nature.
** In summary, making love is a good physics lesson. Enjoy the rest of your day.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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B
1085 An attempt to explain the lack of women in physics is made in M
W
,
Pythagoras’ Trousers – God, Physics and the Gender Wars, Fourth Estate, . Cited on
page .
1086 e consequences of memory loss in this case are already told by V la mémoire, . Cited on page .
, Aventure de
1087 A picture of objects in a red hot oven and at room temperature is shown in C.H. B
,
Demons, Engines and the Second Law, Scienti c American 255, pp. – , November
. Cited on page .
1088
e famous quote is found at the beginning of chapter XI, ‘ e Physical Universe’, in A -
E
, e Philosophy of Physical Science, Cambridge, . Cited on page
.
1089 See the rst intermezzo for details and references. Cited on page .
1090 Details can be found in Chapter XI of this text, a reworked version of the pages published
in French as C. S
, Le vide di ère-t-il de la matière? in E. G
& S. D ,
editeurs, Le vide – Univers du tout et du rien – Des physiciens et des philosophes s’interrogent,
Les Éditions de l’Université de Bruxelles, . Cited on pages and .
1091 An introduction to the sense of time as result of signals in the ganglia and in the temporal
lobe of the brain is found in R.B. I & R. S
, e neural representation of time,
Current Opinion in Neurobiology 14, pp. – , . Cited on page .
1092 For details of he arguments leading to duality, see section . It also includes suggestions supporting the notion that the universe is not a even a set, thus proposing a solution for Hilbert’s sixth problem. Cited on page .
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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.
–
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
e principle of maximum force or power allows us to summarize special relativity, quantum theory and general relativity in one fundamental limit principle each. e three principles are fully equivalent to the standard formulation of the theories. In particular, we show that the maximum force c G implies and is equivalent to the eld equations of general relativity. With the speed limit of special relativity and the action–angular momentum limit of quantum theory, the three fundamental principles imply a bound for every physical observable, from acceleration to size. e new, precise limit values di er from the usual Planck values by numerical prefactors of order unity.* Continuing this approach, we show that every observable in nature has both a lower and an upper limit value.
As a result, a maximum force and thus a minimum length imply that the noncontinuity of space-time is an inevitable consequence of the uni cation of quantum theory and relativity. Furthermore, the limits are shown to imply the maximum entropy bound, including the correct numerical prefactor. e limits also imply a maximum measurement precision possible in nature, thus showing that any description by real numbers is approximate. Finally, the limits show that nature cannot be described by sets. As a result, the limits point to a solution to Hilbert’s sixth problem. e limits also show that vacuum and matter cannot be fully distinguished in detail; they are two limit cases of the same entity. ese fascinating results provide the basis for any search for a uni ed theory of motion.
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F
At dinner parties physicists are regularly asked to summarize physics in a few sentences. It is useful to be able to present a few simple statements answering the request.** Here we propose such a set of statements. We are already familiar with each statement from what we have found out so far in our exploration of motion. But by putting them all together we will get direct insight into the results of modern research on uni cation.
e main lesson of modern physics is the following: Nature limits the possibilities of motion. ese limits are the origin of special relativity, general relativity and quantum theory. In fact, we will nd out that nature poses limits to every aspect of motion; but let us put rst things rst.
S
e step from everyday or Galilean physics to special relativity can be summarized in a single limit statement on motion. It was popularized by Hendrik Antoon Lorentz: ere is a maximum energy speed in nature. For all physical systems and all observers,
v c.
(722)
* ese limits were given for the rst time in 2003, in this section of the present text. ** Stimulating discussions with Saverio Pascazio, Corrado Massa and Steve Carlip helped shaping this section.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Page 282
A few well-known remarks set the framework for the discussion that follows. e speed v
is smaller than or equal to the speed of light c for all physical systems;* in particular, this
limit is valid both for composed systems as well as for elementary particles. e speed
limit statement is also valid for all observers; no exception to the statement is known. In-
deed, only a maximum speed ensures that cause and e ect can be distinguished in nature,
or that sequences of observations can be de ned. e opposite statement, implying the
existence of (real) tachyons, has been explored and tested in great detail; it leads to nu-
merous con icts with observations.
e maximum speed forces us to use the concept of space-time to describe nature.
Maximum speed implies that space and time mix. e existence of a maximum speed in
nature also implies observer-dependent time and space coordinates, length contraction,
time dilation, mass–energy equivalence and all other e ects that characterize special re-
lativity. Only the existence of a maximum speed leads to the principle of maximum ageing
that governs special relativity; only a maximum speed leads to the principle of least ac-
tion at low speeds. In addition, only a nite speed limit makes it to de ne a unit of speed
that is valid at all places and at all times. If a global speed limit did not exist, no natural
measurement standard for speed, independent of all interactions, would exist in nature;
speed would not then be a measurable quantity.
Special relativity also limits the size of systems, independently of whether they are
composed or elementary. Indeed, the limit speed implies that acceleration a and size l
cannot be increased independently without bounds, because the two ends of a system
must not interpenetrate. e most important case concerns massive systems, for which
we have
l
c a
.
(723)
is size limit is induced by the speed of light c; it is also valid for the displacement d of a system, if the acceleration measured by an external observer is used. Finally, the limit implies an ‘indeterminacy’ relation
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∆l ∆a c
(724)
for the length and acceleration indeterminacies. You might want to take a minute to deChallenge 1483 n duce it from the time-frequency indeterminacy. All this is standard knowledge.
Q
In the same way, the di erence between Galilean physics and quantum theory can be Ref. 1093 summarized in a single statement on motion, due to Niels Bohr: ere is a minimum Page 704 action in nature. For all physical systems and all observers,
S ħ.
(725)
* A physical system is a region of space-time containing mass–energy, the location of which can be followed over time and which interacts incoherently with its environment. With this de nition, images, geometrical points or incomplete entangled situations are excluded from the de nition of system.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 1094 Page 704
Half the Planck constant ħ is the smallest observable action or angular momentum. is statement is valid for both composite and elementary systems. e action limit is used less frequently than the speed limit. It starts from the usual de nition of the action, S =
∫ (T − U)dt, and states that between two observations performed at times t and t + ∆t,
even if the evolution of a system is not known, the action is at least ħ . Physical action is de ned to be the quantity that measures the amount of change in the state of a physical system. In other words, there is always a minimum change of state taking place between two observations of a system. In this way, the quantum of action expresses the well-known fundamental fuzziness of nature at a microscopic scale.
It can easily be checked that no observation results in a smaller value of action, irrespective of whether photons, electrons or macroscopic systems are observed. No exception to the statement is known. A minimum action has been observed for fermions, bosons, laser beams and matter systems and for any combination of these. e opposite statement, implying the existence of change that is arbitrary small, has been explored in detail; Einstein’s long discussion with Bohr, for example, can be seen as a repeated attempt by Einstein to nd experiments that make it possible to measure arbitrarily small changes in nature. In every case, Bohr found that this aim could not be achieved.
e minimum value of action can be used to deduce the indeterminacy relation, the tunnelling e ect, entanglement, permutation symmetry, the appearance of probabilities in quantum theory, the information theory aspect of quantum theory and the existence of elementary particle reactions. e minimum value of action implies that, in quantum theory, the three concepts of state, measurement operation and measurement result need to be distinguished from each other; a so-called Hilbert space needs to be introduced. e minimal action is also part of the Einstein–Brillouin–Keller quantization. e details of these connections can be found in the chapter on quantum theory.
Obviously, the existence of a quantum of action has been known right from the beginning of quantum theory. e quantum of action is at the basis of all descriptions of quantum theory, including the many-path formulation and the information-theoretic descriptions. e existence of a minimum quantum of action is completely equivalent to all standard developments of quantum theory.
We also note that only a nite action limit makes it possible to de ne a unit of action. If an action limit did not exist, no natural measurement standard for action would exist in nature; action would not then be a measurable quantity.
e action bound S pd mcd, together with the quantum of action, implies a limit on the displacement d of a system between any two observations:
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d
ħ mc
.
(726)
Page 708
In other words, (half) the (reduced) Compton wavelength of quantum theory is recovered as lower limit on the displacement of a system, whenever gravity plays no role. Since the quantum displacement limit applies in particular to an elementary system, the limit is also valid for the size of a composite system. However, the limit is not valid for the size of elementary particles.
e action limit of quantum theory also implies Heisenberg’s well-known indeterminacy relation for the displacement d and momentum p of systems:
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∆d ∆p ħ .
(727)
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is is valid for both massless and massive systems. All this is textbook knowledge, of course.
G
Least known of all is the possibility of summarizing the step from Galilean physics to general relativity in a single statement on motion: ere is a maximum force or power in nature. For all physical systems and all observers,
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F
c G
=
.
ë
N or P
c G
=
.ë
W.
(728)
Ref. 1095 Page 349
Challenge 1484 e Challenge 1485 e
Page 349
e limit statements contain both the speed of light c and the constant of gravitation G; they thus indeed qualify as statements concerning relativistic gravitation. Like the previous limit statements, they are valid for all observers. is formulation of general relativity is not common; in fact, it seems that it was only discovered years a er the theory of general relativity had rst been proposed.* A detailed discussion is given in the chapter on general relativity.
e value of the force limit is the energy of a Schwarzschild black hole divided by its diameter; here the ‘diameter’ is de ned as the circumference divided by π. e power limit is realized when such a black hole is radiated away in the time that light takes to travel along a length corresponding to the diameter.
Force is change of momentum; power is the change of energy. Since momentum and energy are conserved, force or power can be pictured as the ow of momentum or energy through a given surface. e value of the maximum force is the mass–energy of a black hole divided by its diameter. It is also the surface gravity of a black hole times its mass.
e force limit thus means that no physical system of a given mass can be concentrated in a region of space-time smaller than a (non-rotating) black hole of that mass. In fact, the mass–energy concentration limit can be easily transformed by simple algebra into the force limit; both are equivalent.
It is easily checked that the maximum force is valid for all systems observed in nature, whether they are microscopic, macroscopic or astrophysical. Neither the ‘gravitational force’ (as long as it is operationally de ned) nor the electromagnetic or the nuclear interactions are ever found to exceed this limit.
e next aspect to check is whether a system can be imagined that does exceed the limit. An extensive discussion shows that this is impossible, if the proper size of observers or test masses is taken into account. Even for a moving observer, when the force value is increased by the (cube of the) relativistic dilation factor, or for an accelerating observer, when the observed acceleration is increased by the acceleration of the observer itself, the force limit must still hold. However, no situations allow the limit to be exceeded, because for high accelerations a, horizons appear at distance a c ; since a mass m has a minimum
* It might be that the present author was the rst to point it out, in this very textbook. Also Gary Gibbons found this result independently.
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diameter given by l Gm c , we are again limited by the maximum force.
e exploration of the force and power limits shows that they are achieved only at
horizons; the limits are not reached in any other situation. e limits are valid for all
observers and all interactions.
As an alternative to the maximum force and power limits, we can use as basic principle
the statement: ere is a maximum mass change in nature. For all systems and observers,
one has
dm dt
c G
=
.
ë
kg s .
(729)
Equivalently, we can state: ere is a maximum mass per length ratio in nature. For all systems and observers, one has
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dm dl
c G
=
.
ë
kg m .
(730)
Page 349 Page 349
In detail, both the force and the power limits state that the ow of momentum or of energy through any physical surface – a term de ned below – of any size, for any observer, in any coordinate system, never exceeds the limit values. Indeed, as a result to the lack of nearby black holes or horizons, neither limit value is exceeded in any physical system found so far. is is the case at everyday length scales, in the microscopic world and in astrophysical systems. In addition, even Gedanken experiments do not allow to the limits to be exceeded. However, the limits become evident only when in such Gedanken experiments the size of observers or of test masses is taken into account. If this is not done, apparent exceptions can be constructed; however, they are then unphysical.
D
*
Ref. 1096
In order to elevate the force or power limit to a principle of nature, we have to show that, in the same way that special relativity results from the maximum speed, general relativity results from the maximum force.
e maximum force and the maximum power are only realized at horizons. Horizons are regions of space-time where the curvature is so high that it limits the possibility of observation. e name ‘horizon’ is due to a certain analogy with the usual horizon of everyday life, which also limits the distance to which one can see. However, in general relativity horizons are surfaces, not lines. In fact, we can de ne the concept of horizon in general relativity as a region of maximum force; it is then easy to prove that it is always a two-dimensional surface, and that it has all properties usually associated with it.
e connection between horizons and the maximum force or power allows a simple deduction of the eld equations. We start with a simple connection. All horizons show energy ow at their location. is implies that horizons cannot be planes. An in nitely extended plane would imply an in nite energy ow. To characterize the nite extension of a given horizon, we use its radius R and its total area A.
e energy ow through a horizon is characterized by an energy E and a proper length L of the energy pulse. When such an energy pulse ows perpendicularly through a hori-
* is section can be skipped at rst reading.
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zon, the momentum change dp dt = F is given by
F
=
E L
.
(731)
For a horizon, we need to insert the maximum possible values. With the horizon area A and radius R, we can rewrite the limit case as
c G
=
E A
πR
L
(732) Dvipsbugw
where the maximum force and the maximum possible area πR of a horizon of (maximum local) radius R were introduced. e fraction E A is the energy per area owing through the horizon. O en, horizons are characterized by the so-called surface gravity a instead of the radius R. In the limit case, two are related by a = c R. is leads to
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
E = πG a A L .
(733)
Ref. 1097
Special relativity shows that horizons limit the product aL between proper length and
acceleration to the value c . is leads to the central relation for the energy ow at
horizons:
E=
c πG
a
A
.
(734)
is equation makes three points. First, the energy owing through a horizon is limited. Second, this energy is proportional to the area of the horizon. ird, the energy ow is proportional to the surface gravity. ese results are fundamental statements of general relativity. No other part of physics makes comparable statements.
For the di erential case the last relation can be rewritten as
δE =
c πG
a
δA
.
(735)
Ref. 1098
In this way, the result can also be used for general horizons, such as horizons that are curved or time-dependent.*
In a well known paper, Jacobson has given a beautiful proof of a simple connection: if energy ow is proportional to horizon area for all observers and all horizons, then
* Relation (735) is well known, though with di erent names for the observables. Since no communication
is possible across a horizon, the detailed fate of energy owing through a horizon is also unknown. Energy
whose detailed fate is unknown is o en called heat. Relation (735) therefore states that the heat owing
through a horizon is proportional to the horizon area. When quantum theory is introduced into the discus-
sion, the area of a horizon can be called ‘entropy’ and its surface gravity can be called ‘temperature’; relation
(735) can then be rewritten as
δQ = TδS .
(736)
However, this translation of relation (735), which requires the quantum of action, is unnecessary here. We only cite it to show the relation between horizon behaviour and quantum gravity.
Dvipsbugw
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general relativity holds. To see the connection to general relativity, we generalize relation ( ) to general coordinate systems and general energy- ow directions. is is achieved by introducing tensor notation.
To realize this at horizons, one introduces the general surface element dΣ and the local boost Killing vector eld k that generates the horizon (with suitable norm). Jacobson uses them to rewrite the le hand side of relation ( ) as
∫ δE = Tab kadΣb ,
(737) Dvipsbugw
where Tab is the energy–momentum tensor. is rewrites the energy for arbitrary coordinate systems and arbitrary energy ow directions. Jacobson’s main result is that the
essential part of the right hand side of relation ( ) can be rewritten, using the (purely
geometric) Raychaudhuri equation, as
∫ aδA = c Rab kadΣb ,
(738)
where Rab is the Ricci tensor describing space-time curvature. Combining these two steps, we nd that the energy–area relation (
can be rewritten as
∫ ∫ Tab kadΣb =
c πG
RabkadΣb .
) for horizons (739)
Jacobson shows that this equation, together with local conservation of energy (i.e., vanishing divergence of the energy–momentum tensor), can only be satis ed if
Tab =
c πG
Rab − ( R + Λ) ab
,
(740)
where Λ is a constant of integration whose value is not speci ed by the problem. ese are the full eld equations of general relativity, including the cosmological constant Λ. e
eld equations are thus shown to be valid at horizons. Since it is possible, by choosing a suitable coordinate transformation, to position a horizon at any desired space-time event, the eld equations must be valid over the whole of space-time.
It is possible to have a horizon at every event in space-time; therefore, at every event in nature there is the same maximum possible force (or power). is maximum force (or power) is a constant of nature.
In other words, we just showed that the eld equations of general relativity are a direct consequence of the limited energy ow at horizons, which in turn is due to the existence of a maximum force or power. Maximum force or power implies the eld equations. One can thus speak of the maximum force principle. In turn, the eld equations imply maximum force. Maximum force and general relativity are equivalent.
e bounds on force and power have important consequences. In particular, they imply statements on cosmic censorship, the Penrose inequality, the hoop conjecture, the non-existence of plane gravitational waves, the lack of space-time singularities, new experimental tests of the theory, and on the elimination of competing theories of relativistic
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Ref. 1099 gravitation. ese consequences are presented elsewhere.
D
Challenge 1486 e
Universal gravitation can be derived from the force limit in the case where forces and speeds are much smaller than the maximum values. e rst condition implies
GMa  c , the second v  c and al  c . Let us apply this to a speci c case. We study a satellite circling a central mass M at distance R with acceleration a. is system, with length l = R, has only one characteristic speed. Whenever this speed v is much
smaller than c, v must be proportional both to al = aR and to GMa . If they are taken together, they imply that a = f GM R , where the numerical factor f is not yet
xed. A quick check, for example using the observed escape velocity values, shows that f = . Forces and speeds much smaller than the limit values thus imply that the inverse square law of gravity describes the interaction between systems. In other words, nature’s limit on force implies the universal law of gravity, as is expected.
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T
General relativity, like the other theories of modern physics, provides a limit on the size
of systems: there is a limit to the amount of matter that can be concentrated into a small
volume.
l
Gm c
.
(741)
Ref. 1100
e size limit is only achieved for black holes, those well-known systems which swallow everything that is thrown into them. It is fully equivalent to the force limit. All composite systems in nature comply with the lower size limit. Whether elementary particles ful l or even achieve this limit remains one of the open issues of modern physics. At present, neither experiment nor theory allow clear statements on their size. More about this issue below.
General relativity also implies an ‘indeterminacy relation’:
∆E ∆l
c G
.
(742)
Since experimental data are available only for composite systems, we cannot say yet whether this inequality also holds for elementary particles. e relation is not as popular as the previous. In fact, testing the relation, for example with binary pulsars, may lead to new tests that would distinguish general relativity from competing theories.
A
e maximum force is central to the theory of general relativity. at is the reason why the value of the force (adorned with a factor π) appears in the eld equations. e importance of a maximum force becomes clearer when we return to our old image of space-time as a deformable mattress. Like any material body, a mattress is characterized by a material constant that relates the deformation values to the values of applied energy. Similarly, a mattress, like any material, is characterized by the maximum stress it can bear before
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breaking. Like mattresses, crystals also have these two values. In fact, for perfect crystals (without dislocations) these two material constants are the same.
Empty space-time somehow behaves like a perfect crystal or a perfect mattress: it has a deformation-energy constant that at the same time is the maximum force that can be applied to it. e constant of gravitation thus determines the elasticity of space-time. Now, crystals are not homogeneous, but are made up of atoms, while mattresses are made up of foam bubbles. What is the corresponding structure of space-time? is is a central question in the rest of our adventure. One thing is sure: vacuum has no preferred directions, in complete contrast to crystals. In fact, all these analogies even suggest that the appearance of matter might be nature’s way of preventing space-time from ripping apart. We have to patient for a while, before we can judge this option. A rst step towards the answer to the question appears when we put all limits together.
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U
e existence of a maximum force in nature is equivalent to general relativity. As a result, physics can now be seen as making three simple statements on motion that is found in nature:
quantum theory on action:
Sħ
special relativity on speed: general relativity on force:
vc
F
c G
.
(743)
Challenge 1487 e
e limits are valid for all physical systems, whether composed or elementary, and are valid for all observers. We note that the limit quantities of special relativity, quantum theory and general relativity can also be seen as the right-hand sides of the respective indeterminacy relations. Indeed, the set ( , , ) of indeterminacy relations or the set ( , , ) of length limits is each fully equivalent to the three limit statements ( ). Each set of limits can be taken as a (somewhat selective) summary of twentieth century physics.
If the three fundamental limits are combined, a limit on a number of physical observables arise. e following limits are valid generally, for both composite and elementary systems:
time interval:
t
time distance product: td
acceleration:
a
angular frequency:
ω
Għ c
=
Għ c
=
c Għ
=
π
c Għ
=
.ë − s . ë − sm . ë ms .ë s
(744) (745) (746) (747)
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With the additional knowledge that, in nature, space and time can mix, we get
distance: area: volume curvature: mass density:
d
Għ c
=
.ë − m
A
Għ c
= .ë − m
V
Għ c
= .ë − m
K
c Għ
= .ë
m
ρ
c Għ
= .ë
kg m
(748) (749) (750) (751) (752)
Ref. 1101, Ref. 1102 Ref. 1103
Of course, speed, action, angular momentum, power and force are also limited, as has already been stated. Except for a small numerical factor, for every physical observable these limits correspond to the Planck value. ( e limit values are deduced from the commonly used Planck values simply by substituting G for G and ħ for ħ.) ese values are the true natural units of nature. In fact, the most aesthetically pleasing solution is to rede ne the usual Planck values for every observable to these extremal values by absorbing the numerical factors into the respective de nitions. In the following, we call the rede ned limits the (corrected) Planck limits and assume that the factors have been properly included. In other words, every natural unit or (corrected) Planck unit is at the same
time the limit value of the corresponding physical observable. Most of these limit statements are found throughout the literature, though the numer-
ical factors are o en di erent. Each limit has a string of publications attached to it. e existence of a smallest measurable distance and time interval of the order of the Planck values is discussed in quantum gravity and string theory. e largest curvature has been studied in quantum gravity; it has important consequences for the ‘beginning’ of the universe, where it excludes any in nitely large or small observable. e maximum mass density appears regularly in discussions on the energy of vacuum.
With the present deduction of the limits, two results are achieved. First of all, the various arguments used in the literature are reduced to three generally accepted principles. Second, the confusion about the numerical factors is solved. During the history of Planck units, the numerical factors have varied greatly. For example, the fathers of quantum theory forgot the / in the de nition of the quantum of action. Similarly, the specialists of relativity did not emphasize the factor . With the present framework, the issue of the correct factors in the Planck units can be considered as settled.
We also note that the dimensional independence of the three limits in nature also means that quantum e ects cannot be used to overcome the force limit; similarly, the power limit cannot be used to overcome the speed limit. e same is valid for any other combination of limits: they are independent and consistent at the same time.
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L
Ref. 1104 Ref. 1102, Ref. 1105
Ref. 1106 Ref. 1107
Ref. 1102
e three limits of nature ( ) result in a minimum distance and a minimum time interval. ese minimum intervals directly result from the uni cation of quantum theory and relativity. ey do not appear if the theories are kept separate. In short, uni cation implies that there is a smallest length in nature. e result is important: the formulation of
physics as a set of limit statements shows that the continuum description of space and time is not correct. Continuity and manifolds are only approximations; they are valid for large values of action, low speeds and low values of force. e reformulation of general relativity and quantum theory with limit statements makes this result especially clear. e result is thus a direct consequence of the uni cation of quantum theory and general relativity. No other assumptions are needed.
In fact, the connection between minimum length and gravity is not new. In , Andrei Sakharov pointed out that a minimum length implies gravity. He showed that regularizing quantum eld theory on curved space with a cut-o will induce counter-terms that include to lowest order the cosmological constant and then the Einstein Hilbert action.
e existence of limit values for the length observable (and all others) has numerous consequences discussed in detail elsewhere. In particular, the existence of a smallest length – and a corresponding shortest time interval – implies that no surface is physical if any part of it requires a localization in space-time to dimensions smaller that the minimum length. (In addition, a physical surface must not cross any horizon.) Only by stipulation of this condition can unphysical examples that contravene the force and power limits be eliminated. For example, this condition has been overlooked in Bousso’s earlier discussion of Bekenstein’s entropy bound – though not in his more recent ones.
e corrected value of the Planck length should also be the expression that appears in the so-called theories of ‘doubly special relativity’. ese then try to expand special relativity in such a way that an invariant length appears in the theory.
A force limit in nature implies that no physical system can be smaller than a Schwarzschild black hole of the same mass. e force limit thus implies that point particles do not exist. So far, this prediction has not been contradicted by observations, as the predicted sizes are so small that they are outside experimental reach. If quantum theory is taken into account, this bound is sharpened. Because of the minimum length, elementary particles are now predicted to be larger than the corrected Planck length. Detecting the sizes of elementary particles would thus make it possible to check the force limit directly, for example with future electric dipole measurements.
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M
Mass plays a special role in all these arguments. e set of limits ( ) does not make it possible to extract a limit statement on the mass of physical systems. To nd one, the aim has to be restricted.
e Planck limits mentioned so far apply for all physical systems, whether they are composed or elementary. Additional limits can only be found for elementary systems. In quantum theory, the distance limit is a size limit only for composed systems. A particle is
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elementary if the system size l is smaller than any conceivable dimension:
for elementary particles: l
ħ mc
.
(753)
By using this new limit, valid only for elementary particles, the well-known mass, energy and momentum limits are found:
for elementary particles: m for elementary particles: E for elementary particles: p
ħc G
= . ë − kg = . ë
GeV c
ħc G
=.ë
J= . ë
GeV
ħc G
= . kg m s = . ë
GeV c
(754)
Ref. 1108
ese single-particle limits, corresponding to the corrected Planck mass, energy and momentum, were already discussed in by Andrei Sakharov, though again with di erent numerical factors. ey are regularly cited in elementary particle theory. Obviously, all known measurements comply with the limits. It is also known that the uni cation of the electromagnetic and the two nuclear interactions takes place at an energy near, but still clearly below the maximum particle energy.
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V
–
Page 315 Page 759
In fact, there are physical systems that exceed all three limits. Nature does contain systems that move faster than light, that show action values below half the quantum of action and that experience forces larger than the force limit. e systems in question are called virtual particles.
We know from special relativity that the virtual particles exchanged in collisions move faster than light. We know from quantum theory that virtual particle exchange implies action values below the minimum action. Virtual particles also imply an instantaneous change of momentum; they thus exceed the force limit. Virtual particles are thus those particles that exceed all the limits that hold for (real) physical systems.
L
ermodynamics can also be summarized in a single statement on motion:
smallest entropy in nature.
S k.
ere is a (755)
e entropy S is limited by half the Boltzmann constant k. e result is almost years Ref. 1109 old; it was stated most clearly by Leo Szilard, though with a di erent numerical factor.
In the same way as in the other elds of physics, this result can also be phrased as a
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indeterminacy relation:
∆ T ∆U k .
(756)
Ref. 1110
is relation was given by Bohr and discussed by Heisenberg and many others (though with k instead of k ). It is mentioned here in order to complete the list of indeterminacy relations and fundamental constants. With the single-particle limits, the entropy limit leads to an upper limit for temperature:
T
ħc Gk
=. ë
K.
(757)
is corresponds to the temperature at which the energy per degree of freedom is given by the (corrected) Planck energy. A more realistic value would have to take into account the number of degrees of freedom of a particle at Planck energy. is would change the numerical factor.
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E
e discussion of limits can be extended to include electromagnetism. Using the (lowenergy) electromagnetic coupling constant α, one gets the following limits for physical systems interacting electromagnetically:
electric charge: electric eld: magnetic eld: voltage: inductance:
q
πε αcħ = e = . aC
E
c πεoαħG
=
c Ge
=
.
ë
Vm
B
c πε αħG
=
c Ge
=
.
ë
T
U
c πε
αG
=e
ħc G
=
.
ë
V
L πεoα
ħG c
=
e
ħG c
=
.
ë
−
H
(758) (759) (760) (761) (762)
With the additional assumption that in nature at most one particle can occupy one Planck volume, one gets
charge density: capacitance:
ρe
πεoα G
c ħ
=e
c Għ
=.ë
Cm
C πε α
ħG c
=e
G cħ
=
.
ë
−
F
(763) (764)
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For the case of a single conduction channel, one gets
electric resistance:
R
πε αc =
ħ e
=
. kΩ
electric conductivity: G
πε αc =
e ħ
=
.
mS
electric current:
I
πε αc G
=e
c ħG
=
.
ë
A
(765) (766) (767)
Ref. 1111 Ref. 1112
Many of these limits have been studied already. e magnetic eld limit plays a role in the discussion of extreme stars and black holes. e maximum electric eld plays a role in the theory of gamma ray bursters. e studies of limit values for current, conductivity and resistance in single channels are well known; the values and their e ects have been studied extensively in the s and s. ey will probably win a Nobel prize in the future.
e observation of quarks and of collective excitations in semiconductors with charge e does not necessarily invalidate the charge limit for physical systems. In neither case is there is a physical system – de ned as localized mass–energy interacting incoherently with the environment – with charge e .
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V
–
–
Page 996 Page 1032
Ref. 1102, Ref. 1105 Page 349
In this discussion we have found that there is a xed limit to every physical observable. Many consequences have been discussed already in previous sections. ere we saw that the existence of a limit to all observables implies that at Planck scales no physical observable can be described by real numbers and that no low-energy symmetry is valid.
One conclusion is especially important for the rest of our adventure. We saw that there is a limit for the precision of length measurements in nature. e limit is valid both for the length measurements of empty space and for the length measurements of matter (or radiation). Now let us recall what we do when we measure the length of a table with a ruler. To nd the ends of the table, we must be able to distinguish the table from the surrounding air. In more precise terms, we must be able to distinguish matter from vacuum. But we have no way to perform this distinction at Planck energy. In these domains, the intrinsic measurement limitations of nature do not allow us to say whether we are measuring vacuum or matter. ere is no way, at Planck scales, to distinguish the two.
We have explored this conclusion in detail above and have shown that it is the only consistent description at Planck scales. e limitations in length measurement precision, in mass measurement precision and in the precision of any other observable do not allow to tell whether a box at Planck scale is full or empty. You can pick any observable you want to distinguish vacuum from matter. Use colour, mass, size, charge, speed, angular momentum, or anything you want. At Planck scales, the limits to observables also lead to limits in measurement precision. At Planck scales, the measurement limits are of the same size as the observable to be measured. As a result, it is impossible to distinguish matter and vacuum at Planck scales.
We put the conclusion in the sharpest terms possible: Vacuum and matter do not di er
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Ref. 1113
at Planck scales. is counter-intuitive result is one of the charms of theoretical high energy physics. is result alone inspired many researchers in the eld and induced them to write best-sellers. Brian Greene was especially successful in presenting this side of quantum geometry to the wider public.
However, at this point of our adventure, most issues are still open. e precise manner in which a minimum distance leads to a homogeneous and isotropic vacuum is unclear.
e way to describe matter and vacuum with the same concepts has to be found. And of course, the list of open questions in physics, given above, still waits. However, the conceptual results give hope; there are interesting issues awaiting us.
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P
e (corrected) Planck limits are statements about properties of nature. ere is no way to measure values exceeding these limits, whatever experiment is performed. As can be expected, such a claim provokes the search for counter-examples and leads to many paradoxes.
**
e minimum angular momentum may surprise at rst, especially when one thinks about particles with spin zero. However, the angular momentum of the statement is total angular momentum, including the orbital part with respect to the observer. e total angular momentum is never smaller than ħ .
**
If any interaction is stronger than gravity, how can the maximum force be determined by gravity alone, which is the weakest interaction? It turns out that in situations near the maximum force, the other interactions are negligible. is is the reason that gravity must be included in a uni ed description of nature.
Challenge 1488 ny
**
At rst sight, it seems that electric charge can be used in such a way that the acceleration of a charged body towards a charged black hole is increased to a value exceeding the force limit. However, the changes in the horizon for charged black holes prevent this.
**
e general connection that to every limit value in nature there is a corresponding inde-
terminacy relation is also valid for electricity. Indeed, there is an indeterminacy relation
for capacitors of the form
∆C ∆U e
(768)
where e is the positron charge, C capacity and U potential di erence, and one between
electric current I and time t
∆I ∆t e
(769)
Ref. 1114 and both relations are found in the literature.
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**
e gravitational attraction between two masses never yields force values su ciently high to exceed the force limit. Why? First of all, masses m and M cannot come closer than the sum of their horizon radii. Using F = GmM r with the distance r given by the (naive) sum of the two black hole radii as r = G(M + m) c , one gets
F
c Mm G (M + m)
,
(770)
which is never larger than the force limit. Even two attracting black holes thus do not exceed the force limit – in the inverse square approximation of universal gravity. e minimum size of masses does not allow to exceed the a maximum force.
Ref. 1115
**
It is well known that gravity bends space. To be fully convincing, the calculation needs to be repeated taking space curvature into account. e simplest way is to study the force generated by a black hole on a test mass hanging from a wire that is lowered towards a black hole horizon. For an unrealistic point mass, the force would diverge on the horizon. Indeed, for a point mass m lowered towards a black hole of mass M at (conventionally de ned radial) distance d, the force would be
F = GMm .
d
−
GM dc
(771)
e expression diverges at d = , the location of the horizon. However, even a test mass cannot be smaller than its own gravitational radius. If we want to reach the horizon with a realistic test mass, we need to chose a small test mass m; only a small – and thus light – mass can get near the horizon. For vanishingly small masses however, the resulting force tends to zero. Indeed, letting the distance tend to the smallest possible value by letting d = G(m + M) c d = GM c requires m , which makes the force F(m, d) vanish. If on the other hand, we remain away from the horizon and look for the maximum force by using a mass as large as can possibly t into the available distance (the calculation is straightforward algebra) again the force limit is never exceeded. In other words, for realistic test masses, expression (771) is never larger than c G. Taking into account the minimal size of test masses thus prevents that the maximum force is exceeded in gravitational systems.
**
An absolute power limit implies a limit on the energy that can be transported per time unit through any imaginable surface. At rst sight, it may seem that the combined power emitted by two radiation sources that each emit 3/4 of the maximum value should give 3/2 times the upper value. However, the combination forms a black hole or at least prevents part of the radiation to be emitted by swallowing some of it between the sources.
**
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One possible system that actually achieves the power limit is the nal stage of black hole evaporation. But even in this case the power limit is not exceeded.
Ref. 1095
**
e maximum force limit states that the stress–energy tensor, when integrated over any physical surface, does not exceed the limit value. No such integral, over any physical surface whatsoever, of any tensor component in any coordinate system, can exceed the force limit, provided that it is measured by a nearby observer or a test body with a realistic proper size. e maximum force limit thus applies to any component of any force vector, as well as to its magnitude. It applies to gravitational, electromagnetic, and nuclear forces. It applies to all realistic observers. Whether the forces are real or ctitious is not important. It also plays no role whether we discuss 3-forces of Galilean physics or 4-forces of special relativity. Indeed, the force limit applied to the 0-th component of the 4-force is the power limit.
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**
Translated to mass ows, the power limit implies that ow of water through a tube is limited in throughput. Also this limit seems unknown in the literature.
**
e force limit cannot be overcome with Lorentz boosts. A Lorentz boost of any nonvanishing force value seems to allow exceeding the force limit at high speeds. However, such a transformation would create a horizon that makes any point with a potentially higher force value inaccessible.
**
e power limits is of interest if applied to the universe as a whole. Indeed, it can be used to explain Olber’s paradox. e sky is dark at night because the combined luminosity of all light sources in the universe cannot be brighter than the maximum value.
**
One notes that the negative energy volume density −Λc πG introduced by the positive cosmological constant Λ corresponds to a negative pressure (both quantities have the same dimensions). When multiplied with the minimum area it yields a force value
F = Λħc = . ë − N . π
(772)
is is also the gravitational force between two corrected Planck masses located at the
cosmological distance π Λ . If we make the (wishful) assumption that this is the smallest possible force in nature (the numerical prefactor is not nalized yet), we get the fascinating conjecture that the full theory of general relativity, including the cosmological constant, is de ned by the combination of a maximum and a minimum force in nature.
Another consequence of the limits merits a dedicated section.
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U
Page 349 Page 1068
In our quest to understand motion, we have focused our attention to the limitations it is subjected to. Special relativity poses a limit to speed, namely the speed of light c. General relativity limits force and power respectively by c G and c G, and quantum theory introduces a smallest value ħ for angular momentum or action. We saw that nature poses a limit on entropy and a limit on electrical charge. Together, all these limits provide an extreme value for every physical observable. e extremum is given by the corresponding (corrected) Planck value. We have explored the list of all Planck limits in the previous section. e question may now arise whether nature provides a limit for physical observables also on the opposite end of the measurement scale. For example, there is a highest force and a highest power in nature; is there also a lowest force and a lowest power? Is there also a lowest speed?
We show in the following that there indeed are such limits, for all observables. We give the general method to generate such bounds and explore several examples.* e exploration will lead us along an interesting scan across modern physics.
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S
Looking for additional limits in nature, we directly note a fundamental property. Any upper limit for angular momentum or any lower limit for power must be system dependent. Such limits will not be absolute, but will depend on properties of the system. Now, any physical system is a part of nature characterized by a boundary and its content.** e simplest properties all systems share are thus their size (characterized in the following by the diameter) L and energy E. With these characteristics we can enjoy deducing systemdependent limits for every physical observable. e general method is straightforward. We take the known inequalities for speed, action, power, charge and entropy and then extract a limit for any observable, by inserting length and energy as required. We then have to select the strictest of the limits we nd.
A
,
It only takes a moment to note that the ratio of angular momentum D to mass times length has the dimension of speed. Since speeds are limited by the speed of light, we get
D c LE .
(773)
Ref. 1116
Indeed, there do not seem to be any exceptions to this limit in nature. No known system has a larger angular momentum value, from atoms to molecules, from ice skaters to galaxies. For example, the most violently rotating objects, the so-called extremal black holes, are also limited in angular momentum by D LE c. (In fact, this limit is correct only if the energy is taken as the irreducible mass times c ; if the usual mass is used, the limit is too large by a factor .) One remarks that the limit deduced from general relativity, given
* is section was added in June 2004. ** We mention here that quantum theory narrows down this de nition as a part of nature that in addition Page 800 interacts incoherently with its environment. We assume that this condition is realized in the following.
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Challenge 1489 ny
by D L c G is not stricter than the one just given. In addition, no system-dependent lower limit for angular momentum can be deduced.
e maximum angular momentum value is also interesting when it is seen as action limit. Action is the time integral of the di erence between kinetic and potential energy. In fact, since nature always minimizes action W, we are not used to search for systems which maximize its value. You might check by yourself that the action limit W LE c is not exceeded in any physical process.
Similarly, speed times mass times length is an action. Since action values in nature are limited from below by ħ , we get
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v ħc LE .
(774)
Challenge 1490 ny Challenge 1491 e
is relation is a rewritten form of the indeterminacy relation of quantum theory and is no news. No system of energy E and diameter L has a smaller speed than this limit. Even the slowest imaginable processes show this speed value. For example, the extremely slow radius change of a black hole by evaporation just realizes this minimal speed. Continuing with the method just outlined, one also nds that the limit deduced from general relativity, v (c G)(L E), gives no new information. erefore, no system-dependent upper speed limit exists.
Incidentally, the limits are not unique. Additional limits can be found in a systematic way. Upper limits can be multiplied, for example, by factors of (L E)(c G) or (LE)( ħc) yielding additional, but less strict upper limits. A similar rule can be given for lower limits.*
With the same approach we can now systematically deduce all size and energy dependent limits for physical observables. We have a tour of the most important ones.
F,
We saw that force and power are central to general relativity. Due to the connection W = FLT between action W, force, distance and time, we can deduce
F
ħ cT
.
(775)
Experiments do not reach this limit. e smallest forces measured in nature are those in atomic force microscopes, where values as small as aN are observed. However, the values are all above the lower force limit.
e power P emitted by a system of size L and mass M is limited by
c
M L
P
ħG
M L
(776)
e le , upper limit gives the upper limit for any engine or lamp deduced from relativity; not even the universe exceeds it. e right, lower limit gives the minimum power emit-
* e strictest upper limits are thus those with the smallest possible exponent for length, and the strictest lower limits are those with the largest sensible exponent of length.
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ted by any system due to quantum gravity e ects. Indeed, no system is completely tight. Even black holes, the systems with the best ability in nature to keep components inside their enclosure, nevertheless radiate. e power radiated by black holes should just saturate this limit, provided the length L is taken to be the circumference of the black hole. (However, present literature values of the numerical factors in the black hole power are not yet consistent). e claim of the quantum gravity limit is thus that the power emitted by a black hole is the smallest power that is emitted by any composed system of the same surface gravity.
A
When the acceleration of a system is measured by a nearby inertial observer, the acceleration a of a system of size L and mass M is limited by
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cL a
G
M L
(777)
Ref. 1117
e lower, right limit gives the acceleration due to universal gravity. Indeed, the relative acceleration between a system and an observer has at least this value. e le , upper limit to acceleration is the value due to special relativity. No exception to either of these limits has ever been observed. Using the limit to the size of masses, the upper limit can be transformed into the equivalent acceleration limit
c ħ
M
a
(778)
Challenge 1492 ny
which has never been approached either, despite many attempts. e upper limit to acceleration is thus a quantum limit, the lower one a gravitational limit.
e acceleration of the radius of a black hole due to evaporation can be much slower than the limit a GM L . Why is this not a counter-example?
M e momentum p of a system of size L is limited by
c G
L
p
ħ L
(779)
e lower limit is obviously due to quantum theory; experiments con rmed it for all radiation and matter. e upper limit for momentum is due to general relativity. It has never been exceeded.
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L
,
Time is something special. What are the limits to time measurements? Like before, we nd that any measured time interval t in a system in thermal equilibrium is limited by
ħ ML
t
G c
M
(780)
e lower time limit is the gravitational time. No clock can measure a smaller time than this value. Similarly, no system can produce signals shorter than this duration. Black holes, for example, emit radiation with a frequency given by this minimum value. e upper time limit is expected to be the exact lifetime of a black hole. ( ere is no consensus in the literature on the numerical factor yet.)
e upper limit to time measurements is due to quantum theory. It leads to a question: What happens to a single atom in space a er the limit time has passed by? Obviously, an atom is not a composed system comparable with a black hole. e lifetime expression assumes that decay can take place in the most tiny energy steps. As long as there is no decay mechanism, the life-time formula does not apply. e expression ( ) thus does not apply to atoms.
Distance limits are straightforward.
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c ħ
ML
d
G c
M
(781)
Since curvature is an inverse square distance, curvature of space-time is also limited.
M e mass change dM dt of a system of size L and mass M is limited by
c L dM ħ
G M dt
L
(782)
e limits apply to systems in thermal equilibrium. e le , upper limit is due to general relativity; it is never exceeded. e right, lower limit is due to quantum theory. Again, all experiments are consistent with the limit values.
M
Limits for rest mass make only sense if the system is characterized by a size L only. We
then have
c G
L
M
ħ cL
(783)
e upper limit for mass was discussed in general relativity. Adding mass or energy to a black hole always increases its size. No system can show higher values than this value, and indeed, no such system is known or even imaginable. e lower limit on mass is obviously due to quantum theory; it follows directly from the quantum of action.
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e mass density a system of size L is limited by
c GL
ρ
ħ cL
(784)
Page 1086
e upper limit for mass density, due to general relativity, is only achieved for black holes. e lower limit is the smallest density of a system due to quantum theory. It also applies to the vacuum, if a piece of vacuum of site L is taken as a physical system. We note again that many equivalent (but less strict) limits can be formulated by using the transformations rules mentioned above.
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T
Ref. 1118
In , Bekenstein discovered a famous limit that connects the entropy S of a physical system with its size and mass.
No system has a larger entropy than one bounded by a horizon. e larger the horizon surface, the larger the entropy. We write
S Slimit
A Alimit
(785)
which gives
S
kc Għ
A
,
(786)
Ref. 1119 Page 245
where A is the surface of the system. Equality is realized only for black holes. e old question of the origin of the factor in the entropy of black holes is thus answered here in he following way: it is due to the factor in the force or power bound in nature (provided that the factors from the Planck entropy and the Planck action cancel). e future will tell whether this explanation will stand the winds of time. Stay tuned.
We can also derive a more general relation if we use a mysterious assumption that we discuss a erwards. We assume that the limits for vacuum are opposite to those for matter. We can then write c G M L for the vacuum. is gives
S
πkc ħ
ML
=
πkc ħ
MR
.
(787)
In other words, we used
S Scorr. Planck
M
A
Lcorr. Planck .
Mcorr. Planck Acorr. Planck
L
(788)
Ref. 1106
Expression ( ) is called Bekenstein’s entropy bound. Up to today, no exception has been found or constructed, despite many attempts. Again, the limit value itself is only realized for black holes.
We still need to explain the strange assumption used above. We are exploring the entropy of a horizon. Horizons are not matter, but limits to empty space. e entropy of
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Ref. 1120
horizons is due to the large amount of virtual particles found there. In order to deduce the maximum entropy of expression ( ) one therefore has to use the properties of the vacuum. In other words, either ( ) we use a mass to length ratio for vacuum above the Planck limit or ( ) we use the Planck entropy as maximum value for vacuum.
Other, equivalent limits for entropy can be found if other variables are introduced. For example, since the ratio of shear viscosity η to the volume density of entropy (times k) has the dimensions of an action, we can directly write
S
k ħ
ηV
(789)
Challenge 1493 ny Challenge 1494 e
Again, equality is only reached in the case of black holes. With time, the list of similar bounds will grow longer and longer.
Is there also a smallest entropy limit? So far, there does not seem to be a systemdependent minimum value for entropy; the approach gives no expression that is larger than k.
e entropy limit is an important step in making the description of motion consistent. If space-time can move, as general relativity maintains, it also has an entropy. How could entropy be limited if space-time is continuous? Clearly, due to the minimum distance and a minimum time in nature, space-time is not continuous, but has a nite number of degrees of freedom. e number of degrees of freedom and thus the entropy of spacetime is thus nite.
In addition, the Bekenstein limit also allows some interesting speculations. Let us speculate that the universe itself, being surrounded by a horizon, saturates the Bekenstein bound. e entropy bound gives a bound to all degrees of freedom inside a system; it tells us that the number Nd.o.f. of degrees of freedom of the universe is roughly
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Nd.o.f.
.
(790)
is compares with the number NPl. vol. of Planck volumes in the universe
NPl. vol.
(791)
and with the number Npart. of particles in the universe
Npart.
.
(792)
In other words, particles are only a tiny fraction of what moves around. Most motion must be that of space-time. At the same time, space-time moves much less than naively expected. Finding out how all this happens is the challenge of the uni ed description of motion.
T
A lower limit for the temperature of a thermal system can be found using the idea that the number of degrees of freedom of a system is limited by its surface, more precisely, by
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the ratio between the surface and the Planck surface. One gets the limit
T
Għ M πkc L
(793)
Challenge 1495 n Page 310
Challenge 1496 ny
Alternatively, using the method given above, one can use the limit on the thermal energy kT ħc ( πL) (the thermal wavelength must be smaller than the size of the system) together with the limit on mass c G M L and deduce the same result.
We know the limit already: when the system is a black hole, it gives the temperature of the emitted radiation. In other words, the temperature of black holes is the lower limit for all physical systems for which a temperature can be de ned, provided they share the same boundary gravity. As a criterion, boundary gravity makes sense: boundary gravity is accessible from the outside and describes the full physical system, since it makes use both of its boundary and its content. So far, no exception to this claim is known. All systems from everyday life comply with it, as do all stars. Even the coldest known systems in the universe, namely Bose–Einstein condensates and other cold matter in various laboratories, are much hotter than the limit, and thus much hotter than black holes of the same surface gravity. (Since a consistent Lorentz transformation for temperature is not possible, as we saw earlier, the limit of minimum temperature is only valid for an observer at the same gravitational potential and at zero relative speed to the system under consideration.)
ere seems to be no consistent way to de ne an upper limit for a system-dependent temperature. However, limits for other thermodynamic quantities can be found, but are not discussed here.
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E
When electromagnetism plays a role, the involved system also needs to be characterized
by a charge Q. e method used so far then gives the following lower limit for the electric
eld E:
E
Ge
M QL
(794)
Challenge 1497 ny
We write the limit using the elementary charge e, though writing it using the ne structure constant via e = πε αħc would be more appropriate. Experimentally, this limit is not exceeded in any system in nature. Can you show whether it is achieved by maximally charged black holes?
For the magnetic eld we get
B
Ge M c QL
(795)
Challenge 1498 ny
Again, this limit is satis ed all known systems in nature. Similar limits can be found for the other electromagnetic observables. In fact, several
of the earlier limits are modi ed when electrical charge is included. Can you show how the size limit changes when electric charge is taken into account? In fact, a dedicated research eld is concerned only with the deduction of the most general limits valid in
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nature.
C
e limits yield a plethora of interesting paradoxes that can be discussed in lectures and student exercises. All paradoxes can be solved by carefully taking into account the combination of e ects from general relativity and quantum theory. All apparent violations only appear when one of the two aspects is somehow forgotten. We study a few examples.
**
Several black hole limits are of importance to the universe itself. For example, the observed average mass density of the universe is not far from the corresponding limit. Also the lifetime limit is obviously valid for the universe and provides an upper limit for its age. However, the age of the universe is far from that limit by a large factor. In fact, since the universe’s size and age still increase, the age limit is pushed further into the future with every second that passes. e universe evolves in a way to escape its own decay.
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Challenge 1499 r
**
e content of a system is not only characterized by its mass and charge, but also by its strangeness, isospin, colour charge, charge and parity. Can you deduce the limits for these quantities?
Challenge 1500 n
**
In our discussion of black hole limits, we silently assumed that they interact, like any thermal system, in an incoherent way with the environment. What changes in the results of this section when this condition is dropped? Which limits can be overcome?
** Challenge 1501 e Can you nd a general method to deduce all limits?
Page 1068
**
Brushing some important details aside, we can take the following summary of our study of nature. Galilean physics is that description for which the di erence between composed and elementary systems does not exist. Quantum theory is the description of nature with no (really large) composed systems; general relativity is the description of nature with no elementary systems. is distinction leads to the following interesting conclusion. A uni ed theory of nature has to unify quantum theory and general relativity. Since the
rst theory a rms that no (really large) composed systems exist, while the other that no elementary systems exist, a uni ed theory should state that no systems exist. is strange result indeed seems to be one way to look at the issue. e conclusion is corroborated by the result that in the uni ed description of nature, the observables time, space and mass cannot be distinguished clearly from each other, which implies that systems cannot be clearly distinguished from their surroundings. To be precise, systems thus do not really exist at uni cation energy.
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L
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Challenge 1502 r Ref. 1102
Ref. 1102
We now know that in nature, every physical measurement has a lower and an upper bound. One of the bounds is size-dependent, the other is absolute. So far, no violation of these claims is known. e smallest relative measurement error possible in nature is thus the ratio of the two bounds. In short, a smallest length, a highest force and a smallest action, when taken together, imply that all measurements are limited in precision.
A fundamental limit to measurement precision is not a new result any more. But it raises many issues. If the mass and the size of any system are themselves imprecise, can you imagine or deduce what happens then to the limit formulae given above?
Due to the fundamental limits to measurement precision, the measured values of physical observables do not require the full set of real numbers. In fact, limited precision implies that no observable can be described by real numbers. We thus recover again a result that appears whenever quantum theory and gravity are brought together.
In addition, we found that measurement errors increase when the characteristic measurement energy approaches the Planck energy. In that domain, the measurement errors of any observable are comparable with the measurement values themselves.
Limited measurement precision thus implies that at the Planck energy it is impossible to speak about points, instants, events or dimensionality. Limited measurement precision also implies that at the Planck length it is impossible to distinguish positive and negative time values: particle and anti-particles are thus not clearly distinguished at Planck scales. A smallest length in nature thus implies that there is no way to de ne the exact boundaries of objects or elementary particles. However, a boundary is what separates matter from vacuum. In short, a minimum measurement error means that, at Planck scales, it is impossible to distinguish objects from vacuum with complete precision. To put it bluntly, at Planck scales, time and space do not exist.
e mentioned conclusions are the same as those that are drawn by modern research on uni ed theories. e force limit, together with the other limits, makes it possible to reach the same conceptual results found by string theory and the various quantum gravity approaches. To show the power of the maximum force limit, we now explore a few conclusions which go beyond present approaches.
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M
Ref. 1102, Ref. 1105
e impossibility of completely eliminating measurement errors has an additional and important consequence. In physics, it is assumed that nature is a set of components. ese components are assumed to be separable from each other. is tacit assumption is introduced in three main situations: it is assumed that matter consists of separable particles, that space-time consists of separable events or points, and that the set of states consists of separable initial conditions. So far, all of physics has thus built its complete description of nature on the concept of set.
A fundamentally limited measurement precision implies that nature is not a set of such separable elements. A limited measurement precision implies that distinguishing physical entities is possible only approximately. e approximate distinction is only possible at energies much lower than the Planck energy. As humans we do live at such smaller energies; thus we can safely make the approximation. Indeed, the approximation is excel-
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Ref. 1102 Ref. 1102 Ref. 1105
lent; we do not notice any error when performing it. But the discussion of the situation at Planck energies shows that a perfect separation is impossible in principle. In particular, at the cosmic horizon, at the big bang, and at Planck scales any precise distinction between two events or two particles becomes impossible.
Another way to reach this result is the following. Separation of two entities requires different measurement results, such as di erent positions, di erent masses, di erent velocities, etc. Whatever observable is chosen, at the Planck energy the distinction becomes impossible, due to the large measurements errors. Only at everyday energies is a distinction approximately possible. Any distinction between two physical systems, such as between a toothpick and a mountain, is thus possible only approximately; at Planck scales, a boundary cannot be drawn.
A third argument is the following. In order to count any entities in nature – a set of particles, a discrete set of points, or any other discrete set of physical observables – the entities have to be separable. e inevitable measurement errors, however, contradict separability. At the Planck energy, it is thus impossible to count physical objects with precision.
Nature has no parts. In short, at Planck energies a perfect separation is impossible in principle. We cannot
distinguish observations at Planck energies. In short, at Planck scale it is impossible to split nature into separate entities. ere are no mathematical elements of any kind – or of any set – in nature. Elements of sets cannot be de ned. As a result, in nature, neither discrete nor continuous sets can be constructed. Nature does not contain sets or elements.
Since sets and elements are only approximations, the concept of ‘set’, which assumes separable elements, is already too specialized to describe nature. Nature cannot be described at Planck scales – i.e., with full precision – if any of the concepts used for its description presupposes sets. However, all concepts used in the past twenty- ve centuries to describe nature – space, time, particles, phase space, Hilbert space, Fock space, particle space, loop space or moduli space – are based on sets. ey all must be abandoned at Planck energy. No approach used so far in theoretical physics, not even string theory or the various quantum gravity approaches, satis es the requirement to abandon sets. Nature is one and has no parts. Nature must be described by a mathematical concept that does not contain any set. is requirement must guide us in the future search for the uni cation of relativity and quantum theory.
“Es ist fast unmöglich, die Fackel der Wahrheit durch ein Gedränge zu tragen, ohne jemandem den Bart zu sengen.*
Georg Christoph Lichtenberg (
–
”)
W
?
Certain papers on quantum theory give the impression that observers are indispensable for quantum physics. We have debunked this belief already, showing that the observer in quantum theory is mainly a bath with a de nite interaction. O en, humans are observers.
At Planck scales, observers also play a role. At Planck scales, quantum theory is mandatory. In these domains, observers must realize an additional requirement: they must
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* ‘It is almost impossible to carry the torch of truth through a crowd without scorching somebody’s beard.’
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function at low energies. Only at low energy, an observer can introduce sets for the description of nature. Introducing observers is thus the same as introducing sets.
To put it in another way, the limits of human observers is that they cannot avoid using sets. However, human observers share this limitation with video recorders, cameras, computers and pencil and paper. Nothing singles out humans in this aspect.
In simple terms, observers are needed to describe nature at Planck scales only in so far as they are and use sets. We should not get too bloated about our own importance.
A
H
’
Ref. 1121 Ref. 1105
In the year , David Hilbert gave a well-known lecture in which he listed twentythree of the great challenges facing mathematics in the twentieth century. Most problems provided challenges to many mathematicians for decades a erwards. Of the still unsolved ones, Hilbert’s sixth problem challenges mathematicians and physicists to nd an axiomatic treatment of physics. e challenge has remained in the minds of many physicists since that time.
Since nature does not contain sets, we can deduce that such an axiomatic description of nature does not exist! e reasoning is simple; we only have to look at the axiomatic systems found in mathematics. Axiomatic systems de ne mathematical structures. ese structures are of three main types: algebraic systems, order systems or topological systems. Most mathematical structures – such as symmetry groups, vector spaces, manifolds or elds – are combinations of all three. But all mathematical structures contain sets. Mathematics does not provide axiomatic systems that do not contain sets. e underlying reason is that every mathematical concept contains at least one set.
Furthermore, all physical concepts used so far in physics contain sets. For humans, it is di cult even simply to think without rst de ning a set of possibilities. However, nature is di erent; nature does not contain sets. erefore, an axiomatic formulation of physics is impossible. Of course, this conclusion does not rule out uni cation in itself; however, it does rule out an axiomatic version of it. e result surprises, as separate axiomatic treatments of quantum theory or general relativity (see above) are possible. Indeed, only their uni cation, not the separate theories, must be approached without an axiomatic systems. Axiomatic systems in physics are always approximate. e requirement to abandon axiomatic systems is one of the reasons for the di culties in reaching the uni ed description of nature.
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O
Physics can be summarized in a few limit statements. ey imply that in nature every physical observable is limited by a value near the Planck value. e speed limit is equivalent to special relativity, the force limit to general relativity, and the action limit to quantum theory. Even though this summary could have been made (or at least conjectured) by Planck, Einstein or the fathers of quantum theory, it is much more recent. e numerical factors for most limit values are new. e limits provoke interesting Gedanken experiments, none of which leads to violations of the limits. On the other hand, the force limit is not yet within direct experimental reach.
e existence of limit values to all observables implies that the description of spacetime with a continuous manifold is not correct at Planck scales; it is only an approx-
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Page 28
imation. For the same reason, is predicted that elementary particles are not point-like. Nature’s limits also imply the non-distinguishability of matter and vacuum. As a result, the structure of particles and of space-time remains to be clari ed. So far, we can conclude that nature can be described by sets only approximately. e limit statements show that Hilbert’s sixth problem cannot be solved and that uni cation requires fresh approaches, taking unbeaten paths into unexplored territory.
We saw that at Planck scales there is no time, no space, and there are no particles. Motion is a low energy phenomenon. Motion only appears for low-energy observers. ese are observers who use sets. e (inaccurate) citation of Zeno at the beginning of our walk, stating that motion is an illusion, turns out to be correct! erefore, we now need to nd out how motion actually arises.
e discussion so far hints that motion appears as soon as sets are introduced. To check this hypothesis, we need a description of nature without sets. e only way to avoid the use of sets seems a description of empty space-time, radiation and matter as being made of the same underlying entity. e inclusion of space-time dualities and of interaction dualities is most probably a necessary step. Indeed, both string theory and modern quantum gravity attempt this, though possibly not yet with the full radicalism necessary. Realizing this uni cation is our next task.
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B
1093 Bohr explained the indivisibilty of the quantum of action in his famous Como lecture. N.
B , Atomtheorie und Naturbeschreibung, Springer, Berlin, . More statements about
the indivisibility of the quantum of action can be found in N. B , Atomic Physics and
Human Knowledge, Science Editions, New York, . For summaries of Bohr’s ideas by
others see M J
, e Philosophy of Quantum Mechanics, Wiley, rst edition, ,
pp. – , and J H
, e Description of Nature – Niels Bohr and the Philosophy
of Quantum Physics, Clarendon Press, , p. . Cited on page .
1094 For an overview of the quantum of action as basis of quantum theory, see the introduction
to quantum theory in this textbook, available as by C. S
, An appetizer – quantum
theory for poets and lawyers, http://www.motionmountain.net/C- -QEDA.pdf. Cited on
page .
1095 e rst published statements of the principle of maximum force were in the present text-
book and independently in the paper by G.W. G
, e maximum tension principle
in general relativity, Foundations of Physics 32, pp. – , , or http://www.arxiv.
org/abs/hep-th/
. Cited on pages and .
1096 Maximal luminosity of o en mentioned in studies around gravitational wave detection;
nevertheless, the general maximum is never mentioned. See for example L. J , D.G.
B
& C. Z , Detection of gravitational waves, Reports on Progress in Physics 63,
pp. – , . See also C. M
, K. T
& J.A. W
, Gravitation,
Freeman, , page . Cited on page .
1097 See for example W
R
, Relativity – Special, General and Cosmological,
Oxford University Press, , p. , or R ’I
Introducing Einstein’s Relativ-
ity, Clarendon Press, , p. . Cited on page .
1098 T. J
, ermodynamics of spacetime: the Einstein equation of state, Physical Re-
view Letters 75, pp. – , , or preprint http://www.arxiv.org/abs/gr-qc/
Cited on page .
1099 C. S
, Maximum force: a simple principle encompassing general relativity, part
of this text and downloadable as http://www.motionmountain.net/C- -MAXF.pdf. Cited
on page .
1100 Indeterminacy relations in general relativity are discussed in C.A. M , Possible con-
nection between gravitation and fundamental length, Physical Review B 135, pp. – ,
. See also P.K. T
, Small-scale structure of space-time as the origin of the
gravitational constant, Physical Review D 15, pp. – , , or the paper by M.-
T. J
& S. R
, Gravitational quantum limit for length measurement, Physics
Letters A 185, pp. – , . Cited on page .
1101 Minimal length and minimal time intervals are discussed for example by G. A
–
C
, Limits on the measurability of space-time distances in (the semiclassical ap-
proximation of) quantum gravity, Modern Physics Letters A 9, pp. – , , and by
Y.J. N & H. V D , Limit to space-time measurement, Modern Physics Letters A 9,
pp. – , . Many other authors have explored the topic. Cited on page .
1102 A pedagogical summary is C. S
, http://www.motionmountain.net/C -QMGR.
pdf or C. S
, Le vide di ère-t-il de la matière? in E. G
& S. D , ed-
iteurs, Le vide – Univers du tout et du rien – Des physiciens et des philosophes s’interrogent,
Les Editions de l’Université de Bruxelles, . An older English version is also available as
C. S
, Does matter di er from vacuum? http://www.arxiv.org/abs/gr-qc/
.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Dvipsbugw
Cited on pages , , , , and .
1103 Maximal curvature, as well as area and volume quantization are discussed by A.
A
, Quantum geometry and gravity: recent advances, http://www.arxiv.org/
abs/gr-qc/
and by A. A
, Quantum geometry in action: big bang and
black holes, preprint downloadable at http://www.arxiv.org/abs/math-ph/
. Cited
on page .
1104 is was kindly pointed out by Achim Kempf. e story is told in A.D. S
, Gen-
eral Relativity and Gravitation 32, p. , , a reprint of a paper of . Cited on page
.
1105 About set de nition and Hilbert’s sixth problem see C. S
, Nature at large scales
– is the universe something or nothing? http://www.motionmountain.net/C -LGSM.pdf.
Cited on pages , , , , and .
1106 Several incorrect counterclaims to the entropy limit were made in R. B
, e holo-
graphic principle, Review of Modern Physics 74, pp. – , , also available as down-
loadable preprint at http://www.arxiv.org/abs/hep-th/
. Bousso has changed pos-
ition in the meantime. Cited on pages and .
1107 G. A
–C
, Doubly-special relativity: rst results and key open problems,
International Journal of Modern Physics 11, pp. – , , electronic preprint avail-
able as http://www.arxiv.org/abs/gr-qc/
. See also J. M
& L. S
,
Lorentz invariance with an invariant energy scale, Physical Review Letters 88, p.
,
, or http://www.arxiv.org/abs/hep-th/
. Cited on page .
1108 Maximons, elementary particles of Planck mass, are discussed by A.D. S
, Va-
cuum quantum uctuations in curved space and the theory of gravitation, Soviet Physics
– Doklady, 12, pp. – , . Cited on page .
1109 Minimal entropy is discussed by L. S
, Über die Entropieverminderung in einem
thermodynamischen System bei Eingri en intelligenter Wesen, Zeitschri für Physik 53,
pp. – , . is classic paper can also be found in English translation in his collec-
ted works. Cited on page .
1110 See for example A.E. S
-M
& A.Y . T
, Generalized
uncertainty relation in thermodynamics, http://www.arxiv.org/abs/gr-qc/
, or J.
U
& J.
L-
D , ermodynamic uncertainty relations, Foundations
of Physics 29, pp. – , . Cited on page .
1111 Gamma ray bursts are discussed by G. P
, R. R
& S.-S. X , e dya-
dosphere of black holes and gamma-ray bursts, Astronomy and Astrophysics 338, pp. L –
L , , and C.L. B
, R. R
& S.-S. X , e elementary spike produced
by a pure e+e− pair-electromagnetic pulse from a black hole: the PEM pulse, Astronomy
and Astrophysics 368, pp. – , . Cited on page .
1112 See for example the review in C.W.J. B
& al., Quantum transport in semicon-
ductor nanostructures, pp. – , in H. E
& D. T
, eds., Solid State
Physics, 44, Academic Press, Cambridge, . e prediction of a future Nobel Prize was
made to the authors by C. Schiller in . Cited on page .
1113 B
G
, e Elegant Universe - Superstrings, Hidden Dimensions, and the Quest
for the Ultimate eory, Vintage, , Cited on page .
1114 A discussion of a di erent electric uncertainty relation, namely between current and
charge, can be found in Y-Q. L & B. C , Quantum theory for mesoscopic electronic
circuits and its applications, http://www.arxiv.org/abs/cond-mat/
. Cited on page
.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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1115 W Press,
R
, Relativity – Special, General and Cosmological, Oxford University
, p. . Cited on page .
1116 H C. O
&R R
. Cited on page .
, Gravitation and Spacetime, W.W. Norton & Co.,
1117 A maximal acceleration for microscopic systems was stated by E.R. C
, Lettere
al Nuovo Cimento 41, p. , . It is discussed by G. P
, Shadows of a maximal
acceleration, http://www.arxiv.org/abs/gr-qc/
. Cited on page .
1118 e entropy limit for black holes is discussed by J.D. B
, Entropy bounds
and black hole remnants, Physical Review D 49, pp. – , . See also J.D. B -
, Universal upper bound on the entropy-to-energy ration for bounded systems,
Physical Review D 23, pp. – , . Cited on page .
1119 Private communications with Jos U nk and Bernard Lavenda. Cited on page .
1120 P. K
, D.T. S & A.O. S
, A viscosity bound conjecture, preprint
found at http://www.arxiv.org/abs/hep-th/
. Cited on page .
1121 H. W
& P.S. A
, (eds.) Die Hilbertschen Probleme, Akademische Ver-
lagsgesellscha Geest & Portig, , or B H. Y
, e Honours Class: Hilbert’s
Problems and their Solvers, A.K. Peters, . Cited on page .
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
–
.
–
Nil tam di cile est, quin quaerendo investigari possiet.
“ Terence* ” e expressions for the Compton wavelength λ = h mc and for the Schwarzschild ra-
dius rs = Gm c imply a number of arguments which lead to the conclusion that at Planck energies, space-time points and point particles must be described, in contrast to their name, by extended entities. ese arguments point towards a connection between microscopic and macroscopic scales, con rming the present results of string theory and quantum gravity. At the same time, they provide a pedagogical summary of this aspect of present day theoretical physics.**
– It is shown that any experiment trying to measure the size or the shape of an elementary particle with high precision inevitably leads to the result that at least one dimension of the particle is of macroscopic size.
– ere is no data showing that empty space-time is continuous, but enough data showing that it is not. It is then argued that in order to build up an entity such as the vacuum, that is extended in three dimensions, one necessarily needs extended building blocks
– e existence of minimum measurable distances and time intervals is shown to imply the existence of space-time duality, i.e. a symmetry between very large and very small distances. Space-time duality in turn implies that the fundamental entities that make up vacuum and matter are extended.
– It is argued that the constituents of the universe and thus of space-time, matter and radiation cannot form a set. As a consequence any precise description of nature must use extended entities.
– e Bekenstein–Hawking expression for the entropy of black holes, in particular its surface dependence, con rms that both space-time and particles are composed of extended entities.
– Trying to extend statistical properties to Planck scales shows that both particles and space-time points behave as braids at high energies, a fact which also requires extended entities.
– e Dirac construction for spin provides a model for fermions, without contradiction with experiments, that points to extended entities.
An overview of other arguments in favour of extended entities provided by present research e orts is also given. To complete the discussion, experimental and theoretical checks for the idea of extended building blocks of nature are presented.
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* ‘Nothing is so di cult that it could not be investigated.’ Terence is Publius Terentius Afer (c. 190–159 ), important roman poet. He writes this in his play Heauton Timorumenos, verse 675. ** is section describes a research topic and as such is not a compendium of generally accepted results (yet). It was written between December 2001 and May 2002.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•
I
–
Ref. 1122, Ref. 1123
“Our greatest pretenses are built up not to hide the evil and the ugly in us, but our emptyness. e hardest thing to hide is something that is
not there.
” Eric Ho er, e Passionate State of Mind.
Only the separation of the description of nature into general relativity and quantum the-
ory allows us to use continuous space-time and point particles as basic entities. When
the two theories are united, what we use to call ‘point’ turns out to have quite counter-
intuitive properties. We explore a few of them in the following.
Above, we have given a standard argument showing that points do not exist in nature.
e Compton wavelength and the Schwarzschild radius together determine a minimal
length and a minimal time interval in nature. ey are given (within a factor of order
one, usually omitted in the following) by the Planck length and the Planck time, with the
values
lPl = ħG c tPl = ħG c
= .ë − m = . ë − s.
(796)
Page 1013
e existence of a minimal length and space interval in nature implies that points in space, time or space-time have no experimental backing and that we are forced to part from the traditional idea of continuity. Even though, properly speaking, points do not exist, and thus space points, events or point particles do not exist either, we can still ask what happens when we study these entities in detail. e results provide many fascinating surprises.
Using a simple Gedanken experiment, we have found above that particles and spacetime cannot be distinguished from each other at Planck scales. e argument was the following. e largest mass that can be put in a box of size R is a black hole with a Schwarzschild radius of the same value. at is also the largest possible mass measurement error. But any piece of vacuum also has a smallest mass measurement error.
e issue of a smallest mass measurement error is so important that it merits special attention. Mass measurement errors always prevent humans to state that a region of space has zero mass. In exactly the same way, also nature cannot ‘know’ that the mass of a region is zero, provided that this error is due to quantum indeterminacy. Otherwise nature would circumvent quantum theory itself. Energy and mass are always unsharp in quantum theory. We are not used to apply this fact to vacuum itself, but at Planck scales we have to. We remember from quantum eld theory that the vacuum, like any other system, has a mass; of course, its value is zero for long time averages. For nite measuring times, the mass value will be uncertain and thus di erent from zero. Not only limitations in time, but also limitations in space lead to mass indeterminacy for the vacuum. ese indeterminacies in turn lead to a minimum mass errors for vacuum regions of nite size. Quantum theory implies that nobody, not even nature, knows the exact mass value of a system or of a region of empty space.
A box is empty if it does not contain anything. But emptiness is not well de ned for photons with wavelength of the size R of the box or larger. us the mass measurement error for an ‘empty’ box – corresponding to what we call vacuum – is due to the indeterm-
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inacy relation and is given by that mass whose Compton wavelength matches the size of the box. As shown earlier on, the same mass value is found by every other Gedanken experiment: trying to determine the gravitational mass by weighing the ‘piece’ of vacuum or by measuring its kinetic energy gives the same result. Another, but in the end equivalent way to show that a region of vacuum has a nite mass error is to study how the vacuum energy depends on the position indeterminacy of the border of the region. Any region is de ned through its border. e position indeterminacy of the border will induce a mass error for the contents of the box, in the same way that a time limit does. Again, the resulting mass error value for a region of vacuum is the one for which the box size is the Compton wavelength.
Summarizing, for a box of size R, nature allows only mass values and mass measurement error value m between two limits:
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(full box)
cR G
m
ħ cR
(empty box) .
(797)
We see directly that for sizes R of the order of the Planck length, the two limits coincide; they both give the Planck mass
MPl
=
ħ c lPl
=
ħc G
− kg
GeV c .
(798)
In other words, for boxes of Planck size, we cannot distinguish a full box from an empty one. is means that there is no di erence between vacuum and matter at Planck scales. Of course, a similar statement holds for the distinction between vacuum and radiation. At Planck scales, vacuum and particles cannot be distinguished.
H
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e impossibility to distinguish vacuum from particles is a strong statement. A strong statement needs additional proof.
Mass can also be measured by probing its inertial aspect, i.e. by colliding the unknown
mass M with known velocity V with a known probe particle of mass m and momentum
p = mv. We then have
M=
∆p ∆V
,
(799)
where the di erences are taken between the values before and a er the collision. e error δM of such a measurement is simply given by
δM M
=
δ∆v ∆v
+
δm m
+
δ∆V ∆V
.
(800)
At Planck scales we have δ∆v ∆v , because the velocity error is always, like the velocities themselves, of the order of the speed of light. In other words, at Planck scales the mass measurement error is so large that we cannot determine whether a mass is di erent
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from zero: vacuum is indistinguishable from matter. e same conclusion arises if we take light with a wavelength λ as the probe particle.
In this case, expression ( ) leads to a mass error
δM M
=
δ∆λ ∆λ
+
δ∆V ∆V
.
(801)
Challenge 1503 ny Ref. 1124
In order that photon scattering can probe Planck dimensions, we need a wavelength of
the order of the Planck value; but in this case the rst term is approximately unity. Again
we nd that at Planck scales the energy indeterminacy is always of the same size as the
energy value to be measured. Measurements cannot distinguish between vanishing and
non-vanishing mass M at Planck scales. In fact, this result appears for all methods of mass
measurement that can be imagined. At Planck scales, matter cannot be distinguished
from vacuum.
Incidentally, the same arguments are valid if instead of the mass of matter we meas-
ure the energy of radiation. In other words, it is also impossible to distinguish radiation
from vacuum at high energies. In short, no type of particle di ers from vacuum at high
energies.
e indistinguishability of particles and vacuum, together with the existence of min-
imum space-time intervals, suggest that space, time, radiation and matter are macro-
scopic approximations of an underlying, common and discrete structure. is structure
is o en called quantum geometry. How do the common constituents of the two aspects
of nature look like? We will provide several arguments showing that these constituents
are extended and uctuating.
“Also, die Aufgabe ist nicht zu sehen, was noch nie jemand gesehen hat, sondern über dasjenige was jeder schon gesehen hat zu denken was
noch nie jemand gedacht hat.*
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” Erwin Schrödinger
Size is the length of vacuum taken by an object. is de nition comes natural in everyday life, quantum theory and relativity. However, approaching Planck energy, vacuum and matter cannot be distinguished: it is impossible to de ne the boundary between the two, and thus it is impossible to de ne the size of an object. As a consequence, every object becomes as extended as the vacuum. Care is therefore required.
What happens if Planck energy is approached, advancing step by step to higher energy? Every measurement requires comparison with a standard. A standard is made of matter and comparison is performed using radiation (see Figure ). us any measurement requires to distinguish between matter, radiation and space-time. However, the distinction between matter and radiation is possible only up to the (grand) uni cation energy, which is about an th of the Planck energy. Measurements do not allow us to prove that particles are point-like. Let us take a step back and check whether measurements allow us to say whether particles can at least be contained inside small spheres.
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* ‘Our task is not to see what nobody has ever seen, but to think what nobody has ever thought about that which everybody has seen already.’
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: object
comparison
standard F I G U R E 387 Measurement requires matter and radiation
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e rst and simplest way to determine the size of a compact particle such as a sphere or something akin to it, is to measure the size of a box it ts in. To be sure that the particle ts inside, we rst of all must be sure that the box is tight. is is done by checking whether something, such as matter or radiation, can leave the box. However, nature does not provide a way to ensure that a box has no holes. Potential hills cannot get higher than the maximum energy, namely the Planck energy. e tunnelling e ect cannot be ruled out. In short, there is no way to make fully tight boxes.
In addition, already at the uni cation energy there is no way to distinguish between the box and the object enclosed in it, as all particles can be transformed from any one type into any other.
Let us cross-check this result. In everyday life, we call particles ‘small’ because they can be enclosed. Enclosure is possible because in daily life walls are impenetrable. However, walls are impenetrable for matter particles only up to roughly MeV and for photons only up to keV. In fact, boxes do not even exist at medium energies. We thus cannot extend the idea of ‘box’ to high energies at all.
In summary, we cannot state that particles are compact or of nite size using boxes. We need to try other methods.
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G
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e Greeks deduced the existence of atoms by noting that division of matter must end. In contrast, whenever we think of space (or space-time) as made of points, we assume that it can be subdivided without end. Zeno noted this already long time ago and strongly criticized this assumption. He was right: at Planck energy, in nite subdivision is impossible. Any attempt to divide space stops at Planck dimensions at the latest. e process of cutting is the insertion of a wall. Knifes are limited in the same ways that walls are. e limits of walls imply limits to size determination.
In particular, the limits to walls and knives imply that at Planck energies, a cut does not necessarily lead to two separate parts. One cannot state that the two parts have been really separated; a thin connection between can never be excluded. In short, cutting objects at Planck scales does not prove compactness.
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-
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Ref. 1125
Particles are not point like. Particles are not compact. Are particles are at least of nite size?
To determine particle size, we can take try to determine their departure from pointlikeness. At high energy, detecting this departure requires scattering. For example, we can suspend the particle in some trap and then shoot some probe at it. What happens in a scattering experiment at high energies? e question has been studied already by Leonard Susskind and his coworkers. When shooting at the particle with a high energy probe, the scattering process is characterized by an interaction time. Extremely short interaction times imply sensitivity to the size and shape uctuations due to the quantum of action. An extremely short interaction time provides a cut-o for high energy shape and size
uctuations and thus determines the measured size. As a result, the size measured for any microscopic, but extended object increases when the probe energy is increased towards the Planck value.
In summary, even though at experimentally achievable energies the size is always smaller than measurable, when approaching the Planck energy, size increases above all bounds. As a result, at high energies we cannot give a limit to sizes of particles! In other words, since particles are not point-like at everyday energy, at Planck energies they are enormous: particles are extended.
at is quite a deduction. Right at the start of our mountain ascent, we distinguished objects from their environment. Objects are by de nition localized, bounded and compact. All objects have a boundary, i.e. a surface which itself does not have a boundary. Objects are also bounded in abstract ways; boundedness is also a property of the symmetries of any object, such as its gauge group. In contrast, the environment is not localized, but extended and unbounded. All these basic assumptions disappear at Planck scales. At Planck energy, it is impossible to determine whether something is bounded or compact. Compactness and localisation are only approximate properties; they are not correct at high energies. e idea of a point particle is a low energy, approximated concept.
Particles at Planck scales are as extended as the vacuum. Let us perform another check of this conclusion.
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Ref. 1126 Ref. 1122, Ref. 1127
“Καιρὸν γνῶθι.*
” Pittacus.
Humans or any other types of observers can only observe a part of the world with nite
resolution in time and in space. In this, humans resemble a lm camera. e highest pos-
sible resolution has (almost) been discovered in : the Planck time and the Planck
length. No human, no lm camera and no measurement apparatus can measure space or
time intervals smaller than the Planck values. But what would happen if we took photo-
graphs with shutter times approaching the Planck time?
Imagine that you have the world’s best shutter and that you are taking photographs
at increasingly shorter times. Table gives a rough overview of the possibilities. For
check the greek * ‘Recognize the right moment.’ also rendered as: ‘Recognize thine opportunity.’ Pittacus (Πιττακος) of Mitylene, (c. 650–570 BCE) was the Lesbian tyrant that was also one of the ancient seven sages.
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TA B L E 79 Effects of various camera shutter times on photographs
D
B
O
h s
ms ms
. ms
µs c. ps
fs zs
shorter times
−s
high high lower lower
lower
very low lowest
higher high
very high
highest
Ability to see faint quasars at night if motion is compensated
Everyday motion is completely blurred Interruption by eyelids; impossibility to see small changes
E ective eye/brain shutter time; impossibility to see tennis ball when hitting it
Shortest commercial photographic camera shutter time; allows to photograph fast cars
Ability to photograph ying bullets; requires strong ashlight Study of molecular processes; ability to photograph ying light pulses; requires laser light to get su cient illumination
Light photography becomes impossible due to wave e ects
X-ray photography becomes impossible; only γ-ray imaging is le over
Photographs get darker as illumination gets dimmer; gravitational e ects start playing a role imaging makes no sense
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shorter and shorter shutter times, photographs get darker and darker. Once the shutter time reaches the oscillation time of light, strange things happen: light has no chance to pass undisturbed; signal and noise become impossible to distinguish; in addition, the moving shutter will produce colour shi s. In contrast to our intuition, the picture would get blurred at extremely short shutter times. Photography is not only impossible at long but also at short shutter times.
e di culty of taking photographs is independent of the used wavelength. e limits move, but do not disappear. A short shutter time τ does not allow photons of energy lower than ħ τ to pass undisturbed. e blur is small when shutter times are those of everyday life, but increases when shutter times are shortened towards Planck times. As a result, there is no way to detect or con rm the existence of point objects by taking pictures. Points in space, as well as instants of time, are imagined concepts; they do not allow a precise description of nature.
At Planck shutter times, only signals with Planck energy can pass through the shutter. Since at these energies matter cannot be distinguished from radiation or from empty space, all objects, light and vacuum look the same. As a result, it becomes impossible to say how nature looks at shortest times.
But the situation is much worse: a Planck shutter does not exist at all, as it would need to be as small as a Planck length. A camera using it could not be built, as lenses do not work at this energy. Not even a camera obscura – without any lens – would work, as di raction e ects would make image production impossible. In other words, the idea that at short shutter times, a photograph of nature shows a frozen version of everyday life, like a stopped lm, is completely wrong! Zeno criticized this image already in ancient Greece, in his discussions about motion, though not so clearly as we can do now. Indeed,
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at a single instant of time nature is not frozen at all.* At short times, nature is blurred and fuzzy. is is also the case for point particles.
In summary, whatever the intrinsic shape of what we call a ‘point’ might be, we know that being always blurred, it is rst of all a cloud. Whatever method to photograph of a point particle is used, it always shows an extended entity. Let us study its shape in more detail.
W
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Ref. 1122
Given that particles are not point-like, they have a shape. How can we determine it? Everyday object shape determination is performed by touching the object from all sides. is works with plants, people or machines. It works with molecules, such as water molecules. We can put them (almost) at rest, e.g. in ice, and then scatter small particles o them. Scattering is just a higher energy version of touching. However, scattering cannot determine shapes smaller than the wavelength of the used probes. To determine a size as small as that of an electron, we need the highest energies available. But we already know what happens when approaching Planck scales: the shape of a particle becomes the shape of all the space surrounding it. Shape cannot be determined in this way.
Another method to determine the shape is to build a tight box lled of wax around the system under investigation. We let the wax cool and and observe the hollow part. However, near Planck energies boxes do not exist. We are unable to determine the shape in this way.
A third way to measure shapes is cutting something into pieces and then study the pieces. But cutting is just a low-energy version of a scattering process. It does not work at high energies. Since the term ‘atom’ means ‘uncuttable’ or ‘indivisible’, we have just found out that neither atoms nor indivisible particles can exist. Indeed, there is no way to prove this property. Our everyday intuition leads us completely astray at Planck energies.
A fourth way to measure shapes could appear by distinguishing transverse and longitudinal shape, with respect to the direction of motion. However, for transverse shape we get the same issues as for scattering; transverse shape diverges for high energy. To determine longitudinal shape, we need at least two in nitely high potential walls. Again, we already know that this is impossible.
A further, indirect way of measuring shapes is the measurement of the moment of inertia. A nite moment of inertia means a compact, nite shape. However, when the measurement energy is increased, rotation, linear motion and exchange become mixed up. We do not get meaningful results.
Still another way to determine shapes is to measure the entropy of a collection of particles we want to study. is allows to determine the dimensionality and the number of internal degrees of freedom. At high energies, a collection of electrons would become a black hole. We study the issue separately below, but again we nd no new information.
Are these arguments water-tight? We assumed three dimensions at all scales, and assumed that the shape of the particle itself is xed. Maybe these assumptions are not valid at Planck scales. Let us check the alternatives. We have already shown above that due
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* In fact, a shutter does not exist even at medium energy, as shutters, like walls, stop existing at around MeV.
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:
Ref. 1122
to the fundamental measurement limits, the dimensionality of space-time cannot be determined at Planck scales. Even if we could build perfect three-dimensional boxes, holes could remain in other dimensions. It does not take long to see that all the arguments against compactness work even if space-time has additional dimensions.
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Page 761 Ref. 1128
Only an object composed of localized entities, such as a house or a molecule, can have a xed shape. e smaller a system gets, the more quantum uctuations play a role. Any entity with a nite size, thus also an elementary particle, cannot have a xed shape. Every Gedanken experiment leading to nite shape also implies that the shape itself uctuates. But we can say more.
e distinction between particles and environment resides in the idea that particles have intrinsic properties. In fact, all intrinsic properties, such as spin, mass, charge and parity, are localized. But we saw that no intrinsic property is measurable or de nable at Planck scales. It is impossible to distinguish particles from the environment, as we know already. In addition, at Planck energy particles have all properties the environment also has. In particular, particles are extended.
In short, we cannot prove by experiments that at Planck energy elementary particles are nite in size in all directions. In fact, all experiments one can think of are compatible with extended particles, with ‘in nite’ size. We can also say that particles have tails. More precisely, a particle always reaches the borders of the region of space-time under exploration.
Not only are particles extended; in addition, their shape cannot be determined by the methods just explored. e only possibility le over is also suggested by quantum theory:
e shape of particles is uctuating. We note that for radiation particles we reach the same conclusions. e box argument shows that also radiation particles are extended and uctuating. In our enthusiasm we have also settled an important detail about elementary particles. We saw above that any particle which is smaller than its own Compton wavelength must be elementary. If it were composite, there would be a lighter component inside it; this lighter particle would have a larger Compton wavelength than the composite particle. is is impossible, since the size of a composite particle must be larger than the Compton wavelength of its components.* However, an elementary particle can have constituents, provided that they are not compact. e di culties of compact constituents were already described by Sakharov in the
s. But if the constituents are extended, they do not fall under the argument, as extended entities have no localized mass. As a result, a ying arrow, Zeno’s famous example, cannot be said to be at a given position at a given time, if it is made of extended entities. Shortening the observation time towards the Planck time makes an arrow disappear in the same cloud that also makes up space-time.**
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Ref. 1129
* Examples are the neutron, positronium, or the atoms. Note that the argument does not change when the elementary particle itself is unstable, such as the muon. Note also that the possibility that all components be heavier than the composite, which would avoid this argument, does not seem to lead to satisfying physical properties; e.g. it leads to intrinsically unstable composites. ** us at Planck scales there is no quantum Zeno e ect any more.
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Ref. 1122 Challenge 1504 e
In summary, only the idea of points leads to problems at Planck scales. If space-time and matter are imagined to be made, at Planck scales, of extended and uctuating entities, all problems disappear. We note directly that for extended entities the requirement of a non-local description is realized. Similarly, the entities being uctuating, the requirement of a statistical description of vacuum is realized. Finally, the argument forbidding composition of elementary particles is circumvented, as extended entities have no clearly de ned mass. us the concept of Compton wavelength cannot be de ned or applied. Elementary particles can thus have constituents if they are extended. But if the components are extended, how can compact ‘point’ particles be formed with them? A few options will be studied shortly.
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Ref. 1130 Ref. 1122
“ us, since there is an impossibility that [ nite] quantities are built from contacts and points, it is necessary that there be indivisible material elements and [ nite] quantities. ” Aristotle, Of Generation and Corruption.
We are used to think that empty space is made of spatial points. Let us check whether this is true at high energy. At Planck scales no measurement can give zero length, zero mass, zero area or zero volume. ere is no way to state that something in nature is a point without contradicting experimental results. In addition, the idea of a point is an extrapolation of what is found in small empty boxes getting smaller and smaller. However, we just saw that at high energies small boxes cannot be said to be empty. In fact, boxes do not exist at all, as they are never tight and do not have impenetrable walls at high energies.
Also the idea of a point as a continuous subdivision of empty space is untenable. At small distances, space cannot be subdivided, as division requires some sort of dividing wall, which does not exist.
Even the idea of repeatedly putting a point between two others cannot be applied. At high energy, it is impossible to say whether a point is exactly on the line connecting the outer two points; and near Planck energy, there is no way to nd a point between them at all. In fact, the term ‘in between’ makes no sense at Planck scales.
We thus nd that space points do not exist, in the same way that point particles do not exist. But there is more; space cannot be made of points for additional reasons. Common sense tells us that points need to be kept apart somehow, in order to form space. Indeed, mathematicians have a strong argument stating why physical space cannot be made of mathematical points: the properties of mathematical spaces described by the Banach– Tarski paradox are quite di erent from that of the physical vacuum. e Banach–Tarski paradox states states that a sphere made of mathematical points can be cut into pieces which can be reassembled into two spheres each of the same volume as the original sphere. Mathematically, volume makes no sense. Physically speaking, we can say that the concept of volume does not exist for continuous space; it is only de nable if an intrinsic length exists. is is the case for matter; it must also be the case for vacuum. But any concept with an intrinsic length, also the vacuum, must be described by one or several extended components.* In summary, we need extended entities to build up space-time!
Ref. 1131 * Imagining the vacuum as a collection of entities with Planck size in all directions, such as spheres, would
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Not only is it impossible to generate a volume with mathematical points; it is also impossible to generate exactly three physical dimensions with mathematical points. Mathematics shows that any compact one-dimensional set has as many points as any compact three-dimensional set. And the same is true for any other pair of dimension values. To build up the physical three-dimensional vacuum we need entities which organize their neighbourhood. is cannot be done with purely mathematical points. e fundamental entities must possess some sort of bond forming ability. Bonds are needed to construct or
ll three dimensions instead of any other number. Bonds require extended entities. But also a collection of tangled entities extending to the maximum scale of the region under consideration would work perfectly. Of course the precise shape of the fundamental entities is not known at this point in time. In any case we again nd that any constituents of physical three-dimensional space must be extended.
In summary, we need extension to de ne dimensionality and to de ne volume. We are not surprised. Above we deduced that the constituents of particles are extended. Since vacuum is not distinguishable from matter, we expect the constituents of vacuum to be extended as well. Stated simply, if elementary particles are not point-like, then space-time points cannot be either.
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To check whether space-time constituents are extended, let us perform a few additional Gedanken experiments. First, let us measure the size of a point of space-time. e clearest de nition of size is through the cross-section. How can we determine the cross-section of a point? We can determine the cross section of a piece of vacuum and determine the number of points inside it. From the two determinations we can deduce the cross-section of a single point. At Planck energies however, we get a simple result: the cross-section of a volume of empty space is depth independent. At Planck energies, vacuum has a surface, but no depth. In other words, at Planck energy we can only state that a Planck layer covers the surface of a volume. We cannot say anything about its interior. One way to picture the result is to say that space points are long tubes.
Another way to determine the size of a point is to count the points found in a given volume of space-time. One approach is to count the possible positions of a point particle in a volume. However, point particles are extended at Planck energies and indistinguishable from vacuum. At Planck energy, the number of points is given by surface area of the volume divided by the Planck area. Again, the surface dependence suggests that particles are long tubes.
Another approach to count the number of points in a volume is to ll a piece of vacuum with point particles.
W ?
e maximum mass that ts into a piece of vacuum is a black hole. But also in this case, the maximum mass depends only on the surface of the given vacuum piece. e maximum
avoid the Banach–Tarski paradox, but would not allow to deduce the numbers of dimensions of space and Challenge 1505 n time. It would also contradict all other results of this section. We therefore do not explore it further.
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Page 223 Ref. 1132
mass increases less rapidly than the volume. In other words, the number of points in a volume is only proportional to the surface area of that volume. ere is only one solution: vacuum must be made of extended entities crossing the whole volume, independently of the shape of the volume.
Two thousand years ago, the Greek argued that matter must be made of particles because salt can be dissolved in water and because sh can swim through water. Now that we know more about Planck scales, we have to reconsider the argument. Like sh through water, particles can move through vacuum; but since vacuum has no bounds and since it cannot be distinguished from matter, vacuum cannot be made of particles. However, there is another possibility that allows for motion of particles through vacuum: both vacuum and particles can be made of a web of extended entities. Let us study this option in more detail.
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,
Ref. 1135
“I could be bounded in a nutshell and count myself a king of in nite space, were it not that I have bad dreams. ” William Shakespeare, Hamlet.
If two observables cannot be distinguished, there is a symmetry transformation connecting them. For example, when switching observation frame, an electric eld may change into a magnetic one. A symmetry transformation means that we can change the viewpoint (i.e. the frame of observation) with the consequence that the same observation is described by one quantity from one viewpoint and by the other quantity from the other viewpoint.
When measuring a length at Planck scales it is impossible to say whether we are measuring the length of a piece of vacuum, the Compton wavelength of a body, or the Schwarzschild diameter of a body. For example, the maximum size for an elementary object is its Compton wavelength. e minimum size for an elementary object is its Schwarzschild radius. e actual size of an elementary object is somewhere in between. If we want to measure the size precisely, we have to go to Planck energy: but then all these quantities are the same. In other words, at Planck scales, there is a symmetry transformation between Compton wavelength and Schwarzschild radius. In short, at Planck scales there is a sym-
metry between mass and inverse mass. As a further consequence, at Planck scales there is a symmetry between size and in-
verse size. Matter–vacuum indistinguishability means that there is a symmetry between length and inverse length at Planck energies. is symmetry is called space-time duality or T-duality in the literature of superstrings.* Space-time duality is a symmetry between situations at scale n lPl and at scale f lPl n, or, in other words, between R and ( f lPl) R, where the experimental number f has a value somewhere between and .
Duality is a genuine non-perturbative e ect. It does not exist at low energy, since duality automatically also relates energy E and energy EPl E = ħc GE, i.e. it relates energies below and above Planck scale. Duality is a quantum symmetry. It does not exist in everyday life, as Planck’s constant appears in its de nition. In addition, it is a general relativistic
* ere is also an S-duality, which connects large and small coupling constants, and a U-duality, which is the combination of S- and T-duality.
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e ect, as it includes the gravitational constant and the speed of light. Let us study duality in more detail.
S
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Ref. 1133
“[Zeno of Elea maintained:] If the existing are many, it is necessary that they are at the same time small and large, so small to have no size, and so large to be without limits. Simplicius, Commentary on the Physics of ” Aristotle, , .
To explore the consequences of duality, we can compare it to the π rotation symmetry in everyday life. Every object in daily life is symmetrical under a full rotation. For the rotation of an observer, angles make sense only as long as they are smaller than π. If a rotating observer would insist on distinguishing angles of , π, π etc., he would get a new copy of the universe at each full turn.
Similarly, in nature, scales R and lPl R cannot be distinguished. Lengths make no sense when they are smaller than lPl. If however, we insist on using even smaller values and insist on distinguishing them from large ones, we get a new copy of the universe at those small scales. Such an insistence is part of the standard continuum description of motion, where it is assumed that space and time are described by the real numbers, which are de ned for arbitrary small intervals. Whenever the (approximate) continuum description with in nite extension is used, the R lPl R symmetry pops up.
Duality implies that di eomorphism invariance is only valid at medium scales, not at extremal ones. At extremal scales, quantum theory has to be taken into account in the proper manner. We do not know yet how to do this.
Space-time duality means that introducing lengths smaller than the Planck length (like when one de nes space points, which have size zero) means at the same time introducing things with very large (‘in nite’) value. Space-time duality means that for every small enough sphere the inside equals outside.
Duality means that if a system has a small dimension, it also has a large one. And vice versa. ere are thus no small objects in nature. As a result, space-time duality is consistent
with the idea that the basic entities are extended.
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So far, we have shown that at Planck energy, time and length cannot be distinguished. Duality has shown that mass and inverse mass cannot be distinguished. As a consequence, length, time and mass cannot be distinguished from each other. Since every observable is a combination of length, mass and time, space-time duality means that there is a symmetry between all observables. We call it the total symmetry.*
* A symmetry between size and Schwarzschild radius, i.e. a symmetry between length and mass, will lead to general relativity. Additionally, at Planck energy there is a symmetry between size and Compton wavelength. In other words, there is a symmetry between length and 1/mass. It means that there is a symmetry between coordinates and wave functions. Note that this is a symmetry between states and observables. It leads to quantum theory.
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Page 556 Ref. 1134 Ref. 1135
Challenge 1506 d Challenge 1507 e
Total symmetry implies that there are many types of speci c dualities, one for each pair of quantities under investigation. Indeed, in string theory, the number of duality types discovered is increasing every year. It includes, among others, the famous electric–magnetic duality we rst encountered in the chapter on electrodynamics, coupling constant duality, surface–volume duality, space-time duality and many more. All this con rms that there is an enormous symmetry at Planck scales. Similar statements are also well-known right from the beginning of string theory.
Most importantly, total symmetry implies that gravity can be seen as equivalent to all other forces. Space-time duality shows that uni cation is possible. Physicist have always dreamt about uni cation. Duality tells us that this dream can indeed be realized.
It may seem that total symmetry is in complete contrast with what was said in the previous section, where we argued that all symmetries are lost at Planck scales. Which result is correct? Obviously, both of them are.
At Planck scales, all low energy symmetries are indeed lost. In fact, all symmetries that imply a xed energy are lost. Duality and its generalizations however, combine both small and large dimensions, or large and small energies. Most symmetries of usual physics, such as gauge, permutation and space-time symmetries, are valid at each xed energy separately. But nature is not made this way. e precise description of nature requires to take into consideration large and small energies at the same time. In everyday life, we do not do that. Everyday life is a low and xed energy approximation of nature. For most of the twentieth century, physicists aimed to reach higher and higher energies. We believed that precision increases with increasing energy. But when we combine quantum theory and gravity we are forced to change this approach; to achieve high precision, we must take both high and low energy into account at the same time.*
e large di erences in phenomena at low and high energies are the main reason why uni cation is so di cult. So far, we were used to divide nature along the energy scale. We thought about high energy physics, atomic physics, chemistry, biology, etc. e di erences between these sciences is the energy of the processes involved. But now we are not allowed to think in this way any more. We have to take all energies into account at the same time. at is not easy, but we do not have to despair. Important conceptual progress has been achieved in the last decade of the twentieth century. In particular, we now know that we need only one single concept for all things which can be measured. Since there is only one concept, there are many ways to study it. We can start from any (low-energy) concept in physics and explore how it looks and behaves when we approach Planck scales. In the present section, we are looking at the concept of point. Obviously, the conclusions must be the same, independently of the concept we start with, be it electric eld, spin, or any other. Such studies thus provide a check for the results in this section.
Uni cation thus implies to think using duality and using concepts which follow from it. In particular, we need to understand what exactly happens to duality when we restrict ourselves to low energy only, as we do in everyday life. is question is le for the next section.
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* Renormalization energy does connect di erent energies, but not in the correct way; in particular, it does not include duality.
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Page 681
Pluralitas non est ponenda sine necessitate.*
“ ” William of Occam
Another argument, independent of the ones given above, underlines the correctness of a model of nature made of extended entities. Let us take a little broader view. Any concept for which we can distinguish parts is described by a set. We usually describe nature as a set of objects, positions, instants, etc. e most famous set description of nature is the oldest known, given by Democritus: ‘ e world is made of indivisible particles and void.’
is description was extremely successful in the past; there were no discrepancies with observations yet. However, a er years, the conceptual di culties of this approach are obvious.
We now know that Democritus was wrong, rst of all, because vacuum and matter cannot be distinguished at Planck scales. us the word ‘and’ in his sentence is already mistaken. Secondly, due to the existence of minimal scales, the void cannot be made of ‘points,’ as we usually assume nowadays. irdly, the description fails because particles are not compact objects. Fourth, the total symmetry implies that we cannot distinguish parts in nature; nothing can really be distinguished from anything else with complete precision, and thus the particles or points in space making up the naive model of void cannot exist.
In summary, quantum theory and general relativity together show that in nature, all di erences are only approximate. Nothing can really be distinguished from anything else with complete precision. In other words, there is no way to de ne a ‘part’ of nature, neither for matter, nor for space, nor for time, nor for radiation. Nature cannot be a set.
e conclusion that nature is not a set does not come as a surprise. We have already encountered another reason to doubt that nature is a set. Whatever de nition we use for the term ‘particle’, Democritus cannot be correct for a purely logical reason. e description he provided is not complete. Every description of nature de ning nature as a set of parts necessarily misses certain aspects. Most importantly, it misses the number of these parts. In particular, the number of particles and the number of dimensions of space-time must be speci ed if we describe nature as made from particles and vacuum. Above we saw that it is rather dangerous to make fun of the famous statement by Arthur Eddington
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Ref. 1136
I believe there are , , , , , , , , , , , , , , , , , , , , , , , , , , protons in the universe and the same number of electrons.
In fact, practically all physicists share this belief; usually they either pretend to favour some other number, or worse, they keep the number unspeci ed. We have seen during
* ‘Multitude should not be introduced without necessity.’ is famous principle is commonly called Occam’s razor. William of Ockham (b. 1285/1295 Ockham, d. 1349/50 München), or Occam in the common Latin spelling, was one of the great thinkers of his time. In his famous statement he expresses that only those concepts which are strictly necessary should be introduced to explain observations. It can be seen as the requirement to abandon beliefs when talking about nature. In addition, at this stage of our mountain ascent it has an even more direct interpretation.
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Ref. 1138
our walk that in modern physics many specialized sets are used to describe nature. We have used vector spaces, linear spaces, topological spaces and Hilbert spaces. But very consistently we refrained, like all physicists, from asking about the origin of their sizes (mathematically speaking of their dimensionality or their cardinality). In fact, it is equally unsatisfying to say that the universe contains some speci c number of atoms as it is to say that space-time is made of point-like events arranged in + dimensions. Both statements are about set sizes in the widest sense. In a complete, i.e. in a uni ed description of nature the number of smallest particles and the number of space-time points must not be added to the description, but must result from the description. Only in this case is uni cation achieved.
Requiring a complete explanation of nature leads to a simple consequence. Any part of nature is by de nition smaller than the whole of nature and di erent from other parts. As a result, any description of nature by a set cannot possibly yield the number of particles nor space-time dimensionality. As long as we insist in using space-time or Hilbert spaces for the description of nature, we cannot understand the number of dimensions or the number of particles.
Well, that is not too bad, as we know already that nature is not made of parts. We know that parts are only approximate concepts. In short, if nature were made of parts, it could not be a unity, or a ‘one.’ If however, nature is a unity, a one, it cannot have parts.* Nature cannot be separable exactly. It cannot be made of particles.
To sum up, nature cannot be a set. Sets are lists of distinguishable elements. When general relativity and quantum theory are uni ed, nature shows no elements: nature stops being a set at Planck scales. e result con rms and clari es a discussion we have started in classical physics. ere we had discovered that matter objects were de ned using space and time, and that space and time were de ned using objects. Including the results of quantum theory, this implies that in modern physics particles are de ned with the help of the vacuum and the vacuum with particles. at is not a good idea. We have just seen that since the two concepts are not distinguishable from each other, we cannot de ne them with each other. Everything is the same; in fact, there is no ‘every’ and no ‘thing.’ Since nature is not a set, the circular reasoning is dissolved.
Space-time duality also implies that space is not a set. Duality implies that events cannot be distinguished from each other. ey thus do not form elements of some space. Phil Gibbs has given the name event symmetry to this property of nature. is thoughtprovoking term, even though still containing the term ‘event’, underlines that it is impossible to use a set to describe space-time.
In summary, nature cannot be made of vacuum and particles. at is bizarre. People propagating this idea have been persecuted for years. is happened to the atomists from Democritus to Galileo. Were their battles it all in vain? Let us continue to clarify our thoughts.
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Ref. 1137
* As a curiosity, practically the same discussion can already be found, in Plato’s Parmenides, written in the fourth century . ere, Plato musically ponders di erent arguments on whether nature is or can be a unity or a multiplicity, i.e. a set. It seems that the text is based on the real visit by Parmenides and Zeno in Athens, where they had arrived from their home city Elea, which lies near Naples. Plato does not reach a conclusion. Modern physics however, does.
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Stating that the universe contains something implies that we are able to distinguish the universe from its contents. However, we now know that precise distinctions are impossible. If nature is not made of parts, it is wrong to say that the universe contains something.
Let us go further. As nothing can be distinguished, we need a description of nature which allows to state that at Planck energies nothing can be distinguished from anything else. For example, it must be impossible to distinguish particles from each other or from the vacuum. ere is only one solution: everything, or at least what we call ‘everything’ in everyday life, must be made of the same single entity. All particles are made of one same ‘piece.’ Every point in space, every event, every particle and every instant of time must be made of the same single entity.
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A theory of nothing describing everything is better than a theory of everything describing
“nothing. ” We found that parts are approximate concepts. e parts of nature are not strictly smaller
than nature itself. As a result, any ‘part’ must be extended. Let us try to extract more information about the constituents of nature.
e search for a uni ed theory is the search for a description in which all concepts appearing are only approximately parts of the whole. us we need an entity Ω, describing nature, which is not a set but which can be approximated by one. is is unusual. We all are convinced very early in our life that we are a part of nature. Our senses provide us with this information. We are not used to think otherwise. But now we have to.
Let us eliminate straight away a few options for Ω. One concept without parts is the empty set. Perhaps we need to construct a description of nature from the empty set? We could be inspired by the usual construction of the natural numbers from the empty set. However, the empty set makes only sense as the opposite of some full set. at is not the case here. e empty set is not a candidate for Ω.
Another possibility to de ne approximate parts is to construct them from multiple copies of Ω. But in this way we would introduce a new set through the back door. In addition, new concepts de ned in this way would not be approximate.
We need to be more imaginative. How can we describe a whole which has no parts, but which has parts approximately? Let us recapitulate. e world must be described by a single entity, sharing all properties of the world, but which can be approximated into a set of parts. For example, the approximation should yield a set of space points and a set of particles. We also saw that whenever we look at any ‘part’ of nature without any approximation, we should not be able to distinguish it from the whole world. In other words, composed entities are not always larger than constituents. On the other hand, composed entities must usually appear to be larger than their constituents. For example, space ‘points’ or ‘point’ particles are tiny, even though they are only approximations. Which concept without boundaries can be at their origin? Using usual concepts the world is everywhere at the same time; if nature is to be described by a single constituent, this entity must be
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extended. e entity has to be a single one, but it must seem to be multiple, i.e. it has to be multiple
approximately, as nature shows multiple aspects. e entity must be something folded. It must be possible to count the folds, but only approximately. (An analogy is the question of how many tracks are found on an LP or a CD; depending on the point of view, local or global, one gets di erent answers.) Counting folds would correspond to a length measurement.
e simplest model would be the use of a single entity which is extended, uctuating, going to in nity and allowing approximate localization, thus allowing approximate de nition of parts and points.* In more vivid imagery, nature could be described by some deformable, folded and tangled up entity: a giant, knotted amoeba. An amoeba slides between the ngers whenever one tries to grab a part of it. A perfect amoeba ows around any knife trying to cut it. e only way to hold it would be to grab it in its entirety. However, for an actor himself made of amoeba strands this is impossible. He can only grab it approximately, by catching part of it and approximately blocking it, e.g. using a small hole so that the escape takes a long time.
To sum up, nature is modelled by an entity which is a single unity (to eliminate distinguishability), extended (to eliminate localizability) and uctuating (to ensure approximate continuity). A far-reaching, uctuating fold, like an amoeba. e tangled branches of the amoeba allow a de nition of length via counting of the folds. In this way, discreteness of space, time and particles could also be realized; the quantization of space-time, matter and radiation thus follows. Any exible and deformable entity is also a perfect candidate for the realization of di eomorphism invariance, as required by general relativity.
A simple candidate for the extended fold is the image of a uctuating, exible tube. Counting tubes implies to determine distances or areas. e minimum possible count of one gives the minimum distance, and thus allows us to deduce quantum theory. In fact, we can use as model any object which has exibility and a small dimension, such as a tube, a thin sheet, a ball chain or a woven collection of rings. ese options give the di erent but probably equivalent models presently explored in simplicial quantum gravity, in Ashtekar’s variables and in superstrings.
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We are still collecting arguments to determine particle shape. For a completely di erent
way to explore the shape of particles it is useful to study situations where they appear in
large numbers. Collections of high numbers of constituents behave di erently if they are
point-like or extended. In particular, their entropy is di erent. Studying large-number en-
tropy thus allows to determine component shape. e best approach is to study situations
in which large numbers of particles are crammed in a small volume. is leads to study
the entropy of black holes. A black hole is a body whose gravity is so strong that even light
cannot escape. Black holes tell us a lot about the fundamental entities of nature. It is eas-
ily deduced from general relativity that any body whose mass m ts inside the so-called
Schwarzschild radius
rS = Gm c
(802)
Challenge 1508 ny * Is this the only method to describe nature? Is it possible to nd another description, in particular if space and time are not used as background? e answers are unclear at present.
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Ref. 1139, Ref. 1140
is a black hole. A black hole can be formed when a whole star collapses under its own weight. A black hole is thus a macroscopic body with a large number of constituents. For black holes, like for every macroscopic body, an entropy can be de ned. e entropy S of a macroscopic black hole was determined by Bekenstein and Hawking and is given by
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
S
=
k lPl
A
or
S=k
πGm ħc
(803)
Ref. 1141
Ref. 1142 Ref. 1143 Ref. 1145 Ref. 1146
where k is the Boltzmann constant and A = πrS is the surface of the black hole horizon. is important result has been derived in many di erent ways. e various derivations
also con rm that space-time and matter are equivalent, by showing that the entropy value
can be seen both as an entropy of matter and as one of space-time. In the present context,
the two main points of interest are that the entropy is nite, and that it is proportional to
the area of the black hole horizon.
In view of the existence of minimum lengths and times, the niteness of entropy is not
surprising any more. A nite black hole entropy con rms the idea that matter is made of
a nite number of discrete entities per volume. e existence of an entropy also shows
that these entities behave statistically; they uctuate. In fact, quantum gravity leads to
a nite entropy for any object, not only for black holes; Bekenstein has shown that the
entropy of any object is always smaller than the entropy of a (certain type of) black hole
of the same mass.
e entropy of a black hole is also proportional to its horizon area. Why is this the
case? is question is the topic of a stream of publications up to this day.* A simple way
to understand the entropy–surface proportionality is to look for other systems in nature
with the property that entropy is proportional to system surface instead of system volume.
In general, the entropy of any collection of one-dimensional exible objects, such as poly-
mer chains, shows this property. Indeed, the expression for the entropy of a polymer chain
made of N monomers, each of length a, whose ends are kept a distance r apart, is given
by
S(r) = k
r Na
for Na
Na r .
(804)
e formula is derived in a few lines from the properties of a random walk on a lattice, using only two assumptions: the chains are extended, and they have a characteristic internal length a given by the smallest straight segment. Expression ( ) is only valid if the polymers are e ectively in nite, i.e. if the length Na of the chain and their e ective
average size, the elongation a N , are much larger than the radius r of the region of interest; if the chain length is comparable or smaller than the region of interest, one gets the usual extensive entropy, ful lling S r . us only exible extended entities yield a S r dependence.
However, there is a di culty. From the entropy expression of a black hole we deduce
that the elongation a N is given by a N lPl; thus it is much smaller than the radius of a general, macroscopic black hole which can have diameters of several kilometres. On
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Ref. 1144 * e result can be derived from quantum statistics alone. However, this derivation does not yield the proportionality coe cient.
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Ref. 1125
the other hand, the formula for long entities is only valid when the chains are longer than the distance r between the end points.
is di culty disappears once we remember that space near a black hole is strongly curved. All lengths have to be measured in the same coordinate system. It is well known that for an outside observer, any object of nite size falling into a black hole seems to cover the complete horizon for long times (whereas it falls into the hole in its original size for an observer attached to the object). In short, an extended entity can have a proper length of Planck size but still, when seen by an outside observer, be as long as the horizon of the black hole in question. We thus nd that black holes are made of extended entities.
Another viewpoint can con rm the result. Entropy is (proportional to) the number of yes/no questions needed to know the exact state of the system. is view of black holes has been introduced by Gerard ’t Hoo . But if a system is de ned by its surface, like a black hole is, its components must be extended.
Finally, imagining black holes as made of extended entities is also consistent with the so-called no-hair theorem: black holes’ properties do not depend on what material falls into them, as all matter and radiation particles are made of the same extended components. e nal state only depends on the number of entities and on nothing else. In short, the entropy of a black hole is consistent with the idea that it is made of a big tangle of extended entities, uctuating in shape.
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We are still collecting arguments for the extension of fundamental entities in nature. Let us focus on their exchange behaviour. We saw above that points in space have to be eliminated in favour of continuous, uctuating entities common to space, time and matter. Is such a space ‘point’ or space entity a boson or a fermion? If we exchange two points of empty space in everyday life, nothing happens. Indeed, quantum eld theory is based – among others – on the relation
[x, y] = x y − yx =
(805)
between any two points with coordinates x and y, making them bosons. But at Planck
scale, due to the existence of minimal distances and areas, this relation is at least changed
to
[x, y] = lPl + ... .
(806)
Ref. 1122
Ref. 1147 Ref. 1147
is means that ‘points’ are neither bosons nor fermions.* ey have more complex exchange properties. In fact, the term on the right hand side will be energy dependent, with an e ect increasing towards Planck scales. In particular, we saw that gravity implies that double exchange does not lead back to the original situation at Planck scales. Entities following this or similar relations have been studied in mathematics for many decades: braids. In summary, at Planck scales space-time is not made of points, but of braids or some of their generalizations. us quantum theory and general relativity taken together again show that vacuum must be made of extended entities.
* e same reasoning destroys the fermionic or Grassmann coordinates used in supersymmetry.
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Particles behave in a similar way. We know that at low, everyday energies, particles of
the same type are identical. Experiments sensitive to quantum e ects show that there is
no way to distinguish them: any system of several identical particles obeys permutation
symmetry. On the other hand we know that at Planck energy all low-energy symmetries
disappear. We also know that, at Planck energy, permutation cannot be carried out, as it
implies exchanging positions of two particles. At Planck energy, nothing can be distin-
guished from vacuum; thus no two entities can be shown to have identical properties.
Indeed, no two particles can be shown to be indistinguishable, as they cannot even be
shown to be separate.
What happens when we slowly approach Planck energy? At everyday energies, per-
mutation symmetry is de ned by commutation or anticommutation relations of any two
particle creation operators
a†b† b†a† = .
(807)
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Ref. 1122 Ref. 1147
At Planck energies this cannot be correct. At those energies, quantum gravity e ects appear and modify the right hand side; they add an energy dependent term that is negligible at experimentally accessible energies, but which becomes important at Planck energy. We know from our experience with Planck scales that exchanging particles twice cannot lead back to the original situation, in contrast to everyday life. It is impossible that a double exchange at Planck energy has no e ect, because at planck energy such statements are impossible. e simplest extension of the commutation relation ( ) satisfying the requirement that the right side does not vanish is again braid symmetry. us Planck scales suggest that particles are also made of extended entities.
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In the last argument, we will show that even at everyday energy, the extension of particles makes sense. Any particle is a part of the universe. A part is something which is di erent from anything else. Being ‘di erent’ means that exchange has some e ect. Distinction means possibility of exchange. In other words, any part of the universe is described by its exchange behaviour. Everyday life tells us that exchange can be seen as composed of rotation. In short, distinguishing parts are described by their rotation behaviour. For this reason, for microscopic particles, exchange behaviour is speci ed by spin. Spin distinguishes particles from vacuum.*
We note that volume does not distinguish vacuum from particles; neither does rest mass or charge: there are particles without rest mass or without charge, such as photons.
e only candidate observables to distinguish particles from vacuum are spin and momentum. In fact, linear momentum is only a limiting case of angular momentum. We again nd that rotation behaviour is the basic aspect distinguishing particles from vacuum.
If spin is the central property distinguishing particles from vacuum, nding a model for spin is of central importance. But we do not have to search for long. An well-known model for spin / is part of physics folklore. Any belt provides an example, as we dis-
* With a at (or other) background, it is possible to de ne a local energy–momentum tensor. us particles can be de ned. Without background, this is not possible, and only global quantities can be de ned. Without background, even particles cannot be de ned! erefore, we assume that we have a slowly varying spacetime background in this section.
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particle position
flexible bands in unspecified number reaching the border of space
J=1/2
F I G U R E 388 A possible model for a spin 1/2 particle
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Page 771 Ref. 1148
cussed in detail in chapter VI on permutation symmetry. Any localized structure with any number of long tails attached to it – and reaching the border of the region of space under consideration – has the same properties as a spin / particle. It is a famous exercise to show that such a model, shown in Figure , is indeed invariant under π rotation but not under π rotations, that two such particles get entangled when exchanged, but get untangled when exchanged twice. e model has all properties of spin / particles, independently of the precise structure of the central region, which remains unknown at this point. e tail model even has the same problems with highly curved space as real spin particles have. We explore the idea in detail shortly.
e tail model thus con rms that rotation is partial exchange. More interestingly, it shows that rotation implies connection with the border of space-time. Extended particles can be rotating. Particles can have spin provided that they have tails going to the border of space-time. If the tails do not reach the border, the model does not work. Spin
thus even seems to require extension. We again reach the conclusion that extended entities are a good description for particles.
P
Ref. 1132
To understand is to perceive patterns.
“ ” Isaiah Berlin
e Greek deduced the existence of atoms because sh can swim through water. ey argued that only if water is made of atoms, can a sh nd its way through it by pushing the atoms aside. We can ask a similar question when a particle ies through vacuum: why are particles able to move through vacuum at all? Vacuum cannot be a uid or a solid of small entities, as this would not x its dimensionality. Only one possibility remains: both vacuum and particles are made of a web of extended entities.
Describing matter as composed of extended entities is an idea from the s. Describing nature as composed of ‘in nitely’ extended entities is an idea from the s. Indeed, in addition to the arguments presented so far, present research provides several other approaches that arrive at the same conclusion.
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Ref. 1149
**
Bosonization, the construction of fermions using an in nite number of bosons, is a central property of modern uni cation attempts. It also implies coupling duality, and thus the extension of fundamental constituents.
Ref. 1150
**
String theory and in particular its generalization to membranes are explicitly based on extended entities, as the name already states. e fundamental entities are indeed assumed to reach the limits of space-time.
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**
Ref. 1151 Research into quantum gravity, in particular the study of spin networks and spin foams, has shown that the vacuum must be thought as a collection of extended entities.
Ref. 1152
**
In the 1990s, Dirk Kreimer has shown that high-order QED diagrams are related to knot theory. He thus proved that extension appears through the back door even when electromagnetism is described using point particles.
Ref. 1153
**
A recent discovery in particle physics, ‘holography’, connects the surface and volume of physical systems at high energy. Even if it were not part of string theory, it would still imply extended entities.
**
Other fundamental nonlocalities in the description of nature, such as wave function Page 805 collapse, can be seen as the result of extended entities.
**
e start of the twenty- rst century has brought forwards a number of new approaches, such as string net condensation or knotted particle models. All these make use of extended entities.
C
Is nature really described by extended entities? e idea is taken for granted by all present approaches in theoretical physics. How can we be sure about this result? e arguments Challenge 1509 ny presented above provide several possible checks.
— As Ed Witten likes to say, any uni ed model of nature must be supersymmetric and dual. e idea of extended entities would be dead as soon as it is shown not to be compatible with these requirements.
— Any model of nature must be easily extendible to a model for black holes. If not, it cannot be correct.
— Showing that the results on quantum gravity, such as the results on the area and volume quantization, are in contradiction with extended entities would directly invalidate the
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model. — e same conclusion of extended entities must appear if one starts from any physical
(low-energy) concept – not only from length measurements – and continues to study how it behaves at Planck scales. If the conclusion were not reached, the idea of extension would not be consistent and thus incorrect. — Showing that any conclusion of the idea of extension is in contrast with string theory or with M-theory would lead to strong doubts. — Showing that the measurement of length cannot be related to the counting of folds would invalidate the model. — Finding a single Gedanken experiment invalidating the extended entity idea would prove it wrong.
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Physics is an experimental science. What kind of data could show that extension is incorrect?
Ref. 1154 Ref. 1155
— Observing a single particle in cosmic rays with energy above the Planck energy would bring this approach to a sudden stop. ( e present record is a million times lower.)
— Finding any property of nature not consistent with extended entities would spell the end of the idea.
— Finding an elementary particle of spin 0 would invalidate the idea. In particular, nding the Higgs boson and showing that it is elementary, i.e. that its size is smaller than its own Compton wavelength, would invalidate the model.
— Most proposed experimental checks of string theory can also yield information on the ideas presented. For example, Paul Mende has proposed a number of checks on the motion of extended objects in space-time. He argues that an extended object moves di erently from a mass point; the di erences could be noticeable in scattering or dispersion of light near masses.
— In July 2002, the Italian physicist Andrea Gregori has made a surprising prediction valid for any model using extended entities that reach the border of the universe: if particles are extended in this way, their mass should depend on the size of the universe. Particle masses should thus change with time, especially around the big bang. is completely new point is still a topic of research.
Incidentally, most of these scenarios would spell the death penalty for almost all present uni cation attempts.
P
— e best con rmation would be to nd a concrete model for the electron, muon, tau and for their neutrinos. In addition, a concrete model for photons and gravitons is needed. With these models, nding a knot-based de nition for the electrical charge and the lepton number would be a big step ahead. All quantum numbers should be topological quantities deduced from these models and should behave as observed.
— Estimating the coupling constants and comparing them with the experimental values is of course the main dream of modern theoretical physics.
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Nature's energy scale
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Eeveryday
EPl
F I G U R E 389 Planck effects make the energy axis an approximation
— Proving in full detail that extended entities imply exactly three plus one space-time dimensions is still necessary.
— Estimating the total number of particles in the visible universe would provide the nal check of any extended entity model.
Generally speaking, the only possible con rmations are those from the one-page table of Page 959 unexplained properties of nature given in Chapter X. No other con rmations are possible.
e ones mentioned here are the main ones.
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C
No problem is so formidable that you can’t walk away from it.
“ Charles Schultz ” Even though this section already provided su cient food for thought, here is some more.
** If measurements become impossible near Planck energy, we cannot even draw a diagram Challenge 1510 ny with an energy axis reaching that value. (See Figure 389) Is this conclusion valid in all cases?
** Quantum theory implies that even if tight walls would exist, the lid of such a box can Challenge 1511 n never be tightly shut. Can you provide the argument?
** Challenge 1512 n Is it correct that a detector able to detect Planck mass particles would be of in nite size?
What about a detector to detect a particle moving with Planck energy?
** Challenge 1513 e Can you provide an argument against the idea of extended entities?*
** Does duality imply that the cosmic background uctuations (at the origin of galaxies and Challenge 1514 ny clusters) are the same as vacuum uctuations?
* If so, please email it to the author
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
•
**
Does duality imply that a system with two small masses colliding is the same as one with Challenge 1515 ny two large masses gravitating?
**
It seems that in all arguments so far we assumed and used continuous time, even though Challenge 1516 d we know it is not. Does this change the conclusions so far?
**
Duality also implies that large and small masses are equivalent in some sense. A mass m in a radius r is equivalent to a mass mPl m in a radius lPl r. In other words, duality transforms mass density from ρ to ρPl ρ. Vacuum and maximum density are equivalent. Vacuum is thus dual to black holes.
**
Duality implies that initial conditions for the big bang make no sense. Duality again shows the uselessness of the idea, as minimal distance did before. As duality implies a symmetry between large and small energies, the big bang itself becomes an unclearly de ned concept.
**
e total symmetry, as well as space-time duality, imply that there is a symmetry between Challenge 1517 n all values an observable can take. Do you agree?
**
Can supersymmetry be an aspect or special case of total symmetry or is it something Challenge 1518 n else?
Challenge 1519 n
**
Any description is a mapping from nature to mathematics, i.e. from observed di erences (and relations) to thought di erences (and relations). How can we do this accurately, if di erences are only approximate? Is this the end of physics?
** Challenge 1520 d What do extended entities imply for the big-bang?
**
Can you show that going to high energies or selecting a Planck size region of space-time Challenge 1521 d is equivalent to visiting the big-bang?
Ref. 1155, Ref. 1156 Challenge 1522 ny
**
Additionally, one needs a description for the expansion of the universe in terms of extended entities. First approaches are being explored; no nal conclusions can be drawn yet. Can you speculate about the solution?
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A
Ref. 1157
“Wir müssen wissen, wir werden wissen.* David Hilbert,
”.
Many e orts for uni cation advance by digging deeper and deeper into details of
quantum eld theory and general relativity. Here we took the opposite approach: we took
a step back and looked at the general picture. Guided by this idea we found several argu-
ments, all leading to the same conclusion: space-time points and point particles are made
of extended entities.
Somehow it seems that the universe is best described by a uctuating, multi-branched
entity, a crossing between a giant amoeba and a heap of worms. Another analogy is a big
pot of boiling and branched spaghetti. Such an extended model of quantum geometry
is beautiful and simple, and these two criteria are o en taken as indication, before any
experimental tests, of the correctness of a description. We toured topics such as the exist-
ence of Planck limits, -dimensionality, curvature, renormalization, spin, bosonization,
the cosmological constant problem, as well as the search for a background free descrip-
tion of nature. We will study and test speci c details of the model in the next section. All
these tests concern one of only three possible topics: the construction of space-time, the
construction of particles and the construction of the universe. ese are the only issues
remaining on our mountain ascent of Motion Mountain. We will discuss them in the next
section. Before we do so, we enjoy two small thoughts.
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S
Page 223 Ref. 1158
Fluctuating entities can be seen to answer an old and not so serious question. When nature was de ned as made of tiny balls moving in vacuum, we described this as a typically male idea. Suggesting that it is male implies that the female part is missing. Which part would that be?
From the present point of view, the female part of physics might be the quantum description of the vacuum. e unravelling of the structure of the vacuum, as extended container of localized balls, could be seen as the female half of physics. If women had developed physics, the order of the discoveries would surely have been di erent. Instead of studying matter, as men did, women might have studied the vacuum rst.
In any case, the female and the male approaches, taken together, lead us to the description of nature by extended entities. Extended entities, which show that particles are not balls and that the vacuum is not a container, transcend the sexist approaches and lead to the uni ed description. In a sense, extended entities are thus the politically correct approach to nature.
A
Ref. 1159, Ref. 1160
To ‘show’ that we are not far from the top of Motion Mountain, we give a less serious argument as nal curiosity. Salecker and Wigner and then Zimmerman formulated the fundamental limit for the measurement precision τ attainable by a clock of mass M. It is
given by τ = ħT Mc , where T is the time to be measured. We can then ask what time
* ‘We must know, we will know.’ is was Hilbert’s famous personal credo.
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•
T can be measured with a precision of a Planck time tPl, given a clock of the mass of the whole universe. We get a maximum time of
T=
tPlc ħ
M.
(808)
Challenge 1523 e
Inserting numbers, we nd rather precisely that the time T is the present age of the universe. With the right dose of humour we can see the result as an omen for the belief that time is now ripe, a er so much waiting, to understand the universe down to the Planck scale. We are getting nearer to the top of Motion Mountain. Be prepared for even more fun.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
.
–
String theory, superstring theory or membrane theory – all are names for related approaches – can be characterized as the most investigated theory ever. String theory explores the idea of particles as extended entities. String theory contains a maximum speed, a minimum action and a maximum force. It thus incorporates special relativity, quantum theory and general relativity.
In its attempt to achieve a uni ed description, string theory uses four ideas that go beyond usual general relativity and quantum theory: extension, supersymmetry, higher dimensions and dualities.
Ref. 1161
— String theory describes particles as extended entities. In a further generalization, Mtheory describes particles as higher-dimensional membranes.
— String theory uses supersymmetry. Supersymmetry is the symmetry that relates matter to radiation, or equivalently, fermions to bosons. Supersymmetry is the most general local interaction symmetry that is known.
— String theory uses higher dimensions to introduce extended entities and to unify interactions. A number of dimensions higher than + seems to be necessary for a consistent quantum eld theory. At present however, the topology and size of the dimensions above 4 is still unclear. (In fact, to be honest, the topology of the rst 4 dimensions is also unknown.)
— String theory makes heavy use of dualities. Dualities, in the context of high energy physics, are symmetries that map large to small values of physical observables. Important examples are space-time duality and coupling constant duality. Dualities are global interaction and space-time symmetries. ey are essential to include gravitation in the description of nature. With dualities, string theory integrates the equivalence between space-time and matter-radiation.
By incorporating these four ideas, string theory acquires a number of appealing properties. First of all, the theory is unique. It has no adjustable parameters. Furthermore, as expected from any theory with extended entities,string theory contains gravity. In addition, string theory describes interactions. It does describe gauge elds such as the electromagnetic eld. e theory is thus an extension of quantum eld theory. All essential points of quantum eld theory are retained.
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–
String theory has many large symmetries, a consequence of its many dualities. ese symmetries connect many wildly di erent situations; that makes the theory fascinating but also di cult to picture. Finally, string theory shows special cancellations of anomalies and inconsistencies. Historically, the rst example was the Green-Schwarz anomaly cancellation; however, many other mathematical problems of quantum eld theory are solved.
In short, string theory research has shown that general relativity and quantum eld theory can be uni ed using extended entities.
S
–
What is string theory? ere are two answers. e rst answer is to follow its historical development. Unfortunately, that is almost the hardest of all possible ways to learn and understand string theory.* us we only give a short overview of this approach here.
e full description of string theory along historical lines starts from classical strings, then proceeds to string eld theory, incorporating dualities, higher dimensions and supersymmetry on the way. Each step can be described by a Lagrangian. Let us have a look at them.
e rst idea of string theory, as the name says, is the use of strings to describe particles. Strings replace the idea of point particles. e simplest form of string theory can be given if the existence of a background space-time is already taken for granted.
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– CS – more to be added in the future – CS –
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
M
M-
One of the most beautiful results of string theory research is the fact that string theory can be deduced from an extremely small number of fundamental ideas. e approach is to take a few basic ideas and to deduce M-theory form them. M-theory is the most recent formulation of string theory.
M-theory assumes that the basic entities of nature are membranes of di erent dimensions.M-theory incorporates dualities and holography. e existence of interactions can been deduced. Supersymmetry and, in particular, super Yang–Mills theory can be deduced. Also the entropy of black holes follows from M-theory. Finally, supergravity has been deduced from M-theory. is means that the existence of space-time follows from M-theory. M-theory is thus a promising candidate for a theory of motion.
– CS – the rest of the chapter will be made available in the future – CS –
In other words, M-theory complies with all conceptual requirements that a uni ed theory must ful l. Let us now turn to experiment.
* e history of string theory was characterized by short periods of excitement followed by long periods of disappointment. e main reason for this ine ective evolution was that most researchers only studied topics that everybody else was also studying. Due to the fear of unemployment of young researchers and out of the fear of missing out something, people were afraid to research topics that nobody else was looking at. us string theory had an extremely di cult birth.
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•
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
M
One of the main results of QCD, the theory of strong interactions, is the explanation of Ref. 1166 mass relations such as
mproton
e−k
m αunif
Pl
and
k=
π , αunif =
.
(809)
Ref. 1167
Here, the value of the coupling constant αunif is taken at the unifying energy. is energy is known to be a factor of about below the Planck energy. In other words, a general
understanding of masses of bound states of the strong interaction, such as the proton,
requires almost only a knowledge of the uni cation energy and the coupling constant at
that energy. e approximate value αunif = value, using experimental data.
is an extrapolation from the low energy
Any uni ed theory must allow one to calculate the coupling constants as function of
energy, including the value αunif at the uni cation energy itself. At present, researchers state that the main di culty of string theory and of M-theory is the search for the vacuum
state. Without the vacuum state, no calculations of the masses and coupling constants are
possible.
e vacuum state of string theory or M-theory is expected by researchers to be one of
an extremely involved set of topologically distinct manifolds. At present, is it estimated
that there are around
candidate vacuum states. One notes that the error alone is
as large as the number of Planck -volumes in the history of the universe. is poorness
of the estimate re ects the poor understanding of the theory.
On can describe the situation also in the following way. String and M-theory predict
states with Planck mass and with zero mass. e zero mass particles are then thought to
get a tiny mass (compared to the Planck mass) due to the Higgs mechanism. However,
the Higgs mechanism, with all its coupling constants, has not yet been deduced from
string theory. In other words, so far, string theory predicts no masses for elementary
particles. is disappointing situation is the reason that several scientists are dismissing
string theory altogether. Future will show.
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O
It is estimated that
man-years have been invested in the search for string theory;
compare this with about for Maxwell’s electrodynamics, for general relativity and a
similar number for the foundation of quantum theory.
Historically, the community of string theorists took over years to understand that
strings were not the basic entities of string theory. e fundamental entities are mem-
branes. en string theorists took another years and more to understand that mem-
branes are not the most practical fundamental entities for calculation. However, the
search for the most practical entities is still ongoing. It is probable that knotted mem-
branes must be taken into consideration.
Why is M-theory so hard? Doing calculations on knotted membranes is di cult. In
fact, it is extremely di cult. Mathematicians and physicists all over the world are still
struggling to nd simple ways for such calculations.
In short, a new approach for calculations with extended entities is needed. We then can
ask the following question. M-theory is based on several basic ideas: extension, higher di-
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-
Page 1101
mensions, supersymmetry and duality. Which of these basic assumptions is confusing? Dualities seem to be related to extension, for which experimental and theoretical evidence exists. However, supersymmetry and higher dimensions are completely speculative. We thus leave them aside, and continue our adventure.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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B
1122 C. S
, Does matter di er from vacuum? http://www.arxiv.org/abs/gr-qc/
.
A French language version has been published as C. S
, Le vide di ère-t-il de la
matière? in E. G
& S. D , editeurs, Le vide – Univers du tout et du rien – Des
physiciens et des philosophes s’interrogent, Les Éditions de l’Université de Bruxelles, .
Cited on pages , , , , , , , and .
1123 Most arguments for minimal temporal and spatial intervals are due to M. S
, Oper-
ative time de nition and principal uncertainty, preprint http://www.arxiv.org/abs/gr-qc/
, and to T. P
, Limitations on the operational de nition of space-
time events and quantum gravity, Classical and Quantum Gravity 4, pp. L –L , ;
see also his earlier papers referenced there. Cited on page .
1124 A similar point of view, o en called monism, was proposed by B
S
, Ethics
Demonstrated in Geometrical Order, , originally in Latin; an a ordable French edition
is B. S
, L’Ethique, Folio-Gallimard, . For a discussion of his ideas, especially
his monism, see D G
, ed., e Cambridge Companion to Spinoza, Cambridge
University Press, , or any general text on the history of philosophy. Cited on page .
1125 See L. S
& J. U
, Black holes, interactions, and strings, http://www.arxiv.
org/abs/hep-th/
, or L. S
, String theory and the principle of black hole
complementarity, Physical Review Letters 71, pp. – , , and M. K
, I.
K
& L. S
, Size and shape of strings, International Journal of Modern
Physics A 3, pp. – , , as well as L. S
, Structure of hadrons implied by
duality, Physical Review D 1, pp. – , . Cited on pages and .
1126 M. P
, Über irreversible Strahlungsvorgänge, Sitzungsberichte der Kaiserlichen
Akademie der Wissenscha en zu Berlin, pp. – , . Today it is commonplace to
use ħ = h π instead of Planck’s h, which he originally called b in his paper. Cited on page
.
1127 C.A. M , Possible connection between gravitation and fundamental length, Physical Review B 135, pp. – , . Cited on page .
1128 A.D. S
, Vacuum quantum uctuations in curved space and the theory of grav-
itation, Soviet Physics - Doklady 12, pp. – , . Cited on page .
1129 P. F
& S. P
, Quantum Zeno and inverse quantum Zeno e ects, pp. –
, in E. W , editor, Progress in Optics, 42, . Cited on page .
1130 A
, Of Generation and Corruption, I, II, a . See J -P D
, Les
écoles présocratiques, Folio Essais, Gallimard, p. , . Cited on page .
1131 See for example the speculative model of vacuum as composed of Planck-size spheres pro-
posed by F. W
, Zeitschri für Naturforschung 52a, p. , . Cited on page
.
1132 e Greek salt and sh arguments are given by Lucrece, or Titus Lucretius Carus, De natura
rerum, around
. Cited on pages and .
1133 Cited in J -P D . Cited on page .
, Les écoles présocratiques, Folio Essais, Gallimard, p. ,
1134 D. O & C. M
, Magnetic monopoles as gauge particles, Physics Letters 72B,
pp. – , . Cited on page .
1135 J.H. S
, e second superstring revolution, http://www.arxiv.org/abs/hep-th/
. More on dualities in ... Cited on pages and .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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1136
e famous quote is found at the beginning of chapter XI, ‘ e Physical Universe’, in A -
E
, e Philosophy of Physical Science, Cambridge, . Cited on page
.
1137 P
, Parmenides, c.
. It has been translated into most languages. Reading it
aloud, like a song, is a beautiful experience. A pale re ection of these ideas is Bohm’s
concept of ‘unbroken wholeness’. Cited on page .
1138 P. G , Event-symmetric physics, http://www.arxiv.org/abs/hep-th/
; see also
his http://www.weburbia.com/pg/contents.htm website. Cited on page .
1139 J.D. B
, Black holes and entropy, Physical Review D 7, pp. – , .
Cited on page .
1140 S.W. H
, Particle creation by black holes, Communications in Mathematical Phys-
ics 43, pp. – , ; see also S.W. H
, Black hole thermodynamics, Physical
Review D13, pp. – , . Cited on page .
1141 J.B. H
& S.W. H
, Path integral derivation of black hole radiance, Physical
Review D 13, pp. – , . See also A. S
& C. V , Microscopic ori-
gin of Bekenstein–Hawking entropy, Physics Letters B379, pp. – , , or the down-
loadable version found at http://www.arxiv.org/abs/hep-th/
. For another deriva-
tion of black hole entropy, see G T. H
& J. P
, A correspond-
ence principle for black holes and strings, Physical Review D 55, pp. – , , or
http://www.arxiv.org/abs/hep-th/
. Cited on page .
1142 J.D. B pp. – ,
, Entropy bounds and black hole remnants, Physical Review D 49, . Cited on page .
1143 J. M
, When entropy does not seem extensive, Nature 365, p. , . Other ref-
erences on the proportionality of entropy and surface are ... Cited on page .
1144 L. B
, R.K. K , J. L & R.D. S
, Quantum source of entropy of black
holes, Physical Review D 34, pp. – , . Cited on page .
1145 e analogy between polymers and black holes is by G. W boundaries, Nature 365, p. , . Cited on page .
, ermodynamics at
1146 P.-G. G
, Scaling Concepts in Polymer Physics, Cornell University Press, .
Cited on page .
1147 See for example S. M , Introduction to braided geometry and q-Minkowski space, pre-
print found at http://www.arxiv.org/abs/hep-th/
or S. M , Duality principle
and braided geometry, preprint http://www.arxiv.org/abs/gr-qc/
. Cited on pages
and .
1148 e relation between spin and statistics has been studied recently by M.V. B
& J.M.
R
, Quantum indistinguishability: spin–statistics without relativity or eld theory?,
in R.C. H
& G.M. T , editors, Spin–Statistics Connection and Commutation
Relations, American Institute of Physics CP
. Cited on page .
1149 An example is given by A.A. S
, Fermi–Bose duality via extra dimension, pre-
print http://www.arxiv.org/abs/hep-th/
. See also the standard work by M
S
, editor, Bosonization, World Scienti c, . Cited on page .
1150 A popular account making the point is the bestseller by the US-american string theorist
B
G
, e Elegant Universe – Superstrings, Hidden Dimensions, and the Quest
for the Ultimate eory, Jonathan Cape . Cited on page .
1151 e weave model of space-time appears in certain approaches of quantum gravity, as told
by A. A
, Quantum geometry and gravity: recent advances, December ,
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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preprint found at http://www.arxiv.org/abs/gr-qc/
. See also A. A
,
Quantum geometry in action: big bang and black holes, http://www.arxiv.org/abs/
math-ph/
. On a slightly di erent topic, see also S.A. M , A spin network
primer, http://www.arxiv.org/abs/gr-qc/
. Cited on page .
1152 A good introduction into his work is the paper D. K
, New mathematical struc-
tures in renormalisable quantum eld theories, http://www.arxiv.org/abs/hep-th/
to appear in Annals of Physics Cited on page .
1153 Introductions to holography are E. A
, J. C
& L. H
, Rudi-
ments of Holography, http://www.arxiv.org/abs/hep-th/
or R. B
, e holo-
graphic principle, Review of Modern Physics 74, pp. – , , also at http://www.arxiv.
org/abs/hep-th/
. (However, this otherwise good review has some argumentation
errors, as explained on page .) e importance of holography in theoretical high en-
ergy physics was underlined by the discovery of J. M
, e large N limit of
superconformal eld theories and supergravity, http://www.arxiv.org/abs/hep-th/
.
Cited on page .
1154 P.F. M
, String theory at short distance and the principle of equivalence, preprint
available at http://www.arxiv.org/abs/hep-th/
. Cited on page .
1155 A. G
, Entropy, string theory and our world, http://www.arxiv.org/abs/hep-th/
preprint. Cited on pages and .
1156 String cosmology is a passtime for many. Examples are N.E. M
, String
cosmology, http://www.arxiv.org/abs/hep-th/
and N.G. S
, New develop-
ments in string gravity and string cosmology - a summary report, http://www.arxiv.org/
abs/hep-th/
. Cited on page .
1157 Searches for background free approaches are described by E. W .
, .... Cited on page
1158 Not even the most combative feminists have ever said how women-centered physics should
look like. is lack of alternatives pervades also the agressive text by M
W-
, Pythagoras’ Trousers – God, Physics and the Gender Wars, Fourth Estate, , and
thus makes its author look rather silly. Cited on page .
1159 H. S
& E.P. W
, Quantum limitations of the measurement of space-time
distances, Physical Review 109, pp. – , . Cited on page .
1160 E.J. Z 30, pp. – ,
, e macroscopic nature of space-time, American Journal of Physics . Cited on page .
1161 See A. S , An introduction to duality symmetries in string theory, in Les Houches Summer School: Unity from Duality: Gravity, Gauge eory and Strings, (Les Houches, France, ), Springer Verlag, 76, pp. – , . Cited on page .
1162 See for example C. R
, Notes for a brief history of quantum gravity, electronic pre-
print available at http://www.arxiv.org/abs/gr-qc/
. No citations.
1163 See the http://www.nuclecu.unam.mx/~alberto/physics/stringrev.html website for an upto date list of string theory and M theory reviews. No citations.
1164 W.G. U
, Notes on black hole evaporation, Physical Review D 14, pp. – , .
W.G. U
& R.M. W , What happens when an accelerating observer detects a
Rindler particle, Physical Review D 29, pp. – , . No citations.
1165 e most prominent proposer of the idea that particles might be knots was William om-
son. See W. T
, On vortex motion, Transactions of the Royal Society in Edinburgh
pp. – , . is paper stimulated work on knot theory, as it proposed the idea that
di erent particles might be di erently knotted vortices in the aether. No citations.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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1166 F. W
, Getting its from bits, Nature 397, pp. – , . Cited on page .
1167 See the well-known blog by P edu/~woit/blog. Cited on page
W , Not even wrong, at http://www.math.columbia. .
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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– CS – this chapter will appear in the near future – CS – Dvipsbugw
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
(NOT YET AVAI L ABLE)
UNIFICATION
XII
C
– CS – this chapter will appear in the near future – CS – Dvipsbugw
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
(NOT YET AVAI L ABLE)
THE TOP OF THE MOUNTAIN
XIII
C
Fourth Part
A
Where necessary reference information for mountain ascents is provided, preparing the reader for this and any other future adventure.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
A
A
NOTATION AND CONVENTIONS
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N introduced and de ned concepts in this text are indicated by italic typeface. ew de nitions are also referred to in the index by italic page numbers. We naturally use SI units throughout the text; these are de ned in Appendix B. Experimental results are cited with limited precision, usually only two digits, as this is almost always su cient for our purposes. High-precision reference values can be found in Appendices B and C.
In relativity we use the time convention, where the metric has the signature (+ − − − ). Ref. 1168 is is used in about % of the literature worldwide. We use indices i, j, k for -vectors
and indices a, b, c, etc. for -vectors. Other conventions speci c to general relativity are explained as they arise in the text.
TL
What is written without e ort is in general read without pleasure.
“ Samuel Johnson ” Books are collections of symbols. Writing was probably invented between and
by the Sumerians in Mesopotamia (though other possibilities are also discussed). It
then took over a thousand years before people started using symbols to represent sounds
instead of concepts: this is the way in which the rst alphabet was created. is happened
between and
(possibly in Egypt) and led to the Semitic alphabet. e use
of an alphabet had so many advantages that it was quickly adopted in all neighbouring
cultures, though in di erent forms. As a result, the Semitic alphabet is the forefather of
all alphabets used in the world.
is text is written using the Latin alphabet. At rst sight, this seems to imply that its
pronunciation cannot be explained in print, in contrast to the pronunciation of other al-
phabets or of the International Phonetic Alphabet (IPA). ( ey can be explained using the
alphabet of the main text.) However, it is in principle possible to write a text that describes
exactly how to move lips, mouth and tongue for each letter, using physical concepts where
necessary. e descriptions of pronunciations found in dictionaries make indirect use of
this method: they refer to the memory of pronounced words or sounds found in nature.
Historically, the Latin alphabet was derived from the Etruscan, which itself was a de-
rivation of the Greek alphabet. ere are two main forms.
Dvipsbugw
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
e ancient Latin alphabet, used from the sixth century
onwards:
ABCDEFZ HIKLMNOPQRSTVX
e classical Latin alphabet, used from the second century
until the eleventh century:
ABCDEFGHIKLMNOPQRSTVXYZ
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Ref. 1169
e letter G was added in the third century by the rst Roman to run a fee-paying school, Spurius Carvilius Ruga. He added a horizontal bar to the letter C and substituted the letter Z, which was not used in Latin any more, for this new letter. In the second century , a er the conquest of Greece, the Romans included the letters Y and Z from the Greek alphabet at the end of their own (therefore e ectively reintroducing the Z) in order to be able to write Greek words. is classical Latin alphabet was stable for the next thousand years.*
e classical Latin alphabet was spread around Europe, Africa and Asia by the Romans during their conquests; due to its simplicity it began to be used for writing in numerous other languages. Most modern ‘Latin’ alphabets include a few other letters. e letter W was introduced in the eleventh century in French and was then adopted in most European languages. e letters J and U were introduced in the sixteenth century in Italy, to distinguish certain sounds which had previously been represented by I and V. In other languages, these letters are used for other sounds. e contractions æ and œ date from the Middle Ages. Other Latin alphabets include more letters, such as the German sharp s, written ß, a contraction of ‘ss’ or ‘sz’, the and the Nordic letters thorn, written Þ or þ, and eth, written Ð or ð, taken from the futhorc,** and other signs.
Lower-case letters were not used in classical Latin; they date only from the Middle Ages, from the time of Charlemagne. Like most accents, such as ê, ç or ä, which were also rst used in the Middle Ages, lower-case letters were introduced to save the then expensive paper surface by shortening written words.
Outside a dog, a book is a man’s best friend. Inside a dog, it’s too dark to read.
“ ” Groucho Marx
TG
e Latin alphabet was derived from the Etruscan; the Etruscan from the Greek. e Greek alphabet was itself derived from the Phoenician or a similar northern Semitic al-
Ref. 1170
* To meet Latin speakers and writers, go to http://www.alcuinus.net/. ** e Runic script, also called Futhark or futhorc, a type of alphabet used in the Middle Ages in Germanic, Anglo–Saxon and Nordic countries, probably also derives from the Etruscan alphabet. e name derives from the rst six letters: f, u, th, a (or o), r, k (or c). e third letter is the letter thorn mentioned above; it is o en written ‘Y’ in Old English, as in ‘Ye Olde Shoppe.’ From the runic alphabet Old English also took the letter wyn to represent the ‘w’ sound, and the already mentioned eth. ( e other letters used in Old English – not from futhorc – were the yogh, an ancient variant of g, and the ligatures æ or Æ, called ash, and œ or Œ, called ethel.)
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TA B L E 80 The ancient and classical Greek alphabets, and the correspondence with Latin and Indian digits
A . C .N
C
.
Α
Α α alpha a
1
Β
Β β beta
b
2
Γ
Γ γ gamma g, n 3
∆
∆ δ delta
d
4
Ε
Ε ε epsilon e
5
F ϝ, Ϛ
digamma, w 6
stigma
Ζ
Ζ ζ zeta
z
7
Η
Η η eta
e
8
Θ
Θ θ theta th 9
Ι
Ι ι iota
i, j 10
Κ
Κ κ kappa k
20
Λ
Λ λ lambda l
30
Μ
Μ µ mu
m 40
A . C .N
C
.
v
P
Ν Ξ Ο Π
Ϟ, Ρ Σ Τ
ΛϠ
Νν Ξξ Οο Ππ
Ρρ Σ σ, ς Ττ Υυ Φφ Χχ Ψψ Ωω
nu
n
50
xi
x
60
omicron o
70
pi
p
80
qoppa q
90
rho
r, rh 100
sigma s
200
tau
t
300
upsilon y, u 400
phi
ph, f 500
chi
ch 600
psi
ps 700
omega o
800
sampi s
900
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e regional archaic letters yot, sha and san are not included in the table. e letter san was the ancestor of sampi. . Only if before velars, i.e. before kappa, gamma, xi and chi. . ‘Digamma’ is the name used for the F-shaped form. It was mainly used as a letter (but also sometimes, in its lower-case form, as a number), whereas the shape and name ‘stigma’ is used only for the number. Both names were derived from the respective shapes; in fact, the stigma is a medieval, uncial version of the digamma.
e name ‘stigma’ is derived from the fact that the letter looks like a sigma with a tau attached under it – though unfortunately not in all modern fonts. e original letter name, also giving its pronunciation, was ‘waw’. . e version of qoppa that looks like a reversed and rotated z is still in occasional use in modern Greek. Unicode calls this version ‘koppa’.
. e second variant of sigma is used only at the end of words. . Uspilon corresponds to ‘u’ only as the second letter in diphthongs. . In older times, the letter sampi was positioned between pi and qoppa.
Ref. 1171
phabet in the tenth century . e Greek alphabet, for the rst time, included letters also for vowels, which the Semitic alphabets lacked (and o en still lack). In the Phoenician alphabet and in many of its derivatives, such as the Greek alphabet, each letter has a proper name. is is in contrast to the Etruscan and Latin alphabets. e rst two Greek letter names are, of course, the origin of the term alphabet itself.
In the tenth century , the Ionian or ancient (eastern) Greek alphabet consisted of the upper-case letters only. In the sixth century several letters were dropped, while a few new ones and the lower-case versions were added, giving the classical Greek alphabet. Still later, accents, subscripts and breathings were introduced. Table also gives the values signi ed by the letters took when they were used as numbers. For this special use, the obsolete ancient letters were kept during the classical period; thus they also acquired
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lower-case forms. e Latin correspondence in the table is the standard classical one, used for writing
Greek words. e question of the correct pronunciation of Greek has been hotly debated in specialist circles; the traditional Erasmian pronunciation does not correspond either to the results of linguistic research, or to modern Greek. In classical Greek, the sound that sheep make was βη–βη. (Erasmian pronunciation wrongly insists on a narrow η; modern Greek pronunciation is di erent for β, which is now pronounced ‘v’, and for η, which is now pronounced as ‘i ’ – a long ‘i’.) Obviously, the pronunciation of Greek varied from region to region and over time. For Attic Greek, the main dialect spoken in the classical period, the question is now settled. Linguistic research has shown that chi, phi and theta were less aspirated than usually pronounced in English and sounded more like the initial sounds of ‘cat’, ‘perfect’ and ‘tin’; moreover, the zeta seems to have been pronounced more like ‘zd’ as in ‘buzzed’. As for the vowels, contrary to tradition, epsilon is closed and short whereas eta is open and long; omicron is closed and short whereas omega is wide and long, and upsilon is really a ‘u’ sound as in ‘boot’, not like a French ‘u’ or German ‘ü.’
e Greek vowels can have rough or smooth breathings, subscripts, and acute, grave, circum ex or diaeresis accents. Breathings – used also on ρ – determine whether the letter is aspirated. Accents, which were interpreted as stresses in the Erasmian pronunciation, actually represented pitches. Classical Greek could have up to three of these added signs per letter; modern Greek never has more than one.
Another descendant of the Greek alphabet* is the Cyrillic alphabet, which is used with slight variations, in many Slavic languages, such as Russian and Bulgarian. However, there is no standard transcription from Cyrillic to Latin, so that o en the same Russian name is spelled di erently in di erent countries or even in the same country on di erent occasions.
TA B L E 81 The beginning of the Hebrew abjad
L
N
C
ℵ
aleph a
1
beth
b
2
‫ג‬
gimel g
3
daleth d
4
etc.
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* e Greek alphabet is also the origin of the Gothic alphabet, which was de ned in the fourth century by Wul la for the Gothic language, using also a few signs from the Latin and futhorc scripts.
e Gothic alphabet is not to be confused with the so-called Gothic letters, a style of the Latin alphabet used all over Europe from the eleventh century onwards. In Latin countries, Gothic letters were replaced in the sixteenth century by the Antiqua, the ancestor of the type in which this text is set. In other countries, Gothic letters remained in use for much longer. ey were used in type and handwriting in Germany until 1941, when the National Socialist government suddenly abolished them, in order to comply with popular demand. ey remain in sporadic use across Europe. In many physics and mathematics books, Gothic letters are used to denote vector quantities.
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TH
Ref. 1172 Page 649
Ref. 1173
e phoenician alphabet is also the origin of the Hebrew consonant alphabet or abjad. Its rst letters are given in Table . Only the letter aleph is commonly used in mathematics, though others have been proposed.
Around one hundred writing systems are in use throughout the world. Experts classify them into ve groups. Phonemic alphabets, such as Latin or Greek, have a sign for each consonant and vowel. Abjads or consonant alphabets, such as Hebrew or Arabic, have a sign for each consonant (sometimes including some vowels, such as aleph), and do not write (most) vowels; most abjads are written from right to le . Abugidas, also called syllabic alphabets or alphasyllabaries, such as Balinese, Burmese, Devanagari, Tagalog, ai, Tibetan or Lao, write consonants and vowels; each consonant has an inherent vowel which can be changed into the others by diacritics. Syllabaries, such as Hiragana or Ethiopic, have a sign for each syllable of the language. Finally, complex scripts, such as Chinese, Mayan or the Egyptian hieroglyphs, use signs which have both sound and meaning. Writing systems can have text owing from right to le , from bottom to top, and can count book pages in the opposite sense to this book.
Even though there are about languages on Earth, there are only about one hundred writing systems used today. About y other writing systems have fallen out of use. * For physical and mathematical formulae, though, the sign system used in this text, based on Latin and Greek letters, written from le to right and from top to bottom, is a standard the world over. It is used independently of the writing system of the text containing it.
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D
Both the digits and the method used in this text to write numbers originated in India. ey were brought to the Mediterranean by Arabic mathematicians in the Middle Ages. e number system used in this text is thus much younger than the alphabet.** e In-
dian numbers were made popular in Europe by Leonardo of Pisa, called Fibonacci,*** in his book Liber Abaci or ‘Book of Calculation’, which he published in . at book revolutionized mathematics. Anybody with paper and a pen (the pencil had not yet been invented) was now able to calculate and write down numbers as large as reason allowed, or even larger, and to perform calculations with them. Fibonacci’s book started:
Novem gure indorum he sunt
. Cum his itaque novem g-
* A well-designed website on the topic is http://www.omniglot.com. e main present and past writing
systems are encoded in the Unicode standard, which at present contains 52 writing systems. See http://
www.unicode.org.
** e story of the development of the numbers is told most interestingly by G. I
, Histoire universelle
des chi res, Seghers, 1981, which has been translated into several languages. He sums up the genealogy in ten
beautiful tables, one for each digit, at the end of the book. However, the book contains many factual errors,
as explained in the http://www.ams.org/notices/200201/rev-dauben.pdf and http://www.ams.org/notices/
200202/rev-dauben.pdf review.
It is not correct to call the digits 0 to 9 Arabic. Both the actual Arabic digits and the digits used in Latin
texts such as this one derive from the Indian digits. Only the digits 0, 2, 3 and 7 resemble those used in
Arabic writing, and then only if they are turned clockwise by °.
*** Leonardo di Pisa, called Fibonacci (b. c. 1175 Pisa, d. 1250 Pisa), Italian mathematician, and the most
important mathematician of his time.
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uris, et cum hoc signo , quod arabice zephirum appellatur, scribitur quilibet numerus, ut inferius demonstratur.*
e Indian method of writing numbers consists of a large innovation, the positional sys-
tem, and a small one, the digit zero. e positional system, as described by Fibonacci, was
so much more e cient that it completely replaced the previous Roman number system,
which writes as IVMM or MCMIVC or MCMXCVI, as well as the Greek number system,
in which the Greek letters were used for numbers in the way shown in Table , thus
writing as ͵αϠϞϚʹ. Compared to these systems, the Indian numbers are a much bet-
ter technology. Indeed, the Indian system proved so practical that calculations done on
paper completely eliminated the need for the abacus, which therefore fell into disuse. e
abacus is still in use only in those countries which do not use a positional system to write
numbers. ( e Indian system also eliminated the need r systems to represent numbers
with ngers. Such systems, which could show numbers up to
and more, have le
only one trace: the term ‘digit’ itself, which derives from the Latin word for nger.) Simil-
arly, only the positional number system allows mental calculations and made – and still
makes – calculating prodigies possible.**
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T
Ref. 1172 Ref. 1174
“To avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe o en in woorke use, a paire of paralleles, or Gemowe lines of one lengthe, thus: = , bicause noe . . thynges, can be moare equalle. ” Robert Recorde***
Besides text and numbers, physics books contain other symbols. Most symbols have been developed over hundreds of years, so that only the clearest and simplest are now in use. In this mountain ascent, the symbols used as abbreviations for physical quantities are all taken from the Latin or Greek alphabets and are always de ned in the context where they are used. e symbols designating units, constants and particles are de ned in Appendices B and C. ere is an international standard for them (ISO ), but it is virtually inaccessible; the symbols used in this text are those in common use.
e mathematical symbols used in this text, in particular those for operations and relations, are given in the following list, together with their origins. e details of their history have been extensively studied in the literature.
Ref. 1174
* ‘ e nine gures of the Indians are: 9 8 7 6 5 4 3 2 1. With these nine gures, and with this sign 0 which in
Arabic is called zephirum, any number can be written, as will be demonstrated below.’
** Currently, the shortest time for nding the thirteenth (integer) root of a hundred-digit number, a result
with 8 digits, is 11.8 seconds. For more about the stories and the methods of calculating prodigies, see the
fascinating book by S
B. S
, e Great Mental Calculators – e Psychology, Methods and Lives
of the Calculating Prodigies, Columbia University Press, 1983. e book also presents the techniques that
they use, and that anybody else can use to emulate them.
*** Robert Recorde (c. 1510–1558), English mathematician and physician; he died in prison, though not for
his false pretence to be the inventor of the ‘equal’ sign, which he took over from his Italian colleagues, but
for a smaller crime, namely debt. e quotation is from his e Whetstone of Witte, 1557. An image showing
the quote in manuscript can be found at the http://members.aol.com/je 94100/witte.jpg website.
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S
M
O
+, −
plus, minus
J. Regiomontanus ; the plus sign is derived from Latin ‘et’
read as ‘square root’
used by C. Rudol in ; the sign stems from a deformation of the letter ‘r’, initial of the Latin radix
=
equal to
Italian mathematicians, early sixteenth century, then brought to England by R. Recorde
{ }, [ ], ( )
grouping symbols
use starts in the sixteenth century
,<
larger than, smaller than
T. Harriot
multiplied with, times
W. Oughtred
divided by
G. Leibniz
ë
multiplied with, times
G. Leibniz
an
power
R. Descartes
x, y, z
coordinates, unknowns
R. Descartes
ax +by + c = constants and equations for unknowns R. Descartes
d dx, d x, derivative, di erential, integral ∫ y dx
G. Leibniz
φx
function of x
J. Bernoulli
f x, f (x)
function of x
L. Euler
∆x,
di erence, sum
L. Euler
is di erent from
L. Euler eighteenth century
∂ ∂x
partial derivative, read like ‘d dx’
it was derived from a cursive form of ‘d’
or of the letter ‘dey’ of the Cyrillic alpha-
bet by A. Legendre in
∆
Laplace operator
R. Murphy
x
absolute value
K. Weierstrass
∇
read as ‘nabla’ (or ‘del’)
introduced by William Hamilton in and P.G. Tait in , named a er the shape of an old Egyptian musical instrument
[x]
the measurement unit of a quantity x twentieth century
in nity
J. Wallis
π
arctan
H. Jones
e
n= n! = limn ( + n)n
L. Euler
i
+−
L. Euler
,
set union and intersection
G. Peano
element of
G. Peano
empty set
André Weil as member of the N. Bourbaki group in the early twentieth century
ψ, ψ
bra and ket state vectors
Paul Dirac
dyadic product or tensor product or unknown outer product
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Ref. 1175 Ref. 1176 Ref. 1177 Ref. 1178
Ref. 1179
Other signs used here have more complicated origins. e & sign is a contraction of Latin et meaning ‘and’, as is o en more clearly visible in its variations, such as &, the common italic form.
Each of the punctuation signs used in sentences with modern Latin alphabets, such as , . ; : ! ? ‘ ’ » « – ( ) ... has its own history. Many are from ancient Greece, but the question mark is from the court of Charlemagne, and exclamation marks appear rst in the sixteenth century.* e @ or at-sign probably stems from a medieval abbreviation of Latin ad, meaning ‘at’, similarly to how the & sign evolved from Latin et. In recent years, the smiley :-) and its variations have become popular. e smiley is in fact a new version of the ‘point of irony’ which had been formerly proposed, without success, by A. de Brahm ( – ).
e section sign § dates from the thirteenth century in northern Italy, as was shown by the German palaeographer Paul Lehmann. It was derived from ornamental versions of the capital letter C for capitulum, i.e. ‘little head’ or ‘chapter.’ e sign appeared rst in legal texts, where it is still used today, and then spread into other domains.
e paragraph sign ¶ was derived from a simpler ancient form looking like the Greek letter Γ, a sign which was used in manuscripts from ancient Greece until well into the Middle Ages to mark the start of a new text paragraph. In the Middle Ages it took the modern form, probably because a letter c for caput was added in front of it.
One of the most important signs of all, the white space separating words, was due to Celtic and Germanic in uences when these people started using the Latin alphabet. It became commonplace between the ninth and the thirteenth century, depending on the language in question.
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C
e many ways to keep track of time di er greatly from civilization to civilization. e most common calendar, and the one used in this text, is also one of the most absurd, as it is a compromise between various political forces who tried to shape it.
In ancient times, independent localized entities, such as tribes or cities, preferred lunar calendars, because lunar timekeeping is easily organized locally. is led to the use of the month as a calendar unit. Centralized states imposed solar calendars, based on the year. Solar calendars require astronomers, and thus a central authority to nance them. For various reasons, farmers, politicians, tax collectors, astronomers, and some, but not all, religious groups wanted the calendar to follow the solar year as precisely as possible. e compromises necessary between days and years are the origin of leap days. e compromises necessary between months and year led to the varying lengths are di erent in di erent calendars. e most commonly used year–month structure was organized over
years ago by Gaius Julius Ceasar, and is thus called the Julian calendar. e system was destroyed only a few years later: August was lengthened to days when it was named a er Augustus. Originally, the month was only days long; but in order to show that Augustus was as important as Caesar, a er whom July is named, all
* On the parenthesis see the beautiful book by J. L
, But I Digress, Oxford University Press, 1991.
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Page 144
month lengths in the second half of the year were changed, and February was shortened
by an additional day.
e week is an invention of Babylonia, from where it was spread through the world
by various religious groups. ( e way astrology and astronomy cooperated to determine
the order of the weekdays is explained in the section on gravitation.) Although it is about
three thousand years old, the week was fully included into the Julian calendar only around
the year , towards the end of the Western Roman Empire. e nal change in the
Julian calendar took place between and (depending on the country), when more
precise measurements of the solar year were used to set a new method to determine leap
days, a method still in use today. Together with a reset of the date and the xation of
the week rhythm, this standard is called the Gregorian calendar or simply the modern
calendar. It is used by a majority of the world’s population.
Despite its complexity, the modern calendar does allow you to determine the day of
the week of a given date in your head. Just execute the following six steps:
. take the last two digits of the year, and divide by , discarding any fraction;
. add the last two digits of the year;
. subtract for January or February of a leap year;
. add for s or s, for s or s,
for s and s, and for s or s;
. add the day of the month;
. add the month key value, namely
for JFM AMJ JAS OND.
e remainder a er division by gives the day of the week, with the correspondence - -
- - - - meaning Sunday-Monday-Tuesday-Wednesday- ursday-Friday-Saturday.*
When to start counting the years is a matter of choice. e oldest method not attached
to political power structures was that used in ancient Greece, when years were counted
from the rst Olympic games. People used to say, for example, that they were born in the
rst year of the twenty-third Olympiad. Later, political powers always imposed the count-
ing of years from some important event onwards.** Maybe reintroducing the Olympic
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* Remembering the intermediate result for the current year can simplify things even more, especially since the dates 4.4, 6.6, 8.8, 10.10, 12.12, 9.5, 5.9, 7.11, 11.7 and the last day of February all fall on the same day of the week, namely on the year’s intermediate result plus 4. ** e present counting of years was de ned in the Middle Ages by setting the date for the foundation of Rome to the year 753 , or 753 before the Common Era, and then counting backwards, so that the years behave almost like negative numbers. However, the year 1 follows directly a er the year 1 : there was no year 0.
Some other standards set by the Roman Empire explain several abbreviations used in the text: - c. is a Latin abbreviation for circa and means ‘roughly’; - i.e. is a Latin abbreviation for id est and means ‘that is’; - e.g. is a Latin abbreviation for exempli gratia and means ‘for the sake of example’; - ibid. is a Latin abbreviation for ibidem and means ‘at that same place’; - inf. is a Latin abbreviation for infra and means ‘(see) below’; - op. cit. is a Latin abbreviation for opus citatum and means ‘the cited work’; - et al. is a Latin abbreviation for et alii and means ‘and others’. By the way, idem means ‘the same’ and passim means ‘here and there’ or ‘throughout’. Many terms used in physics, like frequency, acceleration, velocity, mass, force, momentum, inertia, gravitation and temperature, are derived from Latin. In fact, it is arguable that the language of science has been Latin for over two thousand years. In Roman times it was Latin vocabulary with Latin grammar, in modern times it switched to Latin vocabulary with French grammar, then for a short time to Latin vocabulary with German grammar, a er
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counting is worth considering?
A
?
Sentences like the following are the scourge of modern physics:
e EPR paradox in the Bohm formulation can perhaps be resolved using the GRW approach, using the WKB approximation of the Schrödinger equation.
Ref. 1181
Using such vocabulary is the best way to make language unintelligible to outsiders. First of all, it uses abbreviations, which is a shame. On top of this, the sentence uses people’s names to characterize concepts, i.e. it uses eponyms. Originally, eponyms were intended as tributes to outstanding achievements. Today, when formulating radical new laws or variables has become nearly impossible, the spread of eponyms intelligible to a steadily decreasing number of people simply re ects an increasingly ine ective drive to fame.
Eponyms are a proof of scientist’s lack of imagination. We avoid them as much as possible in our walk and give common names to mathematical equations or entities wherever possible. People’s names are then used as appositions to these names. For example, ‘Newton’s equation of motion’ is never called ‘Newton’s equation’; ‘Einstein’s eld equations’ is used instead of ‘Einstein’s equations’; and ‘Heisenberg’s equation of motion’ is used instead of ‘Heisenberg’s equation’.
However, some exceptions are inevitable: certain terms used in modern physics have no real alternatives. e Boltzmann constant, the Planck scale, the Compton wavelength, the Casimir e ect, Lie groups and the Virasoro algebra are examples. In compensation, the text makes sure that you can look up the de nitions of these concepts using the index. In addition, it tries to provide pleasurable reading.
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Ref. 1180
which it changed to Latin vocabulary with British/American grammar. Many units of measurement also date from Roman times, as explained in the next appendix. Even the
infatuation with Greek technical terms, as shown in coinages such as ‘gyroscope’, ‘entropy’ or ‘proton’, dates from Roman times.
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B
1168 For a clear overview of the various sign conventions in general relativity, see the front cover
of C
W. M
, K S. T
& J A. W
, Gravitation, Free-
man, . We use the gravitational sign conventions of H C. O
&R
R
, Gravitazione e spazio-tempo, Zanichelli, . Cited on page .
1169 For more information about the letters thorn and eth, have a look at the extensive report to be found on the website http://www.everytype.com/standards/wynnyogh/thorn.html. Cited on page .
1170 For a modern history of the English language, see D C Allen Lane, . Cited on page .
, e Stories of English,
1171 H J
, Die Schri , Berlin, , translated into English as Sign, Symbol and Script:
an Account of Man’s E orts to Write, Putnam’s Sons, . Cited on page .
1172 D
R. L , editor, CRC Handbook of Chemistry and Physics, th edition, CRC Press,
. is classic reference work appears in a new edition every year. e full Hebrew al-
phabet is given on page - . e list of abbreviations of physical quantities for use in
formulae approved by ISO, IUPAP and IUPAC can also be found there.
However, the ISO standard, which de nes these abbreviations, costs around a thou-
sand euro, is not available on the internet, and therefore can safely be ignored, like any
standard that is supposed to be used in teaching but is kept inaccessible to teachers. Cited
on pages and .
1173 See the mighty text by P
T. D
&W
B
Systems, Oxford University Press, . Cited on page .
, e World’s Writing
1174 See for example the article ‘Mathematical notation’ in the Encyclopedia of Mathematics,
volumes, Kluwer Academic Publishers, – . But rst all, have a look at the in-
formative and beautiful http://members.aol.com/je /mathsym.html website. e main
source for all these results is the classic and extensive research by F
C
,A
History of Mathematical Notations, volumes, e Open Court Publishing Co., – .
e square root sign is used in C
R
, Die Coss, Vuol us Cephaleus
Joanni Jung: Argentorati, . ( e full title was Behend vnnd Hubsch Rechnung durch
die kunstreichen regeln Algebre so gemeinlicklich die Coss genent werden. Darinnen alles so
treülich an tag gegeben, das auch allein auss vleissigem lesen on allen mündtliche vnterricht
mag begri en werden, etc.) Cited on page .
1175 J. T .
, Formenwamdlungen der et-Zeichen, Stempel AG, . Cited on page
1176 M
B. P
, Pause and E ect: An Introduction to the History of Punctuation
in the West, University of California Press, . Cited on page .
1177 is is explained by B Cited on page .
LU
, Ancient Writing and its In uence, .
1178 P L
, Erforschung des Mittelalters – Ausgewählte Abhandlungen und Aufsätze,
Anton Hiersemann, , pp. – . Cited on page .
1179 B
B
, Paläographie des römischen Altertums und des abendländischen
Mittelalters, Erich Schmidt Verlag, , pp. – . Cited on page .
1180 e connections between Greek roots and many French words – and thus many English
ones – can be used to rapidly build up a vocabulary of ancient Greek without much study, as
shown by the practical collection by J. C
, Quelques racines grecques, Wetteren
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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– De Meester, . See also D
M. A , English Words from Latin and Greek
Elements, University of Arizona Press, . Cited on page .
1181 To write well, read W , , or W S
Jahr, . Cited on page
S
& E.B. W , e Elements of Style, Macmillan,
, Deutsch für Kenner – Die neue Stilkunde, Gruner und
.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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A
B
UNITS, MEASUREMENTS AND
C O N S TA N T S
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
M are comparisons with standards. Standards are based on a unit. any di erent systems of units have been used throughout the world. ost standards confer power to the organization in charge of them. Such power
can be misused; this is the case today, for example in the computer industry, and was so
in the distant past. e solution is the same in both cases: organize an independent and
global standard. For units, this happened in the eighteenth century: to avoid misuse by
authoritarian institutions, to eliminate problems with di ering, changing and irreprodu-
cible standards, and – this is not a joke – to simplify tax collection, a group of scientists,
politicians and economists agreed on a set of units. It is called the Système Interna-
tional d’Unités, abbreviated SI, and is de ned by an international treaty, the ‘Convention
du Mètre’. e units are maintained by an international organization, the ‘Conférence
Générale des Poids et Mesures’, and its daughter organizations, the ‘Commission Interna-
tionale des Poids et Mesures’ and the ‘Bureau International des Poids et Mesures’ (BIPM),
Ref. 1182 which all originated in the times just before the French revolution.
All SI units are built from seven base units, whose o cial de nitions, translated from
French into English, are given below, together with the dates of their formulation:
‘ e second is the duration of
periods of the radiation corresponding
to the transition between the two hyper ne levels of the ground state of the caesium
atom.’ ( )*
‘ e metre is the length of the path travelled by light in vacuum during a time interval
of /
of a second.’ ( )
‘ e kilogram is the unit of mass; it is equal to the mass of the international prototype
of the kilogram.’ ( )*
‘ e ampere is that constant current which, if maintained in two straight parallel
conductors of in nite length, of negligible circular cross-section, and placed metre apart
in vacuum, would produce between these conductors a force equal to ë − newton per
metre of length.’ ( )
‘ e kelvin, unit of thermodynamic temperature, is the fraction / . of the ther-
modynamic temperature of the triple point of water.’ ( )*
‘ e mole is the amount of substance of a system which contains as many elementary
entities as there are atoms in . kilogram of carbon .’ ( )*
‘ e candela is the luminous intensity, in a given direction, of a source that emits
monochromatic radiation of frequency ë hertz and has a radiant intensity in that
direction of ( / ) watt per steradian.’ ( )*
* e respective symbols are s, m, kg, A, K, mol and cd. e international prototype of the kilogram is a
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,
Note that both time and length units are de ned as certain properties of a standard example of motion, namely light. is is an additional example making the point that the observation of motion as the fundamental type of change is a prerequisite for the de nition and construction of time and space. By the way, the use of light in the de nitions had been proposed already in by Jacques Babinet.*
From these basic units, all other units are de ned by multiplication and division. us, all SI units have the following properties:
SI units form a system with state-of-the-art precision: all units are de ned with a precision that is higher than the precision of commonly used measurements. Moreover, the precision of the de nitions is regularly being improved. e present relative uncertainty of the de nition of the second is around − , for the metre about − , for the kilogram about − , for the ampere − , for the mole less than − , for the kelvin − and for the candela − .
SI units form an absolute system: all units are de ned in such a way that they can be reproduced in every suitably equipped laboratory, independently, and with high precision. is avoids as much as possible any misuse by the standard-setting organization. ( e kilogram, still de ned with help of an artefact, is the last exception to this requirement; extensive research is under way to eliminate this artefact from the de nition – an international race that will take a few more years. ere are two approaches: counting particles, or xing ħ. e former can be achieved in crystals, the latter using any formula where ħ appears, such as the formula for the de Broglie wavelength or that of the Josephson e ect.)
SI units form a practical system: the base units are quantities of everyday magnitude. Frequently used units have standard names and abbreviations. e complete list includes the seven base units, the supplementary units, the derived units and the admitted units.
e supplementary SI units are two: the unit for (plane) angle, de ned as the ratio of arc length to radius, is the radian (rad). For solid angle, de ned as the ratio of the subtended area to the square of the radius, the unit is the steradian (sr).
e derived units with special names, in their o cial English spelling, i.e. without capital letters and accents, are:
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 1183
platinum–iridium cylinder kept at the BIPM in Sèvres, in France. For more details on the levels of the caesium atom, consult a book on atomic physics. e Celsius scale of temperature θ is de ned as: θ °C = T K− . ; note the small di erence with the number appearing in the de nition of the kelvin. SI also states: ‘When the mole is used, the elementary entities must be speci ed and may be atoms, molecules, ions, electrons, other particles, or speci ed groups of such particles.’ In the de nition of the mole, it is understood that the carbon 12 atoms are unbound, at rest and in their ground state. In the de nition of the candela, the frequency of the light corresponds to . nm, i.e. green colour, around the wavelength to which the eye is most sensitive. * Jacques Babinet (1794–1874), French physicist who published important work in optics.
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N
A
N
A
hertz pascal watt volt ohm weber henry lumen becquerel sievert
Hz = s Pa = N m = kg m s W = kg m s V = kg m As Ω = V A = kg m A s Wb = Vs = kg m As H = Vs A = kg m A s lm = cd sr Bq = s Sv = J kg = m s
newton
N = kg m s
joule
J = Nm = kg m s
coulomb
C = As
farad
F = As V = A s kg m
siemens
S= Ω
tesla
T = Wb m = kg As = kg Cs
degree Celsius °C (see de nition of kelvin)
lux
lx = lm m = cd sr m
gray
Gy = J kg = m s
katal
kat = mol s
Challenge 1524 ny
We note that in all de nitions of units, the kilogram only appears to the powers of ,
and - . e nal explanation for this fact appeared only recently. Can you try to formulate
the reason?
e admitted non-SI units are minute, hour, day (for time), degree = π rad,
minute ′ = π
rad, second ′′ = π
rad (for angles), litre and tonne. All
other units are to be avoided.
All SI units are made more practical by the introduction of standard names and abbre-
viations for the powers of ten, the so-called pre xes:*
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P
N
deca da hecto h kilo k Mega M Giga G Tera T Peta P
P
N
− deci d − centi c − milli m − micro µ − nano n − pico p − femto f
P
N
Exa
E
Zetta
Z
Yotta
Y
uno cial:
Xenta X
Wekta W
Vendekta V
Udekta U
P
N
− − −
Ref. 1184 − − − −
atto
a
zepto z
yocto y
xenno x weko w vendeko v udeko u
SI units form a complete system: they cover in a systematic way the complete set of observables of physics. Moreover, they x the units of measurement for all other sciences as well.
Challenge 1525 e
* Some of these names are invented (yocto to sound similar to Latin octo ‘eight’, zepto to sound similar to Latin septem, yotta and zetta to resemble them, exa and peta to sound like the Greek words for six and ve, the uno cial ones to sound similar to the Greek words for nine, ten, eleven and twelve); some are from Danish/Norwegian (atto from atten ‘eighteen’, femto from femten ‘ een’); some are from Latin (from mille ‘thousand’, from centum ‘hundred’, from decem ‘ten’, from nanus ‘dwarf ’); some are from Italian (from piccolo ‘small’); some are Greek (micro is from µικρός ‘small’, deca/deka from δέκα ‘ten’, hecto from ἑκατόν ‘hundred’, kilo from χίλιοι ‘thousand’, mega from µέγας ‘large’, giga from γίγας ‘giant’, tera from τέρας ‘monster’).
Translate: I was caught in such a tra c jam that I needed a microcentury for a picoparsec and that my car’s fuel consumption was two tenths of a square millimetre.
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Page 1104
SI units form a universal system: they can be used in trade, in industry, in commerce, at home, in education and in research. ey could even be used by extraterrestrial civilizations, if they existed.
SI units form a coherent system: the product or quotient of two SI units is also an SI unit. is means that in principle, the same abbreviation, e.g. ‘SI’, could be used for every unit.
e SI units are not the only possible set that could ful l all these requirements, but they are the only existing system that does so.*
Remember that since every measurement is a comparison with a standard, any measurement requires matter to realize the standard (yes, even for the speed standard), and radiation to achieve the comparison. e concept of measurement thus assumes that matter and radiation exist and can be clearly separated from each other.
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P
’
Challenge 1526 e
Since the exact form of many equations depends on the system of units used, theoretical physicists o en use unit systems optimized for producing simple equations. e chosen units and the values of the constants of nature are related. In microscopic physics, the system of Planck’s natural units is frequently used. ey are de ned by setting c = , ħ = , G = , k = , ε = π and µ = π. Planck units are thus de ned from combinations of fundamental constants; those corresponding to the fundamental SI units are given in Table .** e table is also useful for converting equations written in natural units back to SI units: just substitute every quantity X by X XPl.
TA B L E 83 Planck’s (uncorrected) natural units
N
D
V
Basic units the Planck length the Planck time the Planck mass the Planck current the Planck temperature
lPl
=
tPl =
mPl =
IPl =
TPl =
ħG c ħG c ħc G πε c G ħc Gk
= . ( )ë − m = . ( )ë − s = . ( ) µg = . ( )ë A = . ( )ë K
* Most non-SI units still in use in the world are of Roman origin. e mile comes from milia passum, which used to be one thousand (double) strides of about mm each; today a nautical mile, once de ned as minute of arc on the Earth’s surface, is exactly m). e inch comes from uncia/onzia (a twel h – now of a foot). e pound (from pondere ‘to weigh’) is used as a translation of libra – balance – which is the origin of its abbreviation lb. Even the habit of counting in dozens instead of tens is Roman in origin. ese and all other similarly funny units – like the system in which all units start with ‘f ’, and which uses furlong/fortnight as its unit of velocity – are now o cially de ned as multiples of SI units. ** e natural units xPl given here are those commonly used today, i.e. those de ned using the constant ħ, and not, as Planck originally did, by using the constant h = πħ. e electromagnetic units can also be de ned with other factors than πε in the expressions: for example, using πε α, with the ne structure constant α, gives qPl = e. For the explanation of the numbers between brackets, the standard deviations, see page 1164.
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N
D
Trivial units
the Planck velocity the Planck angular momentum the Planck action the Planck entropy
vPl = c LPl = ħ SaPl = ħ SePl = k
Composed units
the Planck mass density
ρPl = c G ħ
the Planck energy
EPl =
ħc G
the Planck momentum
pPl =
ħc G
the Planck force the Planck power
FPl = c G PPl = c G
the Planck acceleration
aPl =
c ħG
the Planck frequency
fPl =
c ħG
the Planck electric charge
qPl =
πε cħ
the Planck voltage
UPl =
c πε G
the Planck resistance
RPl =
πε c
the Planck capacitance
CPl = πε ħG c
the Planck inductance
LPl = ( πε ) ħG c
the Planck electric eld
EPl =
c πε ħG
the Planck magnetic ux density BPl =
c πε ħG
V
= . Gm s = . ë − Js = . ë − Js = . yJ K
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= . ë kg m = . GJ = . ë eV
= . Ns = .ë N = .ë W = . ë ms = . ë Hz
= . aC = . e = .ë V = .Ω = .ë − F = .ë − H = . ë Vm = .ë T
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Page 1076 Challenge 1527 n
e natural units are important for another reason: whenever a quantity is sloppily called ‘in nitely small (or large)’, the correct expression is ‘as small (or as large) as the corresponding corrected Planck unit’. As explained throughout the text, and especially in the third part, this substitution is possible because almost all Planck units provide, within a correction factor of order , the extremal value for the corresponding observable – some an upper and some a lower limit. Unfortunately, these correction factors are not yet widely known. e exact extremal value for each observable in nature is obtained when G is substituted by G, ħ by ħ , k by k and πε by πε α in all Planck quantities. ese extremal values, or corrected Planck units, are the true natural units. To exceeding the extremal values is possible only for some extensive quantities. (Can you nd out which ones?)
O
A central aim of research in high-energy physics is the calculation of the strengths of all interactions; therefore it is not practical to set the gravitational constant G to unity, as in the Planck system of units. For this reason, high-energy physicists o en only set
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
c = ħ = k = and µ = ε = π,* leaving only the gravitational constant G in the equations.
In this system, only one fundamental unit exists, but its choice is free. O en a standard length is chosen as the fundamental unit, length being the archetype of a measured quantity. e most important physical observables are then related by
[l ] = [E] = [F] = [B] = [Eelectric] , [l] = [E] = [m] = [p] = [a] = [ f ] = [I] = [U] = [T] ,
= [v] = [q] = [e] = [R] = [Saction] = [Sentropy] = ħ = c = k = [α] , [l] = [E] = [t] = [C] = [L] and
[l] = [E] = [G] = [P]
(810) where we write [x] for the unit of quantity x. Using the same unit for time, capacitance and inductance is not to everybody’s taste, however, and therefore electricians do not use this system.**
O en, in order to get an impression of the energies needed to observe an e ect under study, a standard energy is chosen as fundamental unit. In particle physics the most common energy unit is the electronvolt (eV), de ned as the kinetic energy acquired by an electron when accelerated by an electrical potential di erence of volt (‘protonvolt’ would be a better name). erefore one has eV = . ë − J, or roughly
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eV aJ
(811)
Challenge 1529 e Ref. 1187
which is easily remembered. e simpli cation c = ħ = yields G = . ë − eV− and
allows one to use the unit eV also for mass, momentum, temperature, frequency, time and
length, with the respective correspondences eV . ë − kg . ë − Ns
THz
. kK and eV− . fs . µm.
To get some feeling for the unit eV, the following relations are useful. Room temper-
ature, usually taken as °C or K, corresponds to a kinetic energy per particle of
. eV or . zJ. e highest particle energy measured so far belongs to a cosmic ray
with an energy of ë eV or J. Down here on the Earth, an accelerator able to pro-
duce an energy of about GeV or nJ for electrons and antielectrons has been built,
Ref. 1185
Ref. 1186 Challenge 1528 n
* Other de nitions for the proportionality constants in electrodynamics lead to the Gaussian unit system o en used in theoretical calculations, the Heaviside–Lorentz unit system, the electrostatic unit system, and the electromagnetic unit system, among others. ** In the list, l is length, E energy, F force, Eelectric the electric and B the magnetic eld, m mass, p momentum, a acceleration, f frequency, I electric current, U voltage, T temperature, v speed, q charge, R resistance, P power, G the gravitational constant.
e web page http://www.chemie.fu-berlin.de/chemistry/general/units_en.html provides a tool to convert various units into each other.
Researchers in general relativity o en use another system, in which the Schwarzschild radius rs = Gm c is used to measure masses, by setting c = G = . In this case, mass and length have the same dimension, and ħ has the dimension of an area.
Already in the nineteenth century, George Stoney had proposed to use as length, time and mass units
the quantities lS = Ge (c πε ) = . ë − m, tS = Ge (c πε ) = . ë − s and mS =
e (G πε ) = . µg. How are these units related to the Planck units?
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,
Ref. 1188
and one able to produce an energy of TeV or . µJ for protons will be nished soon. Both are owned by CERN in Geneva and have a circumference of km.
e lowest temperature measured up to now is pK, in a system of rhodium nuclei held inside a special cooling system. e interior of that cryostat may even be the coolest point in the whole universe. e kinetic energy per particle corresponding to that temperature is also the smallest ever measured: it corresponds to feV or . vJ = . ë − J. For isolated particles, the record seems to be for neutrons: kinetic energies as low as
− eV have been achieved, corresponding to de Broglie wavelengths of nm.
C Here are a few facts making the concept of physical unit more vivid.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
**
Not using SI units can be expensive. In 1999, NASA lost a satellite on Mars because some so ware programmers had used imperial units instead of SI units in part of the code. As a result, the Mars Climate Orbiter crashed into the planet, instead of orbiting it; the loss was around 100 million euro.*
Ref. 1189
**
A gray is the amount of radioactivity that deposits J on kg of matter. A sievert is the unit of radioactivity adjusted to humans by weighting each type of human tissue with a factor representing the impact of radiation deposition on it. Four to ve sievert are a lethal dose to humans. In comparison, the natural radioactivity present inside human bodies leads to a dose of . mSv per year. An average X-ray image implies an irradiation of mSv; a CAT scan mSv.
Challenge 1530 e
**
Are you confused by the candela? e de nition simply says that cd = lm sr corresponds to W sr. e candela is thus a unit for light power per (solid) angle, except that it is corrected for the eye’s sensitivity: the candela measures only visible power per angle. Similarly, lm = cd sr corresponds to W. So both the lumen and the watt measure power, or energy ux, but the lumen measures only the visible part of the power.
is di erence is expressed by inserting ‘radiant’ or ‘luminous’: thus, the Watt measures radiant ux, whereas the lumen measures luminous ux.
e factor 683 is historical. An ordinary candle emits a luminous intensity of about a
candela. erefore, at night, a candle can be seen up to a distance of 10 or 20 kilometres. A W incandescent light bulb produces lm, and the brightest light emitting diodes
about lm. Cinema projectors produce around Mlm, and the brightest ashes, like lightning, Mlm.
e irradiance of sunlight is about W m on a sunny day; on the other hand, the illuminance is only klm m = klux or W m . (A cloud-covered summer day or a clear winter day produces about klux.) ese numbers show that most of the
energy from the Sun that reaches the Earth is outside the visible spectrum.
* is story revived an old (and false) urban legend that states that only three countries in the world do not use SI units: Liberia, the USA and Myanmar.
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,
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ref. 1190
On a glacier, near the sea shore, on the top of a mountain, or in particular weather condition the brightness can reach klux. e brightest lamps, those used during surgical operations, produce klux. Humans need about lux for comfortable reading.
Museums are o en kept dark because water-based paintings are degraded by light above lux, and oil paintings by light above lux. e full moon produces . lux, and
the sky on a dark moonless night about mlux. e eyes lose their ability to distinguish colours somewhere between . lux and . lux; the eye stops to work below nlux. Technical devices to produce images in the dark, such as night goggles, start to work at µlux. By the way, the human body itself shines with about plux, a value too small to be detected
with the eye, but easily measured with specialized apparatus. e origin of this emission
is still a topic of research.
Ref. 1191
**
e highest achieved light intensities are in excess of W m , more than 15 orders of magnitude higher than the intensity of sunlight. ey are produced by tight focusing of pulsed lasers. e electric eld in such light pulses is of the same order as the eld inside atoms; such a beam therefore ionizes all matter it encounters.
**
e luminous density is a quantity o en used by light technicians. Its unit is cd m , uno cially called 1 Nit and abbreviated nt. Eyes see purely with rods from . µcd m to mcd m ; they see purely with cones above cd m ; they see best between 100 and
cd m ; and they get completely overloaded above Mcd m : a total range of 15 orders of magnitude.
Ref. 1192
**
e Planck length is roughly the de Broglie wavelength λB = h mv of a man walking comfortably (m = kg, v = . m s); this motion is therefore aptly called the ‘Planck stroll.’
**
e Planck mass is equal to the mass of about human embryo at about ten days of age.
protons. is is roughly the mass of a
**
e second does not correspond to 1/86 400th of the day any more, though it did in the
year 1900; the Earth now takes about
. s for a rotation, so that the International
Earth Rotation Service must regularly introduce a leap second to ensure that the Sun is at
the highest point in the sky at 12 o’clock sharp.* e time so de ned is called Universal
Time Coordinate. e speed of rotation of the Earth also changes irregularly from day to
day due to the weather; the average rotation speed even changes from winter to summer
because of the changes in the polar ice caps; and in addition that average decreases over
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* eir website at http://hpiers.obspm.fr gives more information on the details of these insertions, as does http://maia.usno.navy.mil, one of the few useful military websites. See also http://www.bipm.fr, the site of the BIPM.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
,
time, because of the friction produced by the tides. e rate of insertion of leap seconds is therefore higher than once every 500 days, and not constant in time.
**
e most precisely measured quantities in nature are the frequencies of certain millisecond pulsars,* the frequency of certain narrow atomic transitions, and the Rydberg constant of atomic hydrogen, which can all be measured as precisely as the second is de ned. e caesium transition that de nes the second has a nite linewidth that limits the achievable precision: the limit is about 14 digits.
Ref. 1193 Ref. 1194
**
e most precise clock ever built, using microwaves, had a stability of − during a running time of s. For longer time periods, the record in 1997 was about − ; but values around − seem within technological reach. e precision of clocks is limited for short measuring times by noise, and for long measuring times by dri s, i.e. by systematic e ects. e region of highest stability depends on the clock type; it usually lies between
ms for optical clocks and s for masers. Pulsars are the only type of clock for which this region is not known yet; it certainly lies at more than 20 years, the time elapsed at the time of writing since their discovery.
Ref. 1195 Challenge 1531 n
**
e shortest times measured are the lifetimes of certain ‘elementary’ particles. In particular, the lifetime of certain D mesons have been measured at less than − s. Such times are measured using a bubble chamber, where the track is photographed. Can you estimate how long the track is? ( is is a trick question – if your length cannot be observed with an optical microscope, you have made a mistake in your calculation.)
Ref. 1196
**
e longest times encountered in nature are the lifetimes of certain radioisotopes, over years, and the lower limit of certain proton decays, over years. ese times are
thus much larger than the age of the universe, estimated to be fourteen thousand million years.
Page 1167
**
e least precisely measured of the fundamental constants of physics are the gravitational constant G and the strong coupling constant αs. Even less precisely known are the age of the universe and its density (see Table 86).
Challenge 1532 n
**
e precision of mass measurements of solids is limited by such simple e ects as the adsorption of water. Can you estimate the mass of a monolayer of water – a layer with thickness of one molecule – on a metal weight of kg?
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* An overview of this fascinating work is given by J.H. T
, Pulsar timing and relativistic gravity, Philo-
sophical Transactions of the Royal Society, London A 341, pp. 117–134, 1992.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
**
Ref. 1197 Ref. 1198
Variations of quantities are o en much easier to measure than their values. For example, in gravitational wave detectors, the sensitivity achieved in 1992 was ∆l l = ë − for lengths of the order of m. In other words, for a block of about a cubic metre of metal
it is possible to measure length changes about 3000 times smaller than a proton radius.
ese set-ups are now being superseded by ring interferometers. Ring interferometers measuring frequency di erences of − have already been built; and they are still being
improved.
Ref. 1199 Challenge 1533 n
**
e Swedish astronomer Anders Celsius (1701–1744) originally set the freezing point of water at 100 degrees and the boiling point at 0 degrees. Later the scale was reversed. However, this is not the whole story. With the o cial de nition of the kelvin and the degree Celsius, at the standard pressure of . Pa, water boils at . °C. Can you explain why it is not °C any more?
**
In the previous millennium, thermal energy used to be measured using the unit calorie, written as cal. 1 cal is the energy needed to heat g of water by K. To confuse matters,
kcal was o en written Cal. (One also spoke of a large and a small calorie.) e value of kcal is . kJ.
**
SI units are adapted to humans: the values of heartbeat, human size, human weight, human temperature and human substance are no more than a couple of orders of magnitude near the unit value. SI units thus (roughly) con rm what Protagoras said 25 centuries ago: ‘Man is the measure of all things.’
Challenge 1534 n
**
e table of SI pre xes covers 72 orders of magnitude. How many additional pre xes will be needed? Even an extended list will include only a small part of the in nite range of possibilities. Will the Conférence Générale des Poids et Mesures have to go on forever, de ning an in nite number of SI pre xes? Why?
**
e French philosopher Voltaire, a er meeting Newton, publicized the now famous story that the connection between the fall of objects and the motion of the Moon was discovered by Newton when he saw an apple falling from a tree. More than a century later, just before the French Revolution, a committee of scientists decided to take as the unit of force precisely the force exerted by gravity on a standard apple, and to name it a er the English scientist. A er extensive study, it was found that the mass of the standard apple was . g; its weight was called 1 newton. Since then, visitors to the museum in Sèvres near Paris have been able to admire the standard metre, the standard kilogram and the standard apple.*
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* To be clear, this is a joke; no standard apple exists. It is not a joke however, that owners of several apple
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,
N number of measurements
standard deviation
full width at half maximum (FWHM)
limit curve for a large number of measurements
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x average value
x measured values
F I G U R E 390 A precision experiment and its measurement distribution
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
P
As explained on page , precision means how well a result is reproduced when the measurement is repeated; accuracy is the degree to which a measurement corresponds to the actual value. Lack of precision is due to accidental or random errors; they are best measured by the standard deviation, usually abbreviated σ; it is de ned through
n
σ
= n−
(xi − x¯) ,
i=
(812)
Challenge 1535 n
where x¯ is the average of the measurements xi. (Can you imagine why n − is used in the formula instead of n?)
For most experiments, the distribution of measurement values tends towards a normal
distribution, also called Gaussian distribution, whenever the number of measurements is
increased. e distribution, shown in Figure , is described by the expression
N(x)
e . − (x−x¯) σ
(813)
Challenge 1536 e Ref. 1201
e square σ of the standard deviation is also called the variance. For a Gaussian distribution of measurement values, . σ is the full width at half maximum.
Lack of accuracy is due to systematic errors; usually these can only be estimated. is estimate is o en added to the random errors to produce a total experimental error, sometimes also called total uncertainty.
e following tables give the values of the most important physical constants and particle properties in SI units and in a few other common units, as published in the stand-
trees in Britain and in the US claim descent, by rerooting, from the original tree under which Newton had Ref. 1200 his insight. DNA tests have even been performed to decide if all these derive from the same tree. e result
was, unsurprisingly, that the tree at MIT, in contrast to the British ones, is a fake.
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,
Ref. 1202
Challenge 1537 ny Page 1035
ard references. e values are the world averages of the best measurements made up to the present. As usual, experimental errors, including both random and estimated systematic errors, are expressed by giving the standard deviation in the last digits; e.g. . ( ) means – roughly speaking – . . . In fact, behind each of the numbers in the following tables there is a long story which is worth telling, but for which there is not enough room here.*
What are the limits to accuracy and precision? ere is no way, even in principle, to measure a length x to a precision higher than about digits, because ∆x x lPl dhorizon = − . (Is this valid also for force or for volume?) In the third part of our text, studies of clocks and metre bars strengthen this theoretical limit.
But it is not di cult to deduce more stringent practical limits. No imaginable machine can measure quantities with a higher precision than measuring the diameter of the Earth within the smallest length ever measured, about − m; that is about digits of precision. Using a more realistic limit of a m sized machine implies a limit of digits. If, as predicted above, time measurements really achieve digits of precision, then they are nearing the practical limit, because apart from size, there is an additional practical restriction: cost. Indeed, an additional digit in measurement precision o en means an additional digit in equipment cost.
Dvipsbugw
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B
Ref. 1202 In principle, all quantitative properties of matter can be calculated with quantum theory. For example, colour, density and elastic properties, can be predicted using the values of the following constants using the equations of the standard model of high-energy physics.
TA B L E 84 Basic physical constants
Q
S
V
SI
U
.a
number of space-time dimensions
+
b
vacuum speed of lightc
c
vacuum permeabilityc
µ
vacuum permittivityc
ε = µc
original Planck constant
h
reduced Planck constant
ħ
positron charge
e
Boltzmann constant
k
gravitational constant
G
gravitational coupling constant κ = πG c
ne structure constant,d e.m. coupling constant
α=
e πε ħc
= em(me c )
ms
πë − H m
=.
... µH m
.
... pF m
.
( ) ë − Js
.ë −
.
( ) ë − Js . ë −
.
( ) aC
.ë −
.
( )ë − J K . ë −
. ( ) ë − Nm kg
.ë −
. ( ) ë − s kg m
.ë −
.
()
.ë −
=.
()
.ë −
* Some of the stories can be found in the text by N.W. W , e Values of Precision, Princeton University
Press, 1994. e eld of high-precision measurements, from which the results on these pages stem, is a
world on its own. A beautiful introduction to it is J.D. F
, B.S. D
, C.W. E
& P.F.
M
, eds., Near Zero: Frontiers of Physics, Freeman, 1988.
Dvipsbugw
,
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Q
Fermi coupling constant,d weak coupling constant weak mixing angle weak mixing angle
strong coupling constantd
S
V
SI
GF (ħc)
.
αw(MZ) = w π
sin θW(MS)
.
sin θW (on shell) .
= − (mW mZ)
αs(MZ) = s π .
( ) ë − GeV− .( )
() ()
()
U
.a
.ë − ë− .ë − .ë −
ë−
Page 860
a. Uncertainty: standard deviation of measurement errors. b. Only down to − m and up to m.
c. De ning constant.
d. All coupling constants depend on the -momentum transfer, as explained in the section on
renormalization. Fine structure constant is the traditional name for the electromagnetic coupling con-
stant em in the case of a -momentum transfer of Q = me c , which is the smallest one possible. At higher
momentum transfers it has larger values, e.g. em(Q = MWc )
. e strong coupling constant has
higher values at lower momentum transfers; e.g., αs( GeV) = . ( ).
Why do all these constants have the values they have? e answer is di erent in each case. For any constant having a dimension, such as the quantum of action ħ, the numerical value has only historical meaning. It is . ë − Js because of the SI de nition of the joule and the second. e question why the value of a dimensional constant is not larger or smaller always requires one to understand the origin of some dimensionless number giving the ratio between the constant and the corresponding natural unit. Understanding the sizes of atoms, people, trees and stars, the duration of molecular and atomic processes, or the mass of nuclei and mountains, implies understanding the ratios between these values and the corresponding natural units. e key to understanding nature is thus the understanding of all ratios, and thus of all dimensionless constants. e story of the most important ratios is told in the third part of the adventure.
e basic constants yield the following useful high-precision observations.
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TA B L E 85 Derived physical constants
Q
S
V
SI
Vacuum wave resistance Avogadro’s number Rydberg constant a conductance quantum magnetic ux quantum Josephson frequency ratio von Klitzing constant Bohr magneton cyclotron frequency
of the electron classical electron radius Compton wavelength
Z= µ ε NA R = mecα h G=e h φ =h e eh h e =µ c α µB = eħ me fc B = e πme
. .
. .
. .
. .
... Ω
( )ë
.
( ) m−
( ) µS
( ) pWb
( ) THz V
( )Ω
( ) yJ T
( ) GHz T
re = e πε mec .
λc = h mec
.
( ) fm ( ) pm
U
.
.ë − .ë − .ë − .ë − .ë − .ë − .ë − .ë −
.ë − .ë −
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,
Q
S
V
SI
U
.
of the electron Bohr radius a nuclear magneton proton–electron mass ratio Stefan–Boltzmann constant
λc = ħ mec = re α a = re α µN = eħ mp mp me σ=π k ħ c
Wien displacement law constant b = λmaxT bits to entropy conversion const.
TNT energy content
.
( ) pm
.
( ) pm
.
( )ë − J T
.
()
.
( ) nW m K
.
( ) mmK
bit = .
( )J K
. to . MJ kg
.ë − .ë − .ë − .ë − .ë − .ë − .ë − ë−
Dvipsbugw
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
a. For in nite mass of the nucleus.
Some properties of nature at large are listed in the following table. (If you want a challenge, Challenge 1538 n can you determine whether any property of the universe itself is listed?)
TA B L E 86 Astrophysical constants
Q
S
V
gravitational constant
G
.
( ) ë − m kg s
cosmological constant
Λ
c. ë − m−
tropical year a
a
.s
tropical year
a
.s
mean sidereal day
d
h ′.
′′
light year
al
.
... Pm
astronomical unit b
AU
. ( ) km
parsec
pc
.
Pm = .
al
age of the universe c
t
. ( )ë s= . ( . )ë a
(determined from space-time, via expansion, using general relativity)
age of the universe c
t
over . ( ) ë s = . ( . ) ë a
(determined from matter, via galaxies and stars, using quantum theory)
Hubble parameter c
H
. ( ) ë − s− = . ( ) ë − a−
H = h ë km sMpc = h ë . ë − a−
reduced Hubble parameter c
h
. ()
deceleration parameter
q = −(a¨ a) H − . ( )
universe’s horizon distance c
d = ct
. ( ) ë m = . ( ) Gpc
universe’s topology
unknown
number of space dimensions
, for distances up to m
critical density of the universe
ρc = H πG
h ë.
( ) ë − kg m
= . ( ) ë − kg m
(total) density parameter c
Ω = ρ ρc
. ()
baryon density parameter c
ΩB = ρB ρc
. ()
cold dark matter density parameter c ΩCDM = ρCDM ρc . ( )
neutrino density parameter c
Ων = ρν ρc
. to .
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,
Q
S
dark energy density parameter c dark energy state parameter
ΩX = ρX ρc w = pX ρX
baryon mass
mb
baryon number density
luminous matter density
stars in the universe
ns
baryons in the universe
nb
microwave background temperature d T
photons in the universe photon energy density photon number density
nγ ργ = π k T
density perturbation amplitude
S
gravity wave amplitude
T
mass uctuations on Mpc
σ
scalar index
n
running of scalar index
dn d ln k
Planck length
Planck time
Planck mass instants in history c space-time points
inside the horizon c mass inside horizon
lPl = ħG c
tPl = ħG c
mPl = ħc G t tPl N = (R lPl) ë (t tPl) M
V
. () −. ( )
. ë − kg . () m . ( ) ë − kg m
. ( )K
. ë − kg m . cm or
. ( . )ë − <. S . () . () -. ()
. ë −m
. ë−s . µg . ( . )ë
cm (T . K)
kg
Dvipsbugw
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 1539 n Ref. 1203
Page 451
a. De ning constant, from vernal equinox to vernal equinox; it was once used to de ne the second. (Remem-
ber: π seconds is about a nanocentury.) e value for is about . s less, corresponding to a slowdown
of roughly . ms a. (Watch out: why?) ere is even an empirical formula for the change of the length of
the year over time.
b. Average distance Earth–Sun. e truly amazing precision of m results from time averages of signals
sent from Viking orbiters and Mars landers taken over a period of over twenty years.
c. e index indicates present-day values.
d. e radiation originated when the universe was
years old and had a temperature of about K;
the uctuations ∆T which led to galaxy formation are today about
µK = ( ) ë − T .
Warning: in the third part of this text it is shown that many of the constants in Table are not physically sensible quantities. ey have to be taken with many grains of salt. e more speci c constants given in the following table are all sensible, though.
TA B L E 87 Astronomical constants
Q
S
Earth’s mass
M
V
.
( ) ë kg
Dvipsbugw
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
,
Q
S
V
Earth’s gravitational length Earth radius, equatorial a Earth’s radius, polar a Equator–pole distance a Earth’s attening a Earth’s av. density Earth’s age Moon’s radius Moon’s radius Moon’s mass Moon’s mean distance b Moon’s distance at perigee b
Moon’s distance at apogee b
Moon’s angular size c
Moon’s average density Sun’s mass Sun’s gravitational length Sun’s luminosity solar radius, equatorial Sun’s angular size
l = GM c Req Rp
e ρ
Rmv Rmh Mm dm
ρ M l = GM L R
Sun’s average density
ρ
Sun’s average distance
AU
Sun’s age
solar velocity
vg
around centre of galaxy
solar velocity
vb
against cosmic background
distance to galaxy centre
Milky Way’s age
Milky Way’s size
Milky Way’s mass
Jupiter’s mass
M
Jupiter’s radius, equatorial
R
Jupiter’s radius, polar
R
Jupiter’s average distance from Sun D
most distant galaxy known
IR
. ( ) mm
. ( ) km
. ( ) km
. km (average)
. ()
. Mg m
. Ga = Ps
km in direction of Earth
. km in other two directions
. ë kg
km
typically km
Mm, hist. minimum
typically km
Mm, hist. maximum
average .
= . ′, minimum . ,
maximum - shortens line .
. Mg m
.
( ) ë kg
c.
km
. YW
. ( ) Mm
. average; minimum on fourth of July (aphelion) ′′, maximum on fourth of January (perihelion) ′′
. Mg m
. ( ) km
. Ga
( ) km s
. ( ) km s
. ( ) kpc = . ( . ) kal . Ga c. m or kal
solar masses, c. ë kg . ë kg . Mm
. ( ) Mm km
. ë al = . ë m, red-shi
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Dvipsbugw
,
a. e shape of the Earth is described most precisely with the World Geodetic System. e last edition dates from . For an extensive presentation of its background and its details, see the http://www.wgs .com website. e International Geodesic Union re ned the data in . e radii and the attening given here are those for the ‘mean tide system’. ey di er from those of the ‘zero tide system’ and other systems by about . m. e details constitute a science in itself. b. Measured centre to centre. To nd the precise position of the Moon at a given date, see the http://www. fourmilab.ch/earthview/moon_ap_per.html page. For the planets, see the page http://www.fourmilab.ch/ solar/solar.html and the other pages on the same site.
c. Angles are de ned as follows: degree = = π rad, ( rst) minute = ′ =
, second (minute)
= ′′ = ′ . e ancient units ‘third minute’ and ‘fourth minute’, each th of the preceding, are not in
use any more. (‘Minute’ originally means ‘very small’, as it still does in modern English.)
Dvipsbugw
U
Ref. 1204 Challenge 1540 n
π
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 375105
e
2.71828 18284 59045 23536 02874 71352 66249 77572 47093 699959
γ
0.57721 56649 01532 86060 65120 90082 40243 10421 59335 939923
ln 2 0.69314 71805 59945 30941 72321 21458 17656 80755 00134 360255
ln 10 2.30258 50929 94045 68401 79914 54684 36420 76011 01488 628772
10 3.16227 76601 68379 33199 88935 44432 71853 37195 55139 325216
If the number π is normal, i.e. if all digits and digit combinations in its decimal expansion
appear with the same limiting frequency, then every text ever written or yet to be written,
as well as every word ever spoken or yet to be spoken, can be found coded in its sequence.
e property of normality has not yet been proven, although it is suspected to hold. Does
this mean that all wisdom is encoded in the simple circle? No. e property is nothing
special: it also applies to the number .
... and many
others. Can you specify a few examples?
By the way, in the graph of the exponential function ex , the point ( , ) is the only
point with two rational coordinates. If you imagine painting in blue all points on the
plane with two rational coordinates, the plane would look quite bluish. Nevertheless, the
graph goes through only one of these points and manages to avoid all the others.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Dvipsbugw
B
Challenge 1541 e
1182 Le Système International d’Unités, Bureau International des Poids et Mesures, Pavillon de
Breteuil, Parc de Saint Cloud,
Sèvres, France. All new developments concerning SI
units are published in the journal Metrologia, edited by the same body. Showing the slow
pace of an old institution, the BIPM launched a website only in ; it is now reachable at
http://www.bipm.fr. See also the http://www.utc.fr/~tthomass/ emes/Unites/index.html
website; this includes the biographies of people who gave their names to various units.
e site of its British equivalent, http://www.npl.co.uk/npl/reference, is much better; it
provides many details as well as the English-language version of the SI unit de nitions.
Cited on page .
1183 e bible in the eld of time measurement is the two-volume work by J. V
& C.
A
, e Quantum Physics of Atomic Frequency Standards, Adam Hilge, . A pop-
ular account is T J , Splitting the Second, Institute of Physics Publishing, .
e site http://opdaf .obspm.fr/www/lexique.html gives a glossary of terms used in the
eld. On length measurements, see ... On electric current measurements, see ... On mass
and atomic mass measurements, see page . On high-precision temperature measure-
ments, see page . Cited on page .
1184 e uno cial pre xes were rst proposed in the s by Je K. Aronson of University of Oxford, and might come into general usage in the future. Cited on page .
1185 For more details on electromagnetic unit systems, see the standard text by J D
J
, Classical Electrodynamics, rd edition, Wiley, . Cited on page .
1186 G.J. S
, On the physical units of nature, Philosophical Magazine 11, pp. – ,
. Cited on page .
1187 D.J. B & al., Evidence for correlated changes in the spectrum and composition of cosmic rays at extremely high energies, Physical Review Letters 71, pp. – , . Cited on page .
1188 P.J. H
, R.T. V
& J.E. M
, Nuclear antiferromagnetism
in rhodium metal at positive and negative nanokelvin temperatures, Physical Review Let-
ters 70, pp. – , . See also his article in Scienti c American, January . Cited
on page .
1189 G. C
& R.L. G
, e DARI, Europhysics News 33, pp. – , Janu-
ary/February . Cited on page .
1190 Measured values and ranges for physical quantities are collected in H
V&
P
A
, Die Welt in Zahlen, Spektrum Akademischer Verlag, . Cited
on page .
1191 See e.g. K. C
& L.J. F
, Coulomb explosion of simple molecules in in-
tense laser elds, Contemporary Physics 35, pp. – , . Cited on page .
1192 A. Z
, e Planck stroll, American Journal of Physics 58, p. ,
nd another similar example? Cited on page .
. Can you
1193 e most precise clock built in , a caesium fountain clock, had a precision of one
part in . Higher precision has been predicted to be possible soon, among others by M.
T
, F.-L. H , R. H
& H. K
, An optical lattice clock, Nature
435, pp. – , . Cited on page .
1194 J. B
, ed., Proceedings of the Fi h Symposium on Frequency Standards and Met-
rology, World Scienti c, . Cited on page .
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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,
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1195 See the information on Ds mesons from the particle data group at http://pdg.web.cern.ch/ pdg. Cited on page .
1196 About the long life of tantalum , see D. B & al., Photoactivation of Tam and its implications for the nucleosynthesis of nature’s rarest naturally occurring isotope, Physical Review Letters 83, pp. – , December . Cited on page .
1197 See the review by L. J , D.G. B
& C. Z , e detection of gravitational waves,
Reports on Progress in Physics 63, pp. – , . Cited on page .
1198 See the clear and extensive paper by G.E. S
, Ring laser tests of fundamental
physics and geophysics, Reports on Progress in Physics 60, pp. – , . Cited on
page .
1199 Following a private communication by Richard Rusby, this is the value of , whereas it
was estimated as . °C in , as reported by G
J
&R
R,
O cial: water boils at . °C, Physics World 2, pp. – , September , and R.L.
R , Ironing out the standard scale, Nature 338, p. , March . For more details
on temperature measurements, see page . Cited on page .
1200 J. S
, Newton’s apples fall from grace, New Scientist, 2098, p. , September .
More details can be found in R.G. K
, e history of Newton’s apple tree, Contem-
porary Physics 39, pp. – , . Cited on page .
1201 e various concepts are even the topic of a separate international standard, ISO , with
the title Accuracy and precision of measurement methods and results. A good introduction is
J R. T
, An Introduction to Error Analysis: the Study of Uncertainties in Physical
Measurements, nd edition, University Science Books, Sausalito, . Cited on page .
1202 P.J. M & B.N. T
, CODATA recommended values of the fundamental physical
constants: , Reviews of Modern Physics 59, p. , . is is the set of constants
resulting from an international adjustment and recommended for international use by
the Committee on Data for Science and Technology (CODATA), a body in the International
Council of Scienti c Unions, which brings together the International Union of Pure and
Applied Physics (IUPAP), the International Union of Pure and Applied Chemistry (IUPAC)
and other organizations. e website of IUPAC is http://www.iupac.org. Cited on page .
1203 e details are given in the well-known astronomical reference, P. K
S
-
, Explanatory Supplement to the Astronomical Almanac, . Cited on page .
1204 For information about the number π, as well as about other constants, the website http://
oldweb.cecm.sfu.ca/pi/pi.html provides lots of data and references. It also has a link to the
overview at http://mathworld.wolfram.com/Pi.html and to many other sites on the topic.
Simple formulae for π are
π+ =
nn
n
n= n
(814)
or the beautiful formula discovered in by Bailey, Borwein and Plou e
Dvipsbugw
π=
n=
n( n+ − n+ − n+ − n+ ).
(815)
e site also explains the newly discovered methods for calculating speci c binary digits of π without having to calculate all the preceding ones. By the way, the number of (consecutive) digits known in was over . million million, as told in Science News 162, p. , December . ey pass all tests of randomness, as the http://mathworld.wolfram.com/ PiDigits.html website explains. However, this property, called normality, has never been
Dvipsbugw
Challenge 1542 r Challenge 1543 n
proven; it is the biggest open question about π. It is possible that the theory of chaotic dy-
namics will lead to a solution of this puzzle in the coming years.
Another method to calculate π and other constants was discovered and published by
D.V. C
& G.V. C
, e computation of classical constants, Pro-
ceedings of the National Academy of Sciences (USA) 86, pp. – , . e Chud-
nowsky brothers have built a supercomputer in Gregory’s apartment for about
euros, and for many years held the record for calculating the largest number of digits of
π. ey have battled for decades with Kanada Yasumasa, who held the record in , cal-
culated on an industrial supercomputer. New formulae to calculate π are still occasionally
discovered.
For the calculation of Euler’s constant γ see also D.W. D T
, A quicker conver-
gence to Euler’s constant, e Mathematical Intelligencer, pp. – , May .
Note that little is known about the basic properties of some numbers; for example, it is
still not known whether π + e is a rational number or not! (It is believed that it is not.) Do
you want to become a mathematician? Cited on page .
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A
C
PARTICLE PROPERTIES
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T following table lists the known and predicted elementary particles. he list has not changed since the mid- s, mainly because of the ine cient use hat was made of the relevant budgets across the world.
TA B L E 88 The elementary particles
Radiation
electromagnetic interaction
γ
weak interaction strong interaction
W+ , W−
...
Z
photon
intermediate vector bosons
gluons
Radiation particles are bosons with spin 1. W− is the antiparticle of W+; all others are their own antiparticles.
Matter Leptons
Quarks (each in three colours)
generation 1 e νe
generation 2 µ νµ
generation 3 τ ντ
d
s
t
u
c
b
Matter particles are fermions with spin 1/2; all have a corresponding antiparticle.
Hypothetical matter and radiation
Higgs boson
H
Supersymmetric partners se ... h
predicted to have spin 0 and to be elementary
one partner for each of the above
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e following table lists all properties for the elementary particles (for reasons of space, the colour quantity is not given explicitely).* e table and its header thus allow us, in principle, to deduce a complete characterization of the intrinsic properties of any composed moving entity, be it an object or an image.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
TA B L E 89 Elementary particle properties
P
M ma
L
τI ,b
I, J, c
P,
C
C
,L
,&
-
e
,c
-
,
L B,
, R-
:
QISCBT
Elementary radiation (bosons)
photon γ
(< − kg)
stable
I(JPC) =
,,
, ( −−)
W
. ( ) GeV c
. ( ) GeV J =
,,
. ( )% hadrons,
. ( )% l+ν
Z
. ( ) GeV c
J=
,,
. ( ) GeV c
. ( )% hadrons
. ( )%
l+l−
gluon
stable
I(JP) = ( −)
,,
Elementary matter (fermions): leptons
electron e .
( )ë
ë s J=
−
,,
− kg = .
( ) pJ c
=.
( ) MeV c = .
( )u
gyromagnetic ratio µe µB = − . electric dipole moment d = (− .
() . )ë − emf
muon µ
.
( ) yg . ( ) µs J =
−
,,
% e−ν¯e νµ
=.
( ) MeV c = .
( )u
gyromagnetic ratio µµ (eħ mµ) = − .
()
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* e o cial reference for all these data, worth a look for every physicist, is the massive collection of informa-
tion compiled by the Particle Data Group, with the website http://pdg.web.cern.ch/pdg containing the most
recent information. A printed review is published about every two years, with updated data, in one of the
major journals on elementary particle physics. See for example S. E
& al., Review of Particle Phys-
ics, Physics Letters B 592, p. 1, 2004. For many measured properties of these particles, the o cial reference
is the set of CODATA values. e most recent list was published by P.J. M & B.N. T
, Reviews of
Modern Physics 59, p. 351, 2000.
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P tau τ
M ma
L
τI
I,
J, c
,b
P,
C
electric dipole moment d = ( . . ) ë − e m
.
( ) GeV c
. ( . ) fs J =
C
,L
,&
-
e
,c
-
,
L B,
, R-
:
QISCBT
−
,,
el. neutrino < eV c
J=
,,
νe
muon
< . MeV c
J=
,,
neutrino νµ
tau neutrino < . MeV c
J=
,,
ντ
Elementary matter (fermions): quarks
up u down d strange s charm c bottom b top t
. to MeV c to MeV c
to MeV c . ( ) GeV c . ( ) GeV c
. ( . ) GeV c
see proton see proton
τ = . ( ) ps
I(JP) = ( +)
I(JP) = ( +) − −
I(JP) = ( +) − −
I(JP) = ( +)
+
I(JP) = ( +) − −
I(JP) = ( +)
+
Hypothetical, maybe elementary (boson)
Higgs h H
. GeV c
J=
Hypothetical elementary radiation (bosons)
Selectron h
J=
Smuon h
J=
Stauon h
J=
Sneutrinos h
J=
Squark h
J=
R=− R=− R=− R=− R=−
Hypothetical elementary matter (fermions)
Higgsino(s) h
J=
Wino h (a chargino)
J=
Zino h (a neutralino)
J=
Photino h
J=
Gluino h
J=
R=− R=− R=− R=− R=−
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Notes: a. See also the table of SI pre xes on page . About the eV c mass unit, see page . b. e energy width Γ of a particle is related to its lifetime τ by the indeterminacy relation Γτ = ħ. ere
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Ref. 1205, Ref. 1206
is a di erence between the half-life t and the lifetime τ of a particle: they are related by t = τ ln ,
where ln
.
; the half-life is thus shorter than the lifetime. e uni ed atomic mass unit u is
de ned as / of the mass of a carbon atom at rest and in its ground state. One has u = m( C) =
.
( ) yg.
c. To keep the table short, the header does not explicitly mention colour, the charge of the strong interactions.
is has to be added to the list of basic object properties. Quantum numbers containing the word ‘parity’
are multiplicative; all others are additive. Time parity T (not to be confused with topness T), better called
motion inversion parity, is equal to CP. e isospin I (or IZ) is de ned only for up and down quarks and their
composites, such as the proton and the neutron. In the literature one also sees references to the so-called
G-parity, de ned as G = (− )IC .
d. ‘Beauty’ is now commonly called bottomness; similarly, ‘truth’ is now commonly called topness. e signs
of the quantum numbers S, I, C, B, T can be de ned in di erent ways. In the standard assignment shown
here, the sign of each of the non-vanishing quantum numbers is given by the sign of the charge of the
corresponding quark.
e. R-parity is a quantum number important in supersymmetric theories; it is related to the lepton number L,
the baryon number B and the spin J through the de nition R = (− ) B+L+ J . All particles from the standard
model are R-even, whereas their superpartners are odd.
f . e electron radius is less than − m. It is possible to store single electrons in traps for many months.
. See page for the precise de nition and meaning of the quark masses.
h. Currently a hypothetical particle.
Challenge 1544 n
Using the table of elementary particle properties, together with the standard model and the fundamental constants, in principle all properties of composite matter and radiation can be deduced, including all those encountered in everyday life. (Can you explain how the size of an object follows from them?) In a sense, this table contains all our knowledge of matter and radiation, including materials science, chemistry and biology.
e most important examples of composites are grouped in the following table.
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TA B L E 90 Properties of selected composites
C
M m,
L
τ,
mesons (hadrons, bosons) (selected from over known types)
pion π (uu¯ − dd¯) pion π+(ud¯)
. ( ) MeV c
( ) as, γ .
IG (JPC ) = −( −+), S = C = B =
.
( ) MeV c
. ( ) ns,
( )%
µ+νµ . IG (JP) = −( −), S = C = B =
( )%
kaon KS kaon KL kaon K (us¯, u¯s)
mKS mKS + .
.(
( ) µeV c ) MeV c
. ( ) ps . ( ) ns . ( ) ns,
µ+νµ . ( )% π+π . ( )%
kaon K (d¯s) ( % KS , KL)
all kaons K , K , KS , KL
% . ( ) MeV c I(JP) = ( −), S =
n.a. ,B=C=
baryons (hadrons, fermions) (selected from over known types)
S
(
.)
fm fm
fm fm fm
fm
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C
M m,
L
τ,
S
(
.)
proton p or N+ (uud) neutron n or N (udd) omega Ω− (sss)
.
( ) yg
τtotal . ë a,
=.
( ) u τ(p e+π ) . ë a
=.
( ) MeV c
I(JP) = ( +), S =
gyromagnetic ratio µp µN = . electric dipole moment d = (−
() )ë − em
electric polarizability αe = . ( . ) ë − fm magnetic polarizability αm = . ( . ) ë − fm
.
( ) yg
. ( . ) s, pe−ν¯e %
=.
( )u= .
( ) MeV c
I(JP) = ( +), S =
gyromagnetic ratio µn µN = − . electric dipole moment dn = (− . electric polarizability α = . ( ) ë
() . )ë − em − fm
. ( ) MeV c
. ( . ) ps,
ΛK− . ( )%,
Ξ π− . ( )%
gyromagnetic ratio µΩ µN = − . ( )
. ( ) fm
Ref. 1207
fm
fm
composite radiation: glueballs
glueball f ( )
( ) MeV
full width ( ) MeV
fm
IG (JPC ) = +( ++)
atoms (selected from known elements with over
hydrogen ( H) [lightest]
antihydrogen
helium ( He) [smallest]
carbon ( C)
bismuth ( Bi ) [shortest living and rarest]
tantalum ( Ta) [second longest living radioactive]
bismuth ( Bi) [longest living radioactive]
francium ( Fr) [largest]
atom ( Uuh) [heaviest]
. . u= .
. u= .
u
( )u= . yg ( )u= .
( ) yg
u
u
u u
known nuclides) Ref. 1208
yg
ë pm
ë pm
yg
ë pm
ë pm
. ps Ref. 1210
a Ref. 1209
. ( ) a Ref. 1210
min . ms
ë . nm
molecules (selected from over known types)
hydrogen (H )
u
a
water (H O)
u
a
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C
M m,
ATP
(adenosinetriphosphate)
human Y chromosome
u ë base pairs
other composites
blue whale nerve cell
kg
cell (red blood)
. ng
cell (sperm)
pg
cell (ovule)
cell (E. coli) adult human
µg
pg kg < m < kg
heaviest living thing: colony . ë kg of aspen trees
larger composites
See the table on page
L
τ,
a a
S
(
.)
c. nm
c. mm (uncoiled)
a plus days not fecundated: d
fecundated: over million years million years
τ . ë s Ref. 1211 million breaths million heartbeats
< a, % H O and % dust a
m µm
length µm,
head µm µm µm
body: µm .m
km
.
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Page 1176 Ref. 1212
Ref. 1213
Ref. 1214
Notes (see also those of the previous table): G-parity is de ned only for mesons and given by G = (− )L+S+I = (− )IC. In , experiments provided candidates for tetraquarks, namely the X( ), Ds( ), f( )
and Ds( ), and for pentaquarks, namely the Θ+( ) particle. Time will tell whether these interpretations are correct.
Neutrons bound in nuclei have a lifetime of at least years. e f ( ) resonance is a candidate for the glueball ground state and thus for a radiation
composite. e Y ( ) resonance is a candidate for a hybrid meson, a composite of a gluon and a quark–
antiquark pair. is prediction of seems to have been con rmed in . In , the rst evidence for the existence of tetra-neutrons was published by a French group.
However, more recent investigations seem to have refuted the claim. e number of existing molecules is several orders of magnitude larger than the number of
molecules that have been analysed and named. Some nuclei have not yet been observed; in the known nuclei ranged from to , but
and were still missing. e rst anti-atoms, made of antielectrons and antiprotons, were made in January at CERN
in Geneva. All properties of antimatter checked so far are consistent with theoretical predictions. e charge parity C is de ned only for certain neutral particles, namely those that are di erent
from their antiparticles. For neutral mesons, the charge parity is given by C = (− )L+S, where L
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is the orbital angular momentum. P is the parity under space inversion r −r. For mesons, it is related to the orbital angular
momentum L through P = (− )L+ .
e electric polarizability, de ned on page , is predicted to vanish for all elementary
particles.
Ref. 1217
e most important matter composites are the atoms. eir size, structure and interactions determine the properties and colour of everyday objects. Atom types, also called elements in chemistry, are most usefully set out in the so-called periodic table, which groups together atoms with similar properties in rows and columns. It is given in Table and results from the various ways in which protons, neutrons and electrons can combine to form aggregates.
Comparable to the periodic table of the atoms, there are tables for the mesons (made of two quarks) and the baryons (made of three quarks). Neither the meson nor the baryon table is included here; they can both be found in the cited Review of Particle Physics at http://pdg.web.cern.ch/pdg. In fact, the baryon table still has a number of vacant spots. However, the missing particles are extremely heavy and short-lived (which means expensive to make and detect), and their discovery is not expected to yield deep new insights.
e atomic number gives the number of protons (and electrons) found in an atom of a given element. is number determines the chemical behaviour of an element. Most – but not all – elements up to are found on Earth; the others can be produced in laboratories.
e highest element discovered is element . (In a famous case of research fraud, a scientist in the s tricked two whole research groups into claiming to have made and observed elements and . Element was independently made and observed by another group later on.) Nowadays, extensive physical and chemical data are available for every element.
Elements in the same group behave similarly in chemical reactions. e periods de ne the repetition of these similarities. More elaborate periodic tables can be found on the http://chemlab.pc.maricopa.edu/periodic website. e most beautiful of them all can be found on page of this text.
Group are the alkali metals (though hydrogen is a gas), group the Earth-alkali metals. Actinoids, lanthanoids and groups to are metals; in particular, groups to
are transition or heavy metals. e elements of group are called chalkogens, i.e. oreformers; group are the halogens, i.e. the salt-formers, and group are the inert noble gases, which form (almost) no chemical compounds. e groups , and contain metals, semimetals, a liquid and gases; they have no special name. Groups and to are central for the chemistry of life; in fact, % of living matter is made of C, O, N, H;* almost % of P, S, Ca, K, Na, Cl; trace elements such as Mg, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Cd, Pb, Sn, Li, Mo, Se, Si, I, F, As, B form the rest. Over elements are known to be essential for animal life. e full list is not yet known; candidate elements to extend this list are Al, Br, Ge and W.
Many elements exist in versions with di erent numbers of neutrons in their nucleus, and thus with di erent mass; these various isotopes – so called because they are found at the same place in the periodic table – behave identically in chemical reactions. ere are
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* e ‘average formula’ of life is approximately C H O N.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
TA B L E 91 The periodic table of the elements known in 2006, with their atomic numbers
Group
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
I II IIIa IVa Va VIa VIIa VIIIa
Ia IIa III IV V VI VII VIII
Period
11
2
H
He
23
4
Li Be
5
6
7
8
9
10
B C N O F Ne
3
11
12
Na Mg
13 14 15 16 17 18
Al Si P S Cl Ar
4
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr
5
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe
6
55
56
Cs Ba
72 73 74 75 76 77 78 79 80 81 82 83 84 85 86
Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn
7
87
88
Fr Ra
104 105 106 107 108 109 110 111 112 113 114 115 116 117 118
Rf Db Sg Bh Hs Mt Ds Uuu Uub Uut Uuq Uup Uuh
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Lanthanoids Actinoids
57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
89 90 91 92 93 94 95 96 97 98 99 100 101 102 103
Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr
Ref. 1208, Ref. 1215 over
of them.
TA B L E 92 The elements, with their atomic number, average mass, atomic radius and main properties
N
S -A . A .
A -M
.
a
e
(
), -
,(
)h -
Actiniumb
Ac
( . ( )) ( ) highly radioactive metallic rare Earth
. ( )a
(Greek aktis ray) , used as alpha-
emitting source
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N
S -A . A .
A -M
.
a
e
(
), -
,(
)h -
Aluminium Al
Americiumb Am
Antimony
Sb
Argon
Ar
Arsenic
As
Astatineb
At
Barium
Ba
Berkeliumb Bk
Beryllium
Be
Bismuth
Bi
Bohriumb
Bh
Boron
B
Bromine
Br
.
()
stable
( . ( )) . ( ) ka
. ( )f stable
. ( )f stable
.
()
stable
( . ( )) . ( )h
. () stable
( . ( ))
. ( ) ka
.
()
stable
.
()
stable
( . ( )) .s
. ( )f stable
. () stable
c, m () c, m, v
( n)
c, v () m
n.a. c, m m, v
n.a.
c
c, v
light metal (Latin alumen alum) , used in machine construction and living beings
radioactive metal (Italian America from Amerigo) , used in smoke detectors
toxic semimetal (via Arabic from Latin stibium, itself from Greek, Egyptian for one of its minerals) antiquity, colours rubber, used in medicines, constituent of enzymes
noble gas (Greek argos inactive, from anergos without energy) , third component of air, used for welding and in lasers
poisonous semimetal (Greek arsenikon tamer of males) antiquity, for poisoning pigeons and doping semiconductors
radioactive halogen (Greek astatos unstable) , no use
Earth-alkali metal (Greek bary heavy) , used in vacuum tubes, paint, oil in-
dustry, pyrotechnics and X-ray diagnosis
made in lab, probably metallic (Berkeley, US town) , no use because rare
toxic Earth-alkali metal (Greek beryllos, a mineral) , used in light alloys, in nuclear industry as moderator
diamagnetic metal (Latin via German weisse Masse white mass) , used in magnets, alloys, re safety, cosmetics, as catalyst, nuclear industry
made in lab, probably metallic (a er Niels Bohr) , found in nuclear reactions, no use
semimetal, semiconductor (Latin borax, from Arabic and Persian for brilliant)
, used in glass, bleach, pyrotechnics, rocket fuel, medicine
red-brown liquid (Greek bromos strong odour) , fumigants, photography, water puri cation, dyes, medicines
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N
S -A . A .
A -M
.
a
e
(
), -
,(
)h -
Cadmium
Cd
Caesium
Cs
Calcium
Ca
Californiumb Cf
Carbon
C
Cerium
Ce
Chlorine
Cl
Chromium Cr
Cobalt
Co
Copper
Cu
Curiumb
Cm
. ( )f
m
stable
. stable
() m
. ( )f
m
stable
( . ( )) n.a. . ( ) ka
. ( )f
c
stable
. ( )f
m
stable
. ( )f
c,
stable
v
. ()
m
stable
.
()
m
stable
. ( )f
m
stable
( . ( )) n.a. . ( ) Ma
heavy metal, cuttable and screaming (Greek kadmeia, a zinc carbonate mineral where it was discovered) , electroplating, solder, batteries, TV phosphors, dyes
alkali metal (Latin caesius sky blue) , getter in vacuum tubes, photoelectric cells, ion propulsion, atomic clocks
Earth-alkali metal (Latin calcis chalk) antiquity, pure in , found in stones and bones, reducing agent, alloying
made in lab, probably metallic, strong neutron emitter (Latin calor heat and fornicare have sex, the land of hot sex :-) , used as neutron source, for well logging
makes up coal and diamond (Latin carbo coal) antiquity, used to build most life forms
rare Earth metal (a er asteroid Ceres, Roman goddess) , cigarette lighters, incandescent gas mantles, glass manufacturing, self-cleaning ovens, carbon-arc lighting in the motion picture industry, catalyst, metallurgy
green gas (Greek chloros yellow-green) , drinking water, polymers, paper,
dyes, textiles, medicines, insecticides, solvents, paints, rubber
transition metal (Greek chromos colour) , hardens steel, makes steel stain-
less, alloys, electroplating, green glass dye, catalyst
ferromagnetic transition metal (German Kobold goblin) , part of vitamin B , magnetic alloys, heavy-duty alloys, enamel dyes, ink, animal nutrition
red metal (Latin cuprum from Cyprus island) antiquity, part of many enzymes, electrical conductors, bronze, brass and other alloys, algicides, etc.
highly radioactive, silver-coloured (a er Pierre and Marie Curie) , used as radioactivity source
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N
S -A . A .
A -M
.
a
e
(
), -
,(
)h -
Darmstadtiumb Ds
Dubniumb
Db
Dysprosium Dy
Einsteiniumb Es
Erbium
Er
Europium
Eu
Fermiumb
Fm
Fluorine
F
Franciumb
Fr
Gadolinium Gd
Gallium
Ga
Germanium Ge
Gold
Au
Hafnium
Hf
Hassiumb
Hs
Helium
He
( ) . min n.a.
( . ( )) n.a. ( )s
. ( )f
m
stable
( . ( )) ( )d
. ( )f stable
. ( )f stable
n.a. m m
( . ( )) n.a.
. ( )d
.
( ) c,
stable
v
( . ( )) . ( ) min
. ( )f stable
. () stable
()
m
c, m
. ()
c,
stable
v
. stable
() m
. ( )c
m
stable
( ) . min n.a.
.
( ) f ( n)
stable
(a er the German city) , no use
made in lab in small quantities, radioactive (Dubna, Russian city) , no use (once known as Hahnium)
rare Earth metal (Greek dysprositos di cult to obtain) , used in laser materials, as infrared source material, and in nuclear industry
made in lab, radioactive (a er Albert Einstein) , no use
rare Earth metal (Ytterby, Swedish town) , used in metallurgy and optical bres
rare Earth metal (named a er the continent) , used in red screen phosphor for TV tubes
(a er Enrico Fermi) , no use
gaseous halogen (from uorine, a mineral, from Greek uo ow) , used in polymers and toothpaste
radioactive metal (from France) , no use
(a er Johan Gadolin) and phosphors
, used in lasers
almost liquid metal (Latin for both the discoverer’s name and his nation, France)
, used in optoelectronics
semiconductor (from Germania, as opposed to gallium) , used in electronics
heavy noble metal (Sanskrit jval to shine, Latin aurum) antiquity, electronics, jewels
metal (Latin for Copenhagen) incandescent wire
, alloys,
radioactive element (Latin form of German state Hessen) , no use
noble gas (Greek helios Sun) where it was discovered , used in balloons, stars, diver’s gas and cryogenics
Dvipsbugw
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N
S -A . A .
A -M
.
a
e
(
), -
,(
)h -
Holmium
Ho
Hydrogen
H
Indium
In
Iodine
I
Iridium
Ir
Iron
Fe
Krypton
Kr
Lanthanum La
Lawrenciumb Lr
Lead
Pb
Lithium
Li
Lutetium
Lu
Magnesium Mg
Manganese Mn
Meitneriumb Mt Mendeleviumb Md
.
()
m
stable
.
( )f
c
stable
. ()
c,
stable
m
.
()
c,
stable
v
. ()
m
stable
. ()
m
stable
. ( )f stable
( n)
.
( )c, f m
stable
( . ( )) . ( )h
. ( )c, f stable
n.a. m
. ( )f
m
stable
. ( )f
m
stable
. ()
m
stable
.
() m
stable
( . ( )) n.a. .s
( . ( )) n.a. . ( )d
metal (Stockholm, Swedish capital) , alloys
reactive gas (Greek for water-former) , used in building stars and universe
so metal (Greek indikon indigo) , used in solders and photocells
blue-black solid (Greek iodes violet) , used in photography
precious metal (Greek iris rainbow) , electrical contact layers
metal (Indo-European ayos metal, Latin ferrum) antiquity, used in metallurgy
noble gas (Greek kryptos hidden) , used in lasers
reactive rare Earth metal (Greek lanthanein to be hidden) , used in lamps and in special glasses
appears in reactions (a er Ernest Lawrence) , no use
poisonous, malleable heavy metal (Latin plumbum) antiquity, used in car batteries, radioactivity shields, paints
light alkali metal with high speci c heat (Greek lithos stone) , used in batteries, anti-depressants, alloys and many chemicals
rare Earth metal (Latin Lutetia for Paris) , used as catalyst
light common alkaline Earth metal (from Magnesia, a Greek district in essalia)
, used in alloys, pyrotechnics, chemical synthesis and medicine, found in chlorophyll
brittle metal (Italian manganese, a mineral) , used in alloys, colours amethyst and permanganate
appears in nuclear reactions (a er Lise Meitner) , no use
appears in nuclear reactions (a er Dimitri Ivanovitch Mendeleiev) , no use
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N
S -A . A .
A -M
.
a
e
(
), -
,(
)h -
Mercury
Hg
Molybdenum Mo
Neodymium Nd
Neon
Ne
Neptuniumb Np
Nickel
Ni
Niobium
Nb
Nitrogen
N
Nobeliumb No
Osmium
Os
Oxygen
O
Palladium
Pd
Phosphorus P
. () stable
. ( )f stable
. ( )c, f stable
. ( )f stable ( . ( )) . ( ) Ma
. () stable
.
()
stable
. ( )f stable
( . ( )) ( ) min . ( )f
stable
. ( )f stable
. ( )f stable
.
()
stable
m
m m ( n) n.a.
m m
c, v n.a. m c, v
m c, v
liquid heavy metal (Latin god Mercurius, Greek hydrargyrum liquid silver) antiquity, used in switches, batteries, lamps, amalgam alloys
metal (Greek molybdos lead) , used in alloys, as catalyst, in enzymes and lubricants
(Greek neos and didymos new twin)
noble gas (Greek neos new) , used in lamps, lasers and cryogenics
radioactive metal (planet Neptune, a er Uranus in the solar system) , appears in nuclear reactors, used in neutron detection and by the military
metal (German Nickel goblin) , used in coins, stainless steels, batteries, as catalyst
ductile metal (Greek Niobe, mythical daughter of Tantalos) , used in arc welding, alloys, jewellery, superconductors
diatomic gas (Greek for nitre-former) , found in air, in living organisms, Via-
gra, fertilizers, explosives
(a er Alfred Nobel) , no use
heavy metal (from Greek osme odour) , used for ngerprint detection and in
very hard alloys
transparent, diatomic gas (formed from Greek to mean ‘acid former’) , used for combustion, blood regeneration, to make most rocks and stones, in countless compounds, colours auroras red
heavy metal (from asteroid Pallas, a er the Greek goddess) , used in alloys, white gold, catalysts, for hydride storage
poisonous, waxy, white solid (Greek phosphoros light bearer) , fertilizers, glasses, porcelain, steels and alloys, living organisms, bones
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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N
S -A . A .
A -M
.
a
e
(
), -
,(
)h -
Platinum
Pt
Plutonium
Pu
Polonium
Po
Potassium
K
Praeseodymium Pr
Promethiumb Pm Protactinium Pa
. ()
m
stable
( . ( )) n.a. . ( ) Ma
( . ( )) ( ) ( )a
. ()
m
stable
.
()
m
stable
( . ( ))
m
. ( )a
(.
( )) n.a.
. ( ) ka
silvery-white, ductile, noble heavy
metal (Spanish platina little silver)
pre-Columbian, again in
, used
in corrosion-resistant alloys, magnets,
furnaces, catalysts, fuel cells, cathodic
protection systems for large ships and
pipelines; being a catalyst, a ne platinum
wire glows red hot when placed in vapour
of methyl alcohol, an e ect used in hand
warmers
extremely toxic alpha-emitting metal (a er the planet) synthesized , found in nature , used as nuclear explosive, and to power space equipment, such as satellites and the measurement equipment brought to the Moon by the Apollo missions
alpha-emitting, volatile metal (from Poland) , used as thermoelectric power source in space satellites, as neutron source when mixed with beryllium; used in the past to eliminate static charges in factories, and on brushes for removing dust from photographic lms
reactive, cuttable light metal (German Pottasche, Latin kalium from Arabic quilyi, a plant used to produce potash)
, part of many salts and rocks, essential for life, used in fertilizers, essential to chemical industry
white, malleable rare Earth metal (Greek praesos didymos green twin) , used in cigarette lighters, material for carbon arcs used by the motion picture industry for studio lighting and projection, glass and enamel dye, darkens welder’s goggles
radioactive rare Earth metal (from the Greek mythical gure of Prometheus)
, used as beta source and to excite phosphors
radioactive metal (Greek protos rst, as it decays into actinium) , found in nature, no use
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
N
S -A . A .
A -M
.
a
e
(
), -
,(
)h -
Radium
Ra
Radon
Rn
Rhenium
Re
Rhodium
Rh
Roentgeniumb Rg
Rubidium
Rb
Ruthenium Ru
Rutherfordiumb Rf
Samarium
Sm
Scandium
Sc
Seaborgiumb Sg
Selenium
Se
( . ( )) ( )a
( . ( )) . ( )d
. ( )c stable
.
()
stable
( . ( )) . ms
. ( )f stable
. ( )f stable
( . ( )) . min
. ( )c, f stable
.
()
stable
. () s
. ( )f stable
( ) highly radioactive metal (Latin radius ray) , no use any more; once used in
luminous paints and as radioactive source
and in medicine
( n) radioactive noble gas (from its old name
‘radium emanation’) , no use (any
more), found in soil, produces lung can-
cer
m (Latin rhenus for Rhine river) , used in laments for mass spectrographs and
ion gauges, superconductors, thermo-
couples, ash lamps, and as catalyst
m white metal (Greek rhodon rose) , used to harden platinum and palladium
alloys, for electroplating, and as catalyst
n.a.
, no use
m m
n.a. m
m
n.a. c, v
silvery-white, reactive alkali metal (Latin rubidus red) , used in photocells, optical glasses, solid electrolytes
white metal (Latin Rhuthenia for Russia) , used in platinum and palladium al-
loys, superconductors, as catalyst; the tetroxide is toxic and explosive
radioactive transactinide (a er Ernest Rutherford) , no use
silver-white rare Earth metal (from the mineral samarskite, a er Wassily Samarski) , used in magnets, optical glasses, as laser dopant, in phosphors, in high-power light sources
silver-white metal (from Latin Scansia Sweden) , the oxide is used in highintensity mercury vapour lamps, a radioactive isotope is used as tracer
radioactive transurane (a er Glenn Seaborg) , no use
red or black or grey semiconductor (Greek selene Moon) , used in xerography, glass production, photographic toners, as enamel dye
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
N
S -A . A .
A -M
.
a
e
(
), -
,(
)h -
Silicon
Si
Silver
Ag
Sodium
Na
Strontium
Sr
Sulphur
S
Tantalum
Ta
Technetiumb Tc
Tellurium
Te
Terbium
Tb
allium
Tl
orium
. ( )f stable
c, grey, shiny semiconductor (Latin silex v pebble) , Earth’s crust, electronics,
sand, concrete, bricks, glass, polymers, solar cells, essential for life
. ( )f stable
m white metal with highest thermal and electrical conductivity (Latin argentum, Greek argyros) antiquity, used in photography, alloys, to make rain
. stable
() m
light, reactive metal (Arabic souwad soda, Egyptian and Arabic natrium) component of many salts, soap, paper, soda, salpeter, borax, and essential for life
. ( )f stable
m silvery, spontaneously igniting light metal (Strontian, Scottish town) , used in TV tube glass, in magnets, and in optical materials
. ( )f stable
c, yellow solid (Latin) antiquity, used in gunv powder, in sulphuric acid, rubber vulcan-
ization, as fungicide in wine production, and is essential for life; some bacteria use sulphur instead of oxygen in their chemistry
.
()
stable
m heavy metal (Greek Tantalos, a mythical gure) , used for alloys, surgical in-
struments, capacitors, vacuum furnaces, glasses
( . ( )) . ( ) Ma
m radioactive (Greek technetos arti cial) , used as radioactive tracer and in
nuclear technology
. ( )f stable
c, brittle, garlic-smelling semiconductor v (Latin tellus Earth) , used in alloys
and as glass component
.
()
stable
m malleable rare Earth metal (Ytterby, Swedish town) , used as dopant in optical material
. () stable
m so , poisonous heavy metal (Greek thallos branch) , used as poison and for infrared detection
.
( )d, f
. ( ) Ga
m radioactive (Nordic god or, as in ‘ ursday’) , found in nature, heats Earth, used as oxide, in alloys and as coating
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N
S -A . A .
A -M
.
a
e
(
), -
,(
)h -
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
ulium
Tm
Tin
Sn
Titanium
Ti
Tungsten
W
Ununbiumb Uub
Ununtrium Uut
Ununquadiumb Uuq
Ununpentium Uup
Ununhexiumb Uuh
Ununseptium Uus
Ununoctium Uuo
Uranium
U
Vanadium
V
Xenon
Xe
Ytterbium
Yb
Yttrium
Y
Zinc
Zn
Zirconium
Zr
. () stable
m rare Earth metal ( ule, mythical name for Scandinavia) , found in monazite
. ( )f stable
c, grey metal that, when bent, allows one v, to hear the ‘tin cry’ (Latin stannum) anm tiquity, used in paint, bronze and super-
conductors
. () stable
m metal (Greek hero Titanos) fake diamonds
, alloys,
. () stable
m heavy, highest-melting metal (Swedish tung sten heavy stone, German name Wolfram) , lightbulbs
( ) . min n.a.
, no use
n.a.
, no use
( ) . s n.a.
, no use
n.a.
, no use
( ) . ms n.a.
(earlier claim was false), no use
n.a. not yet observed
n.a. not yet observed, but false claim in
.
( )d, f
. ( )ë a
m radioactive and of high density (planet Uranus, a er the Greek sky god) , found in pechblende and other minerals, used for nuclear energy
. () stable
m metal (Vanadis, scandinavian goddess of beauty) , used in steel
. ( ) f ( n) noble gas (Greek xenos foreign) ,
stable
v used in lamps and lasers
. ( )f stable
m malleable heavy metal (Ytterby, Swedish town) , used in superconductors
.
()
stable
m malleable light metal (Ytterby, Swedish town) , used in lasers
. () stable
m heavy metal (German Zinke protuberance) antiquity, iron rust protection
. ( )f stable
m heavy metal (from the mineral zircon, a er Arabic zargum golden colour) , chemical and surgical instruments, nuclear industry
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a. e atomic mass unit is de ned as u = m( C), making u = .
( ) yg. For elements found
on Earth, the average atomic mass for the naturally occurring isotope mixture is given, with the error in the
Ref. 1215 last digit in brackets. For elements not found on Earth, the mass of the longest living isotope is given; as it is
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Ref. 1216 Ref. 1216
Challenge 1545 n
Ref. 1208 Ref. 1215
not an average, it is written in brackets, as is customary in this domain. b. e element is not found on Earth because of its short lifetime. c. e element has at least one radioactive isotope. d. e element has no stable isotopes. e. Strictly speaking, the atomic radius does not exist. Because atoms are clouds, they have no boundary. Several approximate de nitions of the ‘size’ of atoms are possible. Usually, the radius is de ned in such a way as to be useful for the estimation of distances between atoms. is distance is di erent for di erent bond types. In the table, radii for metallic bonds are labelled m, radii for (single) covalent bonds with carbon c, and Van der Waals radii v. Noble gas radii are labelled n. Note that values found in the literature vary by about %; values in brackets lack literature references.
e covalent radius can be up to . nm smaller than the metallic radius for elements on the (lower) le of the periodic table; on the (whole) right side it is essentially equal to the metallic radius. In between, the di erence between the two decreases towards the right. Can you explain why? By the way, ionic radii di er considerably from atomic ones, and depend both on the ionic charge and the element itself.
All these values are for atoms in their ground state. Excited atoms can be hundreds of times larger than atoms in the ground state; however, excited atoms do not form solids or chemical compounds. f . e isotopic composition, and thus the average atomic mass, of the element varies depending on the place where it was mined or on subsequent human treatment, and can lie outside the values given. For example, the atomic mass of commercial lithium ranges between . and . u. e masses of isotopes are known in atomic mass units to nine or more signi cant digits, and usually with one or two fewer digits in kilograms.
e errors in the atomic mass are thus mainly due to the variations in isotopic composition. . e lifetime errors are asymmetric or not well known. h. Extensive details on element names can be found on http://elements.vanderkrogt.net.
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B
1205 e electron radius limit is deduced from the − measurements, as explained in the
Nobel Prize talk by H D
, Experiments with an isolated subatomic particle
at rest, Reviews of Modern Physics 62, pp. – , , or in H D
, Is the
electron a composite particle?, Hyper ne Interactions 81, pp. – , . Cited on page .
1206 G. G
, H. D
& W. K , Observation of a relativistic, bistable hyster-
esis in the cyclotron motion of a single electron, Physical Review Letters 54, pp. – ,
. Cited on page .
1207 e proton charge radius was determined by measuring the frequency of light emitted by
hydrogen atoms to high precision by T. U , A. H
, B. G , J. R
, M.
P
, M. W & T.W. H
, Phase-coherent measurement of the hydro-
gen S– S transition frequency with an optical frequency interval divider chain, Physical
Review Letters 79, pp. – , . Cited on page .
1208 For a full list of isotopes, see R.B. F
, Table of Isotopes, Eighth Edition, 1999
Update, with CD-ROM, John Wiley & Sons, . For a list of isotopes on the web, see the
Korean website by J. C
, http://atom.kaeri.re.kr/. For a list of precise isotope masses,
see the http://csnwww.in p .fr website. Cited on pages , , and .
1209 For information on the long life of tantalum , see D. B & al., Photoactivation of Tam and its implications for the nucleosynthesis of nature’s rarest naturally occurring
isotope, Physical Review Letters 83, pp. – , December . Cited on page .
1210 e ground state of bismuth was thought to be stable until early . It was then
discovered that it was radioactive, though with a record lifetime, as reported by P.
M
, N. C
, G. D
, J. L
& J.-P. M
, Experimental
detection of α-particles from the radioactive decay of natural bismuth, Nature 422, pp. –
, . By coincidence, the excited state MeV above the ground state of the same
bismuth nucleus is the shortest known radioactive nuclear state. Cited on page .
1211 S
J. G
, e Panda’s thumb, W.W. Norton & Co., . is is one of several
interesting and informative books on evolutionary biology by the best writer in the eld.
Cited on page .
1212 A tetraquark is thought to be the best explanation for the f ( ) resonance at MeV.
However, this assignment is still under investigation. Pentaquarks were rst predicted by
Maxim Polyakov, Dmitri Diakonov, and Victor Petrov in . Two experimental groups in
claimed to con rm their existence, with a mass of MeV; see K. H , An exper-
imental review of the Θ+ pentaquark, http://www.arxiv.org/abs/hep-ex/
. Results
from , however, seem to rule out that the MeV particle is a pentaquark. Cited on
page .
1213 F. M
& al., Detection of neutron clusters, Physical Review C 65, p.
,
. Opposite results have been obtained by B.M. S
& C.A. B
,
Proton-tetraneutron elastic scattering, Physical Review C 69, p.
, , and D.V.
A
& al., Search for resonances in the three- and four-neutron systems in
the Li( Li, C) n and Li( Li, C) n reactions, JETP Letters 81, p. , . Cited on
page .
1214 For a good review, see the article by P.T. G 38, pp. – , . Cited on page .
, Antimatter, Contemporary Physics
1215 e atomic masses, as given by IUPAC, can be found in Pure and Applied Chemistry 73, pp. – , , or on the http://www.iupac.org website. For an isotope mass list, see
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the http://csnwww.in p .fr website. Cited on pages , , and .
1216 e metallic, covalent and Van der Waals radii are from N
W. A
, Bond-
ing and Structure, Ellis Horwood, . is text also explains in detail how the radii are
de ned and measured. Cited on page .
1217 Almost everything known about each element and its chemistry can be found in the en-
cyclopaedic G
, Handbuch der anorganischen Chemie, published from onwards.
ere are over volumes, now all published in English under the title Handbook of Inor-
ganic and Organometallic Chemistry, with at least one volume dedicated to each chemical
element. On the same topic, an incredibly expensive book with an equally bad layout is
PE
, Encyclopedia of the Elements, Wiley–VCH, . Cited on page .
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A
D
NUMBERS AND SPACES
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
A mathematician is a machine that transforms co ee into theorems.
“ Paul Erdős (b. Budapest, d. Warsaw) ” concepts can all be expressed in terms of ‘sets’ and ‘relations.’
any fundamental concepts were presented in the rst Intermezzo. Why does
Mathematics, given this simple basis, grow into a passion for certain people? e
Ref. 1218 following pages present a few more advanced concepts as simply* and vividly as possible, for all those who want to smell the passion for mathematics. In particular, in this appendix we shall expand the range of algebraic and of topological structures; the third basic type of mathematical structures, order structures, are not so important in physics. Mathematicians are concerned not only with the exploration of concepts, but also with their classi cation. Whenever a new mathematical concept is introduced, mathematicians try to classify all the possible cases and types. is has been achieved most spectacularly for the di erent types of numbers, for nite simple groups, and for many types of spaces and manifolds.
N
Challenge 1546 ny
“A person who can solve x − y = in less than a year is a mathematician. Brahmagupta (b. Sindh, d. ) (implied: ” solve in integers)
We start with a short introduction to the vocabulary. Any mathematical system with the same basic properties as the natural numbers is called a semi-ring. Any mathematical system with the same basic properties as the integers is called a ring. ( e term is due to David Hilbert. Both structures can also be nite rather than in nite.) More precisely, a ring (R, +, ë) is a set R of elements with two binary operations, called addition and multiplication, usually written + and ë (the latter may simply be understood without notation), for which the following properties hold for all elements a, b, c R:
— R is a commutative group with respect to addition, i.e. a + b R, a + b = b + a, a + = a, a + (−a) = a − a = and a + (b + c) = (a + b) + c;
— R is closed under multiplication, i.e. ab R;
* e opposite approach can have the same e ect: it is taken in the delightful text by C E. L
-
, Mathematics Made Di cult, Wolfe Publishing, 1971.
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— multiplication is associative, i.e. a(bc) = (ab)c; — distributivity holds, i.e. a(b + c) = ab + ac and (b + c)a = ba + ca.
De ning properties such as these are called axioms. Note that axioms are not basic beliefs, as is o en stated; axioms are the basic properties used in the de nition of a concept: in this case, of a ring.
A semi-ring is a set satisfying all the axioms of a ring, except that the existence of neutral and negative elements for addition is replaced by the weaker requirement that if a + c = b + c then a = b.
To incorporate division and de ne the rational numbers, we need another concept. A eld K is a ring with
— an identity 1, such that all elements a obey a = a; — at least one element di erent from zero; and most importantly — a (multiplicative) inverse a− for every element a = .
Ref. 1220
A ring or eld is said to be commutative if the multiplication is commutative. A noncommutative eld is also called a skew eld. Fields can be nite or in nite. (A eld or a ring is characterized by its characteristic p. is is the smallest number of times one has to add to itself to give zero. If there is no such number the characteristic is set to . p is always a prime number or zero.) All nite elds are commutative. In a eld, all equations of the type cx = b and xc = b (c ) have solutions for x; there is a unique solution if b . To sum up sloppily by focusing on the most important property, a eld is a set of elements for which, together with addition, subtraction and multiplication, a division (by non-zero elements) is also de ned. e rational numbers are the simplest eld that incorporates the integers.
e system of the real numbers is the minimal extension of the rationals which is complete and totally ordered.*
However, the concept of ‘number’ is not limited to these examples. It can be generalized in several ways. e simplest generalization is achieved by extending the real numbers to manifolds of more than one dimension.
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* A set is mathematically complete if physicists call it continuous. More precisely, a set of numbers is complete if every non-empty subset that is bounded above has a lest upper bound.
A set is totally ordered if there exists a binary relation between pairs of elements such that for all elements a and b — if a b and b c, then a c; — if a b and b a, then a = b; — a b or b a holds. In summary, a set is totally ordered if there is a binary relation that allows to say about any two elements which one is the predecessor of the other in a consistent way.
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C
A complex number is de ned by z = a + ib, where a and b are real numbers, and i is a new symbol. Under multiplication, the generators of the complex numbers, and i, obey
ë
i
i
i i−
(816)
o en summarized as i = + − . e complex conjugate z , also written z¯, of a complex number z = a + ib is de ned
as z = a − ib. e absolute value z of a complex number is de ned as z = zz =
z z = a + b . It de nes a norm on the vector space of the complex numbers. From wz = w z follows the two-squares theorem
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(a + a )(b + b ) = (a b − a b ) + (a b + a b )
(817)
Challenge 1547 e
valid for all real numbers ai, bi. It was already known, in its version for integers, to Diophantus of Alexandria.
Complex numbers can also be written as
ordered pairs (a, A) of real numbers, with their
addition de ned as (a, A) + (b, B) = (a + b, A +
ic
B) and their multiplication de ned as (a, A) ë
(b, B) = (ab − AB, aB + bA). is notation al-
lows us to identify the complex numbers with the
points on a plane. Translating the de nition of
multiplication into geometrical language allows us to rapidly prove certain geometrical theorems,
ih
=
−
iab c
such as the one of Figure .
Complex numbers a + ib can also be repres-
ented as matrices
a
b
ab −b a
witha, b R .
F I G U R E 391 A property of triangles easily
(818) provable with complex numbers
Challenge 1548 n Page 1204
Challenge 1549 ny
Matrix addition and multiplication then correspond to complex addition and multiplication. In this way, complex numbers can be represented by a special type of real matrix. What is z in matrix language?
e set C of complex numbers with addition and multiplication as de ned above forms both a commutative two-dimensional eld and a vector space over R. In the eld of complex numbers, quadratic equations az + bz + c = for an unknown z always have two solutions (for a and counting multiplicity).
Complex numbers can be used to describe the points of a plane. A rotation around the origin can be described by multiplication by a complex number of unit length. Other twodimensional quantities can also be described with complex numbers. Electrical engineers
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Challenge 1550 e Challenge 1551 n
use complex numbers to describe quantities with phases, such as alternating currents or electrical elds in space.
Writing complex numbers of unit length as cos θ + i sin θ is a useful method for remembering angle addition formulae. Since one has cos nθ + i sin nθ = (cos θ + i sin θ)n, one can easily deduce formulae cos θ = cos θ − sin θ and sin θ = sin θ cos θ.
By the way, there are exactly as many complex numbers as there are real numbers. Can you show this?
“ ” Love is complex: it has real and imaginary parts.
Q
Ref. 1221 Page 1207
e positions of the points on a line can be described by real numbers. Complex numbers can be used to describe the positions of the points of a plane. It is natural to try to generalize the idea of a number to higher-dimensional spaces. However, it turns out that no useful number system can be de ned for three-dimensional space. A new number system, the quaternions, can be constructed which corresponds the points of four-dimensional space, but only if the commutativity of multiplication is sacri ced. No useful number system can be de ned for dimensions other than , and .
e quaternions were discovered by several mathematicians in the nineteenth century, among them Hamilton,* who studied them for much of his life. In fact, Maxwell’s theory of electrodynamics was formulated in terms of quaternions before three-dimensional vectors were used.
Under multiplication, the quaternions H form a -dimensional algebra over the reals with a basis , i, j, k satisfying
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ë
i jk
i jk i i − k −j . j j −k − i k k j −i −
(819)
Challenge 1552 ny
ese relations are also o en written i = j = k = − , i j = − ji = k, jk = −k j = i, ki = −ik = j. e quaternions , i, j, k are also called basic units or generators. e lack of symmetry across the diagonal of the table shows the non-commutativity of quaternionic multiplication. With the quaternions, the idea of a non-commutative product appeared for the rst time in mathematics. However, the multiplication of quaternions is associative. As a consequence of non-commutativity, polynomial equations in quaternions have many more solutions than in complex numbers: just search for all solutions of the equation X + = to convince yourself of it.
Every quaternion X can be written in the form
X = x + x i + x j + x k = x + v = (x , x , x , x ) = (x , v) ,
(820)
* William Rowan Hamilton (b. 1805 Dublin, d. 1865 Dunsink), Irish child prodigy and famous mathematician, named the quaternions a er an expression from the Vulgate (Acts. 12: 4).
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where x is called the scalar part and v the vector part. e multiplication is thus de ned as (x, v)(y, w) = (x y−vëw, xw+yv+v w). e multiplication of two general quaternions can be written as
(a , b , c , d )(a , b , c , d ) = (a a − b b − c c − d d , a b + b a + c d − d c , a c −b d +c a +d b ,a d +b c −c b +d a ). (821)
e conjugate quaternion X is de ned as X = x − v, so that XY = Y X. e norm X of a quaternion X is de ned as X = X X = X X = x + x + x + x = x + v . e norm is multiplicative, i.e. XY = X Y .
Unlike complex numbers, every quaternion is related to its complex conjugate by
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X = − (X + iXi + jXj + kXk) .
(822)
No relation of this type exists for complex numbers. In the language of physics, a complex number and its conjugate are independent variables; for quaternions, this is not the case. As a result, functions of quaternions are less useful in physics than functions of complex variables.
e relation XY = X Y implies the four-squares theorem
(a + a + a + a )(b + b + b + b ) = (a b − a b − a b − a b ) + (a b + a b + a b − a b ) +(a b + a b + a b − a b ) + (a b + a b + a b − a b )
(823)
Challenge 1553 ny
valid for all real numbers ai and bi, and thus also for any set of eight integers. It was discovered in by Leonhard Euler ( – ) when trying to prove that each integer is the sum of four squares. ( at fact was proved only in , by Joseph Lagrange.)
Hamilton thought that a quaternion with zero scalar part, which he simply called a vector (a term which he invented), could be identi ed with an ordinary three-dimensional translation vector; but this is wrong. Such a quaternion is now called a pure, or homogeneous, or imaginary quaternion. e product of two pure quaternions V = ( , v) and W = ( , w) is given by V W = (−v ë w, v w), where ë denotes the scalar product and denotes the vector product. Note that any quaternion can be written as the ratio of two pure quaternions.
In reality, a pure quaternion ( , v) does not behave like a translation vector under coordinate transformations; in fact, a pure quaternion represents a rotation by the angle π or ° around the axis de ned by the direction v = (vx , vy , vz).
It turns out that in three-dimensional space, a general rotation about the origin can be described by a unit quaternion Q, also called a normed quaternion, for which Q = . Such a quaternion can be written as (cos θ , n sin θ ), where n = (nx , ny , nz) is the normed vector describing the direction of the rotation axis and θ is the rotation angle. Such a unit quaternion Q = (cos θ , n sin θ ) rotates a pure quaternion V = ( , v) into
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k
i j
k
1
j
palm
of right
hand
i
back of right hand
F I G U R E 393 The hand and the quaternions
Dvipsbugw
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another pure quaternion W = ( , w) given by
W = QVQ .
(824)
us, if we use pure quaternions such as V or W to describe positions, we can use unit quaternions to describe rotations and to calculate coordinate changes. e concatenation of two rotations is then given by the product of the corresponding unit quaternions. Indeed, a rotation by an angle α about the axis l followed by a rotation by an angle β about the axis m gives a rotation by an angle γ about the axis n, with the values determined by
(cos γ , sin γ n) = (cos α , sin α l)(cos β , sin β m) .
(825)
Ref. 1219 Challenge 1554 e Challenge 1555 n
Page 783
One way to show the result graphically is given in Figure . By drawing a triangle on a unit sphere, and taking care to remember the factor in the angles, the combination of two rotations can be simply determined.
e interpretation of quaternions as rotations is also illustrated, in a somewhat di erent way, in the motion of any hand. To see this, take a green marker and write the letters , i, j and k on your hand as shown in Figure . De ning the three possible ° rotations axes as shown in the gure and taking concatenation as multiplication, the motion of the right hand follows the same ‘laws’ as those of pure unit quaternions. (One still needs to distinguish +i and −i, and the same for the other units, by the sense of the arm twist. And the result of a multiplication is that letter that can be read by a person facing you.) You can show that i = j = k = − , that i = , and all other quaternion relations.)
e model also shows that the rotation angle of the arm is half the rotation angle of the corresponding quaternion. In other words, quaternions can be used to describe the belt trick, if the multiplication V W of two quaternions is taken to mean that rotation V is performed a er rotation W. Quaternions, or the human hand, thus behaves like a spin
Dvipsbugw
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Page 1218
Page 783 Ref. 1222
/ particle. Quaternions and spinors are isomorphic.
e reason for the half-angle behaviour of
rotations can be speci ed more precisely using
mathematical language. e rotations in three
dimensions around a point form the ‘special
α
orthogonal group’ in three dimensions, which
l
π−γ
is called SO( ). But the motions of a hand at-
tached to a shoulder via an arm form a di er-
ent group, isomorphic to the Lie group SU( ).
e di erence is due to the appearance of half
n
angles in the parametrization of rotations; in-
deed, the above parametrizations imply that a rotation by π corresponds to a multiplication
m
β
by − . Only in the twentieth century was it realized that there exist fundamental physical ob-
F I G U R E 392 Combinations of rotations
servables that behaves like hands attached to arms: they are called spinors. More on
spinors can be found in the section on permutation symmetry, where belts are used as an
analogy as well as arms. In short, the group SU( ) of the quaternions is the double cover
of the rotation group SO( ).
e simple representation of rotations and positions with quaternions is used in by
computer programmes in robotics, in astronomy and in ight simulation. In the so -
ware used to create three-dimensional images and animations, visualization so ware,
quaternions are o en used to calculate the path taken by repeatedly re ected light rays
and thus give surfaces a realistic appearance.
e algebra of the quaternions is the only associative, non-commutative. nite-dimen-
sional normed algebra with an identity over the eld of real numbers. Quaternions form
a non-commutative eld, i.e. a skew eld, in which the inverse of a quaternion X is X X .
We can therefore de ne division of quaternions (while being careful to distinguish XY−
and Y− X). erefore quaternions are said to form a division algebra. In fact, the qua-
ternions H, the complex numbers C and the reals R are the only three nite-dimensional
associative division algebras. In other words, the skew- eld of quaternions is the only
nite-dimensional real associative non-commutative algebra without divisors of zero.
e centre of the quaternions, i.e. the set of quaternions that commute with all other qua-
ternions, is just the set of real numbers.
Quaternions can be represented as matrices of the form
Dvipsbugw
ab cd
A −B
B A
with A, B C,
or as
−b −c
a −d c d a −b
with a, b, c, d R, (826)
−d −c b a
where A = a + ib, B = c + id and the quaternion X is X = A + B j = a + ib + jc + kd; matrix addition and multiplication then corresponds to quaternionic addition and multiplication.
Dvipsbugw
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
e generators of the quaternions can be realized as
σ , i −iσ , j −iσ , k −iσ
(827)
where the σn are the Pauli spin matrices.* Real representations are not unique, as the alternative representation
a b −d −c −b a −c d
d c ab c −d −b a
Dvipsbugw (829)
Challenge 1556 ny
Page 545 Ref. 1221 Challenge 1557 n
shows; however, no representation by matrices is possible. ese matrices contain real and complex elements, which pose no special problems.
In contrast, when matrices with quaternionic elements are constructed, care has to be taken, because quaternionic multiplication is not commutative, so that simple relations such as trAB = trBA are not generally valid.
What can we learn from quaternions about the description of nature? First of all, we see that binary rotations are similar to positions, and thus to translations: all are represented by -vectors. Are rotations the basic operations of nature? Is it possible that translations are only ‘shadows’ of rotations? e connection between translations and rotations s is investigated in the third part of our mountain ascent.
When Maxwell wrote down his equations of electrodynamics, he used quaternion notation. ( e now usual -vector notation was introduced later by Hertz and Heaviside.)
e equations can be written in various ways using quaternions. e simplest is achieved when one keeps a distinction between − and the units i, j, k of the quaternions. One then can write all of electrodynamics in a single equation:
dF
=
−Q ε
(830)
where F is the generalized electromagnetic eld and Q the generalized charge. ese are
* e Pauli spin matrices are the complex Hermitean matrices
σ= =
, σ=
, σ=
−i
, σ=
i
−
(828)
Page 1209
all of whose eigenvalues are ; they satisfy the relations [σi , σk]+ = δik and [σi , σk] = i εikl σl . e linear combinations σ = (σ σ ) are also frequently used. By the way, another possible representation of the quaternions is i iσ , j iσ , k iσ .
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de ned by
F = E + − cB E = iEx + jEy + kEz B = iBx + jBy + kBz
d = δ + − ∂t c δ = i∂x + j∂y + k∂z
Q=ρ+ − J c
(831) Dvipsbugw
where the elds E and B and the charge distributions ρ and J have the usual meanings. e content of equation for the electromagnetic eld is exactly the same as the usual
formulation. Despite their charm, quaternions do not seem to be ready for the reformulation of
special relativity; the main reason for this is the sign in the expression for their norm. erefore, relativity and space-time are usually described using real numbers.
O
In the same way that quaternions are constructed from complex numbers, octonions can be constructed from quaternions. ey were rst investigated by Arthur Cayley ( – ). Under multiplication, octonions (or octaves) are the elements of an eightdimensional algebra over the reals with the generators , in with n = . . . satisfying
ë
iiii ii i
iiii ii i
i i − i −i i −i i −i
i i −i − i −i i i −i
i i i −i − i i −i −i
i i −i i −i − i −i i
i i i −i −i −i − i i
i i −i −i i i −i − i
i i i i i −i −i −i −
(832)
Nineteen other, equivalent multiplication tables are also possible. is algebra is called the Cayley algebra; it has an identity and a unique division. e algebra is non-commutative, and also non-associative. It is, however, alternative, meaning that for all elements x and y, one has x(x y) = x y and (x y)y = x y : a property somewhat weaker than associativity. It is the only -dimensional real alternative algebra without zero divisors. Because it is not associative, the set Ω of all octonions does not form a eld, nor even a ring, so that the old designation of ‘Cayley numbers’ has been abandoned. e octonions are the most general hypercomplex ‘numbers’ whose norm is multiplicative. Its generators obey since (in im)il = in(im il ), where the minus sign, which shows the non-associativity, is valid for combinations of indices, such as - - , which are not quaternionic.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Octonions can be represented as matrices of the form
AB −B¯ A¯
where A, B H , or as real
matrices.
(833)
Matrix multiplication then gives the same result as octonionic multiplication. e relation wz = w z allows one to deduce the impressive eight-squares theorem
(a + a + a + a + a + a + a + a )(b + b + b + b + b + b + b + b )
= (a b − a b − a b − a b − a b − a b − a b − a b )
+ (a b + a b + a b − a b + a b − a b + a b − a b )
+ (a b − a b + a b + a b − a b + a b + a b − a b )
+ (a b + a b − a b + a b + a b + a b − a b − a b )
+ (a b − a b + a b − a b + a b + a b − a b + a b )
+ (a b + a b − a b − a b − a b + a b + a b + a b )
+ (a b − a b − a b + a b + a b − a b + a b + a b )
+ (a b + a b + a b + a b − a b − a b − a b + a b )
(834)
Dvipsbugw
valid for all real numbers ai and bi and thus in particular also for all integers. ( ere are many variations of this expression, with di erent possible sign combinations.) e theorem was discovered in by Carl Ferdinand Degen ( – ), and then rediscovered in by John Graves and in by Arthur Cayley. ere is no generalization to higher numbers of squares, a fact proved by Adolf Hurwitz ( – ) in .
e octonions can be used to show that a vector product can be de ned in more than three dimensions. A vector product or cross product is an operation satisfying
u v = −v u anticommutativity (u v)w = u(v w) exchange rule.
(835)
Using the de nition
X Y = (XY − Y X) ,
(836)
Ref. 1220 Challenge 1558 e
the -products of imaginary quaternions, i.e. of quaternions of the type ( , u), are again imaginary, and correspond to the usual vector product, thus ful lling ( ). Interestingly, it is possible to use de nition ( ) for octonions as well. In that case, the product of imaginary octonions is also imaginary, and ( ) is again satis ed. In fact, this is the only other non-trivial example of a vector product. us a vector product exists only in three and in seven dimensions.
Dvipsbugw
O
Ref. 1223 Challenge 1559 n
Ref. 1224 Page 653
Challenge 1560 ny Ref. 1220
e process of constructing new systems of hypercomplex ‘numbers’ or real algebras by
‘doubling’ a given one can be continued ad in nitum. However, octonions, sedenions and
all the following doublings are neither rings nor elds, but only non-associative algebras
with unity. Other nite-dimensional algebras with unit element over the reals, once called
hypercomplex ‘numbers’, can also be de ned: they include the so-called such as ‘dual
numbers’, ‘double numbers’, ‘Cli ord–Lifshitz numbers’ etc. ey play no special role in
physics.
Mathematicians have also de ned number elds which have ‘one and a bit’ dimensions,
such as algebraic number elds. ere is also a generalization of the concept of integers to
the complex domain: the Gaussian integers, de ned as n+im, where n and m are ordinary
integers. Gauss even de ned what are now known as Gaussian primes. (Can you nd out
how?) ey are not used in the description of nature, but are important in number theory.
Physicists used to call quantum-mechanical operators ‘q-numbers.’ But this term has
now fallen out of fashion.
Another way in which the natural numbers can be extended is to include numbers
larger in nite numbers. e most important such classes of trans nite number are the
ordinals, the cardinals and the surreals (which were de ned in the rst intermezzo). e
ordinals are essentially an extension of the integers beyond in nity, whereas the surreals
are a continuous extension of the reals, also beyond in nity. Loosely speaking, among
the trans nites, the ordinals have a similar role as the integers have among the reals; the
surreals ll in all the gaps between the ordinals, like the reals do for integers. Interestingly,
many series that diverge in R converge n the surreals. Can you nd one example?
e surreals include in nitely small numbers, as do the numbers of nonstandard an-
alysis, also called hyperreals. In both the number systems, in contrast to real numbers, the
numbers and .
(with an in nite string of nines) do not coincide, and indeed
are separated by in nitely many other numbers.
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G
With the discovery of supersymmetry, another system of numbers became important, called the Grassmann numbers.* ey are in fact a special type of hypercomplex numbers. In supersymmetric Lagrangians, elds depend on two types of coordinates: on the usual real space-time coordinates and additionally on the Grassmann coordinates.
Grassmann numbers, also called fermionic coordinates, θ have the de ning properties
θ = and θi θ j + θ jθi = .
(837)
Challenge 1561 ny You may want to look for a matrix representation of these numbers.
V
Vector spaces, also called linear spaces, are mathematical generalizations of certain aspects of the intuitive three-dimensional space. A set of elements any two of which can
* Hermann Günther Grassmann (1809–1877), schoolteacher in Stettin, and one of the most profound mathematical thinkers of the nineteenth century.
Dvipsbugw
be added together and any one of which can be multiplied by a number is called a vector space, if the result is again in the set and the usual rules of calculation hold.
More precisely, a vector space over a number eld K is a set of elements, called vectors, for which a vector addition and a scalar multiplication is de ned, such that for all vectors a, b, c and for all numbers s and r from K one has
(a + b) + c = a + (b + c) = a + b + c n+a=a
(−a) + a = n
a=a (s + r)(a + b) = sa + sb + ra + rb
associativity of vector addition
existence of null vector
existence of negative vector
(838)
regularity of scalar multiplication
complete distributivity of scalar multiplication
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Challenge 1562 ny
If the eld K, whose elements are called scalars in this context, is taken to be the real (or complex, or quaternionic) numbers, one speaks of a real (or complex, or quaternionic) vector space. Vector spaces are also called linear vector spaces or simply linear spaces.
e complex numbers, the set of all real functions de ned on the real line, the set of all polynomials, the set of matrices with a given number of rows and columns, all form vector spaces. In mathematics, a vector is thus a more general concept than in physics. (What is the simplest possible mathematical vector space?)
In physics, the term ‘vector’ is reserved for elements of a more specialized type of vector space, namely normed inner product spaces. To de ne these, we rst need the concept of a metric space.
A metric space is a set with a metric, i.e. a way to de ne distances between elements. A real function d(a, b) between elements is called a metric if
d(a, b) positivity of metric d(a, b) + d(b, c) d(a, c) triangle inequality d(a, b) = if and only if a = b regularity of metric
(839)
Challenge 1563 n
A non-trivial example is the following. We de ne a special distance d between cities. If the two cities lie on a line going through Paris, we use the usual distance. In all other cases, we de ne the distance d by the shortest distance from one to the other travelling via Paris. is strange method de nes a metric between all cities in France.
A normed vector space is a linear space with a norm, or ‘length’, associated to each a vector. A norm is a non-negative number a de ned for each vector a with the properties
ra = r a a+b a + b a = only if a =
linearity of norm triangle inequality regularity
(840)
Challenge 1564 ny Usually there are many ways to de ne a norm for a given space. Note that a norm can always be used to de ne a metric by setting
d(a, b) = a − b
(841)
Dvipsbugw
so that all normed spaces are also metric spaces. is is the natural distance de nition (in contrast to unnatural ones like that between French cities).
e norm is o en de ned with the help of an inner product. Indeed, the most special class of linear spaces are the inner product spaces. ese are vector spaces with an inner product, also called scalar product ë (not to be confused with the scalar multiplication!) which associates a number to each pair of vectors. An inner product space over R satis es
aëb=bëa (ra) ë (sb) = rs(a ë b) (a + b) ë c = a ë c + b ë c a ë (b + c) = a ë b + a ë c
aëa a ë a = if and only if a =
commutativity of scalar product bilinearity of scalar product le distributivity of scalar product right distributivity of scalar product positivity of scalar product regularity of scalar product
Dvipsbugw (842)
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
for all vectors a, b, c and all scalars r, s. A real inner product space of nite dimension is also called a Euclidean vector space. e set of all velocities, the set of all positions, or the set of all possible momenta form such spaces.
An inner product space over C satis es*
aëb=bëa=bëa (ra) ë (sb) = rs(a ë b) (a + b) ë c = a ë c + b ë d a ë (b + c) = a ë b + a ë c
aëa a ë a = if and only if a =
Hermitean property sesquilinearity of scalar product le distributivity of scalar product right distributivity of scalar product positivity of scalar product regularity of scalar product
(843)
Page 1195
for all vectors a, b, c and all scalars r, s. A complex inner product space (of nite dimen-
sion) is also called a unitary or Hermitean vector space. If the inner product space is
complete, it is called, especially in the in nite-dimensional complex case, a Hilbert space.
e space of all possible states of a quantum system forms a Hilbert space.
All inner product spaces are also metric spaces, and thus normed spaces, if the metric
is de ned by
d(a, b) = (a − b) ë (a − b) .
(844)
Challenge 1565 ny Challenge 1566 ny
Only in the context of an inner product spaces we can speak about angles (or phase differences) between vectors, as we are used to in physics. Of course, like in normed spaces, inner product spaces also allows us to speak about the length of vectors and to de ne a basis, the mathematical concept necessary to de ne a coordinate system.
e dimension of a vector space is the number of linearly independent basis vectors. Can you de ne these terms precisely?
Which vector spaces are of importance in physics?
* Two inequivalent forms of the sequilinearity axiom exist. e other is (ra) ë (sb) = rs(a ë b). e term sesquilinear is derived from Latin and means for ‘one-and-a-half-linear’.
Dvipsbugw
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
A
Page 1194
e term algebra is used in mathematics with three di erent, but loosely related, meanings. First, it denotes a part of mathematics, as in ‘I hated algebra at school’. Secondly, it denotes a set of formal rules that are obeyed by abstract objects, as in the expression ‘tensor algebra’. Finally – and this is the only meaning used here – an algebra denotes a speci c type of mathematical structure.
Intuitively, an algebra is a set of vectors with a vector multiplication de ned on it. More precisely, an algebra is a vector space (over a eld K) that is also a ring. ( e concept is due to Benjamin Peirce ( – ), father of Charles Sanders Peirce.) A ring is a set for which an addition and a multiplication is de ned – like the integers. us, in an algebra, there are (o en) three types of multiplications:
— scalar multiplication: the c-fold multiple of a vector x is another vector y = cx; — the (main) algebraic multiplication: the product of two vectors x and y is another
vector z = x y;
— if the vector space is a inner product space, the scalar product: the scalar product of two algebra elements (vectors) x and y is a scalar c = x ë y;
A precise de nition of an algebra thus only needs to de ne properties of the (main) multiplication and to specify the number eld K. An algebra is de ned by the following axioms
Dvipsbugw
x(y + z) = x y + xz , (x + y)z = xz + yz distributivity of multiplication
c(x y) = (cx)y = x(c y) bilinearity
(845)
for all vectors x, y, z and all scalars c K. To stress their properties, algebras are also called linear algebras.
For example, the set of all linear transformations of an n-dimensional linear space (such as the translations on a plane, in space or in time) is a linear algebra, if the composition is taken as multiplication. So is the set of observables of a quantum mechanical system.*
An associative algebra is an algebra whose multiplication has the additional property
Challenge 1567 n
* Linear transformations are mappings from the vector space to itself, with the property that sums and scalar multiples of vectors are transformed into the corresponding sums and scalar multiples of the transformed vectors. Can you specify the set of all linear transformations of the plane? And of three-dimensional space? And of Minkowski space?
All linear transformations transform some special vectors, called eigenvectors (from the German word eigen meaning ‘self ’) into multiples of themselves. In other words, if T is a transformation, e a vector, and
T(e) = λe
(846)
Page 744 Page 805
where λ is a scalar, then the vector e is called an eigenvector of T, and λ is associated eigenvalue. e set of all eigenvalues of a transformation T is called the spectrum of T. Physicists did not pay much attention to these mathematical concepts until they discovered quantum theory. Quantum theory showed that observables are transformations in Hilbert space, because any measurement interacts with a system and thus transforms it. Quantum-mechanical experiments also showed that a measurement result for an observable must be an eigenvalue of the corresponding transformation. e state of the system a er the measurement is given by the eigenvector corresponding to the measured eigenvalue. erefore every expert on motion must know what an eigenvalue is.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
that
x(yz) = (x y)z associativity .
(847)
Challenge 1568 e
Challenge 1569 ny Challenge 1570 n
Most algebras that arise in physics are associative.* erefore, in mathematical physics, a linear associative algebra is o en simply called an algebra.
e set of multiples of the unit of the algebra is called the eld of scalars scal(A) of the algebra A. e eld of scalars is also a subalgebra of A. e eld of scalars and the scalars themselves behave in the same way.
We explore a few examples. e set of all polynomials in one variable (or in several variables) forms an algebra. It is commutative and in nite-dimensional. e constant polynomials form the eld of scalars.
e set of n n matrices, with the usual operations, also forms an algebra. It is n dimensional. ose diagonal matrices (matrices with all o -diagonal elements equal to zero) whose diagonal elements all have the same value form the eld of scalars. How is the scalar product of two matrices de ned?
e set of all real-valued functions over a set also forms an algebra. Can you specify the multiplication? e constant functions form the eld of scalars.
A star algebra, also written -algebra, is an algebra over the complex numbers for which there is a mapping A A, x x , called an involution, with the properties
Dvipsbugw
(x ) = x (x + y) = x + y
(cx) = cx for all c C (x y) = y x
(848)
valid for all elements x, y of the algebra A. e element x is called the adjoint of x. Star algebras are the main type of algebra used in quantum mechanics, since quantummechanical observables form a -algebra.
A C -algebra is a Banach algebra over the complex numbers with an involution (a function that is its own inverse) such that the norm x of an element x satis es
x =x x.
(849)
Challenge 1571 n
(A Banach algebra is a complete normed algebra; an algebra is complete if all Cauchy sequences converge.) In short, C -algebra is a nicely behaved algebra whose elements form a continuous set and a complex vector space. e name C comes from ‘continuous functions’. Indeed, the bounded continuous functions form such an algebra, with a properly de ned norm. Can you nd it?
Every C -algebra contains a space of Hermitean elements (which have a real spectrum), a set of normal elements, a multiplicative group of unitary elements and a set of positive elements (with non-negative spectrum).
We should mention one important type of algebra used in mathematics. A division algebra is an algebra for which the equations ax = b and ya = b are uniquely solvable in
* Note that a non-associative algebra does not possess a matrix representation.
Dvipsbugw
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
x or y for all b and all a . Obviously, all type of continuous numbers must be division algebras. Division algebras are thus one way to generalize the concept of a number. One of the important results of modern mathematics states that ( nite-dimensional) division algebras can only have dimension , like the reals, dimension , like the complex numbers, dimension , like the quaternions, or dimension , like the octonions. ere is thus no way to generalize the concept of (continuous) ‘number’ to other dimensions.
And now for some fun. Imagine a ring A which contains a number eld K as a subring (or ‘ eld of scalars’). If the ring multiplication is de ned in such a way that a general ring element multiplied with an element of K is the same as the scalar multiplication, then A is a vector space, and thus an algebra – provided that every element of K commutes with every element of A. (In other words, the subring K must be central.)
For example, the quaternions H are a four-dimensional real division algebra, but although H is a two-dimensional complex vector space, it is not a complex algebra, because i does not commute with j (one has i j = − ji = k). In fact, there are no nite-dimensional complex division algebras, and the only nite-dimensional real associative division algebras are R, C and H.
Now, if you are not afraid of getting a headache, think about this remark: every Kalgebra is also an algebra over its eld of scalars. For this reason, some mathematicians prefer to de ne an (associative) K-algebra simply as a ring which contains K as a central sub eld.
In physics, it is the algebras related to symmetries which play the most important role. We study them next.
Dvipsbugw
L
A Lie algebra is special type of algebra (and thus of vector space). Lie algebras are the most important type of non-associative algebra. A vector space L over the eld R (or C) with an additional binary operation [ , ], called Lie multiplication or the commutator, is called a real (or complex) Lie algebra if this operation satis es
[X, Y] = −[Y , X] [aX + bY , Z] = a[X, Z] + b[Y , Z] [X, [Y ,Z]] + [Y , [Z, X]] + [Z, [X, Y]] =
antisymmetry (le -)linearity Jacobi identity
(850)
Challenge 1572 e
for all elements X, Y , Z L and for all a, b R (or C). (Lie algebras are named a er Sophus Lie.) e rst two conditions together imply bilinearity. A Lie algebra is called commutative if [X, Y] = for all elements X and Y. e dimension of the Lie algebra is the dimension of the vector space. A subspace N of a Lie algebra L is called an ideal* if [L, N] ⊂ N; any ideal is also a subalgebra. A maximal ideal M which satis es [L, M] = is called the centre of L.
A Lie algebra is called a linear Lie algebra if its elements are linear transformations of another vector space V (intuitively, if they are ‘matrices’). It turns out that every nitedimensional Lie algebra is isomorphic to a linear Lie algebra. erefore, there is no loss
Challenge 1573 ny * Can you explain the notation [L, N]? Can you de ne what a maximal ideal is and prove that there is only one?
Dvipsbugw
Page 1220
of generality in picturing the elements of nite-dimensional Lie algebras as matrices.
e name ‘Lie algebra’ was chosen because the generators, i.e. the in nitesimal ele-
ments of every Lie group, form a Lie algebra. Since all important symmetries in nature
form Lie groups, Lie algebras appear very frequently in physics. In mathematics, Lie al-
gebras arise frequently because from any associative nite-dimensional algebra (in which
the symbol ë stands for its multiplication) a Lie algebra appears when we de ne the com-
mutator by
[X, Y] = X ë Y − Y ë X .
(851)
( is fact gave the commutator its name.) Lie algebras are non-associative in general;
but the above de nition of the commutator shows how to build one from an associative
algebra.
Since Lie algebras are vector spaces, the elements Ti of a basis of the Lie algebra always obey a relation of the form:
[Ti , Tj] = cikjTk .
(852)
k
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Page 1201
e
numbers
c
k ij
are
called
the
structure
constants
of
the
Lie
algebra.
ey depend on
the choice of basis. e structure constants determine the Lie algebra completely. For
example, the algebra of the Lie group SU( ), with the three generators de ned by Ta = σ a i, where the σ a are the Pauli spin matrices, has the structure constants Cabc = εabc.*
C
L
Finite-dimensional Lie algebras are classi ed as follows. Every nite-dimensional Lie algebra is the (semidirect) sum of a semisimple and a solvable Lie algebra.
A Lie algebra is called solvable if, well, if it is not semisimple. Solvable Lie algebras have not yet been classi ed completely. ey are not important in physics.
A semisimple Lie algebra is a Lie algebra which has no non-zero solvable ideal. Other
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
* Like groups, Lie algebras can be represented by matrices, i.e. by linear operators. Representations of Lie
algebras are important in physics because many continuous symmetry groups are Lie groups.
e adjoint representation of a Lie algebra with basis a ...an is the set of matrices ad(a) de ned for each
element a by
[a, a j] = ad(a)c j ac .
(853)
c
e de
nition
implies
that
ad(ai )jk
=
c ikj ,
where
c
k ij
are
the
structure
constants
of
the
Lie
algebra.
For
a
real
Lie algebra, all elements of ad(a) are real for all a L.
Note that for any Lie algebra, a scalar product can be de ned by setting
X ë Y = Tr( adX ë adY ) .
(854)
is scalar product is symmetric and bilinear. (Can you show that it is independent of the representation?) e corresponding bilinear form is also called the Killing form, a er the German mathematician Wilhelm Killing (1847–1923), the discoverer of the ‘exceptional’ Lie groups. e Killing form is invariant under the action of any automorphism of the Lie algebra L. In a given basis, one has
X
ë
Y
=
Tr( (adX)
ë
(adY ))
=
c
i l
k
cski
x
l
ys
=
lsxl ys
where
ls
=
c
i l
k
cski
is
called
the
Cartan
metric
tensor
of
L.
(855)
Dvipsbugw
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equivalent de nitions are possible, depending on your taste:
— a semisimple Lie algebra does not contain non-zero abelian ideals; — its Killing form is non-singular, i.e. non-degenerate; — it splits into the direct sum of non-abelian simple ideals (this decomposition is
unique); — every nite-dimensional linear representation is completely reducible; — the one-dimensional cohomology of with values in an arbitrary nite-dimensional
-module is trivial.
Ref. 1225
Finite-dimensional semisimple Lie algebras have been completely classi ed. ey decom-
pose uniquely into a direct sum of simple Lie algebras. Simple Lie algebras can be complex
or real.
e simple nite-dimensional complex Lie algebras all belong to four in nite classes
and to ve exceptional cases. e in nite classes are also called classical, and are: An for n , corresponding to the Lie groups SL(n + ) and their compact ‘cousins’ SU(n + );
Bn for n , corresponding to the Lie groups SO( n + ); Cn for n , corresponding to the Lie groups Sp( n); and Dn for n , corresponding to the Lie groups SO( n).
us An is the algebra of all skew-Hermitean matrices; Bn and Dn are the algebras of the symmetric matrices; and Cn is the algebra of the traceless matrices.
e exceptional Lie algebras are G , F , E , E , E . In all cases, the index gives the
number of roots. e dimensions of these algebras are An n(n+ ); Bn and Cn n( n+ );
Dn n( n − ); G ; F ; E ; E
;E
.
e simple and nite-dimensional real Lie algebras are more numerous; their classi-
cation follows from that of the complex Lie algebras. Moreover, corresponding to each
complex Lie group, there is always one compact real one. Real Lie algebras are not so
important in fundamental physics.
e so-called superalgebras (see below) play a role in systems with supersymmetry.
Of the large number of in nite-dimensional Lie algebras, only few are important in
physics: among them are the Poincaré algebra, the Cartan algebra, the Virasoro algebra
and a few other Kac–Moody algebras.
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L
Lie superalgebras arise when the concept of Lie algebra is extended to the case of supersymmetry. A Lie superalgebra contains even and odd elements; the even elements correspond to bosons and the odd elements to fermions. Supersymmetry applies to systems with anticommuting coordinates. Lie superalgebras are a natural generalization of Lie algebras to supersymmetry, and simply add a Z -grading.
In detail, a Lie superalgebra is a Z -graded algebra over a eld of characteristic – usually R or C – with a product [., .], called the (Lie) superbracket or supercommutator, that has the properties
[x, y] = −(− ) x y [y, x] = (− ) z x [x, [y, z]] + (− ) x y [y, [z, x]] + (− ) y z [z, [x, y]]
(856)
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where x, y and z are algebra elements that are ‘pure’ in the Z -grading. e expression x denotes the degree of the algebra element x and is either or . e second condition is sometimes called the ‘super Jacobi identity’. Obviously, the even subalgebra of a Lie superalgebra forms a Lie algebra; in that case the superbracket becomes the usual Lie bracket.
As in the case of Lie algebras, the simple Lie superalgebras have been completely classi ed. ey fall into ve in nite classes and some special cases, namely
— A(n, m) corresponding to the Lie supergroups SL(n + m + ) and their compact ‘cousins’ SU(N M);
— B(n, m) corresponding to the Lie supergroups OSp( n + m); — D(n, m) corresponding to the Lie supergroups OSp( n m); — P(n); — Q(n); — the exceptional cases Dα( , ), G( ), F( ); — and nally the Cartan superalgebras.
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TV e Virasoro algebra is the in nite algebra of operators Ln satisfying [Lm, Ln] = (m − n)Lm+n + c (m − m)δm,−n
(857)
Challenge 1574 ny
where the number c, which may be zero, is called the central charge, and the factor is introduced by historical convention. is rather speci c algebra is important in physics because it is the algebra of conformal symmetries in two dimensions.* Can you nd a representation in terms of in nite square matrices? Mathematically speaking, the Virasoro algebra is a special case of a Kac–Moody algebra.
K –M
In physics, more general symmetries than Lie groups also appear. is happens in particular when general relativity is taken into account. Because of the symmetries of space-time, the number of generators becomes in nite. e concepts of Lie algebra and of superalgebra have to be extended to take space-time symmetry into account. e corresponding algebras are the Kac–Moody algebras. (‘Kac’ is pronounced like ‘Katz’.) Kac–Moody algebras are a particular class of in nite-parameter Lie algebras; thus they have an in nite number of generators. ey used to be called associated a ne algebras or a ne Lie algebras; sometimes they are also called Z-graded Lie algebras. ey were introduced independently in by Victor Kac and Robert Moody. We present some speci c examples.
Of basic physical importance are those Kac–Moody algebras associated to a symmetry group G C[t], where t is some continuous parameter. e generators M(an) of the corresponding algebra have two indices, one, a, that indexes the generators of the group G and another, n, that indexes the generators of the group of Laurent polynomials C[t]. e
Page 328 * Note that, in contrast, the conformal symmetry group in four dimensions has 15 parameters, and thus its Lie algebra is nite (15) dimensional.
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–
generators form a Kac–Moody algebra if they obey the relations:
[M
(n) a
,
Mb(m)]
=
C
ab
c
M
(n+m) c
for
n, m = , , , . . . ,
.
(858)
For example, if we take G as SU( ) with its generators Tn = σ a i, σ a being the Pauli spin matrices, then the objects
M(an) = Ta
tn , e.g. M(n) = i
tn −tn
(859) Dvipsbugw
form a Kac–Moody algebra with the structure constants Cabc = εabc of SU( ).
T
–
?
Challenge 1575 n
Topology is group theory.
“ ” e Erlangen program
In a simpli ed view of topology that is su cient for physicists, only one type of entity can possess shape: manifolds. Manifolds are generalized examples of pullovers: they are locally at, can have holes, boundaries and can o en be turned inside out.
Pullovers are subtle entities. For example, can you turn your pullover inside out while your hands are tied together? (A friend may help you.) By the way, the same feat is also possible with your trousers, while your feet are tied together. Certain professors like to demonstrate this during topology lectures – of course with a carefully selected pair of underpants.
T
Ref. 1227
e study of shapes requires a good de nition of a set made of ‘points’. To be able to talk about shape, these sets must be structured in such a way as to admit a useful concept of ‘neighbourhood’ or ‘closeness’ between the elements of the set. e search for the most general type of set which allows a useful de nition of neighbourhood has led to the concept of topological space.
A topological space is a nite or in nite set X of elements, called points, together with a neighbourhood for each point. A neighbourhood N of a point x is a collection of subsets Yx of X with the properties that
— x is in N; — if N and M are neighbourhoods, so is N M; — anything containing a neighbourhood of x is itself a neighbourhood of x.
e choice of the subsets is free. All the subsets Yx , for all points x, that were chosen in the de nition are then called open sets. (A neighbourhood and an open set are not necessarily the same; but all open sets are also neighbourhoods.)
One also calls a topological space a ‘set with a topology’. In e ect, a topology speci es the systems of ‘neighbourhoods’ of every point of the set. ‘Topology’ is also the name of the branch of mathematics that studies topological spaces.
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Challenge 1576 e
For example, the real numbers together with the open intervals form the usual topology of R. If one takes all subsets of Ras open sets, one speaks of the discrete topology. If one takes only the full set X and the empty set as open sets, one speaks of the trivial or indiscrete topology.
e concept of topological space allows us to de ne continuity. A mapping from one topological space X to another topological space Y is continuous if the inverse image of every open set in Y is an open set in X. You may verify that this de nition is not satis ed by a real function that makes a jump. You may also check that the term ‘inverse’ is necessary in the de nition; otherwise a function with a jump would be continuous, as such a function may still map open sets to open sets.*
We thus need the concept of topological space, or of neighbourhood, if we want to express the idea that there are no jumps in nature. We also need the concept of topological space in order to be able to de ne limits.
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– CS – more on topological spaces to be added – CS –
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Of the many special kinds of topological spaces that have been studied, one type is particularly important. A Hausdor space is a topological space in which for any two points x and y there are disjoint open sets U and V such that x is in U and y is in V . A Hausdor space is thus a space where, no matter how ‘close’ two points are, they can always be separated by open sets. is seems like a desirable property; indeed, non-Hausdor spaces are rather tricky mathematical objects. (At Planck energy, it seems that vacuum appears to behave like a non-Hausdor space; however, at Planck energy, vacuum is not really a space at all. So non-Hausdor spaces play no role in physics.) A special case of Hausdor space is well-known: the manifold.
M
In physics, the most important topological spaces are di erential manifolds. Loosely speaking, a di erential manifold is a set of points that looks like Rn under the microscope – at small distances. For example, a sphere and a torus are both two-dimensional di erential manifolds, since they look locally like a plane. Not all di erential manifolds are that simple, as the examples of Figure show.
A di erential manifold is called connected if any two points can be joined by a path lying in the manifold. ( e term has a more general meaning in topological spaces. But the notions of connectedness and pathwise connectedness coincide for di erential manifolds.) We focus on connected manifolds in the following discussion. A manifold is called simply connected if every loop lying in the manifold can be contracted to a point. For example, a sphere is simply connected. A connected manifold which is not simply connected is called multiply connected. A torus is multiply connected.
Manifolds can be non-orientable, as the well-known Möbius strip illustrates. Nonorientable manifolds have only one surface: they do not admit a distinction between front
Challenge 1577 ny
* e Cauchy-Weierstass de nition of continuity says that a real function f (x) is continuous at a point a if (1) f is de ned on a open interval containing a, (2) f (x) tends to a limit as x tends to a, and (3) the limit is f (a). In this de nition, the continuity of f is de ned using the intuitive idea that the real numbers form the basic model of a set that has no gaps. Can you see the connection with the general de nition given above?
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–
F I G U R E 394 Examples of orientable and non-orientable manifolds of two dimensions: a disc, a Möbius strip, a sphere and a Klein bottle
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F I G U R E 395 Compact (left) and noncompact (right) manifolds of various dimensions
Challenge 1578 e
and back. If you want to have fun, cut a paper Möbius strip into two along a centre line. You can also try this with paper strips with di erent twist values, and investigate the regularities.
In two dimensions, closed manifolds (or surfaces), i.e. surfaces that are compact and without boundary, are always of one of three types:
— e simplest type are spheres with n attached handles; they are called n-tori or surfaces of genus n. ey are orientable surfaces with Euler characteristic − n.
— e projective planes with n handles attached are non-orientable surfaces with Euler characteristic − n.
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F I G U R E 396 Simply connected (left), multiply connected (centre) and disconnected (right) manifolds of one (above) and two (below) dimensions
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— e Klein bottles with n attached handles are non-orientable surfaces with Euler characteristic − n.
erefore Euler characteristic and orientability describe compact surfaces up to homeomorphism (and if surfaces are smooth, then up to di eomorphism). Homeomorphisms are de ned below.
e two-dimensional compact manifolds or surfaces with boundary are found by removing one or more discs from a surface in this list. A compact surface can be embedded in R if it is orientable or if it has non-empty boundary.
In physics, the most important manifolds are space-time and Lie groups of observables. We study Lie groups below. Strangely enough, the topology of space-time is not known. For example, it is unclear whether or not it is simply connected. Obviously, the reason is that it is di cult to observe what happens at large distances form the Earth. However, a similar di culty appears near Planck scales.
If a manifold is imagined to consist of rubber, connectedness and similar global properties are not changed when the manifold is deformed. is fact is formalized by saying that two manifolds are homeomorphic (from the Greek words for ‘same’ and ‘shape’) if between them there is a continuous, one-to-one and onto mapping with a continuous inverse. e concept of homeomorphism is somewhat more general than that of rubber deformation, as can be seen from Figure .
H,
Only ‘well-behaved’ manifolds play a role in physics: namely those which are orientable and connected. In addition, the manifolds associated with observables, are always compact. e main non-trivial characteristic of connected compact orientable manifolds is that they contain ‘holes’ (see Figure ). It turns out that a proper description of the holes of manifolds allows us to distinguish between all di erent, i.e. non-homeomorphic, types of manifold.
ere are three main tools to describe holes of manifolds and the relations among them: homotopy, homology and cohomology. ese tools play an important role in the study of gauge groups, because any gauge group de nes a manifold.
– CS – more on topology to be added – CS –
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F I G U R E 397 Examples of homeomorphic pairs of manifolds
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F I G U R E 398 The first four two-dimensional compact connected orientable manifolds: 0-, 1-, 2- and 3-tori
Challenge 1579 d
In other words, through homotopy and homology theory, mathematicians can classify manifolds. Given two manifolds, the properties of the holes in them thus determine whether they can be deformed into each other.
Physicists are now extending these results of standard topology. Deformation is a classical idea which assumes continuous space and time, as well as arbitrarily small action. In nature, however, quantum e ects cannot be neglected. It is speculated that quantum e ects can transform a physical manifold into one with a di erent topology: for example, a torus into a sphere. Can you nd out how this can be achieved?
Topological changes of physical manifolds happen via objects that are generalizations of manifolds. An orbifold is a space that is locally modelled by Rn modulo a nite group. Examples are the tear-drop or the half-plane. Orbifolds were introduced by Satake Ichiro in ; the name was coined by William urston. Orbifolds are heavily studied in string theory.
T
Page 192
We introduced groups early on because groups play an important role in many parts of physics, from the description of solids, molecules, atoms, nuclei, elementary particles and forces up to the study of shapes, cycles and patterns in growth processes.
Group theory is also one of the most important branches of modern mathematics, and is still an active area of research. One of the aims of group theory is the classi cation of all groups. is has been achieved only for a few special types. In general, one distinguishes between nite and in nite groups. Finite groups are better understood.
Every nite group is isomorphic to a subgroup of the symmetric group SN , for some number N. Examples of nite groups are the crystalline groups, used to classify crystal structures, or the groups used to classify wallpaper patterns in terms of their symmetries.
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Ref. 1226
e symmetry groups of Platonic and many other regular solids are also nite groups.
Finite groups are a complex family. Roughly speaking, a general ( nite) group can be
seen as built from some fundamental bricks, which are groups themselves. ese fun-
damental bricks are called simple ( nite) groups. One of the high points of twentieth-
century mathematics was the classi cation of the nite simple groups. It was a collaborat-
ive e ort that took around years, roughly from to . e complete list of nite
simple groups consists of
) the cyclic groups Zp of prime group order; ) the alternating groups An of degree n at least ve; ) the classical linear groups, PSL(n; q), PSU(n; q), PSp( n; q) and PΩε(n; q);
) the exceptional or twisted groups of Lie type D (q), E (q), E (q), E (q), E (q),
F (q), F ( n), G (q), G ( n) and B ( n);
) the sporadic groups, namely M , M , M , M , M (the Mathieu groups), J ,
J , J , J (the Janko groups), Co , Co , Co (the Conway groups), HS, Mc, Suz (the Co
‘babies’), Fi , Fi , Fi′ (the Fischer groups), F = M (the Monster), F , F , F , He (= F )
(the Monster ‘babies’), Ru, Ly, and ON.
e classi cation was nished in the s a er over
pages of publications.
e proof is so vast that a special series of books has been started to summarize and
explain it. e rst three families are in nite. e last family, that of the sporadic groups,
is the most peculiar; it consists of those nite simple groups which do not t into the
other families. Some of these sporadic groups might have a role in particle physics: pos-
sibly even the largest of them all, the so-called Monster group. is is still a topic of
research. ( e Monster group has about . ë elements; more precisely, its order is
or ë ë ë
ë ë ë ë ë ë ë ë ë ë ë .)
Of the in nite groups, only those with some niteness condition have been studied. It
is only such groups that are of interest in the description of nature. In nite groups are
divided into discrete groups and continuous groups. Discrete groups are an active area
of mathematical research, having connections with number theory and topology. Con-
tinuous groups are divided into nitely generated and in nitely generated groups. Finitely
generated groups can be nite-dimensional or in nite-dimensional.
e most important class of nitely generated continuous groups are the Lie groups.
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L
In nature, the Lagrangians of the fundamental forces are invariant under gauge transformations and under continuous space-time transformations. ese symmetry groups are examples of Lie groups, which are a special type of in nite continuous group. ey are named a er the great Norwegian mathematician Sophus Lie ( – ). His name is pronounced like ‘Lee’.
A (real) Lie group is an in nite symmetry group, i.e. a group with in nitely many elements, which is also an analytic manifold. Roughly speaking, this means that the elements of the group can be seen as points on a smooth (hyper-) surface whose shape can be described by an analytic function, i.e. by a function so smooth that it can be expressed as a power series in the neighbourhood of every point where it is de ned. e points of the Lie group can be multiplied according to the group multiplication. Furthermore, the
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coordinates of the product have to be analytic functions of the coordinates of the factors, and the coordinates of the inverse of an element have to be analytic functions of the coordinates of the element. In fact, this de nition is unnecessarily strict: it can be proved that a Lie group is just a topological group whose underlying space is a nite-dimensional, locally Euclidean manifold.
A complex Lie group is a group whose manifold is complex and whose group operations are holomorphic (instead of analytical) functions in the coordinates.
In short, a Lie group is a well-behaved manifold in which points can be multiplied (and technicalities). For example, the circle T = z C z = , with the usual complex
multiplication, is a real Lie group. It is abelian. is group is also called S , as it is the onedimensional sphere, or U( ), which means ‘unitary group of one dimension’. e other one-dimensional Lie groups are the multiplicative group of non-zero real numbers and its subgroup, the multiplicative group of positive real numbers.
So far, in physics, only linear Lie groups have played a role – that is, Lie groups which act as linear transformations on some vector space. ( e cover of SL( ,R) or the complex compact torus are examples of non-linear Lie groups.) e important linear Lie groups for physics are the Lie subgroups of the general linear group GL(N,K), where K is a number
eld. is is de ned as the set of all non-singular, i.e. invertible, N N real, complex or quaternionic matrices. All the Lie groups discussed below are of this type.
Every complex invertible matrix A can be written in a unique way in terms of a unitary matrix U and a Hermitean matrix H:
Dvipsbugw
A = UeH .
(860)
Challenge 1580 n (H is given by H = ln A†A, and U is given by U = Ae−H.)
– CS – more on Lie groups to be added – CS –
e simple Lie groups U( ) and SO( ,R) and the Lie groups based on the real and complex numbers are abelian (see Table ); all others are non-abelian.
Lie groups are manifolds. erefore, in a Lie group one can de ne the distance between two points, the tangent plane (or tangent space) at a point, and the notions of integration and di erentiations. Because Lie groups are manifolds, Lie groups have the same kind of structure as the objects of Figures , and . Lie groups can have any number of dimensions. Like for any manifold, their global structure contains important information; let us explore it.
C
It is not hard to see that the Lie groups SU(N) are simply connected for all N = , . . . ; they have the topology of a N-dimensional sphere. e Lie group U( ), having the topology of the -dimensional sphere, or circle, is multiply connected.
e Lie groups SO(N) are not simply connected for any N = , . . . . In general, SO(N,K) is connected, and GL(N,C) is connected. All the Lie groups SL(N,K) are connected; and SL(N,C) is simply connected. e Lie groups Sp(N,K) are connected; Sp( N,C) is simply connected. Generally, all semi-simple Lie groups are connected.
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e Lie groups O(N,K), SO(N,M,K) and GL(N,R) are not connected; they contain two connected components.
Note that the Lorentz group is not connected: it consists of four separate pieces. Like the Poincaré group, it is not compact, and neither is any of its four pieces. Broadly speaking, the non-compactness of the group of space-time symmetries is a consequence of the non-compactness of space-time.
C
Page 1209
A Lie group is compact if it is closed and bounded when seen as a manifold. For a given parametrization of the group elements, the Lie group is compact if all parameter ranges are closed and nite intervals. Otherwise, the group is called non-compact. Both compact and non-compact groups play a role in physics. e distinction between the two cases is important, because representations of compact groups can be constructed in the same simple way as for nite groups, whereas for non-compact groups other methods have to be used. As a result, physical observables, which always belong to a representation of a symmetry group, have di erent properties in the two cases: if the symmetry group is compact, observables have discrete spectra; otherwise they do not.
All groups of internal gauge transformations, such as U( ) and SU(n), form compact groups. In fact, eld theory requires compact Lie groups for gauge transformations. e only compact Lie groups are Tn, O(n), U(n), SO(n) and SU(n), their double cover Spin(n) and the Sp(n). In contrast, SL(n,R), GL(n,R), GL(n,C) and all others are not compact.
Besides being manifolds, Lie groups are obviously also groups. It turns out that most of their group properties are revealed by the behaviour of the elements which are very close (as points on the manifold) to the identity.
Every element of a compact and connected Lie group has the form exp(A) for some A. e elements A arising in this way form an algebra, called the corresponding Lie algebra. For any linear Lie group, every element of the connected subgroup can be expressed as a nite product of exponentials of elements of the corresponding Lie algebra. In short, Lie algebras express the local properties of Lie groups. at is the reason for their importance in physics.
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TA B L E 93 Properties of the most important real and complex Lie groups
L
D
-P
a
L
-D
L
1. Real groups
Rn
Euclidean abelian, simply
Rn
space with connected, not
addition
compact;
π =π =
R
non-zero real abelian, not
R
numbers with connected, not
multiplica- compact; π = Z ,
tion
no π
abelian, thus Lie bracket is zero; not simple
abelian, thus Lie bracket is zero
D-
real n
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L
D
-P
a
L
-D
L
D-
R
positive real abelian, simply
R
numbers with connected, not
multiplica- compact;
tion
π =π =
S = R Z complex
abelian, connected, R
= U( ) = numbers of not simply
T
absolute value connected, compact;
= SO( ) , with multi- π = , π = Z = Spin( ) plication
H
non-zero
simply connected, H
quaternions not compact;
with multi- π = π = plication
S
quaternions simply connected, Im(H)
of absolute compact;
value , with isomorphic to
multiplica- SU( ), Spin( ) and
tion, also
to double cover of
known as Sp( );
SO( ); π = π =
topologically
a -sphere
GL(n, R) general linear not connected, not
group:
compact; π = Z ,
invertible no π
n-by-n real
matrices
M(n, R)
n-by-n real simply connected, M(n, R)
GL+(n, R) matrices with not compact; π = ,
positive
for n = : π = Z, for
determinant n : π = Z ;
GL+( , R)
isomorphic to R
SL(n, R)
special linear simply connected, sl(n, R) group: real not compact if n ; = An− matrices with π = , for n = : determinant π = Z, for n :
π = Z ; SL( , R) is a single point, SL( , R) is isomorphic to
SU( , ) and Sp( , R)
abelian, thus Lie bracket is zero
abelian, thus Lie bracket is zero
quaternions, with Lie bracket the commutator
quaternions with zero real part, with Lie bracket the commutator; simple and semi-simple; isomorphic to real -vectors, with Lie bracket the cross product; also isomorphic to su( ) and to so( ) n-by-n matrices, with n Lie bracket the commutator
n-by-n matrices, with n Lie bracket the commutator
n-by-n matrices with n − trace , with Lie bracket the commutator
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L
D
-P
a
L
-D
L
D-
O(n, R) = O(n)
orthogonal group: real orthogonal matrices; symmetry of hypersphere
not connected, compact; π = Z , no π
so(n, R)
skew-symmetric
n(n − )
n-by-n real matrices,
with Lie bracket the
commutator; so( , R)
is isomorphic to su( )
and to R with the
cross product
SO(n, R) special
connected, compact; so(n, R) skew-symmetric
= SO(n) orthogonal for n not simply = B n− or n-by-n real matrices,
group: real connected; π = , D n
with Lie bracket the
orthogonal for n = : π = Z, for
commutator; for n =
matrices with n : π = Z
and n simple and
determinant
semisimple; SO( ) is
n(n − )
semisimple but not
simple
Spin(n)
spin group; simply connected
double cover for n , compact;
of SO(n); for n = and n
Spin( ) is
simple and
isomorphic to semisimple; for
Q , Spin( ) to n : π = , for
S
n :π =
so(n, R)
skew-symmetric n-by-n real matrices, with Lie bracket the commutator
n(n − )
symplectic Sp( n, R) group: real
symplectic matrices
not compact; π = , sp( n, R) real matrices A that n( n + )
π =Z
= Cn
satisfy JA + AT J =
where J is the standard
skew-symmetric
matrix;b simple and
semisimple
Sp(n) for compact
n
symplectic
group:
quaternionic
n n unitary
matrices
compact, simply connected;
π =π =
sp(n)
n-by-n quaternionic matrices A satisfying A = −A , with Lie bracket the
commutator; simple
and semisimple
n( n + )
U(n)
unitary group: not simply
u(n)
complex
connected, compact;
n n unitary it is not a complex
matrices
Lie group/algebra;
π = , π = Z;
isomorphic to S for
n-by-n complex
n
matrices A satisfying
A = −A , with Lie
bracket the
commutator
n=
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
L
D
-P
a
L
-D
L
D-
SU(n)
special
simply connected,
unitary group: compact; it is not a
complex
complex Lie
n n unitary group/algebra;
matrices with π = π =
determinant
su(n)
n-by-n complex
n−
matrices A with trace
satisfying A = −A ,
with Lie bracket the
commutator; for n
simple and
semisimple
2. Complex groupsc
Cn
group
abelian, simply
Cn
operation is connected, not
addition
compact;
π =π =
C
nonzero
abelian, not simply C
complex
connected, not
numbers with compact; π = ,
multiplica- π = Z
tion
GL(n, C) general linear simply connected, M(n, C)
group:
not compact; π = ,
invertible π = Z; for n =
n-by-n
isomorphic to C
complex
matrices
SL(n, C) special linear simply connected;
group:
for n not
complex
compact;
sl(n, C)
matrices with π = π = ; determinant SL( , C) is
isomorphic to
Spin( , C) and Sp( , C)
projective not compact; π = , sl( , C) PSL( , C) special linear π = Z
group;
isomorphic to the Möbius
group, to the
restricted
Lorentz group
SO+( , , R) and to SO( , C)
abelian, thus Lie bracket is zero
complex n
abelian, thus Lie bracket is zero
n-by-n matrices, with n Lie bracket the commutator
n-by-n matrices with n − trace , with Lie bracket the commutator; simple, semisimple; sl( , C) is isomorphic to su( , C) C
-by- matrices with trace , with Lie bracket the commutator; sl( , C) is isomorphic to su( , C) C
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L
D
-P
a
L
-D
L
D-
O(n, C) orthogonal group: complex orthogonal matrices
SO(n, C) special orthogonal group: complex orthogonal matrices with determinant
symplectic Sp( n, C) group:
complex symplectic matrices
not connected; for n not compact; π = Z , no π
for n not compact; not simply connected; π = , for n = : π = Z, for n :π =Z ; nonabelian for n , SO( , C) is abelian and isomorphic to C not compact; π =π =
so(n, C) so(n, C)
skew-symmetric n-by-n complex matrices, with Lie bracket the commutator
skew-symmetric n-by-n complex matrices, with Lie bracket the commutator; for n = and n simple and semisimple
n(n − ) n(n − )
sp( n, C) complex matrices that n( n + ) satisfy JA + AT J = where J is the standard skew-symmetric matrix;b simple and
semi-simple
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
a. e group of components π of a Lie group is given; the order of π is the number of components of the Lie group. If the group is trivial ( ), the Lie group is connected. e fundamental group π of a connected Lie group is given. If the group π is trivial ( ), the Lie group is simply connected. is table is based on that in the Wikipedia, at http://en.wikipedia.org/wiki/Table_of_Lie_groups. b. e standard skew-symmetric matrix J of rank n is Jkl = δk,n+l − δk+n,l .
c. Complex Lie groups and Lie algebras can be viewed as real Lie groups and real Lie algebras of twice the
dimension.
M
Mathematics is a passion in itself.
**
Mathematics provides many counter-intuitive results. Reading a book on the topic, such
as B
R. G
& J M.H. O
, eorems and Counter-examples
in Mathematics, Springer, 1993, can help you sharpen your mind.
**
e distinction between one, two and three dimensions is blurred in mathematics. is is
well demonstrated in the text H S
, Space Filling Curves, Springer Verlag, 1994.
**
ere are at least seven ways to win a million dollars with mathematical research. e Clay Mathematics Institute at http://www.claymath.org o ers them for major advances
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in seven topics:
— proving the Birch and Swinnerton–Dyer conjecture about algebraic equations; — proving the Poincaré conjecture about topological manifolds; — solving the Navier–Stokes equations for uids; — nding criteria distinguishing P and NP numerical problems; — proving the Riemann hypothesis stating that the nontrivial zeros of the zeta function
lie on a line; — proving the Hodge conjectures; — proving the connection between Yang–Mills theories and a mass gap in quantum eld
theory.
On each of these topics, substantial progress can buy you a house.
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No man but a blockhead ever wrote except for money.
“ ” Samuel Johnson
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
B
1218 A good reference is the Encyclopedia of Mathematics, in volumes, Kluwer Academic Publishers, − . It explains most concepts used in mathematics. Spending an hour with it looking up related keywords is an e cient way to get an introduction into any part of mathematics, especially into the vocabulary and the main connections. Cited on page .
1219 S.L. A
, Rotations, Quaternions and Double Groups, Clarendon Press, , and
also S.L. A
, Hamilton, Rodriguez and the quaternion scandal, Mathematical
Magazine 62, pp. – , . See also J.C. H , G.K. F
& L.H. K
,
Visualzing quaternion rotation, ACM Transactions on Graphics 13, pp. – , . e
latter can be downloaded in several places via the internet. Cited on page .
1220 An excellent introduction into number systems in mathematics, including nonstandard
numbers, quaternions, octonions, p-adic numbers and surreal numbers, is the book H.-D.
E
, H. H
, F. H
, M. K
, K. M
, J. N -
, A. P
& R. R
, Zahlen, rd edition, Springer Verlag, . It is also
available in English, under the title Numbers, Springer Verlag, . Cited on pages ,
, and .
1221 A. W , Quaternions in Electrodynamics, ous websites. Cited on pages and .
. e text can be downloaded from vari-
1222 See the ne book by L
H. K
, Knots and Physics, World Scienti c, nd
edition, , which gives a clear and visual introduction to the mathematics of knots and
their main applications to physics. Cited on page .
1223 Gaussian integers are explored by G.H. H
& E.M. W
, An Introduction to
the eory of Numbers, th edition, Clarendon Press, Oxford, , in the sections . ‘ e
Rational Integers, the Gaussian Integers, and the Integers’, pp. – , and . ‘Properties
of the Gaussian Integers’ pp. – . For challenges relating to Gaussian integers, look at
http://www.mathpuzzle.com/Gaussians.html. Cited on page .
1224 About trans nite numbers, see the delightful paperback by R R
, In nity and
the Mind – the Science and Philosophy of the In nite, Bantam, . Cited on page .
1225 M. F
, P. S
& G. Z
(editors), Applications of Group eory in Phys-
ics and Mathematical Physics, Lectures in applied mathematics, volume , American Math-
ematical Society, . is interesting book was written before the superstring revolution,
so that the topic is missing from the otherwise excellent presentation. Cited on page .
1226 An introduction to the classi cation theorem is R. S
, On nite simple groups
and their classi cation, Notices of the AMS 42, pp. – , , also available on the web
as http://www.ams.org/notices/
/solomon.ps Cited on page .
1227 For an introduction to topology, see for example M N and Physics, IOP Publishing, . Cited on page .
, Geometry, Topology
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
A
E
SOURCES OF INFORMATION ON
MOTION
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No place a ords a more striking conviction of the vanity of human hopes than a public library.
“ Samuel Johnson
In a consumer society there are inevitably two
” kinds of slaves: the prisoners of addiction and “the prisoners of envy.
Ivan Illich*
” the text, outstanding books introducing neighbouring domains are presented
n footnotes. e bibliographies at the end of each chapter collect general material
In order to satisfy further curiosity about what is encountered in this mountain ascent.
All citations can also be found by looking up the author’s name in the index. To nd additional information, either libraries or the internet can help.
In a library, review articles of recent research appear in journals such as Reviews of Modern Physics, Reports on Progress in Physics, Contemporary Physics and Advances in Physics. Good pedagogical introductions are found in the American Journal of Physics, the European Journal of Physics and Physik in unserer Zeit. Another useful resource is Living Reviews in Relativity, found at http://www.livingreviews.org.
Overviews on research trends occasionally appear in magazines such as Physics World, Physics Today, Europhysics Journal, Physik Journal and Nederlands tijdschri voor natuurkunde. For coverage of all the sciences together, the best sources are the magazines Nature, New Scientist, Naturwissenscha en, La Recherche and the cheap but excellent Science News.
Research papers appear mainly in Physics Letters B, Nuclear Physics B, Physical Review D, Physical Review Letters, Classical and Quantum Gravity, General Relativity and Gravitation, International Journal of Modern Physics and Modern Physics Letters. e newest results and speculative ideas are found in conference proceedings, such as the Nuclear Physics B Supplements. Research articles also appear in Fortschritte der Physik, Zeitschri für Physik C, La Rivista del Nuovo Cimento, Europhysics Letters, Communications in Mathematical Physics, Journal of Mathematical Physics, Foundations of Physics, International Journal of eoretical Physics and Journal of Physics G. ere is also the purely electronic New Journal of Physics, which can be found at the http://www.njp.org website.
Papers on the description of motion without time and space which appear a er this text is published can be found via the Scienti c Citation Index. It is published in printed form and as compact disc, and allows, one to search for all publications which cite a given
* Ivan Illich (b. 1926 Vienna, d. 2002 Bremen), Austrian theologian and social and political thinker.
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TA B L E 94 The structure of the Arxiv preprint archive for physics and related topics at http://www.arxiv.org
T
A
general relativity and quantum cosmology astrophysics experimental nuclear physics theoretical nuclear physics theoretical high-energy physics computational high-energy physics phenomenological high-energy physics experimental high-energy physics quantum physics general physics condensed matter physics nonlinear sciences mathematical physics mathematics computer science quantitative biology
gr-qc astro-ph nucl-ex nucl-th hep-th hep-lat hep-ph hep-ex quant-ph physics cond-mat nlin math-ph math CoRR q-bio
To receive preprints by email, send an email to an address of the form gr-qc@arxiv.org (or the corresponding abbreviation in place of ‘grqc’), with a subject line consisting simply of the word ‘help’, without the quotes.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
paper. en, using the bimonthly Physics Abstracts, which also exists in both paper and electronic form, you can look up the abstract of the paper and check whether it is of interest.
But by far the simplest and most e cient way to keep in touch with ongoing research on motion is to use the internet, the international computer network. To anybody with a personal computer connected to a telephone, most theoretical physics papers are available free of charge, as preprints, i.e. before o cial publication and checking by referees, at the http://www.arxiv.org website. Details are given in Table . A service for nding subsequent preprints that cite a given one is also available.
In the last decade of the twentieth century, the internet expanded into a combination of library, media store, discussion platform, order desk, brochure collection and time waster. Today, commerce, advertising and – unfortunately – crime of all kind are also an integral part of the web. With a personal computer, a modem and free browser so ware, one can look for information in millions of pages of documents. e various parts of the documents are located in various computers around the world, but the user does not need to be aware of this.*
* Several decades ago, the provocative book by I I listed four basic ingredients for any educational system:
, Deschooling Society, Harper & Row, 1971,
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To start using the web, ask a friend who knows.* Searching the web for authors, organizations, books, publications, companies or simple keywords using search engines can be a rewarding or a time-wasting experience. A selection of interesting servers are given below.
TA B L E 95 Some interesting servers on the world-wide web
T
W
‘ URL ’
General topics
Wikipedia
http://www.wikipedia.org
Information search
http://www.altavista.com
engines
http://www.metager.de
http://www.google.com
http://www.yahoo.com
Search old usenet articles http://groups.google.com
Frequently asked questions on http://www.faqs.org
physics and other topics
Libraries
http://www.konbib.nl
http://portico.bl.uk
http://www.theeuropeanlibrary.org
http://www.hero.ac.uk/uk/niss/niss_library .cfm
http://www.bnf.fr
http://www.grass-gis.de/bibliotheken
http://www.loc.gov
Physics Research preprints
http://www.arxiv.org – see page
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
1. access to resources for learning, e.g. books, equipment, games, etc. at an a ordable price, for everybody, at any time in their life;
2. for all who want to learn, access to peers in the same learning situation, for discussion, comparison, cooperation and competition;
3. access to elders, e.g. teachers, for their care and criticism towards those who are learning; 4. exchanges between students and performers in the eld of interest, so that the latter can be models for the former. For example, there should be the possibility to listen to professional musicians and reading the works of specialist writers. is also gives performers the possibility to share, advertise and use their skills. Illich develops the idea that if such a system were informal – he then calls it a ‘learning web’ or ‘opportunity web’ – it would be superior to formal, state- nanced institutions, such as conventional schools, for the development of mature human beings. ese ideas are deepened in his following works, Deschooling Our Lives, Penguin, 1976, and Tools for Conviviality, Penguin, 1973. Today, any networked computer o ers one or more of the following: email (electronic mail), ftp ( le transfer to and from another computer), access to Usenet (the discussion groups on speci c topics, such as particle physics), and the powerful world-wide web. (Roughly speaking, each of those includes the ones before.) In a rather unexpected way, all these facilities of the internet have transformed it into the backbone of the ‘opportunity web’ discussed by Illich. However, as in any school, it strongly depends on the user’s discipline whether the internet actually does provide a learning web. * It is also possible to use both the internet and to download les through FTP with the help of email only. But the tools change too o en to give a stable guide here. Ask your friend.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
T
W
‘ URL ’
http://www.slac.stanford.edu/spires
Particle data
http://pdg.web.cern.ch/pdg
Physics news, weekly
http://www.aip.org/physnews/update
Physics news, daily
http://www.innovations-report.de/berichte/physik.php
Physics problems by Yakov http://star.tau.ac.il/QUIZ/ Kantor
Physics problems by Henry http://www.phy.duke.edu/~hsg/physics-challenges/challenges.
Greenside
html
Physics ‘question of the week’ http://www.physics.umd.edu/lecdem/outreach/QOTW/active
Physics ‘miniproblem’
http://www.nyteknik.se/miniproblemet
O cial SI unit site
http://www.bipm.fr
Unit conversion
http://www.chemie.fu-berlin.de/chemistry/general/units.html
‘Ask the experts’
http://www.sciam.com/askexpert_directory.cfm
Abstracts of papers in physics http://www.osti.gov journals
Science News
http://www.sciencenews.org
Nobel Prize winners
http://www.nobel.se/physics/laureates
Pictures of physicists
http://www.if.ufrj.br/famous/physlist.html
Gravitation news
http://www.phys.lsu.edu/mog.html
Living Reviews in Relativity http://www.livingreviews.org
Information on relativity
http://math.ucr.edu/home/baez/relativity.html
Relativistic imaging and lms http://www.tat.physik.uni-tuebingen.de/~weiskopf
Physics organizations
http://www.cern.ch/
http://www.hep.net
http://www.nikhef.nl
http://www.het.brown.edu/physics/review/index.html
Physics textbooks on the web http://www.plasma.uu.se/CED/Book
http://www.biophysics.org/education/resources.htm
http://www.lightandmatter.com
http://www.motionmountain.net
ree beautiful French sets of http://feynman.phy.ulaval.ca/marleau/notesdecours.htm notes on classical mechanics and particle theory
e excellent Radical Freshman http://www.physics.nmt.edu/~raymond/teaching.html Physics by David Raymond
Physics course scripts from http://ocw.mit.edu/OcwWeb/Physics/index.html MIT
Physics lecture scripts in German and English
http://www.akleon.de
‘World lecture hall’
http://www.utexas.edu/world/lecture
Engineering data and formulae http://www.efunda.com/
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T
W
‘ URL ’
Mathematics
‘Math forum’ internet resource http://mathforum.org/library/
collection
Biographies of mathematicians http://www-history.mcs.st-andrews.ac.uk/BiogIndex.html
Purdue math problem of the http://www.math.purdue.edu/academics/pow/
week
Macalester College maths http://mathforum.org/wagon/
problem of the week
Mathematical formulae
http://dlmf.nist.gov
Functions
http://functions.wolfram.com
Symbolic integration
http://www.integrals.com
Weisstein’s World of
http://mathworld.wolfram.com
Mathematics
Curiosities
Minerals
ESA
NASA
Hubble space telescope Sloan Digital Sky Survey
e ‘cosmic mirror’ Solar system simulator Observable satellites Astronomy picture of the day
e Earth from space Current solar data Optical illusions Petit’s science comics Physical toys Physics humour Literature on magic Algebraic surfaces Making paper aeroplanes
Small ying helicopters Ten thousand year clock Gesellscha Deutscher Naturforscher und Ärzte Pseudoscience
http://webmineral.com http://www.mindat.org http://sci.esa.int http://www.nasa.gov http://hubble.nasa.gov http://skyserver.sdss.org http://www.astro.uni-bonn.de/~d scher/mirror http://space.jpl.nasa.gov http://li o .msfc.nasa.gov/RealTime/JPass/ / http://antwrp.gsfc.nasa.gov/apod/astropix.html http://www.visibleearth.nasa.gov http://www.n kl.org/sun http://www.sandlotscience.com http://www.jp-petit.com http://www.e .physik.tu-muenchen.de/~cucke/toylinke.htm http://www.dctech.com/physics/humor/biglist.php http://www.faqs.org/faqs/magic-faq/part / http://www.mathematik.uni-kl.de/~hunt/drawings.html http://www.pchelp.net/paper_ac.htm http://www.ivic.qc.ca/~aleexpert/aluniversite/klinevogelmann. html http://pixelito.reference.be http://www.longnow.org http://www.gdnae.de/
http://suhep.phy.syr.edu/courses/modules/PSEUDO/pseudo_ main.html
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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T
W
‘ URL ’
Crackpots Mathematical quotations
e ‘World Question Center’ Plagiarism Hoaxes
http://www.crank.net http://math.furman.edu/mwoodard/~mquot.html http://www.edge.org/questioncenter.html http://www.plagiarized.com http://www.museumofhoaxes.com
Do you want to study physics without actually going to university? Nowadays it is possible to do so via email and internet, in German, at the University of Kaiserslautern.* In the near future, a nationwide project in Britain should allow the same for English-speaking students. As an introduction, use the latest update of this physics text!
Das Internet ist die o enste Form der geschlossenen Anstalt.**
“ Matthias Deutschmann ” Si tacuisses, philosophus mansisses.*** “ ” A er Boethius.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
* See the http://www.fernstudium-physik.de website. ** ‘ e internet is the most open form of a closed institution.’ *** ‘If you had kept quiet, you would have remained a philosopher.’ A er the story Boethius tells in De consolatione philosophiae, 2.7, 67 .
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A
F
CHALLENGE HINTS AND SOLUTIONS
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Never make a calculation before you know the answer.
“ John Wheeler’s motto ” John Wheeler wanted people to estimate, to try and to guess; but not saying it out loud.
A correct guess reinforces the physics instinct, whereas a wrong one leads to the pleasure of surprise. is text contains 1580 challenges. Let me know the challenge for which you want a hint or a solution to be added next.
Challenge 2, page 23: ese topics are all addressed later in the text. Challenge 3, page 30: ere are many ways to distinguish real motion from an illusion of motion: for example, only real motion can be used to set something else into motion. In addition, the motion illusions of the gures show an important failure; nothing moves if the head and the paper remain xed with respect to each other. In other words, the illusion only ampli es existing motion, it does not create motion from nothing. Challenge 4, page 30: Without detailed and precise experiments, both sides can nd examples to prove their point. Creation is supported by the appearance of mould or bacteria in a glass of water; creation is also supported by its opposite, namely traceless disappearance, such as the disappearance of motion. However, conservation is supported and creation falsi ed by all those investigations that explore assumed cases of appearance or disappearance in full detail. Challenge 6, page 32: Political parties, sects, helping organizations and therapists of all kinds are typical for this behaviour. Challenge 7, page 36: e issue is not yet completely settled for the motion of empty space, such as in the case of gravitational waves. In any case, empty space is not made of small particles of nite size, as this would contradict the transversality of gravity waves. Challenge 8, page 38: e circular de nition is: objects are de ned as what moves with respect to the background, and the background is de ned as what stays when objects change. We shall return to this important issue several times in our adventure. It will require a certain amount of patience to solve it, though. Challenge 9, page 39: Holes are not physical systems, because in general they cannot be tracked. Challenge 10, page 39: See page 815. Challenge 11, page 40: A ghost can be a moving image; it cannot be a moving object, as objects cannot interpenetrate. See page 785. Challenge 12, page 40: Hint: yes, there is such a point. Challenge 13, page 40: Can one show at all that something has stopped moving? Challenge 14, page 40: How would you measure this?
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Challenge 15, page 40: e number of reliable digits of a measurement result is a simple quanti cation of precision.
Challenge 16, page 40: No; memory is needed for observation and measurements.
Challenge 17, page 40: Note that you never have observed zero speed.
Challenge 18, page 41: (264 − 1) = 18 446 744 073 700 551 615 grains of rice, given a world harvest of 500 million tons, are about 4000 years of rice harvests.
Challenge 19, page 41: Some books state that the ame leans inwards. But experiments are not easy, and sometimes the ame leans outwards. Just try it. Can you explain your observations?
Challenge 20, page 41: Accelerometers are the simplest motion detectors. ey exist in form of piezo devices that produce a signal whenever the box is accelerated and can cost as little as one euro. Another accelerometer that might have a future is an interference accelerometer that makes use of the motion of an interference grating; this device might be integrated in silicon. Other, more precise accelerometers use gyroscopes or laser beams running in circles.
Velocimeters and position detectors can also detect motion; they need a wheel or at least an optical way to look out of the box. Tachographs in cars are examples of velocimeters, computer mice are examples of position detectors.
Challenge 21, page 41: e ball rolls towards the centre of the table, as the centre is somewhat lower than the border, shoots over, and then performs an oscillation around that centre. e period is 84 min, as shown in challenge 299.
Challenge 22, page 41: Accelerations can be felt. Many devices measure accelerations and then deduce the position. ey are used in aeroplanes when ying over the atlantic.
Challenge 23, page 41: e necessary rope length is nh, where n is the number of wheels/pulleys.
Challenge 24, page 41: e block moves twice as fast as the cylinders, independently of their radius.
Challenge 25, page 41: is methods is known to work with other fears as well.
Challenge 26, page 42: ree couples require 11 passages. Two couples require 5. For four or more couples there is no solution. What is the solution if there are n couples and n − 1 places on the boat?
Challenge 27, page 42: In everyday life, this is correct; what happens when quantum e ects are taken into account?
Challenge 28, page 43: ere is only one way: compare the velocity to be measured with the speed of light. In fact, almost all physics textbooks, both for schools and for university, start with the de nition of space and time. Otherwise excellent relativity textbooks have di culties avoiding this habit, even those that introduce the now standard k-calculus (which is in fact the approach mentioned here). Starting with speed is the logically cleanest approach.
Challenge 29, page 44: Take the average distance change of two neighbouring atoms in a piece of quartz over the last million years. Do you know something still slower?
Challenge 30, page 45: Equivalently: do points in space exist? e third part studies this issue in detail; see page 1001.
Challenge 31, page 46: All electricity sources must use the same phase when they feed electric power into the net. Clocks of computers on the internet must be synchronized.
Challenge 32, page 46: Note that the shi increases quadratically with time, not linearly.
Challenge 33, page 47: Natural time is measured with natural motion. Natural motion is the motion of light. Natural time is this de ned with the motion of light.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Ref. 35
Challenge 34, page 48: Galileo measured time with a scale (and with other methods). His stop-
watch was a water tube that he kept closed with his thumb, pointing into a bucket. To start the
stopwatch, he removed his thumb, to stop it, he put it back on. e volume of water in the bucket
then gave him a measure of the time interval. is is told in his famous book G
G
,
Discorsi e dimostrazioni matematiche intorno a due nuove scienze attenenti alla mecanica e i movi-
menti locali, usually simply called the ‘Discorsi’, which he published in 1638 with Louis Elsevier
in Leiden, in the Netherlands.
Challenge 35, page 48: ere is no way to de ne a local time at the poles that is consistent with all neighbouring points.
Challenge 37, page 50: e forest is full of light and thus of light rays.
Challenge 38, page 50: One pair of muscles moves the lens along the third axis by deforming the eye from prolate to spherical to oblate.
Challenge 39, page 50: is you can solve trying to think in four dimensions. Try to imagine how to switch the sequence when two pieces cross.
Challenge 40, page 51: Measure distances using light.
Challenge 43, page 54: It is easier to work with the unit torus. Take the unit interval [0, 1] and equate the end points. De ne a set B in which the elements are a given real number b from the interval plus all those numbers who di er from that real by a rational number. e unit circle can be thought as the union of all the sets B. (In fact, every set B is a shi ed copy of the rational numbers Q.) Now build a set A by taking one element from each set B. en build the set family consisting of the set A and its copies Aq shi ed by a rational q. e union of all these sets is the unit torus. e set family is countably in nite. en divide it into two countably in nite set families. It is easy to see that each of the two families can be renumbered and its elements shi ed in such a way that each of the two families forms a unit torus.
Mathematicians say that there is no countably in nitely additive measure of Rn or that sets such as A are non-measurable. As a result of their existence, the ‘multiplication’ of lengths is possible. Later on we shall explore whether bread or gold can be multiplied in this way.
Challenge 44, page 54: Hint: start with triangles.
Challenge 45, page 54: An example is the region between the x-axis and the function which assigns 1 to every transcendental and 0 to every non-transcendental number.
Challenge 46, page 55: We use the de nition of the function of the text. e dihedral angle of a regular tetrahedron is an irrational multiple of π, so the tetrahedron has a non-vanishing Dehn invariant. e cube has a dihedral angle of π 2, so the Dehn invariant of the cube is 0. erefore, the cube is not equidecomposable with the regular tetrahedron.
Challenge 47, page 55: If you think you can show that empty space is continuous, you are wrong. Check your arguments. If you think you can prove the opposite, you might be right – but only if you already know what is explained in the third part of the text. If that is not the case, check your arguments.
Challenge 48, page 56: Obviously, we use light to check that the plumb line is straight, so the two de nitions must be the same. is is the case because the eld lines of gravity are also possible paths for the motion of light. However, this is not always the case; can you spot the exceptions?
Another way to check straightness is along the surface of calm water.
Challenge 49, page 56: e hollow Earth theory is correct if the distance formula is used consistently. In particular, one has to make the assumption that objects get smaller as they approach the centre of the hollow sphere. Good explanations of all events are found on http://www.geocities. com/inversedearth/. Quite some material can be found on the internet, also under the names of
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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paper discs
F I G U R E 399 A simple way to measure bullet speeds
d
b
w
L R
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 400 Leaving a parking space – the outer turning radius
celestrocentric system, inner world theory or concave Earth theory. ere is no way to prefer one description over the other, except possibly for reasons of simplicity or intellectual laziness. Challenge 51, page 57: A hint is given in Figure 399. For the measurement of the speed of light with almost the same method, see page 278. Challenge 52, page 57: Color is a property that applies only to objects, not to boundaries. e question shows that it is easy to ask questions that make no sense also in physics. Challenge 53, page 57: You can do this easily yourself. You can even nd websites on the topic. Challenge 54, page 57: e required gap is
d = (L − b)2 − w2 + 2w R2 − (L − b)2 − L + b ,
as deduced from Figure 400. Challenge 55, page 58: A smallest gap does not exist: any value will do! Can you show this? Challenge 56, page 58: e rst solution sent in will go here. Challenge 57, page 58: Clocks with two hands: 22 times. Clocks with three hands: 2 times. Challenge 58, page 59: For two hands, the answer is 143 times. Challenge 59, page 59: e Earth rotates with 15 minutes per minute. Challenge 60, page 59: You might be astonished, but no reliable data exist on this question. e highest speed of a throw measured so far seems to be a 45 m s cricket bowl. By the way, much more data are available for speeds achieved with the help of rackets. e c. 70 m s of fast badminton smashes seem to be a good candidate for record racket speed; similar speeds are achieved by golf balls.
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F I G U R E 401 A simple drawing proving Pythagoras’ theorem
F I G U R E 402 The trajectory of the middle point between the two ends of the hands of a clock
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Page 154
Challenge 61, page 59: Yes, it can. In fact, many cats can slip through as well.
Challenge 62, page 59: 1.8 km h or 0.5 m s.
Challenge 63, page 59: Nothing, neither a proof nor a disproof.
Challenge 64, page 60: e di erent usage re ects the idea that we are able to determine our position by ourselves, but not the time in which we are. e section on determinism will show how wrong this distinction is.
Challenge 65, page 60: Yes, there is. However, this is not obvious, as it implies that space and time are not continuous, in contrast to what we learn in primary school. e answer will be found in the third part of this text.
Challenge 66, page 60: For a curve, use, at each point, the curvature radius of the circle approximating the curve in that point; for a surface, de ne two directions in each point and use two such circles along these directions.
Challenge 67, page 60: It moves about 1 cm in 50 ms.
Challenge 68, page 60: e surface area of the lung.
Challenge 69, page 60: e nal shape is a full cube without any hole.
Challenge 70, page 60: See page 279.
Challenge 71, page 60: A hint for the solution is given by Figure 401.
Challenge 72, page 60: Because they are or were uid.
Challenge 73, page 60: e shape is shown in Figure 402; it has eleven lobes. Challenge 74, page 61: e cone angle φ is related to the solid angle Ω through Ω = 2π(1 − cos φ 2).
Challenge 76, page 61: See Figure 403.
Challenge 78, page 62: Hint: draw all objects involved.
Challenge 79, page 62: Hint: there is an in nite number of such shapes.
Challenge 80, page 62: e curve is obviously called a catenary, from Latin ‘catena’ for chain. e formula for a catenary is y = a cosh(x a). If you approximate the chain by short straight
segments, you can make wooden blocks that can form an arch without any need for glue. e St. Louis arch is in shape of a catenary. A suspension bridge has the shape of a catenary before it is loaded, i.e. before the track is attached to it. When the bridge is nished, the shape is in between a catenary and a parabola.
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1° 10°
4° 3°
6°
3° 2° 3°
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 403 The angles defined by the hands against the sky, when the arms are extended
Challenge 81, page 63: A limit does not exist in classical physics; however, there is one in nature which appears as soon as quantum e ects are taken into account. Challenge 82, page 63: e inverse radii, or curvatures, obey a2 + b2 + c2 + d2 = (1 2)(a + b + c + d)2. is formula was discovered by René Descartes. If one continues putting circles in the remaining spaces, one gets so-called circle packings, a pretty domain of recreational mathematics.
ey have many strange properties, such as intriguing relations between the coordinates of the circle centres and their curvatures. Challenge 83, page 63: One option: use the three-dimensional analogue of Pythagoras’s theorem. e answer is 9. Challenge 84, page 63: Draw a logarithmic scale, i.e., put every number at a distance corresponding to its natural logarithm. Challenge 85, page 63: Two more. Challenge 86, page 63: e Sun is exactly behind the back of the observer; it is setting, and the rays are coming from behind and reach deep into the sky in the direction opposite to that of the Sun. A slightly di erent situation – equally useful for getting used to perspective drawing – appears when you have a lighthouse in your back. Can you draw it? Challenge 87, page 63: Problems appear when quantum e ects are added. A two-dimensional universe would have no matter, since matter is made of spin 1/2 particles. But spin 1/2 particles do not exist in two dimensions. Can you nd other reasons? Challenge 88, page 65: From x = t2 2 you get the following rule: square the number of seconds, multiply by ve and you get the depth in metres. Challenge 89, page 65: Just experiment. Challenge 90, page 65: e Academicians suspended one cannon ball with a thin wire just in front of the mouth of the cannon. When the shot was released, the second, ying cannon ball
ew through the wire, thus ensuring that both balls started at the same time. An observer from far away then tried to determine whether both balls touched the Earth at the same time. e experiment is not easy, as small errors in the angle and air resistance confuse the results. Challenge 91, page 66: A parabola has a so-called focus or focal point. All light emitted from that point and re ected exits in the same direction: all light ray are emitted in parallel. e name
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‘focus’ – Latin for replace – expresses that it is the hottest spot when a parabolic mirror is illuminated. Where is the focus of the parabola y = 2x? (Ellipses have two foci, with a slightly di erent de nition. Can you nd it?)
Challenge 92, page 66: Neglecting air resistance and approximating the angle by 45°, we get v = d , or about 3.8 m s. is speed is created by a stead pressure build-up, using blood pressure,
which is suddenly released with a mechanical system at the end of the digestive canal. e cited reference tells more about the details.
Challenge 93, page 67: On horizontal ground, for a speed v and an angle from the horizontal α, neglecting air resistance and the height of the thrower, the distance d is d = v2 sin 2α .
Challenge 94, page 67: Walk or run in the rain, measure your own speed v and the angle from the vertical α with which the rain appears to fall. en the speed of the rain is vrain = v tan α.
Challenge 95, page 67: Check your calculation with the information that the 1998 world record is juggling with 9 balls.
Challenge 96, page 67: e long jump record could surely be increased by getting rid of the sand stripe and by measuring the true jumping distance with a photographic camera; that would allow jumpers to run more closely to their top speed. e record could also be increased by a small inclined step or by a spring-suspended board at the take-o location, to increase the takeo angle.
Challenge 97, page 67: It is said so, as rain drops would then be ice spheres and fall with high speed.
Challenge 98, page 67: It seems not too much. But the lead in them can poison the environment.
Challenge 99, page 67: Stones never follow parabolas: when studied in detail, i.e. when the change of with height is taken into account, their precise path turns out to be an ellipse. is shape appears most clearly for long throws, such as throws around the part of the Earth, or for orbiting objects. In short, stones follow parabolas only if the Earth is assumed to be at. If its curvature is taken into account, they follow ellipses.
Challenge 102, page 69: e set of all rotations around a point in a plane is indeed a vector space. What about the set of all rotations around all points in a plane? And what about the threedimensional cases?
Challenge 103, page 70: e scalar product between two vectors a and b is given by
ab = ab cos ∢(a, b) .
(861)
How does this di er form the vector product?
Challenge 104, page 70: Professor to student: What is the derivative of velocity? Acceleration! What is the derivative of acceleration? I don’t know. Jerk! e fourth, h and sixth derivatives of position are sometimes called snap, crackle and pop.
Challenge 105, page 70: A candidate for low acceleration of a physical system might be the accelerations measured by gravitational wave detectors. ey are below 10−13 m s2.
Challenge 107, page 71: One can argue that any source of light must have nite size.
Challenge 109, page 72: What the unaided human eye perceives as a tiny black point is usually about 50 µm in diameter.
Challenge 110, page 72: See page 570.
Challenge 111, page 72: One has to check carefully whether the conceptual steps that lead us to extract the concept of point from observations are correct. It will be shown in the third part of the adventure that this is not the case.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 112, page 73: One can rotate the hand in a way that the arm makes the motion described. See also page 783.
Challenge 113, page 73: Any number, without limit.
Challenge 114, page 73: e blood and nerve supply is not possible if the wheel has an axle. e method shown to avoid tangling up connections only works when the rotating part has no axle: the ‘wheel’ must oat or be kept in place by other means. It thus becomes impossible to make a wheel axle using a single piece of skin. And if a wheel without an axle could be built (which might be possible), then the wheel would periodically run over the connection. Could such a axle-free connection realize a propeller?
By the way, it is still thinkable that animals have wheels on axles, if the wheel is a ‘dead’ object. Even if blood supply technologies like continuous ow reactors were used, animals could not make such a detached wheel grow in a way tuned to the rest of the body and they would have di culties repairing a damaged wheel. Detached wheels cannot be grown on animals; they must be dead.
Challenge 115, page 74: e brain in the skull, the blood factories inside bones or the growth of the eye are examples.
Challenge 116, page 74: One can also add the Sun, the sky and the landscape to the list.
Challenge 117, page 75: Ghosts, hallucinations, Elvis sightings, or extraterrestrials must all be one or the other. ere is no third option. Even shadows are only special types of images.
Challenge 118, page 75: e issue was hotly discussed in the seventeenth century; even Galileo argued for them being images. However, they are objects, as they can collide with other objects, as the spectacular collision between Jupiter and the comet Shoemaker-Levy 9 in 1994 showed. In the meantime, satellites have been made to collide with comets and even to shoot at them (and hitting).
Challenge 119, page 77: e minimum speed is roughly the one at which it is possible to ride without hands. If you do so, and then gently push on the steering wheel, you can make the experience described above. Watch out: too strong a push will make you fall badly.
Challenge 120, page 79: If the ball is not rotating, a er the collision the two balls will depart with a right angle between them.
Challenge 121, page 79: Part of the energy is converted into heat; the rest is transferred as kinetic energy of the concrete block. As the block is heavy, its speed is small and easily stopped by the human body. is e ect works also with anvils, it seems. In another common variation the person does not lie on nails, but on air: he just keeps himself horizontal, with head and shoulders on one chair, and the feet on a second one.
Challenge 122, page 80: Yes, mass works also for magnetism, because the precise condition is not that the interaction be central, but that it realizes a more general condition, which includes accelerations such as those produced by magnetism. Can you deduce the condition from the de nition of mass?
Challenge 123, page 80: e weight decreased due to the evaporated water lost by sweating and, to a minor degree, due to the exhaled carbon bound in carbon dioxide.
Challenge 124, page 80: Rather than using the inertial e ects of the Earth, it is easier to deduce its mass from its gravitational e ects. See challenge 234.
Challenge 128, page 82: At rst sight, relativity implies that tachyons have imaginary mass; however, the imaginary factor can be extracted from the mass–energy and mass–momentum relation, so that one can de ne a real mass value for tachyons; as a result, faster tachyons have smaller energy and smaller momentum. Both momentum and energy can be a negative number of any size.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 129, page 82: Legs are never perfectly vertical; they would immediately glide away. Once the cat or the person is on the oor, it is almost impossible to stand up again.
Challenge 130, page 82: Momentum (or centre of mass) conservation would imply that the environment would be accelerated into the opposite direction. Energy conservation would imply that a huge amount of energy would be transferred between the two locations, melting everything in between. Teleportation would thus contradict energy and momentum conservation.
Challenge 131, page 83: e part of the tides due to the Sun, the solar wind, and the interactions between both magnetic elds are examples of friction mechanisms between the Earth and the Sun.
Challenge 132, page 84: With the factor 1 2, increase of (physical) kinetic energy is equal to the (physical) work performed on a system: total energy is thus conserved only if the factor 1/2 is added.
Challenge 134, page 85: It is a smart application of momentum conservation.
Challenge 135, page 85: Neither. With brakes on, the damage is higher, but still equal for both cars.
Challenge 136, page 86: Heating systems, transport engines, engines in factories, steel plants, electricity generators covering the losses in the power grid, etc. By the way, the richest countries in the world, such as Sweden or Switzerland, consume only half the energy per inhabitant as the USA. is waste is one of the reasons for the lower average standard of living in the USA.
Challenge 138, page 88: If the Earth changed its rotation speed ever so slightly we would walk inclined, the water of the oceans would ow north, the atmosphere would be lled with storms and earthquakes would appear due to the change in Earth’s shape.
Challenge 140, page 89: Just throw it into the air and compare the dexterity needed to make it turn around various axes.
Challenge 141, page 90: Use the de nition of the moment of inertia and Pythagoras’ theorem for every mass element of the body.
Challenge 142, page 90: Hang up the body, attaching the rope in two di erent points. e crossing point of the prolonged rope lines is the centre of mass.
Challenge 143, page 90: Spheres have an orientation, because we can always add a tiny spot on their surface. is possibility is not given for microscopic objects, and we shall study this situation in the part on quantum theory.
Challenge 146, page 91: See Tables 14 and 15.
Challenge 147, page 91: Self-propelled linear motion contradicts the conservation of momentum; self-propelled change of orientation (as long as the motion stops again) does not contradict any conservation law. But the deep, nal reason for the di erence will be unveiled in the third part of our adventure.
Challenge 148, page 91: Yes, the ape can reach the banana. e ape just has to turn around its own axis. For every turn, the plate will rotate a bit towards the banana. Of course, other methods, like blowing at a right angle to the axis, peeing, etc., are also possible.
Challenge 150, page 92: e points that move exactly along the radial direction of the wheel form a circle below the axis and above the rim. ey are the points that are sharp in Figure 38 of page 92.
Challenge 151, page 92: Use the conservation of angular momentum around the point of contact. If all the wheel’s mass is assumed in the rim, the nal rotation speed is half the initial one; it is independent of the friction coe cient.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 155, page 96: A short pendulum of length L that swings in two dimensions (with amplitude ρ and orientation φ) shows two additional terms in the Lagrangian :
=
T
−
V
=
1 2
m
ρ˙2
(1
+
ρ2 L2
)
+
l
2 z
2mρ2
−
1 2
mω20
ρ
2(1
+
ρ2 4 L2
)
(862)
where as usual the basic frequency is ω20 = L and the angular momentum is lz = mρ2φ˙. e two additional terms disappear when L ; in that case, if the system oscillates in an ellipse
with semiaxes a and b, the ellipse is xed in space, and the frequency is ω0. For nite pendulum length L, the frequency changes to
ω
=
ω0 (1
−
a2 + b2 16 L2
)
;
most of all, the ellipse turns with a frequency
(863)
Ω
=
ω
3 8
ab L2
.
(864)
( ese formulae can be derived using the least action principle, as shown by C.G. G , G.
K & V.A. N
, Progress in classical and quantum variational principles, http://www.
arxiv.org/abs/physics/0312071.) In other words, a short pendulum in elliptical motion shows a
precession even without the Coriolis e ect. Since this precession frequency diminishes with 1 L2,
the e ect is small for long pendulums, where only the Coriolis e ect is le over. To see the Coriolis
e ect in a short pendulum, one thus has to avoid that it starts swinging in an elliptical orbit by
adding a suppression method of elliptical motion.
Challenge 156, page 96: e Coriolis acceleration is the reason for the deviation from the straight line. e Coriolis acceleration is due to the change of speed with distance from the rotation axis. Now think about a pendulum, located in Paris, swinging in the North-South direction with amplitude A. At the Southern end of the swing, the pendulum is further from the axis by A sin φ, where φ is the latitude. At that end of the swing, the central support point overtakes the pendulum bob with a relative horizontal speed given by v = 2πA sin φ 24 h. e period of precession is given by TF = v 2πA, where 2πA is the circumference 2πA of the envelope of the pendulum’s path (relative to the Earth). is yields TF = 24 h sin φ.
Challenge 157, page 97: e axis stays xed with respect to distant stars, not with respect to absolute space (which is an entity that cannot be observed at all).
Challenge 158, page 97: Rotation leads to a small frequency and thus colour changes of the circulating light.
Challenge 159, page 97: e weight changes when going east or when moving west due to the Coriolis acceleration. If the rotation speed is tuned to the oscillation frequency of the balance, the e ect is increased by resonance. is trick was also used by Eötvös.
Challenge 160, page 97: e Coriolis acceleration makes the bar turn, as every moving body is de ected to the side, and the two de ections add up in this case. e direction of the de ection depends on whether the experiments is performed on the northern or the southern hemisphere.
Challenge 161, page 98: When rotated by π around an east–west axis, the Coriolis force produces a dri velocity of the liquid around the tube. It has the value
v = 2ωr sin θ,
(865)
as long as friction is negligible. Here ω is the angular velocity of the Earth, θ the latitude and r
the (larger) radius of the torus. For a tube with 1 m diameter in continental Europe, this gives a speed of about 6.3 ë 10−5 m s.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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light source
Ωt
t=0
t = 2πR c
F I G U R E 404 Deducing the expression for the Sagnac effect
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
e measurement can be made easier if the tube is restricted in diameter at one spot, so that the velocity is increased there. A restriction by an area factor of 100 increases the speed by the same factor. When the experiment is performed, one has to carefully avoid any other e ects that lead to moving water, such as temperature gradients across the system.
Challenge 162, page 98: Imagine a circular light path (for example, inside a circular glass bre)
and two beams moving in opposite directions along it, as shown in Figure 404. If the bre path
rotates with rotation frequency Ω, we can deduce that, a er one turn, the di erence ∆L in path
length is
∆L
=
2RΩt
=
4πR2 Ω c
.
(866)
e phase di erence is thus
∆φ
=
8π2 R2 cλ
Ω
(867)
if the refractive index is 1. is is the required formula for the main case of the Sagnac e ect.
Challenge 163, page 101: e original result by Bessel was 0.3136 ′′, or 657.7 thousand orbital radii, which he thought to be 10.3 light years or 97.5 Pm.
Challenge 165, page 104: e galaxy forms a stripe in the sky. e galaxy is thus a attened structure. is is even clearer in the infrared, as shown more clearly in Figure 197 on page 439. From the attening (and its circular symmetry) we can deduce that the galaxy must be rotating.
us other matter must exist in the universe.
Challenge 166, page 104: Probably the ‘rest of the universe’ was meant by the writer. Indeed, a moving a part never shi s the centre of gravity of a closed system. But is the universe closed? Or a system? e third part of the adventure centres on these issues.
Challenge 167, page 105: Hint: an energy per distance is a force.
Challenge 170, page 105: e scale reacts to your heartbeat. e weight is almost constant over time, except when the heart beats: for a short duration of time, the weight is somewhat lowered at each beat. Apparently it is due to the blood hitting the aortic arch when the heart pumps it upwards. e speed of the blood is about 0.3 m s at the maximum contraction of the le ventricle.
e distance to the aortic arch is a few centimetres. e time between the contraction and the reversal of direction is about 15 ms.
Challenge 171, page 105: e conservation of angular momentum saves the glass. Try it.
Challenge 172, page 105: e mass decrease could also be due to expelled air. e issue is still open.
Challenge 173, page 105: Assuming a square mountain, the height h above the surrounding
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crust and the depth d below are related by
h = ρm − ρc
d
ρc
(868)
where ρc is the density of the crust and ρm is the density of the mantle. For the density values given, the ratio is 6.7, leading to an additional depth of 6.7 km below the mountain.
Challenge 177, page 107: e behaviour of the spheres can only be explained by noting that elastic waves propagate through the chain of balls. Only the propagation of these elastic waves, in particular their re ection at the end of the chain, explains that the same number of balls that hit on one side are li ed up on the other. For long times, friction makes all spheres oscillate in phase. Can you con rm this?
Challenge 178, page 107: When the short cylinder hits the long one, two compression waves start to run from the point of contact through the two cylinders. When each compression wave arrives at the end, it is re ected as an expansion wave. If the geometry is well chosen, the expansion wave coming back from the short cylinder can continue into the long one (which is still in his compression phase). For su ciently long contact times, waves from the short cylinder can thus depose much of their energy into the long cylinder. Momentum is conserved, as is energy; the long cylinder is oscillating in length when it detaches, so that not all its energy is translational energy. is oscillation is then used to drive nails or drills into stone walls. In commercial hammer drills, length ratios of 1:10 are typically used.
Challenge 179, page 108: e momentum transfer to the wall is double when the ball rebounds perfectly.
Challenge 180, page 108: If the cork is in its intended position: take the plastic cover o the cork, put the cloth around the bottle (this is for protection reasons only) and repeatedly hit the bottle on the oor or a fall in an inclined way, as shown in Figure 33 on page 83. With each hit, the cork will come out a bit.
If the cork has fallen inside the bottle: put half the cloth inside the bottle; shake until the cork falls unto the cloth. Pull the cloth out: rst slowly, until the cloth almost surround the cork, and then strongly.
Challenge 182, page 108: e atomic force microscope.
Challenge 183, page 108: Use Figure 42 on page 95 for the second half of the trajectory, and think carefully about the rst half.
Challenge 184, page 108: Hint: starting rockets at the Equator saves a lot of energy, thus of fuel and of weight.
Challenge 185, page 109: Running man: E 0.5 ë 80 kg ë (5 m s)2 = 1 kJ; ri e bullet: E 0.5 ë 0.04 kg ë (500 m s)2 = 5 kJ.
Challenge 186, page 109: e ame leans towards the inside.
Challenge 187, page 109: e ball leans in the direction it is accelerated to. As a result, one could imagine that the ball in a glass at rest pulls upwards because the oor is accelerated upwards. We will come back to this issue in the section of general relativity.
Challenge 188, page 109: It almost doubles in size.
Challenge 189, page 109: For your exam it is better to say that centrifugal force does not exist. But since in each stationary system there is a force balance, the discussion is somewhat a red herring.
Challenge 191, page 110: Place the tea in cups on a board and attach the board to four long ropes that you keep in your hand.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 192, page 110: e friction if the tides on Earth are the main cause.
Challenge 195, page 111: An earthquake with Richter magnitude of 12 is 1000 times the energy of the 1960 Chile quake with magnitude 10; the latter was due to a crack throughout the full 40 km of the Earth’s crust along a length of 1000 km in which both sides slipped by 10 m with respect to each other. Only the impact of a meteorite could lead to larger values than 12.
Challenge 196, page 111: is is not easy; a combination of friction and torques play a role. See
for example the article J. S
, E. S
& C. L
, Real-time rigid body simula-
tion of some classical mechanical toys, 10th European Simulation and Symposium and Exhibition
(ESS ’98) 1998, pp. 93–98, or http//www.lennerz.de/paper_ess98.pdf.
Challenge 198, page 111: If a wedding ring rotates on an axis that is not a principal one, angular momentum and velocity are not parallel.
Challenge 199, page 111: Yes; it happens twice a year. To minimize the damage, dishes should be dark in colour.
Challenge 200, page 111: A rocket red from the back would be a perfect defence against planes attacking from behind. However, when released, the rocket is e ectively ying backwards with respect to the air, thus turns around and then becomes a danger to the plane that launched it. Engineers who did not think about this e ect almost killed a pilot during the rst such tests.
Challenge 202, page 111: Whatever the ape does, whether it climbs up or down or even lets himself fall, it remains at the same height as the mass. Now, what happens if there is friction at the wheel?
Challenge 204, page 111: Weigh the bullet and shoot it against a mass hanging from the ceiling. From the mass and the angle it is de ected to, the momentum of the bullet can be determined.
Challenge 206, page 112: Yes, if he moves at a large enough angle to the direction of the boat’s motion.
Challenge 208, page 112:
e
moment
of
inertia
is
Θ
=
2 5
mr2.
Challenge 209, page 112: e moments of inertia are equal also for the cube, but the values are
Θ
=
1 6
ml2.
e e orts required to put a sphere and a cube into rotation are thus di erent.
Challenge 210, page 112: See the article by C. U & H.-J. S
, Faszinierendes
Dynabee, Physik in unserer Zeit 33, pp. 230–231, 2002.
Challenge 211, page 112: See the article by C. U & H.-J. S Büroklammer, Physik in unserer Zeit 36, pp. 33–35, 2005.
, Die kreisende
Challenge 212, page 113: Yes. Can you imagine what happens for an observer on the Equator?
Challenge 213, page 113: A straight line at the zenith, and circles getting smaller at both sides. See an example on the website http://antwrp.gsfc.nasa.gov/apod/ap021115.html.
Challenge 215, page 114: e plane is described in the websites cited; for a standing human the plane is the vertical plane containing the two eyes.
Challenge 216, page 114: As said before, legs are simpler than wheels to grow, to maintain and to repair; in addition, legs do not require at surfaces (so-called ‘streets’) to work.
Challenge 217, page 115: e staircase formula is an empirical result found by experiment, used by engineers world-wide. Its origin and explanation seems to be lost in history.
Challenge 218, page 115: Classical or everyday nature is right-le symmetric and thus requires an even number of legs. Walking on two-dimensional surfaces naturally leads to a minimum of four legs.
Challenge 220, page 116: e length of the day changes with latitude. So does the length of a shadow or the elevation of stars at night, facts that are easily checked by telephoning a friend.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Page 669
Ships appear at the horizon rst be showing only their masts. ese arguments, together with the round shadow of the earth during a lunar eclipse and the observation that everything falls downwards everywhere, were all given already by Aristotle, in his text On the Heavens. It is now known that everybody in the last 2500 years knew that the Earth is s sphere. e myth that many people used to believe in a at Earth was put into the world – as rhetorical polemic – by Copernicus. e story then continued to be exaggerated more and more during the following centuries, because a new device for spreading lies had just been invented: book printing. Fact is that since 2500 years the vast majority of people knew that the Earth is a sphere.
Challenge 221, page 116: Robert Peary had forgotten that on the date he claimed to be at the North Pole, 6th of April 1909, the Sun is very low on the horizon, casting very long shadows, about ten times the height of objects. But on his photograph the shadows are much shorter. (In fact, the picture is taken in such a way to hide all shadows as carefully as possible.) Interestingly, he had even convinced the US congress to o cially declare him the rst man on the North Pole in 1911. (A rival had claimed to have reached it earlier on, but his photograph has the same mistake.)
Challenge 222, page 116: Yes, the e ect has been measured for skyscrapers. Can you estimate the values?
Challenge 223, page 117: e tip of the velocity arrow, when drawn over time, produces a circle around the centre of motion.
Challenge 226, page 118: e value of the product GM for the Earth is 4.0 ë 1014 m3 s2.
Challenge 227, page 118: All points can be reached for general inclinations; but when shooting horizontally in one given direction, only points on the rst half of the circumference can be reached.
Challenge 229, page 119: On the moon, the gravitational acceleration is 1.6 m s2, about one sixth of the value on Earth. e surface values for the gravitational acceleration for the planets can be found on many internet sites.
Challenge 230, page 119: e Atwood machine is the answer: two almost equal weights connected by a string hanging from a well-oiled wheel. e heavier one falls very slowly. Can you determine the acceleration as a function of the two masses?
Challenge 231, page 119: You should absolutely try to understand the origin of this expression. It allows to understand many essential concepts of mechanics. e idea is that for small amplitudes, the acceleration of a pendulum of length l is due to gravity. Drawing a force diagram for a pendulum at a general angle α shows that
ma = −m sin α
ml
d2 α dt2
=
−m
sin α
l
d2 α dt2
=
−
sin α .
For the mentioned small amplitudes (below 15°) we can approximate this to
(869)
l
d2 α dt2
=−
α.
(870)
is is the equation for a harmonic oscillation (i.e., a sinusoidal oscillation). is:
α(t) = A sin(ωt + φ) .
e resulting motion (871)
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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e amplitude A and the phase φ depend on the initial conditions; however, the oscillation frequency is given by the length of the pendulum and the acceleration of gravity (check it!):
ω= l .
(872)
(For arbitrary amplitudes, the formula is much more complex; see the internet or special mechanics books for more details.)
Challenge 232, page 119: Walking speed is proportional to l T, which makes it proportional to l 1 2.
Challenge 234, page 120: Cavendish suspended a horizontal handle with a long metal wire. He then approached a large mass to the handle, avoiding any air currents, and measured how much the handle rotated.
Challenge 235, page 120: e acceleration due to gravity is a = Gm r2 5 nm s2 for a mass
of 75 kg. For a y with mass mfly = 0.1 g landing on a person with a speed of vfly = 1 cm s and deforming the skin (without energy loss) by d = 0.3 mm, a person would be accelerated by a = (v2 d)(mfly m) = 0.4 µm s2. e energy loss of the inelastic collision reduces this value at least by a factor of ten.
Challenge 237, page 122: e easiest way to see this is to picture gravity as a ux emanating form a sphere. is gives a 1 rd−1 dependence for the force and thus a 1 rd−2 dependence of the
potential.
Challenge 239, page 123: Since the paths of free fall are ellipses, which are curves lying in a plane, this is obvious.
Challenge 241, page 124: e low gravitational acceleration of the Moon, 1.6 m s2, implies that gas molecules at usual temperatures can escape its attraction.
Challenge 242, page 126: A ash of light is sent to the Moon, where several Cat’s-eyes have been deposited by the Lunakhod and Apollo missions. e measurement precision of the time a ash take to go and come back is su cient to measure the Moon’s distance change. For more details, see challenge 546.
Challenge 249, page 129: is is a resonance e ect, in the same way that a small vibration of a string can lead to large oscillation of the air and sound box in a guitar.
Challenge 251, page 131: e total angular momentum of the Earth and the Moon must remain constant.
Challenge 257, page 136: e centre of mass of a broom falls with the usual acceleration; the end thus falls faster.
Challenge 258, page 136: Just use energy conservation for the two masses of the jumper and
the string. For more details, including the comparison of experimental measurements and the-
ory, see N. D
& R. B
, De valversnelling bij bungee-jumping, Nederlands
tijdschri voor natuurkunde 69, pp. 316–318, October 2003.
Challenge 259, page 136: About 1 ton.
Challenge 260, page 136: About 5 g.
Challenge 261, page 137: Your weight is roughly constant; thus the Earth must be round. On a at Earth, the weight would change from place to place.
Challenge 262, page 137: Nobody ever claimed that the centre of mass is the same as the centre of gravity! e attraction of the Moon is negligible on the surface of the Earth.
Challenge 264, page 137: at is the mass of the Earth. Just turn the table on its head.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Ref. 98
Challenge 267, page 137: e Moon will be about 1.25 times as far as it is now. e Sun then will slow down the Earth–Moon system rotation, this time due to the much smaller tidal friction from the Sun’s deformation. As a result, the Moon will return to smaller and smaller distances to Earth. However, the Sun will have become a red giant by then, a er having swallowed both the Earth and the Moon.
Challenge 269, page 137: As Galileo determined, for a swing (half a period) the ratio is 2 pi. (See challenge 231). But not more than two, maybe three decimals of π can be determined this way.
Challenge 270, page 138: Momentum conservation is not a hindrance, as any tennis racket has the same e ect on the tennis ball.
Challenge 271, page 138: In fact, in velocity space, elliptic, parabolic and hyperbolic motions are all described by circles. In all cases, the hodograph is a circle.
Challenge 272, page 138: is question is old (it was already asked in Newton’s times) and deep. One reason is that stars are kept apart by rotation around the galaxy. e other is that galaxies are kept apart by the momentum they got in the big bang. Without the big bang, all stars would have collapsed together. In this sense, the big bang can be deduced from the attraction of gravitation and the immobile sky at night. We shall nd out later that the darkness of the night sky gives a second argument for the big bang.
Challenge 273, page 138: Due to the plateau, the e ective mass of the Earth is larger.
Challenge 274, page 138: e choice is clear once you notice that there is no section of the orbit which is concave towards the Sun. Can you show this?
Challenge 275, page 139: It would be a black hole; no light could escape. Black holes are discussed in detail in the chapter on general relativity.
Challenge 276, page 139: A handle of two bodies.
Challenge 279, page 139: Using a maximal jumping height of h =0.5 m on Earth and an estimated asteroid density of ρ =3 Mg m3, we get a maximum radius of R2 = 3 h 4πGρ 703 m.
Challenge 280, page 140: For each pair of opposite shell elements (drawn in yellow), the two attractions compensate.
Challenge 281, page 141: ere is no practical way; if the masses on the shell could move, along the surface (in the same way that charges can move in a metal) this might be possible, provided that enough mass is available.
Challenge 282, page 141: Capture of a uid body if possible if it is split by tidal forces.
Challenge 283, page 141: e tunnel would be an elongated ellipse in the plane of the Equator,
reaching from one point of the Equator to the point at the antipodes. e time of revolution
would not change, compared to a non-rotating Earth. See A.J. S
, Falling down a hole
through the Earth, Mathematics Magazine 77, pp. 171–188, June 2004.
Challenge 285, page 141: e centre of mass of the solar system can be as far as twice the radius from the centre of the Sun; it thus can be outside the Sun.
Challenge 286, page 142: First, during northern summer time the Earth moves faster around the Sun than during northern winter time. Second, shallow Sun’s orbits on the sky give longer days because of light from when the Sun is below the horizon.
Challenge 287, page 142: Apart from the visibility of the moon, no such e ect has ever been detected. Gravity e ects, electrical e ects, magnetic e ects, changes in cosmic rays seem all to be independent of the phase. e locking of the menstrual cycle to the moon phase is a visual e ect.
Challenge 288, page 142: Distances were di cult to measure.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 289, page 142: See the mentioned reference.
Challenge 290, page 142: True.
Challenge 294, page 143: Never. e Moon points always towards the Earth. e Earth changes position a bit, due to the ellipticity of the Moon’s orbit. Obviously, the Earth shows phases.
Challenge 296, page 143: What counts is local verticality; with respect to it, the river always ows downhill.
Challenge 297, page 143: ere are no such bodies, as the chapter of general relativity will show.
Challenge 299, page 146: e oscillation is a purely sinusoidal, or harmonic oscillation, as the restoring force increases linearly with distance from the centre of the Earth. e period T for a homogeneous Earth is T = 2π R3 GM = 84 min.
Challenge 300, page 146: e period is the same for all such tunnels and thus in particular it is
the same as the 84 min valid also for the pole to pole tunnel. See for example, R.H. R
,e
answer is forty-two – many mechanics problems, only one answer, Physics Teacher 41, pp. 286–
290, May 2003.
Challenge 301, page 146: ere is no simple answer: the speed depends on the latitude and on other parameters.
Challenge 303, page 148: In reality muscles keep an object above ground by continuously li ing and dropping it; that requires energy and work.
Challenge 304, page 148: e electricity consumption of a rising escalator indeed increases when the person on it walks upwards. By how much?
Challenge 308, page 149: e lack of static friction would avoid that the uid stays attached to the body; the so-called boundary layer would not exist. One then would have to wing e ect.
Challenge 305, page 148: Knowledge is power. Time is money. Now, power is de ned as work per time. Inserting the previous equations and transforming them yields
money
=
work knowledge
,
(873)
Page 573
which shows that the less you know, the more money you make. at is why scientists have low salaries.
Challenge 310, page 151: True? Challenge 313, page 151: From dv dt = − v2(1 2cw Aρ m) and using the abbreviation c = 1 2cw Aρ, we can solve for v(t) by putting all terms containing the variable v on one side, all terms with t on the other, and integrating on both sides. We get v(t) = m c tanh c m t.
Challenge 315, page 153: e phase space has 3N position coordinates and 3N momentum coordinates.
Challenge 316, page 153: e light mill is an example.
Challenge 317, page 153: Electric charge.
Challenge 318, page 153: If you have found reasons to answer yes, you overlooked something. Just go into more details and check whether the concepts you used apply to the universe. Also de ne carefully what you mean by ‘universe’.
Challenge 320, page 155: A system showing energy or matter motion faster than light would imply that for such systems there are observers for which the order between cause and e ect are reversed. A space-time diagram (and a bit of exercise from the section on special relativity) shows this.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 405 The south-pointing carriage
Challenge 321, page 155: If reproducibility would not exist, we would have di culties in checking observations; also reading the clock is an observation. e connection between reproducibility and time shall become important in the third part of our adventure.
Challenge 322, page 156: Even if surprises were only rare, each surprise would make it impossible to de ne time just before and just a er it.
Challenge 325, page 157: Of course; moral laws are summaries of what others think or will do about personal actions.
Challenge 326, page 157: Space-time is de ned using matter; matter is de ned using spacetime.
Challenge 327, page 157: Fact is that physics has been based on a circular de nition for hundreds of years. us it is possible to build even an exact science on sand. Nevertheless, the elimination of the circularity is an important aim.
Challenge 328, page 161: For example, speed inside materials is slowed, but between atoms, light still travels with vacuum speed.
Challenge 331, page 174: Figure 405 shows the most credible reconstruction of a southpointing carriage.
Challenge 332, page 175: e water is drawn up along the sides of the spinning egg. e fastest way to empty a bottle of water is to spin the water while emptying it.
Challenge 333, page 175: e right way is the one where the chimney falls like a V, not like an inverted V. See challenge 257 on falling brooms for inspiration on how to deduce the answer.
Challenge 341, page 181: In one dimension, the expression F = ma can be written as
−dV dx = md2x dt2.
is can be
rewritten as d(−V )
dx
−d
dt[d
dx˙(
1 2
mx˙
2)]
=
0.
is can be
expanded to ∂
∂
x
(
1 2
m
x˙
2
−
V
(x
))
−
d
[∂
∂
x˙(
1 2
m
x˙
2
−
V
(x
))]
=
0,
which
is
Lagrange’s
equation
for this case.
Challenge 343, page 182: Do not despair. Up to now, nobody has been able to imagine a universe (that is not necessarily the same as a ‘world’) di erent from the one we know. So far, such attempts have always led to logical inconsistencies.
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Page 1046
Challenge 345, page 182: e two are equivalent since the equations of motion follow from the principle of minimum action and at the same time the principle of minimum action follows from the equations of motion.
Challenge 347, page 183: For gravity, all three systems exist: rotation in galaxies, pressure in planets and the Pauli pressure in stars. Against the strong interaction, the Pauli principle acts in nuclei and neutron stars; in neutron stars maybe also rotation and pressure complement the Pauli pressure. But for the electromagnetic interaction there are no composites other than our everyday matter, which is organized by the Pauli principle alone.
Challenge 349, page 187: Angular momentum is the change with respect to angle, whereas rotational energy is again the change with respect to time, as all energy is.
Challenge 350, page 187: Not in this way. A small change can have a large e ect, as every switch shows. But a small change in the brain must be communicated outside, and that will happen roughly with a 1 r2 dependence. at makes the e ects so small, that even with the most sensitive switches – which for thoughts do not exist anyway – no e ects can be realized.
Challenge 354, page 188: e relation is
c1 = sin α1 . c2 α2
(874)
e particular speed ratio between air (or vacuum, which is almost the same) and a material gives
the index of refraction n:
n
=
c1 c0
=
sin α1 α0
(875)
Challenge 355, page 188: Gases are mainly made of vacuum. eir index of refraction is near to one.
Challenge 356, page 188: Diamonds also sparkle because they work as prisms; di erent colours have di erent indices of refraction. us their sparkle is also due to their dispersion; therefore it is a mix of all colours of the rainbow.
Challenge 357, page 188: e principle for the growth of trees is simply the minimum of potential energy, since the kinetic energy is negligible. e growth of vessels inside animal bodies is minimized for transport energy; that is again a minimum principle. e refraction of light is the path of shortest time; thus it minimizes change as well, if we imagine light as moving entities moving without any potential energy involved.
Challenge 358, page 188: Special relativity requires that an invariant measure of the action exist. It is presented later in the walk.
Challenge 359, page 189: e universe is not a physical system. is issue will be discussed in detail later on.
Challenge ??, page ??: Physical elds are properties which vary from point to point. Parities are observables (in the wide sense of the term) but not elds.
Challenge 360, page 189: We talk to a person because we know that somebody understands us. us we assume that she somehow sees the same things we do. at means that observation is
partly viewpoint-independent. us nature is symmetric.
Challenge 361, page 190: Memory works because we recognize situations. is is possible because situations over time are similar. Memory would not have evolved without this reproducibility.
Challenge 362, page 191: Taste di erences are not fundamental, but due to di erent viewpoints and – mainly – to di erent experiences of the observers. e same holds for feelings and judgements, as every psychologist will con rm.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 363, page 192: e integers under addition form a group. Does a painter’s set of oil colours with the operation of mixing form a group?
Challenge 364, page 192: ere is only one symmetry operation: a rotation about π around the central point.
Challenge 370, page 196: Scalar is the magnitude of any vector; thus the speed, de ned as v = v , is a scalar, whereas the velocity v is not. us the length of any vector (or pseudovector), such as force, acceleration, magnetic eld, or electric eld, is a scalar, whereas the vector itself is not a scalar.
Challenge 373, page 197: e charge distribution of an extended body can be seen as a sum of a charge, a charge dipole, a charge quadrupole, a charge octupole, etc. e quadrupole is described by a tensor.
Compare: e inertia against motion of an extended body can be seen as sum of a mass, a mass dipole, a mass quadrupole, a mass octupole, etc. e mass quadrupole is described by the moment of inertia.
Challenge 377, page 199: e conserved charge for rotation invariance is angular momentum.
Challenge 380, page 203: An oscillation has a period in time, i.e. a discrete time translation symmetry. A wave has both discrete time and discrete space translation symmetry.
Challenge 381, page 203: Motion reversal is a symmetry for any closed system; despite the observations of daily life, the statements of thermodynamics and the opinion of several famous physicists (who form a minority though) all ideally closed systems are reversible.
Challenge 389, page 207: e potential energy is due to the ‘bending’ of the medium; a simple displacement produces no bending and thus contains no energy. Only the gradient captures the bending idea.
Challenge 391, page 208: e phase changes by π.
Challenge 393, page 209: Waves can be damped to extremely low intensities. If this is not possible, the observation is not a wave.
Challenge 394, page 209: Page 560 tells how to observe di raction and interference with your naked ngers.
Challenge 399, page 214: If the distances to the loudspeaker is a few metres, and the distance to the orchestra is 20 m, as for people with enough money, the listener at home hears it rst.
Challenge 401, page 214: An ellipse (as for planets around the Sun) with the xed point as centre (in contrast to planets, where the Sun is in a focus of the ellipse).
Challenge 403, page 215: e sound of thunder or of car tra c gets lower and lower in frequency with increasing distance.
Challenge 405, page 215: Neither; both possibilities are against the properties of water: in surface waves, the water molecules move in circles.
Challenge 406, page 216: Swimmers are able to cover 100 m in 48 s, or slightly better than 2 m s. With a body length of about 1.9 m, the critical speed is 1.7 m s. at is why short distance swimming depends on training; for longer distances the technique plays a larger role, as the critical speed has not been attained yet. e formula also predicts that on the 1500 m distance, a 2 m tall swimmer has a potential advantage of over 45 s on one with body height of 1.8 m. In addition, longer swimmers have an additional advantage: they swim shorter distances (why?). It is thus predicted that successful long-distance swimmers will get taller and taller over time. is is a pity for a sport that so far could claim to have had champions of all sizes and body shapes, in contrast to many other sports.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 408, page 217: To reduce noise re ection and thus hall e ects. ey e ectively diffuse the arriving wave fronts.
Challenge 410, page 217: Waves in a river are never elliptical; they remain circular.
Challenge 411, page 217: e lens is a cushion of material that is ‘transparent’ to sound. e speed of sound is faster in the cushion than in the air, in contrast to a glass lens, where the speed of light is slower in the glass. e shape is thus di erent: the cushion must look like a biconcave lens.
Challenge 413, page 217: e Sun is always at a di erent position than the one we observe it to be. What is the di erence, measured in angular diameters of the Sun?
Challenge 414, page 217: e 3 3 3 cube has a rigid system of three perpendicular axes, on which a square can rotate at each of the 6 ends. e other squares are attaches to pieces moving around theses axes. e 4 4 4 cube is di erent though; just nd out. e limit on the segment number seems to be 6, so far. A 7 7 7 cube requires varying shapes for the segments. But more than 5 5 5 is not found in shops. However, the website http://www.oinkleburger.com/Cube/ applet/ allows to play with virtual cubes up to 100 100 100 and more.
Challenge 416, page 218: An overview of systems being tested at present can be found in K.U. G , Energiereservoir Ozean, Physik in unserer Zeit 33, pp. 82–88, Februar 2002. See also Oceans of electricity – new technologies convert the motion of waves into watts, Science News 159, pp. 234–236, April 2001.
Challenge 417, page 218: In everyday life, the assumption is usually justi ed, since each spot can be approximately represented by an atom, and atoms can be followed. e assumption is questionable in situations such as turbulence, where not all spots can be assigned to atoms, and most of all, in the case of motion of the vacuum itself. In other words, for gravity waves, and in particular for the quantum theory of gravity waves, the assumption is not justi ed.
Challenge 419, page 219: ere are many. One would be that the transmission and thus re ection coe cient for waves would almost be independent of wavelength.
Challenge 420, page 220: A drop with a diameter of 3 mm would cover a surface of 7.1 m2 with a 2 nm lm.
Challenge 421, page 221: For jumps of an animal of mass m the necessary energy E is given as E = m h, and the work available to a muscle is roughly speaking proportional to its mass W m.
us one gets that the height h is independent of the mass of the animal. In other words, the speci c mechanical energyof animals is around 1.5 0.7 J kg.
Challenge 423, page 222: e critical height for a column of material is given by hc4rit =
β 4π
m
E ρ2
,
where
β
1.9 is the constant determined by the calculation when a column buckles
under its own weight.
Challenge 425, page 223: One possibility is to describe see particles as extended objects, such as clouds; another is given in the third part of the text.
Challenge 427, page 226: rowing the stone makes the level fall, throwing the water or the piece of wood leaves it unchanged.
Challenge 428, page 226: No metal wire allows to build such a long wire. Only the idea of carbon nanotubes has raised the hope again; some dream of wire material based on them, stronger than any material known so far. However, no such material is known yet. e system faces many dangers, such as fabrication defects, lightning, storms, meteorites and space debris. All would lead to the breaking of the wires – if such wires will ever exist. But the biggest of all dangers is the lack of cash to build it.
Challenge 432, page 226: e pumps worked in suction; but air pressure only allows 10 m of height di erence for such systems.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 434, page 227: is argument is comprehensible only when one remembers that ‘twice the amount’ means ‘twice as many molecules’.
Challenge 435, page 227: e alcohol is frozen and the chocolate is put around it.
Challenge 436, page 227: I suggested in an old edition that a machine should be based on the same machines that throw the clay pigeons used in the sports of trap shooting and skeet. In the meantime, Lydéric Bocquet and Christophe Clanet have built such a machine, but using s di erent design; a picture can be found on the website lpmcn.univ-lyon1.fr/%7Elbocquet/.
Challenge 437, page 227: e third component of air is the noble gas argon, making up about 1 %. e rest is made up by carbon dioxide, water vapour and other gases. Are these percentages volume or weight percentages?
Challenge 438, page 227: It uses the air pressure created by the water owing downwards.
Challenge 439, page 227: Yes. e bulb will not resist two such cars though.
Challenge 442, page 228: None.
Challenge 443, page 228: He brought the ropes into the cabin by passing them through liquid mercury.
Challenge 444, page 228: e pressure destroys the lung.
Challenge 446, page 229: Either they fell on inclined snowy mountain sides, or they fell into high trees, or other so structures. e record was over 7 km of survived free fall.
Challenge 447, page 229: e blood pressure in the feet of a standing human is about 27 kPa, double the pressure at the heart.
Challenge 448, page 229: Calculation gives N = J j = 0.0001 m3 s (7 µm20.0005 m s), or about 6 ë 109; in reality, the number is much larger, as most capillaries are closed at a given instant. e reddening of the face shows what happens when all small blood vessels are opened at the same time.
Challenge 449, page 229: e soap ows down the bulb, making it thicker at the bottom and thinner at the top, until it bursts.
Challenge 450, page 229: A medium-large earthquake would be generated.
Challenge 451, page 229: A stalactite contains a thin channel along its axis through which the water ows, whereas a stalagmite is massive throughout.
Challenge 453, page 230: About 1 part in a thousand.
Challenge 454, page 230: For this to happen, friction would have to exist on the microscopic scale and energy would have to disappear.
Challenge 455, page 230: e longer funnel is empty before the short one. (If you do not believe it, try it out.) In the case that the amount of water in the funnel outlet can be neglected, one can use energy conservation for the uid motion. is yields the famous Bernoulli equation p ρ +
h + v2 2 = const, where p is pressure, ρ the density of water, and is 9.81 m s2. erefore, the speed v is higher for greater lengths h of the thin, straight part of the funnel: the longer funnel empties rst.
But this is strange: the formula gives a simple free fall relation, as the pressure is the same above and below and disappears from the calculation. e expression for the speed is thus independent of whether a tube is present or not. e real reason for the faster emptying of the tube is thus that a tube forces more water to ow out that a lack of tube. Without tube, the diameter of the water ow diminishes during fall. With tube, it stay constant. is di erence leads to the faster emptying.
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Page 1046
Challenge 456, page 230: e eyes of sh are positioned in such a way that the pressure reduction by the ow is compensated by the pressure increase of the stall. By the way, their heart is positioned in such a way that it is helped by the underpressure.
Challenge 458, page 230: Glass shatters, glass is elastic, glass shows transverse sound waves, glass does not ow (in contrast to what many books state), not even on scale of centuries, glass molecules are xed in space, glass is crystalline at small distances, a glass pane supported at the ends does not hang through.
Challenge 459, page 231: is feat has been achieved for lower mountains, such as the Monte Bianco in the Alps. At present however, there is no way to safely hover at the high altitudes of the Himalayas.
Challenge 461, page 231: e iron core of the Earth formed in the way described by collecting the iron from colliding asteroids. However, the Earth was more liquid at that time. e iron will most probably not sink. In addition, there is no known way to make the measurement probe described.
Challenge 462, page 231: Press the handkerchief in the glass, and lower the glass into the water with the opening rst, while keeping the opening horizontal. is method is also used to lower people below the sea. e paper ball in the bottle will y towards you. Blowing into a funnel will keep the ping-pong ball tightly into place, and the more so the stronger you blow. Blowing through a funnel towards a candle will make it lean towards you.
Challenge 465, page 237: In 5000 million years, the present method will stop, and the Sun will become a red giant. abut it will burn for many more years a er that.
Challenge 469, page 240: We will nd out later that the universe is not a physical system; thus the concept of entropy does not apply to it. us the universe is neither isolated nor closed.
Challenge 472, page 241: e answer depends on the size of the balloons, as the pressure is not a monotonous function of the size. If the smaller balloon is not too small, the smaller balloon wins.
Challenge 474, page 241: Measure the area of contact between tires and street (all four) and then multiply by 200 kPa, the usual tire pressure. You get the weight of the car.
Challenge 478, page 243: If the average square displacement is proportional to time, the mat-
ter is made of smallest particles. is was con rmed by the experiments of Jean Perrin. e next
step is to deduce the number number of these particles form the proportionality constant. is
constant, de ned by d2 = 4Dt, is called the di usion constant (the factor 4 is valid for random
motion in two dimensions). e di usion constant can be determined by watching the motion
of a particle under the microscope.
We study a Brownian particle of radius a. In two dimensions, its square displacement is given
by
d2
4kT µ
t
,
(876)
where k is the Boltzmann constant and T the temperature. e relation is deduced by studying the
motion of a particle with drag force −µv that is subject to random hits. e linear drag coe cient
µ of a sphere of radius a is given by
µ = 6πηa .
(877)
In other words, one has
k
=
6πηa 4T
d2 t
.
(878)
All quantities on the right can be measured, thus allowing to determine the Boltzmann constant
k. Since the ideal gas relation shows that the ideal gas constant R is related to the Boltzmann
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constant by R = NAk, the Avogadro constant NA that gives the number of molecules in a mole is also found in this way.
Challenge 485, page 249: Yes, the e ect is easily noticeable.
Challenge 487, page 250: Hot air is less dense and thus wants to rise.
Challenge 490, page 250: e air had to be dry.
Challenge 491, page 250: In general, it is impossible to draw a line through three points.
Challenge 492, page 250: No, as a water molecule is heavier than that. However, if the water is allowed to be dirty, it is possible. What happens if the quantum of action is taken into account?
Challenge 488, page 250: Keep the paper wet.
Challenge 493, page 251: e danger is not due to the amount of energy, but due to the time in which it is available.
Challenge 494, page 251: e internet is full of solutions.
Challenge 496, page 252: Only if it is a closed system. Is the universe closed? Is it a system? is is discussed in the third part of the mountain ascent.
Challenge 499, page 252: For such small animals the body temperature would fall too low. ey could not eat fast enough to get the energy needed to keep themselves warm.
Challenge 508, page 253: It is about 10−9 that of the Earth.
Challenge 510, page 253: e thickness of the folds in the brain, the bubbles in the lung, the density of blood vessels and the size of biological cells.
Challenge 511, page 254: e mercury vapour above the liquid gets saturated.
Challenge 512, page 254: A dedicated NASA project studies this question. Figure 406 gives an example comparison. You can nd more details on their website.
Challenge 513, page 254: e risks due to storms and the nancial risks are too large.
Challenge 514, page 254: e vortex in the tube is
cold in the middle and hot at its outside; the air from
the middle is sent to one end and the air from the out-
side to the other. e heating of the outside is due to
the work that the air rotating inside has to do on the
air outside to get a rotation that eats up angular mo-
mentum. For a detailed explanation, see the beautiful
text by M P. S
, And Yet it Moves: Strange
Systems and Subtle Questions in Physics, Cambridge Uni-
versity Press, 1993, p. 221.
F I G U R E 406 A candle on Earth and in microgravity (NASA)
Challenge 515, page 254: Egg white hardens at 70°C, egg white at 65 to 68°C. Cook an egg at the latter temperature, and the feat is possible.
Challenge 519, page 255: is is also true for the shape human bodies, the brain control of human motion, the growth of owers, the waves of the sea, the formation of clouds, the processes leading to volcano eruptions, etc.
Challenge 524, page 259: ere are many more butter ies than tornadoes. In addition, the belief in the butter y e ect completely neglects an aspect of nature that is essential for selforganization: friction and dissipation. e butter y e ect, assumed it did exist, requires that dissipation is neglected. ere is no experimental basis for the e ect, it has never been observed.
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Challenge 534, page 264: All three statements are hogwash. A drag coe cient implies that the cross area of the car is known to the same precision. is is actually extremely di cult to measure and to keep constant. In fact, the value 0.375 for the Ford Escort was a cheat, as many other measurements showed. e fuel consumption is even more ridiculous, as it implies that fuel volumes and distances can be measured to that same precision. Opinion polls are taken by phoning at most 2000 people; due to the di culties in selecting the right representative sample, that gives a precision of at most 3 %.
Challenge 535, page 264: No. Nature does not allow more than about 20 digits of precision, as we will discover later in our walk. at is not su cient for a standard book. e question whether such a number can be part of its own book thus disappears.
Challenge 537, page 265: Every measurement is a comparison with a standard; every comparison requires light or some other electromagnetic eld. is is also the case for time measurements.
Challenge 538, page 265: Every mass measurement is a comparison with a standard; every comparison requires light or some other electromagnetic eld.
Challenge 539, page 265: Angle measurements have the same properties as length or time measurements.
Challenge 540, page 275: A cone or a hyperboloid also look straight from all directions, provided the positioning is correct. One thus needs not only to turn the object, but also to displace it. e best method to check planarity is to use interference between an arriving and a departing coherent beam of light. If the fringes are straight, the surface is planar. (How do you ensure the wavefront of the light beam is planar?)
Challenge 541, page 276: A fraction of in nity is still in nite.
Challenge 542, page 276: e time at which the Moon Io enters the shadow in the second measurement occurs about 1000 s later than predicted from the rst measurement. Since the Earth is about 3 ë 1011 m further away from Jupiter and Io, we get the usual value for the speed of light.
Challenge 543, page 277: To compensate for the aberration, the telescope has to be inclined along the direction of motion of the Earth; to compensate for parallaxis, against the motion.
Challenge 544, page 277: Otherwise the velocity sum would be larger than c.
Challenge 545, page 277: e drawing shows it. Observer, Moon and Sun form a triangle. When the Moon is half full, the angle at the Moon is a right angle. us the distance ration can be determined, though not easily, as the angle at the observer is very near a right angle as well.
Challenge 546, page 278: ere are Cat’s-eyes on the Moon deposited there during the Apollo and Lunakhod missions. ey are used to re ect laser 35 ps light pulses sent there through telescopes. e timing of the round trip then gives the distance to the Moon. Of course, absolute distance is not know to high precision, but the variations are. e thickness of the atmosphere is the largest source of error. See the http://www.csr.utexas.edu/mlrs and http://ilrs.gsfc.nasa.gov websites.
Challenge 547, page 278: Fizeau used a mirror about 8.6 km away. As the picture shows, he only had to count the teeth of his cog-wheel and measure its rotation speed when the light goes in one direction through one tooth and comes back to the next.
Challenge 548, page 279: e time must be shorter than T = l c, in other words, shorter than 30 ps; it was a gas shutter, not a solid one. It was triggered by a red light pulse (show in the photographed) timed by the one to be photographed; for certain materials, such as the used gas, strong light can lead to bleaching, so that they become transparent. For more details about the shutter and its neat trigger technique, see the paper by the authors.
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Ref. 258 Page 753
Challenge 549, page 279: Just take a photograph of a lightning while moving the camera horizontally. You will see that a lightning is made of several discharges; the whole shows that lightning is much slower than light.
If lightning moved only nearly as fast as light itself, the Doppler e ect would change it colour depending on the angle at which we look at it, compared to its direction of motion. A nearby lightning would change colour from top to bottom.
Challenge 550, page 280: e fastest lamps were subatomic particles, such as muons, which decay by emitting a photon, thus a tiny ash of light. However, also some stars emit fasts jets of matter, which move with speeds comparable to that of light.
Challenge 551, page 281: e speed of neutrinos is the same as that of light to 9 decimal digits, since neutrinos and light were observed to arrive together, within 12 seconds of each other, a er a trip of 170 000 light years from a supernova explosion.
Challenge 553, page 283: is is best discussed by showing that other possibilities make no sense.
Challenge 554, page 283: e spatial coordinate of the event at which the light is re ected is c(k2 − 1)T 2; the time coordinate is (k2 + 1)T 2. eir ratio must be v. Solving for k gives the result.
Challenge 556, page 284: e motion of radio waves, infrared, ultraviolet and gamma rays is also unstoppable. Another past suspect, the neutrino, has been found to have mass and to be thus in principle stoppable. e motion of gravity is also unstoppable.
Challenge 558, page 286: λR λS = γ.
Challenge 559, page 286: To change from bright red (650 nm) to green (550 nm), v = 0.166c is necessary.
Challenge 560, page 286: People measure the shi of spectral lines, such as the shi of the socalled Lyman-α line of hydrogen, that is emitted (or absorbed) when a free electron is captured (or ejected) by a proton. It is one of the famous Fraunhofer lines.
Challenge 561, page 286: e speeds are given by
v
c=
(z + 1)2 − 1 (z + 1)2 + 1
(879)
which implies v(z = −0.1) = 31 Mm s = 0.1c towards the observer and v(z = 5) = 284 Mm s = 0.95c away from the observer.
A red-shi of 6 implies a speed of 0.96c; such speeds appear because, as we will see in the section of general relativity, far away objects recede from us. And high red-shi s are observed only for objects which are extremely far from Earth, and the faster the further they are away. For a red-shi of 6 that is a distance of several thousand million light years.
Challenge 562, page 286: No Doppler e ect is seen for a distant observer at rest with respect to the large mass. In other cases there obviously is a Doppler e ect, but it is not due to the de ection.
Challenge 563, page 287: Sound speed is not invariant of the speed of observers. As a result, the Doppler e ect for sound even con rms – within measurement di erences – that time is the same for observers moving against each other.
Challenge 566, page 288: Inside colour television tubes (they use higher voltages than black and white ones), electrons are described by v c 2 ë 30 511 or v 0.3c.
Challenge 567, page 289: If you can imagine this, publish it. Readers will be delighted to hear the story.
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Page 1068
Challenge 569, page 289: e connection between observer invariance and limit property seems to be generally valid in nature, as shown in section 36.. However, a complete and airtight argument is not yet at hand. If you have one, publish it!
Challenge 572, page 291: If the speed of light is the same for all observers, no observer can pretend to be more at rest than another (as long as space-time is at), because there is no observation from electrodynamics, mechanics or another part of physics that allows to make the statement.
Challenge 575, page 293: Redrawing Figure 137 on page 283 for the other observer makes the point.
Challenge 576, page 293: e human value is achieved in particle accelerators; the value in nature is found in cosmic rays of the highest energies.
Challenge 577, page 294: e set of events behaves like a manifold, because it behaves like a four-dimensional space: it has in nitely many points around any given starting point, and distances behave as we are used to, limits behave as we are used to. It di ers by one added dimension, and by the sign in the de nition of distance; thus, properly speaking, it is a Riemannian manifold.
Challenge 578, page 295: In nity is obvious, as is openness. us the topology equivalence can be shown by imagining that the manifold is made of rubber and wrapped around a sphere.
Challenge 579, page 296: e light cone remains unchanged; thus causal connection as well.
Challenge 580, page 296: In such a case, the division of space-time around an inertial observer into future, past and elsewhere would not hold any more, and the future could in uence the past (as seen from another observer).
Challenge 583, page 299: e ratio predicted by naive reasoning is (1 2)(6.4 2.2) = 0.13.
Challenge 584, page 299: e time dilation factor for v = 0.9952c is 10.2, giving a proper time of 0.62 µs; thus the ratio predicted by special relativity is (1 2)(0.62 2.2) = 0.82.
Challenge 585, page 299: Send a light signal from the rst clock to the second clock and back. Take the middle time between the departure and arrival, and then compare it with the time at the re ection. Repeat this a few times. See also Figure 137.
Challenge 588, page 301: Hint: think about di erent directions of sight.
Challenge 589, page 301: Not with present experimental methods.
Challenge 591, page 301: Hint: be careful with the de nition of ‘rigidity’.
Challenge 593, page 302: e light cannot stay on at any speed, if the glider is shorter than the gap. is is strange, because the bar does not light the lamp even at high speeds, even though in the frame of the bar there is contact at both ends. e reason is that in this case there is not enough time to send the signal to the battery that contact is made, so that the current cannot start
owing. Assume that current ows with speed u, which is of the order of c. en, as Dirk Van de
Moortel showed, the lamp will go o if the glider length lglider and the gap length lgap obey lglider lgap < γ(u + v) u. See also the cited reference.
Why are the debates o en heated? Some people will (falsely) pretend that the problem is unphysical; other will say that Maxwell’s equations are needed. Still others will say that the problem is absurd, because for larger lengths of the glider, the on/o answer depends on the precise speed value. However, this actually is the case in this situation.
Challenge 594, page 302: Yes, the rope breaks; in accelerated cars, distance changes, as shown later on in the text.
Challenge 595, page 302: e submarine will sink. e fast submarine will even be heavier, as his kinetic energy adds to his weight. e contraction e ect would make it lighter, as the captain
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says, but by a smaller amount. e total weight – counting upwards as positive – is given by F = −m (γ − 1 γ).
Challenge 596, page 302: A relativistic submarine would instantly melt due to friction with the water. If not, it would y of the planet because it moves faster than the escape velocity. And produce several other disasters.
Challenge 597, page 305: e question confuses observation of Lorentz contraction and its measurement. A relativistic pearl necklace does get shorter, but the shortening can only be measured, not photographed. e measured sizes of the pearls are attened ellipsoids relativistic speeds. e observed necklace consists of overlapping spheres.
Challenge 598, page 305: e website o ers a prize for a lm checking this issue.
Challenge 601, page 305: Yes, ageing in a valley is slowed compared to mountain tops. However, the proper sensation of time is not changed. e reason for the appearance of grey hair is not known; if the timing is genetic, the proper time at which it happens is the same in either location.
Challenge 602, page 306: ere is no way to put an observer at the speci ed points. Proper velocity can only be de ned for observers, i.e., for entities which can carry a clock. at is not the case for images.
Challenge 603, page 307: Just use plain geometry to show this.
Challenge 604, page 307: Most interestingly, the horizon can easily move faster than light, if you move your head appropriately, as can the end of the rainbow.
Challenge 607, page 311: Relativity makes the arguments of challenge 130 watertight.
Challenge 612, page 313: e lower collision in Figure 161 shows the result directly, from energy conservation. For the upper collision the result also follows, if one starts from momentum conservation γmv = ΓMV and energy conservation ( amma + 1)m = ΓM.
Challenge 614, page 314: Annihilation of matter and antimatter.
Challenge 621, page 317: Just turn the le side of Figure 164 a bit in anti-clockwise direction.
Challenge 622, page 318: In collisions between relativistic charges, part of the energy is radiated away as light, so that the particles e ectively lose energy.
Challenge 623, page 319: Probably not, as all relations among physical quantities are known now. However, you might check for yourself; one might never know. It is worth to mention that the maximum force in nature was discovered (in this text) a er remaining hidden for over 80 years.
Challenge 624, page 321: Write down the four-vectors U′ and U and then extract v′ as function of v and the relative coordinate speed V . en rename the variables.
Challenge 625, page 321: Any motion with light speed.
Challenge 626, page 322: b0 = 0, bi = γ2ai .
Challenge 629, page 323: For ultrarelativistic particles, like for massless particles, one has E = pc.
Challenge 630, page 323: Hint: evaluate P1 and P2 in the rest frame of one particle. Challenge 631, page 324: Use the de nition f = dp dt and the relation KU = 0 = fv − dE dt valid for rest-mass preserving forces.
Challenge 659, page 333: e energy contained in the fuel must be comparable to the rest mass of the motorbike, multiplied by c2. Since fuel contains much more mass than energy, that gives a big problem.
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Challenge 661, page 334: Constant acceleration and gravity are similar in their e ects, as discussed in the section on general relativity.
Challenge 667, page 336: Yes, it is true.
Challenge 668, page 336: It is at, like a plane.
Challenge 670, page 337: Yes; however, the e ect is minimal and depends on the position of the Sun. In fact, what is white at one height is not white at another.
Challenge 672, page 337: Locally, light always moves with speed c.
Challenge 673, page 338: Away from Earth, decreases; it is e ectively zero over most of the distance.
Challenge 674, page 339: Light is necessary to determine distance and to synchronize clocks; thus there is no way to measure the speed of light from one point to another alone. e reverse motion needs to be included. However, some statements on the one-way speed of light can still be made (see http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html). All experiments on the one-way speed of light performed so far are consistent with an isotropic value that is equal to the two-way velocity. However, no experiment is able to rule out a group of theories in which the one-way speed of light is anisotropic and thus di erent from the two-way speed. All theories from this group have the property that the round-trip speed of light is isotropic in any inertial frame, but the one-way speed is isotropic only in a preferred ‘ether’ frame. In all of these theories, in all inertial frames, the e ects of slow clock transport exactly compensate the e ects of the anisotropic one-way speed of light. All these theories are experimentally indistinguishable from special relativity. In practice, therefore, the one-way speed of light has been measured and is constant. But a small option remains.
Challenge 675, page 339: See the cited reference. e factor 2 was forgotten there; can you deduce it?
Challenge 678, page 340: ough there are many publications pretending to study the issue, there are also enough physicists who notice the impossibility. Measuring a variation of the speed of light is not much far from measuring the one way speed of light: it is not possible. However, the debates on the topic are heated; the issue will take long to be put to rest.
Challenge 679, page 349: e inverse square law of gravity does not comply with the maximum speed principle; it is not clear how it changes when one changes to a moving observer.
Challenge 680, page 353: Take a surface moving with the speed of light, or a surface de ned with a precision smaller than the Planck length.
Challenge 681, page 357: Also shadows do not remain parallel in curved surfaces. forgetting this leads to strange mistakes: many arguments allegedly ‘showing’ that men have never been on the moon neglect this fact when they discuss the photographs taken there.
Challenge 684, page 364: If so, publish it; then send it to the author.
Challenge 685, page 366: For example, it is possible to imagine a surface that has such an intricate shape that it will pass all atoms of the universe at almost the speed of light. Such a surface is not physical, as it is impossible to imagine observers on all its points that move in that way all at the same time.
Challenge 693, page 378: ey are accelerated upwards.
Challenge 694, page 378: In everyday life, (a) the surface of the Earth can be taken to be at, (b) the vertical curvature e ects are negligible, and (c) the lateral length e ects are negligible.
Challenge 698, page 379: For a powerful bus, the acceleration is 2 m s2; in 100 m of acceleration, this makes a relative frequency change of 2.2 ë 10−15.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 699, page 379: Yes, light absorption and emission are always lossless conversions of energy into mass.
Challenge 702, page 379: For a beam of light, in both cases the situation is described by an environment in which masses ‘fall’ against the direction of motion. If the Earth and the train walls were not visible – for example if they were hidden by mist – there would not be any way to determine by experiment which situation is which. Or again, if an observer would be enclosed in a box, he could not distinguish between constant acceleration or constant gravity. (Important: this impossibility only applies if the observer has negligible size!)
Challenge 708, page 381: Both fall towards the centre of the Earth. Orbiting particles are also in free fall; their relative distance changes as well, as explained in the text.
Challenge 711, page 383: Such a graph would need four or even 5 dimensions.
Challenge 713, page 385: e energy due to the rotation can be neglected compared with all other energies in the problem.
Challenge 723, page 390: Di erent nucleons, di erent nuclei, di erent atoms and di erent molecules have di erent percentages of binding energies relative to the total mass.
Challenge 725, page 391: In free fall, the bottle and the water remain at rest with respect to each other.
Challenge 726, page 391: Let the device fall. e elastic rubber then is strong enough to pull
the ball into the cup. See M.T. W
, Einsteins verjaardagscadeau, Nederlands tijdschri voor
natuurkunde 69, p. 109, April 2003. e original device also had a spring connected in series to
the rubber.
Challenge 727, page 391: Apart the chairs and tables already mentioned, important antigravity devices are suspenders, belts and plastic bags.
Challenge 733, page 392: ey use a spring scale, and measure the oscillation time. From it they deduce their mass.
Challenge 734, page 392: e apple hits the wall a er about half an hour.
Challenge 738, page 393: With ħ as smallest angular momentum one get about 100 Tm.
Challenge 739, page 393: No. e di raction of the beams does not allow it. Also quantum theory makes this impossible; bound states of massless particles, such as photons, are not stable.
Challenge 741, page 394: e orbital radius is 4.2 Earth radii; that makes c. 38 µs every day.
Challenge 742, page 395: To be honest, the experiments are not consistent. ey assume that some other property of nature is constant – such as atomic size – which in fact also depends on G. More on this issue on page 501.
Challenge 743, page 395: Of course other spatial dimensions could exist which can be detected only with the help of measurement apparatuses. For example, hidden dimensions could appear at energies not accessible in everyday life.
Challenge 753, page 401: Since there is no negative mass, gravitoelectric elds cannot be neutralized. In contrast, electric elds can be neutralized around a metallic conductor with a Faraday cage.
Challenge 766, page 408: One needs to measure the timing of pulses which cross the Earth at di erent gravitational wave detectors on Earth.
Challenge 782, page 415: No; a line cannot have intrinsic curvature. A torus is indeed intrinsically curved; it cannot be cut open to a at sheet of paper.
Challenge 804, page 423: e trace of the Einstein tensor is the negative of the Ricci scalar; it is thus the negative of the trace of the Ricci tensor.
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Challenge 821, page 434: Indeed, in general relativity gravitational energy cannot be localized in space, in contrast to what one expects and requires from an interaction.
Challenge 833, page 442: ere is a good chance that some weak form of a jet exists; but a detection will not be easy.
Challenge 840, page 454: e rabbit observes that all other rabbits seem to move away from him.
Challenge 846, page 458: Stand in a forest in winter, and try to see the horizon. If the forest is very deep, you hit tree trunks in all directions. If the forest is nite in depth, you have chance to see the horizon.
Challenge 867, page 475: Flattening due to rotation requires other masses to provide the background against which the rotation takes place.
Challenge 897, page 487: is happens in the same way that the static electric eld comes out of a charge. In both cases, the transverse elds do not get out, but the longitudinal elds do. Quantum theory provides the deeper reason. Real radiation particles, which are responsible for free, transverse elds, cannot leave a black hole because of the escape velocity. However, virtual particles can, as their speed is not bound by the speed of light. All static, longitudinal elds are produced by virtual particles. In addition, there is a second reason. Classical eld can come out of a black hole because for an outside observer everything making it up is continuously falling, and nothing has actually crossed the horizon. e eld sources thus are not yet out of reach.
Challenge 901, page 487: e description says it all. A visual impression can be found in the room on black holes in the ‘Deutsches Museum’ in München.
Challenge 910, page 493: Any device that uses mirrors requires electrodynamics; without electrodynamics, mirrors are impossible.
Challenge 912, page 496: e hollow Earth theory is correct if usual distance are consistently changed to rhe = RE2arth r. is implies a quantum of action that decreases towards the centre of the hollow sphere. en there is no way to prefer one description over the other, except for
reasons of simplicity.
Challenge 919, page 521: e liquid drops have to detach from the ow exactly inside the metal counter-electrodes. Opel simply earthed the metal piece they had built into the cars without any contact to the rest of the car.
Challenge 920, page 522: A lot of noise while the metal pendulum banged wildly between the two xed bells.
Challenge 923, page 524: e eld at a distance of 1 m from an electron is 1.4 nV m.
Challenge 924, page 525: A simple geometrical e ect: anything owing out homogeneously from a sphere diminishes with the square of the distance.
Challenge
925,
page
525:
One has F
=
α
ħc
N
2 A
4R2
=
3 ë 1012 N, an enormous force, corres-
ponding to the weight of 300 million tons. It shows the enormous forces that keep matter together.
Obviously, there is no way to keep 1 g of positive charge together, as the repulsive forces among
the charges would be even larger.
Challenge 926, page 525: To show the full equivalence of Coulomb’s and Gauss’s ‘laws’, rst show that it holds for a single point charge. en expand the result for more than one point charge. at gives Gauss’s ‘law’ in integral form, as given just before this challenge.
To deduce the integral form of Gauss’s ‘law’ for a single point charge, one has to integrate over the closed surface. e essential point here is to note that the integration can be carried out for an inverse square dependence only. is dependence allows to transform the scalar product between the local eld and the area element into a normal product between the charge and the
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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solid angle Ω:
E
dA =
qdA cos θ 4πε0r2
=
qdΩ 4πε0
.
In case that the surface is closed the integration is then straightforward.
To deduce the di erential form of (the static) Gauss’s ‘law’, namely
(880)
∇E
=
ρ ε0
,
(881)
make use of the de nition of the charge density ρ and of the purely mathematical relation
∫
∫ E dA =
∇E dV ,
closedsurface
enclosedvolume
(882)
is mathematical relation, valid for any vector eld E, is called Gauss’s theorem. It simply states that the ux is the volume integral of the divergence.
To deduce the full form of Gauss’s law, including the time-derivative of the magnetic eld, include relativistic e ects by changing viewpoint to a moving observer.
Challenge 928, page 526: No; batteries only separate charges and pump them around.
Challenge 929, page 526: Uncharged bodies can attract each other if they are made of charged constituents neutralizing each other, and if the charges are constrained in their mobility. e charge uctuations then lead to attraction. Most molecules interact among each other in this way; such forces are also at the basis of surface tension in liquids and thus of droplet formation.
Challenge 931, page 527: e ratio q m of electrons and that of the free charges inside metals is not exactly the same.
Challenge 938, page 534: e correct version of Ampère’s ‘law’ is
∇
B
−
1 c2
∂E ∂t
=
µ0j
(883)
whereas the expression mentioned in the text misses For another way to state the di erence, see R
the
term P. F
∂E ∂t
.
, R.B. L
S
, e Feynman Lectures on Physics, volume II, Addison Wesley, p. 21-1, 1977.
& M.
Challenge 939, page 535: Only boosts with relativistic speeds mix magnetic and electric elds to an appreciable amount.
Challenge 941, page 535: e dual eld F is de ned on page 546.
Challenge 942, page 536: Scalar products of four vectors are always, by construction, Lorentz invariant quantities.
Challenge 948, page 539: Usually, the cables of high voltage lines are too warm to be comfortable.
Challenge 949, page 539: Move them to form a T shape.
Challenge 950, page 539: For four and more switches, on uses inverters; an inverter is a switch with two inputs and two outputs which in one position, connects rst and second input to rst and second output respectively, and in the other position connects the rst input to the second output and vice versa. ( ere are other possibilities, though; wires can be saved using electromagnetic relay switches.) For three switches, there is a simpler solution than with inverters.
Challenge 952, page 540: It is possible; however, the systems so far are not small and are dangerous for human health. e idea to collect solar power in deep space and then beam it to the Earth as microwaves has o en been aired. Finances and dangers have blocked it so far.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 953, page 540: Glue two mirrors together at a right angle. Or watch yourself on TV using a video camera.
Challenge 954, page 540: is is again an example of combined triboluminescence and triboelectricity. See also the websites http://scienceworld.wolfram.com/physics/Triboluminescence. html and http://www.geocities.com/RainForest/9911/tribo.htm.
Challenge 956, page 541: Pepper is lighter than salt, and thus reacts to the spoon before the salt does.
Challenge 957, page 542: For a wavelength of 546.1 nm (standard green), that is a bit over 18 wavelengths.
Challenge 958, page 542: e angular size of the Sun is too large; di raction plays no role here.
Challenge 959, page 542: Just use a high speed camera.
Challenge 960, page 543: e current ows perpendicularly to the magnetic eld and is thus de ected. It pulls the whole magnet with it.
Challenge 961, page 543: Light makes seven turns of the Earth in one second.
Challenge 964, page 543: e most simple equivalent to a coil is a rotating mass being put into rotation by the owing water. A transformer would then be made of two such masses connected through their axis.
Challenge 967, page 545: e charged layer has the e ect that almost only ions of one charge pass the channels. As a result, charges are separated on the two sides of the liquid, and a current is generated.
Challenge 972, page 548: Some momentum is carried away by the electromagnetic eld.
Challenge 973, page 548: Field lines and equipotential surfaces are always orthogonal to each other. us a eld line cannot cross an equipotential surface twice.
Challenge 984, page 554: Just draw a current through a coil with its magnetic eld, then draw the mirror image of the current and redraw the magnetic eld.
Challenge 985, page 555: Other asymmetries in nature include the helicity of the DNA molecules making up the chromosomes and many other molecules in living systems, the right hand preference of most humans, the asymmetry of sh species which usually stay at on the bottom of the seas.
Challenge 986, page 555: is is not possible at all using gravitational or electromagnetic systems or e ects. e only way is to use the weak nuclear interaction, as shown in the chapter on the nucleus.
Challenge 987, page 556: e Lagrangian does not change if one of the three coordinates is changed by its negative value.
Challenge 988, page 556: e image ips up: a 90 degree rotation turns the image by 180 degrees.
Challenge 989, page 556: Imagine E and B as the unite vectors of two axes in complex space. en any rotation of these axes is also a generalized duality symmetry.
Challenge 992, page 561: In every case of interference, the energy is redistributed into other directions. is is the general rule; sometimes it is quite tricky to discover this other direction.
Challenge 993, page 561: e author regularly sees about 7 lines; assuming that the distance is around 20 µm, this makes about 3 µm per line. e wavelength must be smaller than this value and the frequency thus larger than 100 THz. e actual values for various colours are given in the table of the electromagnetic spectrum.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 995, page 562: He noted that when a prism produces a rainbow, a thermometer placed in the region a er the colour red shows a temperature rise.
Challenge 996, page 562: Light re ected form a water surface is partly polarized. Mirages are not.
Challenge 998, page 563: Drawing them properly requires four dimensions; and there is no analogy with two-dimensional waves. It is not easy to picture them. e direction of oscillation of the elds rotates as the wave advances. e oscillation direction thus forms a spiral. Picturing the rest of the wave is not impossible, but not easy.
Challenge 1001, page 567: Such an observer would experience a wavy but static eld, which cannot exist, as the equations for the electromagnetic eld show.
Challenge 1002, page 567: You would never die. Could you reach the end of the universe?
Challenge 1003, page 568: Syrup shows an even more beautiful e ect in the following setting. Take a long transparent tube closed at one end and ll it with syrup. Shine a red helium–neon laser into the tube from the bottom. en introduce a linear polarizer into the beam: the light seen in the tube will form a spiral. By rotating the polarizer you can make the spiral advance or retract. is e ect, called the optical activity of sugar, is due to the ability of sugar to rotate light polarization and to a special property of plants: they make only one of the two mirror forms of sugar.
Challenge 1004, page 568: e thin lens formula is
1 +1 =1. do di f
(884)
It is valid for diverging and converging lenses, as long as their own thickness is negligible. e strength of a lens can thus be measured with the quantity 1 f . e unit 1 m−1 is called a diopter; it is used especially for reading glasses. Converging lenses have positive, diverging lenses negative
values.
Challenge 1005, page 569: A light microscope is basically made of two converging lenses. One lens – or lens system – produces an enlarged real image and the second one produces an enlarged virtual image of the previous real image. Figure 407 also shows that microscopes always turn images upside down. Due to the wavelength of light, light microscopes have a maximum resolution of about 1 µm. Note that the magni cation of microscopes is unlimited; what is limited is their resolution. is is exactly the same behaviour shown by digital images. e resolution is simply the size of the smallest possible pixel that makes sense.
Challenge 1007, page 570: e dispersion at the lens leads to di erent apparent image positions, as shown in Figure 408. For more details on the dispersion in the human eye and the ways of using it to create three-dimensional e ects, see the article by C. U & R. W , Durch Farbe in die dritte Dimension, Physik in unserer Zeit 30, pp. 50–53, 1999.
Challenge 1008, page 570: e 1 mm beam would return 1000 times as wide as the 1 m beam. A perfect 1 m-wide beam of green light would be 209 m wide on the Moon; can you deduce this result from the (important) formula that involves distance, wavelength, initial diameter and nal diameter? Try to guess this beautiful formula rst, and then deduce it. In reality, the values are a few times larger than the theoretical minimum thus calculated. See the http://www.csr.utexas. edu/mlrs and http://ilrs.gsfc.nasa.gov websites.
Challenge 1009, page 570: e answer should lie between one or two dozen kilometres, assuming ideal atmospheric circumstances.
Challenge 1014, page 572: A surface of 1 m2 perpendicular to the light receives about 1 kW of radiation. It generates the same pressure as the weight of about 0.3 mg of matter. at generates 3 µPa for black surfaces, and the double for mirrors.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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focus ocular real intermediate focus image
focus objective object virtual image
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 407 Two converging lenses make a microscope
Challenge 1016, page 573: e shine side gets twice the momentum transfer as the black side, and thus should be pushed backwards. Challenge 1019, page 574: A polarizer can do this. Challenge 1022, page 574: e interference patterns change when colours are changed. Rainbows also appear because di erent colours are due to di erent frequencies. Challenge 1024, page 575: e full rainbow is round like a circle. You can produce one with a garden hose, if you keep the hose in your hand while you stand on a chair, with your back to the evening Sun. (Well, one small part is missing; can you imagine which part?) e circle is due to the spherical shape of droplets. If the droplets were of di erent shape, and if they were all aligned, the rainbow would have a di erent shape than a simple circle. Challenge 1028, page 577: Film a distant supernova explosion and check whether it happens at the same time for each colour separately. Challenge 1030, page 579: e rst part of the forerunner is a feature with the shortest possible e ective wavelength; thus it is given by taking the limit for in nite frequency. Challenge 1031, page 579: e light is pulsed; thus it is the energy velocity. Challenge 1032, page 579: Inside matter, the energy is transferred to atoms, then back to light, then to the next atoms, etc. at takes time and slows down the propagation. Challenge 1034, page 581: is is true even in general relativity, when the bending of the vacuum is studied. Challenge 1035, page 582: Not really; a Cat’s-eye uses two re ections at the sides of a cube. A living cat’s eye has a large number of re ections. e end e ect is the same though: light returns back to the direction it came from.
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Eye lens dispersion
apparent blue position apparent red position real position
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
F I G U R E 408 The relation between the colour effect and the lens dispersion
Challenge 1036, page 583: ere is a blind spot in the eye; that is a region in which images are not perceived. e brain than assumes that the image at that place is the same than at its borders. If a spot falls exactly inside it, it disappears. Challenge 1038, page 585: e eye and brain surely do not switch the up and the down direction at a certain age. Challenge 1039, page 586: e eye and vision system subtract patterns that are constant in time. Challenge 1041, page 587: In fact, there is no way that a hologram of a person can walk around and frighten a real person. A hologram is always transparent; one can always see the background through the hologram. A hologram thus always gives an impression similar to what moving pictures usually show as ghosts. Challenge 1042, page 590: See challenge 566. Challenge 1043, page 590: e electrons move slowly, but the speed of electrical signals is given by the time at which the electrons move. Imagine long queue of cars (representing electrons) waiting in front of a red tra c light. All drivers look at the light. As soon as it turns green, everybody starts driving. Even though the driving speed might be only 10 m s, the speed of tra c ow onset was that of light. It is this latter speed which is the speed of electrical signals.
Water pipes tell the same story. A long hose provides water almost in the same instant as the tap is opened, even if the water takes a long time to arrive from the tap to the end of the hose. e speed with which the water reacts is gives by the speed for pressure waves in water. Also for water hoses the signal speed, roughly given by the sound speed in water, is much higher than the speed of the water ow.
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Challenge 1044, page 591: One can measure current uctuations, or measure smallest charges, showing that they are always multiples of the same unit. e latter method was used by Millikan.
Challenge 1047, page 591: Earth’s potential would be U = −q (4πεoR) = 60 MV, where the number of electrons in water must be taken into account.
Challenge 1049, page 592: Almost no light passes; the intensity of the little light that is transmitted depends exponentially on the ratio between wavelength and hole diameter. One also says that a er the hole there is an evanescent wave.
Challenge 1051, page 592: e angular momentum was put into the system when it was
formed. If we bring a point charge from in nity along a straight line to its nal position close
to a magnetic dipole, the magnetic force acting on the charge is not directed along the line of mo-
tion. It therefore creates a non-vanishing torque about the origin. See J.M. A
& A. H
, e Feynman paradox revisited, European Journal of Physics 2, pp. 168–170,
1981.
Challenge 1053, page 593: Leakage currents change the picture. e long term voltage ratio is given by the leakage resistance ratio V1 V2 = R1 R2, as can be easily veri ed in experiments.
Challenge 1054, page 593: ere is always a measurement error when measuring eld values, even when measuring a ‘vanishing’ electromagnetic eld.
Challenge 1055, page 593: e green surface seen at a low high angle is larger than when seen vertically, where the soil is also seen; the soil is covered by the green grass in low angle observation.
Challenge 1058, page 593: e charges in a metal rearrange in a way that the eld inside remains vanishing. is makes cars and aeroplanes safe against lightning. Of course, if the outside
eld varies so quickly that the rearrangement cannot follow, elds can enter the Faraday cage. (By the way, also elds with long wavelengths penetrate metals; remote controls regularly use frequencies of 25 kHz to achieve this.) However, one should wait a bit before stepping out of a car a er lightning has hit, as the car is on rubber wheels with low conduction; waiting gives the charge time to ow into the ground.
For gravity and solid cages, mass rearrangement is not possible, so that there is no gravity shield.
Challenge 1063, page 595: Of course not, as the group velocity is not limited by special relativity. e energy velocity is limited, but is not changed in this experiments.
Challenge 1065, page 595: e Prussian explorer Alexander von Humboldt extensively checked this myth in the nineteenth century. He visited many mine pits and asked countless mine workers in Mexico, Peru and Siberia about their experiences. He also asked numerous chimney-sweeps. Neither him nor anybody else had ever seen the stars during the day.
Challenge 1066, page 596: e number of photons times the quantum of action ħ.
Challenge 1068, page 596: e charging stops because a negatively charged satellite repels electrons and thus stops any electron collecting mechanism. Electrons are captured more frequently than ions because it is easier for them than for ions to have an inelastic collision with the satellite, due to their larger speed at a given temperature.
Challenge 1069, page 596: Any loss mechanism will explain the loss of energy, such as electrical resistance or electromagnetic radiation. A er a fraction of a second, the energy will be lost.
is little problem is o en discussed on the internet.
Challenge 1071, page 597: Show that even though the radial magnetic eld of a spherical wave is vanishing by de nition, Maxwell’s equations would require it to be di erent from zero. Since electromagnetic waves are transversal, it is also su cient to show that it is impossible to comb a hairy sphere without having a (double) vortex or two simple vortices. Despite these statements,
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quantum theory changes the picture somewhat: the emission probability of a photon from an excited atom in a degenerate state is spherically symmetric exactly.
Challenge 1072, page 598: e human body is slightly conducting and changes the shape of the eld and thus e ectively short circuits it. Usually, the eld cannot be used to generate energy, as
the currents involved are much too small. (Lightning bolts are a di erent story, of course. ey are due – very indirectly – to the eld of the Earth, but they are too irregular to be used consistently. Franklin’s lightning rod is such an example.)
Challenge 1076, page 602: is should be possible in the near future; but both the experiment, which will probably measure brain magnetic eld details, and the precise check of its seriousness will not be simple.
Challenge 1083, page 605: Any new one is worth a publication.
Challenge 1084, page 608: Sound energy is also possible, as is mechanical work.
Challenge 1085, page 610: Space-time deformation is not related to electricity; at least at everyday energies. Near Planck energies, this might be di erent, but nothing has been predicted yet.
Challenge 1087, page 611: Ideal absorption is blackness (though it can be redness or whiteness at higher temperatures).
Challenge 1088, page 611: Indeed, the Sun emits about 4 ë 1026 W from its mass of 2 ë 1030 kg, about 0.2 mW kg. e adult human body (at rest) emits about 100 W (you can check this in bed at night), thus about 1.2 W kg per ton. is is about 6000 times more than the Sun.
Challenge 1089, page 612: e average temperature of the Earth is thus 287 K. e energy from the Sun is proportional to the fourth power of the temperature. e energy is spread (roughly) over half the Earth’s surface. e same energy, at the Sun’s surface, comes from a much smaller surface, given by the same angle as the Earth subtends there. We thus have E 2πRE2arthTE4arth = TS4unRE2arthα2, where α is half the angle subtended by the Sun. As a result, the temperature of the Sun is estimated to be TSun = (TE4arth α2)0.25 = 4 kK. Challenge 1096, page 613: At high temperature, all bodies approach black bodies. e colour is more important than other colour e ects. e oven and the objects have the same temperature.
us they cannot be distinguished from each other. To do so nevertheless, illuminate the scene with powerful light and then take a picture with small sensitivity. us one always needs bright light to take pictures of what happens inside res.
Challenge 1101, page 633: e issue is: is the ‘universe’ a concept? More about this issue in the third part of the text.
Challenge 1103, page 636: When thinking, physical energy, momentum and angular momentum are conserved, and thermodynamic entropy is not destroyed. Any experiment that this would not be so would point to unknown processes. However, there is no evidence for this.
Challenge 1104, page 636: e best method cannot be much shorter than what is needed to describe 1 in 6000 million, or 33 bits. e Dutch and UK post code systems (including the letters NL or UK) are not far from this value and thus can claim to be very e cient.
Challenge 1105, page 636: For complex systems, when the unknowns are numerous, the advance is thus simply given by the increase in answers. For the universe as a whole, the number of open issues is quite low, as shown on page 959; here there has not been much advance in the last years. But the advance is clearly measurable in this case as well.
Challenge 1106, page 637: Is it possible to use the term ‘complete’ when describing nature?
Challenge 1109, page 638: ere are many baths in series: thermal baths in each light-sensitive cell of the eyes, thermal baths inside the nerves towards the brain and thermal baths inside brain cells.
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Challenge 1111, page 638: Yes.
Challenge 1114, page 644: Physicists claim that the properties of objects, of space-time and of interactions form the smallest list possible. However, this list is longer than the one found by linguists! e reason is that physicists have found primitives that do not appear in everyday life. In a sense, the aim of physicists is limited by list of unexplained questions of nature, given on page 959.
Challenge 1115, page 645: Neither has a de ned content, clearly stated limits or a domain of application.
Challenge 1116, page 645: Impossible! at would not be a concept, as it has no content. e solution to the issue must be and will be di erent.
Challenge 1117, page 647: To neither. is paradox shows that such a ‘set of all sets’ does not exist.
Challenge 1118, page 647: e most famous is the class of all sets that do not contain themselves. is is not a set, but a class.
Challenge 1119, page 648: Dividing cakes is di cult. A simple method that solves many – but not all – problems among N persons P1...PN is the following:
— P1 cuts the cake into N pieces. — P2 to PN choose a piece. — P1 keeps the last part. — P2...PN assemble their parts back into one. — en P2...PN repeat the algorithm for one person less.
e problem is much more complex if the reassembly is not allowed. A just method (in nite many steps) for 3 people, using nine steps, was published in 1944 by Steinhaus, and a fully satisfactory method in the 1960s by John Conway. A fully satisfactory method for four persons was found only in 1995; it has 20 steps.
Challenge 1120, page 648: (x, y) = x, x, y .
Challenge 1121, page 649: Hint: show that any countable list of reals misses at least one number. is was proven for the rst time by Cantor. His way was to write the list in decimal expansion and then nd a number that is surely not in the list. Second hint: his world-famous trick is called the diagonal argument.
Challenge 1122, page 649: Hint: all reals are limits of series of rationals.
Challenge 1124, page 651: Yes, provided division by zero is not allowed.
Challenge 1125, page 651: ere are in nitely many of them. But the smallest is already quite large.
Challenge 1126, page 651: 0 = , 1 =
, 2=
etc.
Challenge 1127, page 655: Subtraction is easy. Addition is not commutative only for cases when in nite numbers are involved: ω + 2 2 + ω.
Challenge 1128, page 655: Examples are 1 − ε or 1 − 4ε2 − 3ε3.
Challenge 1129, page 656: e answer is 57; the cited reference gives the details.
Challenge 1130, page 657: 2222 and 4444 .
Challenge 1132, page 658: is is not an easy question. e rst nontrivial numbers are 7, 23,
47, 59, 167 and 179. See R
M
, Maximally periodic reciprocals, Bulletin of the
Institute of Mathematics and its Applications 28, pp. 147–148, 1992. Matthews shows that a number
n for which 1 n generates the maximum of n − 1 decimal digits in the decimal expansion is a
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special sort of prime number that can be deduced from the so-called Sophie Germain primes S; one must have n = 2S + 1, where both S and 2S + 1 must be prime and where S mod 20 must be 3, 9, or 11.
us the rst numbers n are 7, 23, 47, 59, 167 and 179, corresponding to values for S of 3, 11, 23, 29, 83 and 89. In 1992, the largest known S that meets the criteria was
S = (39051 ë 26002) − 1 ,
(885)
Page 1046
a 1812-digit long Sophie Germain prime number that is 3 mod 20. It was discovered by Wilfred Keller. is Sophie Germain prime leads to a prime n with a decimal expansion that is around 101812 digits long before it starts repeating itself. Read your favourite book on number theory to
nd out more. Interestingly, the solution to this challenge is also connected to that of challenge 1125. Can you nd out more?
Challenge 1133, page 658: Klein did not belong to either group. As a result, some of his nastier students concluded that he was not a mathematician at all.
Challenge 1134, page 658: A barber cannot belong to either group; the de nition of the barber is thus contradictory and has to be rejected.
Challenge 1135, page 658: See the http://members.shaw.ca/hdhcubes/cube_basics.htm web page for more information on magic cubes.
Challenge 1136, page 658: Such an expression is derived with the intermediate result (1−22)−1.
e handling of divergent series seems absurd, but mathematicians know how to give the expres-
sion a de ned content. (See G
H. H
, Divergent Series, Oxford University Press,
1949.) Physicists o en use similar expressions without thinking about them, in quantum eld
theory.
Challenge 1137, page 668: ‘All Cretans lie’ is false, since the opposite, namely ‘some Cretans say the truth’ is true in the case given. e trap is that the opposite of the original sentence is usually, but falsely, assumed to be ‘all Cretans say the truth’.
Challenge 1138, page 668: e statement cannot be false, due to the rst half and the ‘or’ construction. Since it is true, the second half must be true and you are an angel.
Challenge 1147, page 669: e light bulb story seems to be correct. e bulb is very weak, so that the wire is not evaporating.
Challenge 1149, page 674: Only induction allows to make use of similarities and thus to de ne concepts.
Challenge 1151, page 676: Yes, as we shall nd out.
Challenge 1152, page 677: Yes, as observation implies interaction.
Challenge 1153, page 677: Lack of internal contradictions means that a concept is valid as a thinking tool; as we use our thoughts to describe nature, mathematical existence is a specialized version of physical existence, as thinking is itself a natural process. Indeed, mathematical concepts are also useful for the description of the working of computers and the like.
Another way to make the point is to stress that all mathematical concepts are built from sets and relations, or some suitable generalizations of them. ese basic building blocks are taken from our physical environment. Sometimes the idea is expressed di erently; many mathematicians have acknowledged that certain mathematical concepts, such as natural numbers, are taken directly from experience.
Challenge 1154, page 677: Examples are Achilles, Odysseus, Mickey Mouse, the gods of polytheism and spirits.
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Challenge 1156, page 679: Torricelli made vacuum in a U-shaped glass tube, using mercury, the same liquid metal used in thermometers. Can you imagine how? A more di cult question: where did he get mercury from?
Challenge 1157, page 680: Stating that something is in nite can be allowed, if the statement is falsi able. An example is the statement ‘ ere are in nitely many mosquitoes.’
Other statements are not falsi able, such as ‘ e universe continue without limit behind the horizon.’ Such a statement is a belief, not a fact.
Challenge 1158, page 682: ey are not sets either and thus not collections of points.
Challenge 1159, page 682: ere is still no possibility to interact with all matter and energy, as this includes oneself.
Challenge 1160, page 688: No. ere is only a generalization encompassing the two.
Challenge 1161, page 689: An explanation of the universe is not possible, as the term explanation require the possibility to talk about systems outside the one under consideration. e universe is not part of a larger set.
Challenge 1162, page 689: Both can in fact be seen as two sides of the same argument: there is no other choice;there is only one possibility. e rest of nature shows that it has to be that way, as everything depends on everything.
Challenge 1163, page 704: Classical physics fails in explaining any material property, such as colour or so ness. Material properties result from nature’s interactions; they are inevitably quantum. Explanations always require particles and their quantum properties.
Challenge 1164, page 705: Classical physics allows any observable to change smoothly with time. ere is no minimum value for any observable physical quantity.
Challenge 1166, page 706: e simplest length is 2Għ c3 . e factor 2 is obviously not xed; it is explained later on. Including it, this length is the smallest length measurable in nature.
Challenge 1167, page 706: e electron charge is special to the electromagnetic interactions; it does not take into account the nuclear interactions. It is also unclear why the length should be of importance for neutral systems or for the vacuum. On the other hand, it turns out that the di erences are not too fundamental, as the electron charge is related to he quantum of action by e = 4πε0αcħ .
Challenge 1168, page 707: On purely dimensional grounds, the radius of an atom must be
Page 221
r
ħ24πε0 me2
(886)
is is about 160 nm; indeed, this guessed equation is simply π times the Bohr radius.
Challenge 1169, page 707: Due to the quantum of action, atoms in all people, be they giants or dwarfs, have the same size. at giants do not exist was shown already by Galilei. e argument is based on the given strength of materials, which thus implies that atoms are the same everywhere.
at dwarfs cannot exist is due to the same reason; nature is not able to make people smaller than usual (except in the womb) as this would require smaller atoms.
Challenge 1181, page 713: Also photons are indistinguishable. See page 732.
Challenge 1184, page 716: e total angular momentum counts, including the orbital angular momentum. e orbital angular momentum L is given, using the radius and the linear momentum, L = r p.
Challenge 1185, page 716: Yes, we could have!
Challenge 1207, page 735: e quantum of action implies that two subsequent observations always di er. us the surface of a liquid cannot be at rest.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 1218, page 751: Use ∆E < E and a ∆t < c.
Challenge 1222, page 751: e di culties to see hydrogen atoms are due to their small size and their small number of electrons. As a result, hydrogen atoms produce only weak contrasts in X-ray images. For the same reasons it is di cult to image them using electrons; the Bohr radius of hydrogen is only slightly larger than the electron Compton wavelength.
Challenge 1226, page 752: r = 86 pm, thus T = 12 eV. at compares to the actual value of
13.6 eV.
e
trick
for
the
derivation
of
the
formula
is
to
use
<
ψ
r
2 x
ψ
=
1 3
<
ψ
rr ψ
, a relation
valid for states with no orbital angular momentum. It is valid for all coordinates and also for the
three momentum observables, as long as the system is non-relativistic.
Challenge 1227, page 752: e elds are crated by neutrons or protons, which have a smaller Compton wavelength.
Challenge 1241, page 763: A change of physical units such that ħ = c = e = 1 would change the value of ε0 in such a way that 4πεo = 1 α = 137.036...
Challenge 1242, page 771: Point particles cannot be marked; nearby point particles cannot be distinguished, due to the quantum of action.
Challenge 1248, page 774: For a large number of particles, the interaction energy will introduce errors. For very large numbers, the gravitational binding energy will do so as well.
Challenge 1250, page 775: Two write two particles on paper, one has to distinguish them, even if the distinction is arbitrary.
Challenge 1254, page 779: Twins di er in the way their intestines are folded, in the lines of their hands and other skin folds. Sometimes, but not always, features like black points on the skin are mirror inverted on the two twins.
Challenge 1260, page 785: ree.
Challenge 1261, page 785: Angels can be distinguished by name, can talk and can sing; thus they are made of a large number of fermions. In fact, many angels are human sized, so that they do not even t on the tip of a pin.
Challenge 1269, page 789: Ghosts, like angels, can be distinguished by name, can talk and can be seen; thus they contain fermions. However, they can pass through walls and they are transparent; thus they cannot be made of fermions, but must be images, made of bosons. at is a contradiction.
Challenge 1271, page 796: e loss of non-diagonal elements leads to an increase in the diagonal elements, and thus of entropy.
Challenge 1274, page 801: e energy speed is given by the advancement of the outer two tails; that speed is never larger than the speed of light.
Challenge 1275, page 804: A photograph requires illumination; illumination is a macroscopic electromagnetic eld; a macroscopic eld is a bath; a bath implies decoherence; decoherence destroys superpositions.
Challenge 1278, page 805: Such a computer requires clear phase relations between components; such phase relations are extremely sensitive to outside disturbances. At present, they do not hold longer than a microsecond, whereas long computer programs require minutes and hours to run.
Challenge 1282, page 812: Any other bath also does the trick, such as the atmosphere, sound vibrations, electromagnetic elds, etc.
Challenge 1283, page 812: e Moon is in contact with baths like the solar wind, falling meteorites, the electromagnetic background radiation of the deep universe, the neutrino ux from the Sun, cosmic radiation, etc.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 1284, page 813: Spatially periodic potentials have the property. Decoherence then leads to momentum diagonalisation.
Challenge 1285, page 816: A virus is an example. It has no own metabolism. (By the way, the ability of some viruses to form crystals is not a proof that they are not living beings, in contrast to what is o en said.)
Challenge 1286, page 817: e navigation systems used by ies are an example.
Challenge 1287, page 820: e thermal energy kT is about 4 zJ and a typical relaxation time is 0.1 ps.
Challenge 1288, page 822: is is not possible at present. If you know a way, publish it. It would help a sad single mother who has to live without nancial help from the father, despite a lawsuit, as it was yet impossible to decide which of the two candidates is the right one.
Challenge 1289, page 822: Also identical twins count as di erent persons and have di erent fates. Imprinting in the womb is di erent, so that their temperament will be di erent. e birth experience will be di erent; this is the most intense experience of every human, strongly determining his fears and thus his character. A person with an old father is also quite di erent from that with a young father. If the womb is not that of his biological mother, a further distinction of the earliest and most intensive experiences is given.
Challenge 1290, page 822: Life’s chemicals are synthesized inside the body; the asymmetry has been inherited along the generations. e common asymmetry thus shows that all life has a common origin.
Challenge 1291, page 823: Well, men are more similar to chimpanzees than to women. More
seriously, the above data, even though o en quoted, are wrong. Newer measurements by Roy Brit-
ten in 2002 have shown that the di erence in genome between humans and chimpanzees is about
5 % (See R.J. B
, Divergence between samples of chimpanzee and human DNA sequences
is 5 %, counting indels, Proceedings of the National Academy of Sciences 99, pp. 13633–13635, 15th
of October, 2002.) In addition, though the di erence between man and woman is smaller than
one whole chromosome, the large size of the X chromosome, compared with the small size of the
Y chromosome, implies that men have about 3 % less genetic material than women. However, all
men have an X chromosome as well. at explains that still other measurements suggest that all
humans share a pool of at least 99.9 % of common genes.
Challenge 1310, page 836: All detectors of light can be called relativistic, as light moves with maximal speed. Touch sensors are not relativistic following the usual sense of the word, as the speeds involved are too small. e energies are small compared to the rest energies; this is the case even if the signal energies are attributed to electrons only.
Challenge 1311, page 836: e noise is due to the photoacoustic e ect; the periodic light periodically heats the air in the jam glass at the blackened surface and thus produces sound. See M. E , Kann man Licht hören?, Physik in unserer Zeit 32, pp. 180–182, 2001.
Challenge 1293, page 824: Since all the atoms we are made of originate from outer space, the answer is yes. But if one means that biological cells came to Earth from space, the answer is no, as cells do not like vacuum. e same is true for DNA.
Challenge 1292, page 824: e rst steps are not known yet.
Challenge 1295, page 824: Chemical processes, including di usion and reaction rates, are strongly temperature dependent. ey a ect the speed of motion of the individual and thus its chance of survival. Keeping temperature in the correct range is thus important for evolved life forms.
Challenge 1296, page 826: Haven’t you tried yet? Physics is an experimental science.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 1298, page 828: Radioactive dating methods can be said to be based on the nuclear interactions, even though the detection is again electromagnetic.
Challenge 1315, page 842: With a combination of the methods of Table 62 it is possible; but whether there will ever be an organization willing to pay for this to happen is another question.
Challenge 1317, page 844: For example, a heavy mountain will push down the Earth’s crust into the mantle, makes it melt on the bottom side, and thus lowers the position of the top.
Challenge 1318, page 844: ese developments are just starting; the results are still far from the original one is trying to copy, as they have to ful l a second condition, in addition to being a ‘copy’ of original feathers or of latex: the copy has to be cheaper than the original. at is o en a much tougher request than the rst.
Challenge 1320, page 845: Since the height of the potential is always nite, walls can always be overcome by tunnelling.
Challenge 1321, page 845: e lid of a box can never be at rest, as is required for a tight closure, but is always in motion, due to the quantum of action.
Challenge 1326, page 850: e one somebody else has thrown away. Energy costs about 10 cents kWh. For new lamps, the uorescence lamp is the best for the environment, even though it is the least friendly to the eye and the brain, due to its ickering.
Challenge 1327, page 853: is old dream depends on the precise conditions. How exible does the display have to be? What lifetime should it have? e newspaper like display is many years away and maybe not even possible.
Challenge 1328, page 853: e challenge here is to nd a cheap way to de ect laser beams in a controlled way. Cheap lasers are already available.
Challenge 1329, page 853: ere is only speculation on the answer; the tendency of most researchers is to say no.
Challenge 1330, page 853: No, as it is impossible because of momentum conservation, because of the no-cloning theorem.
Challenge 1332, page 854: e author predicts that mass-produced goods using this technology (at least 1 million pieces sold) will not be available before 2025.
Challenge 1333, page 854: Maybe, but for extremely high prices.
Challenge 1334, page 856: For example, you could change gravity between two mirrors.
Challenge 1335, page 856: As usual in such statements, either group or phase velocity is cited, but not the corresponding energy velocity, which is always below c.
Challenge 1336, page 858: Echoes do not work once the speed of sound is reached and do not work well when it is approached. Both the speed of light and that of sound have a nite value. Moving with a mirror still gives a mirror image. is means that the speed of light cannot be reached. If it cannot be reached, it must be the same for all observers.
Challenge 1337, page 858: Mirrors do not usually work for matter; in addition, if they did, matter would require much higher acceleration values.
Challenge 1340, page 860: e overhang can have any value whatsoever. ere is no limit. Taking the indeterminacy principle into account introduces a limit as the last brick or card must not allow the centre of gravity, through its indeterminacy, to be over the edge of the table.
Challenge 1341, page 861: A larger charge would lead to eld that spontaneously generate electron positron pairs, the electron would fall into the nucleus and reduce its charge by one unit.
Challenge 1344, page 862: e Hall e ect results from the deviation of electrons in a metal due to an applied magnetic eld. erefore it depends on their speed. One gets values around 1 mm. Inside atoms, one can use Bohr’s atomic model as approximation.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 1345, page 862: e usual way to pack oranges on a table is the densest way to pack spheres.
Challenge 1346, page 863: Just use a paper drawing. Draw a polygon and draw it again at latter times, taking into account how the sides grow over time. You will see by yourself how the faster growing sides disappear over time.
Challenge 1347, page 864: e steps are due to the particle nature of electricity and all other moving entities.
Challenge 1348, page 864: Mud is a suspension of sand; sand is not transparent, even if made of clear quartz, because of the scattering of light at the irregular surface of its grains. A suspension cannot be transparent if the index of refraction of the liquid and the suspended particles is di erent. It is never transparent if the particles, as in most sand types, are themselves not transparent.
Challenge 1349, page 864: No. Bound states of massless particles are always unstable.
Challenge 1350, page 864: e rst answer is probably no, as composed systems cannot be smaller than their own compton wavelength; only elementary systems can. However, the universe is not a system, as it has no environment. As such, its length is not a precisely de ned concept, as an environment is needed to measure and to de ne it. (In addition, gravity must be taken into account in those domains.) us the answer is: in those domains, the question makes no sense.
Challenge 1351, page 865: Methods to move on perfect ice from mechanics:
— if the ice is perfectly at, rest is possible only in one point – otherwise you oscillate around that point, as shown in challenge 21;
— do nothing, just wait that the higher centrifugal acceleration at body height pulls you away; — to rotate yourself, just rotate your arm above your head; — throw a shoe or any other object away; — breathe in vertically, breathing out (or talking) horizontally (or vice versa); — wait to be moved by the centrifugal acceleration due to the rotation of the Earth (and its ob-
lateness); — jump vertically repeatedly: the Coriolis acceleration will lead to horizontal motion; — wait to be moved by the Sun or the Moon, like the tides are; — ‘swim’ in the air using hands and feet; — wait to be hit by a bird, a ying wasp, inclined rain, wind, lava, earthquake, plate tectonics, or
any other macroscopic object (all objects pushing count only as one solution); — wait to be moved by the change in gravity due to convection in Earth’s mantle; — wait to be moved by the gravitation of some comet passing by; — counts only for kids: spit, sneeze, cough, fart, pee; or move your ears and use them as wings.
Note that gluing your tongue is not possible on perfect ice.
Challenge 1352, page 865: Methods to move on perfect ice using thermodynamics and electrodynamics:
— use the radio/TV stations nearby to push you around; — use your portable phone and a mirror; — switch on a pocket lam, letting the light push you; — wait to be pushed around by Brownian motion in air; — heat up one side of your body: black body radiation will push you; — heat up one side of your body, e.g. by muscle work: the changing air ow or the evaporation
will push you; — wait for one part of the body to be cooler than the other and for the corresponding black body
radiation e ects;
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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— wait for the magnetic eld of the Earth to pull on some ferromagnetic or paramagnetic metal piece in your clothing or in your body;
— wait to be pushed by the light pressure, i.e. by the photons, from the Sun or from the stars, maybe using a pocket mirror to increase the e ciency;
— rub some polymer object to charge it electrically and then move it in circles, thus creating a magnetic eld that interacts with the one of the Earth.
Note that perfect frictionless surfaces do not melt.
Challenge 1353, page 865: Methods to move on perfect ice using quantum e ects:
— wait for your wave function to spread out and collapse at the end of the ice surface; — wait for the pieces of metal in the clothing to attract to the metal in the surrounding through
the Casimir e ect; — wait to be pushed around by radioactive decays in your body.
Challenge 1354, page 865: Methods to move on perfect ice using general relativity: — move an arm to emit gravitational radiation; — deviate the cosmic background radiation with a pocket mirror; — wait to be pushed by gravitational radiation from star collapses; — wait to the universe to contract.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 1355, page 865: Methods to move on perfect ice using materials science, geophysics, astrophysics:
— be pushed by the radio waves emitted by thunderstorms and absorbed in painful human joints; — wait to be pushed around by cosmic rays; — wait to be pushed around by the solar wind; — wait to be pushed around by solar neutrinos; — wait to be pushed by the transformation of the Sun into a red giant; — wait to be hit by a meteorite.
Challenge 1356, page 865: A method to move on perfect ice using selforganisation, chaos theory, and biophysics:
— wait that the currents in the brain interact with the magnetic eld of the Earth by controlling your thoughts.
Challenge 1357, page 865: Methods to move on perfect ice using quantum gravity, supersymmetry, and string theory:
— accelerate your pocket mirror with your hand; — deviate the Unruh radiation of the Earth with a pocket mirror; — wait for proton decay to push you through the recoil.
Challenge 1361, page 870: is is easy only if the black hole size is inserted into the entropy bound by Bekenstein. A simple deduction of the black hole entropy that includes the factor 1/4 is not yet at hand.
Challenge 1362, page 871: An entropy limit implies an information limit; only a given information can be present in a given region of nature. is results in a memory limit.
Challenge 1363, page 871: In natural units, the expression for entropy is S = A 4 = 0.25A. If each Planck area carried one bit (degree of freedom), the entropy would be S = ln W = ln(2A) = A ln 2 = 0.693A. is quite near the exact value.
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Challenge 1367, page 875: e universe has about 1022 stars; the Sun has a luminosity of about
1026 W; the total luminosity of the visible matter in the universe is thus about 1048 W. A gamma ray burster emits up to 3 ë 1047 W.
Challenge 1373, page 877: ey are carried away by the gravitational radiation.
Challenge 1385, page 900: Two stacked foils show the same e ect as one foil of the same total thickness. us the surface plays no role.
Challenge 1387, page 902: e electron is held back by the positive charge of the nucleus, if the number of protons in the nucleus is su cient, as is the case for those nuclei we are made of.
Challenge 1389, page 909: e number is small compared with the number of cells. However, it is possible that the decays are related to human ageing.
Challenge 1392, page 911: e radioactivity necessary to keep the Earth warm is low; lava is only slightly more radioactive than usual soil.
Challenge 1394, page 919: By counting decays and counting atoms to su cient precision.
Challenge 1395, page 922: e nuclei of nitrogen and carbon have a high electric charge which strongly repels the protons.
Challenge 1396, page 927: Touching something requires getting near it; getting near means a small time and position indeterminacy; this implies a small wavelength of the probe that is used for touching; this implies a large energy.
Challenge 1397, page 935: Building a nuclear weapon is not di cult. University students can do it, and even have done so once, in the 1980s. e problem is getting or making the nuclear material. at requires either an extensive criminal activity or an vast technical e ort, with numerous large factories, extensive development, coordination of many technological activities. Most importantly, such a project requires a large nancial investment, which poor countries cannot a ord. e problems are thus not technical, but nancial.
Challenge 1400, page 952: Most macroscopic matter properties fall in this class, such as the change of water density with temperature.
Challenge 1403, page 963: Before the speculation can be fully tested, the relation between particles and black holes has to be clari ed rst.
Challenge 1404, page 964: Never expect a correct solution for personal choices. Do what you yourself think and feel is correct.
Challenge 1407, page 970: A mass of 100 kg and a speed of 8 m s require 43 m2 of wing surface.
Challenge 1410, page 980: e in nite sum is not de ned for numbers; however, it is de ned for a knotted string.
Challenge 1413, page 982: is is a simple but hard question. Find out.
Challenge 1383, page 882: No system is known in nature which emits or absorbs only one graviton at a time. is is another point speaking against the existence of gravitons.
Challenge 1419, page 989: Lattices are not isotropic, lattices are not Lorentz invariant.
Challenge 1421, page 991: Large raindrops are pancakes with a massive border bulge. When the size increases, e.g. when a large drop falls through vapour, the drop splits, as the central membrane is then torn apart.
Challenge 1422, page 991: It is a drawing; if it is interpreted as an image of a three-dimensional object, it either does not exist, or is not closed, or is an optical illusion of a torus.
Challenge 1423, page 991: See T. F 2000.
& Y. M , e 85 Ways to Tie a Tie, Broadway Books,
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Challenge 1424, page 991: See T. C
, Laces high, Nature Science Update 5th of Decem-
ber, 2002, or http://www.nature.com/nsu/021202/021202-4.html.
Challenge 1426, page 1001: e other scale is the horizon of the universe, as we will see shortly.
Challenge 1427, page 1003: Sloppily speaking, such a clock is not able to move its hands in such a way to guarantee precise time reading.
Challenge 1430, page 1016: e nal energy E produced by a proton accelerator increases with its radius R roughly as E R1.2; as an example, CERN’s SPS achieves about 450 GeV for a radius of 740 m. us we would get a radius of more than 100 000 light years (larger than our galaxy) for a Planck energy accelerator. An accelerator achieving Planck energy is impossible.
A uni cation energy accelerator would be about 1000 times smaller. Nature has no accelerator of this power, but gets near it. e maximum measured value of cosmic rays, 1022 eV, is about one thousandth of the uni cation energy. e mechanisms of acceleration are obscure. Black holes are no sources for uni cation energy particles, due to their gravitational potential. But also the cosmic horizon is not the source, for some yet unclear reasons. is issue is still a topic of research.
Challenge 1431, page 1016: e Planck energy is EPl = ħc5 G = 2.0 GJ. Car fuel delivers about 43 MJ kg. us the Planck energy corresponds to the energy of 47 kg of car fuel, about a tankful.
Challenge 1432, page 1017: Not really, as the mass error is equal to the mass only in the Planck case.
Challenge 1433, page 1017: It is improbable that such deviations can be found, as they are masked by the appearance of quantum gravity e ects. However, if you do think that you have a prediction for a deviation, publish it.
Challenge 1435, page 1017: ere is no gravitation at those energies and there are no particles. ere is thus no paradox.
Challenge 1437, page 1018: e Planck acceleration is given by aPl = 5.6 ë 1051 m s2.
c7 ħG =
Challenge 1438, page 1019: All mentioned options could be valid at the same time. e issue is not closed and clear thinking about it is not easy.
Challenge 1439, page 1019: e energy is the uni cation energy, about 800 times smaller than the Planck energy.
Challenge 1440, page 1020: is is told in detail in the section on maximum force starting on page 1068.
Challenge 1441, page 1025: Good! Publish it.
Challenge 1442, page 1032: See the table on page 184.
Challenge 1443, page 1033: e cosmic background radiation is a clock in the widest sense of the term.
Challenge 1465, page 1047: For the description of nature this is a contradiction. Nevertheless, the term ‘universe’, ‘set of all sets’ and other mathematical terms, as well as many religious concepts are of this type.
Challenge 1468, page 1048: For concept of ‘universe’.
Challenge 1472, page 1050: Augustine and many theologians have de ned ‘god’ in exactly this way. (See also omas Aquinas, Summa contra gentiles, 1, 30.) ey claim that it possible to say what ‘god’ is not, but that it is not possible to say what it is. ( is statement is also part of the o cial roman catholic catechism: see part one, section one, chapter one, IV, 43.)
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Many legal scholars would also propose a di erent concept that ts the de nition – namely ‘administration’. It is di cult to say what it is, but easy to say what it is not.
More seriously, the properties common to the universe and to ‘god’ suggest the conclusion that the both are the same. Indeed, the analogy between the two concepts can be expanded to a proof. ( is is le to the reader.) In fact, this might be the most interesting of all proofs of the existence of gods. is proof certainly lacks all the problems that the more common ‘proofs’ have. Despite its interest, the present proof is not found in any book on the topic. e reason is obvious: the result of the proof, the equivalence of ‘god’ and the universe, is a heresy for most religions.
If one is ready to explore the analogy nevertheless, one nds that a statement like ‘god created the universe’ translates as ‘the universe implies the universe´. e original statement is thus not a lie any more, but is promoted to a tautology. Similar changes appear for many other – but not all – statements using the term ‘god’. Enjoy the exploration.
Challenge 1473, page 1050: If you nd one, publish it! And send it to the author as well.
Challenge 1475, page 1051: If you nd one, publish it and send it to the present author as well.
Challenge 1479, page 1057: Any change in rotation speed of the Earth would change the sea level.
Challenge 1480, page 1058: Just measure the maximum water surface the oil drop can cover, by looking at the surface under a small angle.
Challenge 1481, page 1058: Keep the ngers less than 1 cm from your eye.
Challenge 1482, page 1065: As vacuum and matter cannot be distinguished, both share the same properties. In particular, both scatter strongly at high energies.
Challenge 1483, page 1069: Take ∆ f ∆t 1 and substitute ∆l = c ∆ f and ∆a = c ∆t.
Challenge 1505, page 1111: e number of spatial dimensions must be given rst, in order to talk about spheres.
Challenge 1506, page 1114: is is a challenge to you to nd out and publish; it is fun, may bring success and would yield an independent check of the results of the section.
Challenge 1511, page 1125: e lid of a box must obey the indeterminacy relation. It cannot be at perfect rest with respect to the rest of the box.
Challenge 1512, page 1125: Of course not, as there are no in nite quantities in nature. e question is whether the detector would be as large as the universe or smaller. What is the answer?
Challenge 1517, page 1126: Yes, as nature’s inherent measurement errors cannot clearly distinguish between them.
Challenge 1518, page 1126: Of course.
Challenge 1516, page 1126: No. Time is continuous only if either quantum theory and point particles or general relativity and point masses are assumed. e argument shows that only the combination of both theories with continuity is impossible.
Challenge 1519, page 1126: We still have the chance to nd the best approximate concepts possible. ere is no reason to give up.
Challenge 1520, page 1126: A few thoughts. beginning of the big bang does not exist, but is given by that piece of continuous entity which is encountered when going backwards in time as much as possible. is has several implications.
— Going backwards in time as far as possible – towards the ‘beginning’ of time – is the same as zooming to smallest distances: we nd a single strand of the amoeba.
— In other words, we speculate that the whole world is one single piece, knotted, branched and uctuating.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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— Going far away into space – to the border of the universe – is like taking a snapshot with a short shutter time: strands everywhere.
— Whenever we sloppily say that extended entities are ‘in nite’ in size, we only mean that they reach the horizon of the universe.
In summary, no starting point of the big bang exists, because time does not exist there. For the same reason, no initial conditions for particles or space-time exist. In addition, this shows there was no creation involved, since without time and without possibility of choice, the term ‘creation’ makes no sense.
Challenge 1521, page 1126: e equivalence follows from the fact that all these processes require Planck energy, Planck measurement precision, Planck curvature, and Planck shutter time.
Challenge 1500, page 1092: e system limits cannot be chosen in other ways; a er the limits have been corrected, the limits given here should still apply.
Challenge 1527, page 1158: Planck limits can be exceeded for extensive observables for which many particle systems can exceed single particle limits, such as mass, momentum, energy or electrical resistance.
Challenge 1531, page 1162: Do not forget the relativistic time dilation.
Challenge 1533, page 1163: Since the temperature of the triple point of water is xed, the temperature of the boiling point is xed as well. Historically, the value of the triple point has not been well chosen.
Challenge 1534, page 1163: Probably the quantity with the biggest variation is mass, where a pre x for 1 eV c2 would be useful, as would be one for the total mass in the universe, which is about 1090 times larger.
Challenge 1535, page 1164: e formula with n − 1 is a better t. Why?
Challenge 1538, page 1167: No, only properties of parts of the universe. e universe itself has no properties, as shown on page 1050.
Challenge 1539, page 1168: e slowdown goes quadratically with time, because every new slowdown adds to the old one!
Challenge 1540, page 1170: e double of that number, the number made of the sequence of all even numbers, etc.
Challenge 1532, page 1162: About 10 µg.
Challenge 1542, page 1173: is could be solved with a trick similar to those used in the irrationality of each of the two terms of the sum, but nobody has found one.
Challenge 1543, page 1173: ere are still many discoveries to be made in modern mathematics, especially in topology, number theory and algebraic geometry. Mathematics has a good future.
Challenge 1544, page 1177: e gauge coupling constants determine the size of atoms, the strength of chemical bonds and thus the size of all things.
Challenge 1545, page 1191: Covalent bonds tend to produce full shells; this is a smaller change on the right side of the periodic table.
Challenge 1548, page 1196: z is the determinant of the matrix z =
a −b
b a
.
Challenge 1551, page 1197: Use Cantor’s diagonal argument, as in challenge 1121.
Challenge 1555, page 1199: Any rotation by an angle 2π is described by −1. Only a rotation by 4π is described by +1; quaternions indeed describe spinors.
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Challenge 1557, page 1201: Just check the result component by component. See also the mentioned reference.
Challenge 1559, page 1204: For a Gaussian integer n + im to be prime, the integer n2 + m2 must be prime, and in addition, a condition on n mod 3 must be satis ed; which one and why?
Challenge 1563, page 1205: e metric is regular, positive de nite and obeys the triangle inequality.
Challenge 1567, page 1207: e solution is the set of all two by two matrices, as each two by two matrix speci es a linear transformation, if one de nes a transformed point as the product of the point and this matrix. (Only multiplication with a xed matrix can give a linear transformation.) Can you recognize from a matrix whether it is a rotation, a re ection, a dilation, a shear, or a stretch along two axes? What are the remaining possibilities?
Challenge 1570, page 1208: e (simplest) product of two functions is taken by point-by-point multiplication.
Challenge 1571, page 1208: its absolute value:
e norm f of a real function f is de ned as the supremum of
f = sup f (x) .
xR
(887)
In simple terms: the maximum value taken by the absolute of the function is its norm. It is also called ‘sup’-norm. Since it contains a supremum, this norm is only de ned on the subspace of bounded continuous functions on a space X, or, if X is compact, on the space of all continuous functions (because a continuous function on a compact space must be bounded).
Challenge 1575, page 1213: Take out your head, then pull one side of your pullover over the corresponding arm, continue pulling it over the over arm; then pull the other side, under the rst, to the other arm as well. Put your head back in. Your pullover (or your trousers) will be inside out.
Challenge 1579, page 1217: e transformation from one manifold to another with di erent topology can be done with a tiny change, at a so-called singular point. Since nature shows a minimum action, such a tiny change cannot be avoided.
Challenge 1580, page 1219: e product M† M is Hermitean, and has positive eigenvalues. us H is uniquely de ned and Hermitean. U is unitary because U†U is the unit matrix.
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So far, out of 1580 challenges, 745 solutions are given; in addition, 241 solutions are too easy to be included. Another 594 solutions need to be written; let the author know which one you want most.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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LIST OF ILLUSTRATIONS
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20 62 21 64 22 65
23 66 24 66
An example of motion observed in nature Motion Mountain and the trail to be followed Illusions of motion: look at the gure on the le and slightly move the page, or look at the white dot at the centre of the gure on the right and move your head back and forward How much water is required to make a bucket hang vertically? At what angle does the pulled reel change direction of motion? (© Luca Gastaldi) A time line of scienti c and political personalities in antiquity (the last letter of the name is aligned with the year of death) An example of transport Transport, growth and transformation (© Philip Plisson) A block and tackle and a di erential pulley Galileo Galilei A typical path followed by a stone thrown through the air Two proofs of the three-dimensionality of space: a knot and the inner ear of a mammal René Descartes A curvemeter or odometer A fractal: a self-similar curve of in nite length (far right), and its construction A polyhedron with one of its dihedral angles (© Luca Gastaldi) A photograph of the Earth – seen from the direction of the Sun A model illustrating the hollow Earth theory, showing how day and night appear (© Helmut Diehl) Leaving a parking space
e de nition of plane and solid angles How the apparent size of the Moon and the Sun changes How the apparent size of the Moon changes during its orbit (© Anthony Ayiomamitis) A vernier/nonius/clavius Anticrepuscular rays (© Peggy Peterson) Two ways to test that the time of free fall does not depend on horizontal velocity Various types of graphs describing the same path of a thrown stone
ree superimposed images of a frass pellet shot away by a caterpillar (© Stanley Caveney)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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25 67 Derivatives
69 Gottfried Leibniz
26 72 Orion (in natural colours) and Betelgeuse
27 73 How an object can rotate continuously without tangling up the connection
to a second object
28 73 Legs and ‘wheels’ in living beings
29 75 In which direction does the bicycle turn?
30 76 Collisions de ne mass
31 76
e standard kilogram (© BIPM)
32 78 Is this dangerous?
78 Antoine Lavoisier
78 Christiaan Huygens
33 83 What happens?
85 Robert Mayer
34 90 Angular momentum and the two versions of the right-hand rule
35 91 How a snake turns itself around its axis
36 91 Can the ape reach the banana?
37 92
e velocities and unit vectors for a rolling wheel
38 92 A simulated photograph of a rolling wheel with spokes
39 93
e measured motion of a walking human (© Ray McCoy)
40 93
e parallaxis – not drawn to scale
41 94 Earth’s deviation from spherical shape due to its rotation
42 95
e deviations of free fall towards the east and towards the Equator due to
the rotation of the Earth
43 96
e turning motion of a pendulum showing the rotation of the Earth
45 97 Showing the rotation of the Earth through the rotation of an axis
46 97 Demonstrating the rotation of the Earth with water
44 97
e gyroscope
47 99
e precession and the nutation of the Earth’s axis
48 101 e continental plates are the objects of tectonic motion
49 102 Changes in the Earth’s motion around the Sun
50 103 e motion of the Sun around the galaxy
51 105 Is it safe to let the cork go?
52 106 A simple model for continents and mountains
53 107 A well-known toy
54 107 An elastic collision that seems not to obey energy conservation
55 107 e centre of mass de nes stability
56 107 How does the ladder fall?
57 109 Observation of sonoluminescence and a diagram of the experimental set-up
58 110 How does the ball move when the jar is accelerated?
59 110 e famous Celtic stone and a version made with a spoon
60 112 How does the ape move?
61 113 A long exposure of the stars at night - over the Mauna Kea telescope in Hawaii
(Gemini)
62 115 A basilisk lizard (Basiliscus basiliscus) running on water, showing how the
propulsing leg pushes into the water (© TERRA)
63 115 A water strider (© Charles Lewallen)
64 118 A physicist’s and an artist’s view of the fall of the Moon: a diagram by Chris-
tiaan Huygens (not to scale) and a marble statue by Auguste Rodin
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83 140 84 145 85 151 86 173
87 173 88 174 89 174
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93 175 94 177
178 95 179 96 184 97 188 98 189 99 193 100 204 101 205 102 206
e measurements that lead to the de nition of the metre (© Ken Alder) e potential and the gradient e shape of the Earth, with exaggerated height scale (© GeoForschungsZentrum Potsdam) e motion of a planet around the Sun, showing its semimajor axis d, which is also the spatial average of its distance from the Sun e change of the moon during the month, showing its libration (© Martin Elsässer) Maps (not photographs) of the near side (le ) and far side (right) of the moon, showing how o en the latter saved the Earth from meteorite impacts (courtesy USGS) e possible orbits due to universal gravity e two stable Lagrangian points Tidal deformations due to gravity e origin of tides A spectacular result of tides: volcanism on Io (NASA) Particles falling side by side approach over time Masses bend light Brooms fall more rapidly than stones (© Luca Gastaldi) e starting situation for a bungee jumper An honest balance? Which of the two Moon paths is correct? e analemma over Delphi, between January and December 2002 (© Anthony Ayiomamitis) e vanishing of gravitational force inside a spherical shell of matter A solar eclipse (11 August 1999, photographed from the Russian Mir station) Shapes and air/water resistance What shape of rail allows the black stone to glide most rapidly from point A to the lower point B? Can motion be described in a manner common to all observers? What happens when one rope is cut? How to draw a straight line with a compass: x point F, put a pencil into joint P and move C with a compass along a circle A south-pointing carriage How and where does a falling brick chimney break? Why do hot-air balloons stay in ated? How can you measure the weight of a bicycle rider using only a ruler? What determines the number of petals in a daisy? De ning a total e ect as an accumulation (addition, or integral) of small e ects over time Joseph Lagrange e minimum of a curve has vanishing slope Aggregates in nature Refraction of light is due to travel-time optimization Forget-me-not, also called Myosotis (Boraginaceae) (© Markku Savela) A Hispano–Arabic ornament from the Governor’s Palace in Sevilla e simplest oscillation Decomposing a general wave or signal into harmonic waves e formation of gravity waves on water
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e six main properties of the motion of waves e electrical signals measured in a nerve A solitary water wave followed by a motor boat, reconstructing the discovery by Scott Russel (© Dugald Duncan) Solitons are stable against encounters Shadows and refraction Floors and mountains as fractals (© Paul Martz) Atoms exist: rotating an aluminium rod leads to brightness oscillations Atomic steps in broken gallium arsenide crystals can be seen under a light microscope e principle, and a simple realization, of an atomic force microscope e atoms on the surface of a silicon crystal mapped with an atomic force microscope e result of moving helium atoms on a metallic surface (© IBM) What is your personal stone-skipping record? Heron’s fountain Which funnel is faster? e basic idea of statistical mechanics about gases Which balloon wins? Daniel Bernoulli Example paths for particles in Brownian motion and their displacement distribution e re pump Can you boil water in this paper cup? e invisible loudspeaker e Wirbelrohr or Ranque–Hilsch vortex tube Examples of self-organization for sand Oscillons formed by shaken bronze balls; horizontal size is about 2 cm (© Paul Umbanhowar) Magic numbers: 21 spheres, when swirled in a dish, behave di erently from non-magic numbers, like 23, of spheres (redrawn from photographs, © Karsten Kötter) Examples of di erent types of motion in con guration space Sensitivity to initial conditions e wavy surface of icicles Water pearls A braiding water stream (© Vakhtang Putkaradze) Rømer’s method of measuring the speed of light e rain method of measuring the speed of light Fizeau’s set-up to measure the speed of light (© AG Didaktik und Geschichte der Physik, Universität Oldenburg) A photograph of a light pulse moving from right to le through a bottle with milky water, marked in millimetres (© Tom Mattick) A consequence of the niteness of the speed of light Albert Einstein A drawing containing most of special relativity Moving clocks go slow e set-up for the observation of the Doppler e ect Lucky Luke
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Experimental values (dots) for the electron velocity v as function of their kinetic energy T, compared with the prediction of Galilean physics (blue) and that of special relativity (red) How to deduce the composition of velocities
e result, the schematics and the cryostat set-up for the most precise Michelson–Morley experiment performed to date (© Stephan Schiller) Two inertial observers and a beam of light Space-time diagrams for light seen from two di erent observers using coordinates (t, x) and (τ, ξ) A space-time diagram for a moving object T seen from an inertial observer O in the case of one and two spatial dimensions
e twin paradox More muons than expected arrive at the ground because fast travel keeps them young
e observations of the pilot and the barn owner e observations of the trap digger and of the snowboarder, as (misleadingly) published in the literature Does the conducting glider keep the lamp lit at large speeds? What happens to the rope? Flying through twelve vertical columns (shown in the two uppermost images) with 0.9 times the speed of light as visualized by Nicolai Mokros and Norbert Dragon, showing the e ect of speed and position on distortions (© Nicolai Mokros) Flying through three straight and vertical columns with 0.9 times the speed of light as visualized by Daniel Weiskopf: on the le with the original colours; in the middle including the Doppler e ect; and on the right including brightness e ects, thus showing what an observer would actually see (© Daniel Weiskopf ) What a researcher standing and one running rapidly through a corridor observe (ignoring colour e ects) (© Daniel Weiskopf) For the athlete on the le , the judge moving in the opposite direction sees both feet o the ground at certain times, but not for the athlete on the right A simple example of motion that is faster than light Another example of faster-than-light motion Hypothetical space-time diagram for tachyon observation If O’s stick is parallel to R’s and R’s is parallel to G’s, then O’s stick and G’s stick are not An inelastic collision of two identical particles seen from two di erent inertial frames of reference A useful rule for playing non-relativistic snooker e dimensions of detectors in particle accelerators are based on the relativistic snooker angle rule Space-time diagram of a collision for two observers ere is no way to de ne a relativistic centre of mass e space-time diagram of a moving object T Energy–momentum is tangent to the world line On the de nition of relative velocity Observers on a rotating object e simplest situation for an inertial and an accelerated observer
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189 409 190 411 191 411 192 413 193 415 194 417 195 419 196 438
197 439 198 440
199 440 200 441 201 441
202 445
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204 448
e hyperbolic motion of an rectilinearly, uniformly accelerating observer Ω e de nitions necessary to deduce the composition behaviour of accelerations Hyperbolic motion and event horizons Clocks and the measurement of the speed of light as two-way velocity e mountain attempt to exceed the maximum mass ow value Inside an accelerating train or bus e necessity of blue- and red-shi of light: why trees are greener at the bottom Tidal e ects: what bodies feel when falling e mattress model of space: the path of a light beam and of a satellite near a spherical mass All paths of ying stones have the same curvature in space-time A puzzle e irring and the irring–Lense e ects e LAGEOS satellites: metal spheres with a diameter of 60 cm, a mass of 407 kg, and covered with 426 retrore ectors e reality of gravitomagnetism A Gedanken experiment showing the necessity of gravity waves E ects on a circular or spherical body due to a plane gravitational wave moving in a direction perpendicular to the page Detection of gravitational waves Comparison between measured time delay for the periastron of the binary pulsar PSR 1913+16 and the prediction due to energy loss by gravitational radiation Calculating the bending of light by a mass Time delay in radio signals – one of the experiments by Irwin Shapiro e orbit around a central body in general relativity e geodesic e ect Positive, vanishing and negative curvature in two dimensions e maximum and minimum curvature of a curved surface Curvature (in two dimensions) and geodesic behaviour e Andromeda nebula M31, our neighbour galaxy (and the 31st member of the Messier object listing) (NASA) How our galaxy looks in the infrared (NASA) e elliptical galaxy NGC 205 (the 205th member of the New Galactic Catalogue) (NASA) e colliding galaxies M51 and M110 (NASA) e X-rays in the night sky, between 1 and 30 MeV (NASA) Rotating clouds emitting jets along their axis; top row: a composite image (visible and infrared) of the galaxy 0313-192, the galaxy 3C296, and the Vela pulsar; bottom row: the star in formation HH30, the star in formation DG Tauri B, and a black hole jet from the galaxy M87 (NASA) e universe is full of galaxies – this photograph shows the Perseus cluster (NASA) An atlas of our cosmic environment: illustrations at scales up to 12.5, 50, 250, 5 000, 50 000, 500 000, 5 million, 100 million, 1 000 million and 14 000 million light years (© Richard Powell, http://www.anzwers.org/free/universe) e relation between star distance and star velocity
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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224 521 225 521 226 527 227 530
228 531 229 531 230 532 231 533
232 536 233 538
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235 542 236 543 237 543 238 544 239 545 240 546 241 548 242 549 243 555
e Hertzsprung–Russell diagram (© Richard Powell) e ranges for the Ω parameters and their consequences e evolution of the universe’s scale a for di erent values of its mass density e long-term evolution of the universe’s scale factor a for various parameters e uctuations of the cosmic background radiation (WMAP/NASA) e absorption of the atmosphere (NASA) How one star can lead to several images e Zwicky-Einstein ring B1938+666, seen in the radio spectrum (le ) and in the optical domain (right) (NASA) Multiple blue images of a galaxy formed by the yellow cluster CL0024+1654 (NASA) e light cones in the equatorial plane around a non-rotating black hole, seen from above Motions of uncharged objects around a non-rotating black hole – for di erent impact parameters and initial velocities Motions of light passing near a non-rotating black hole e ergosphere of a rotating black hole Motion of some light rays from a dense body to an observer A ‘hole’ in space A model of the hollow Earth theory (© Helmut Diehl) Objects surrounded by elds: amber, lodestone and mobile phone How to amaze kids Lightning: a picture taken with a moving camera, showing its multiple strokes (© Steven Horsburgh) A simple Kelvin generator Franklin’s personal lightning rod Consequences of the ow of electricity e magentotactic bacterium Magnetobacterium bavaricum with its magnetosomes (© Marianne Hanzlik) An old and a newer version of an electric motor An electrical current always produces a magnetic eld Current makes a metal rods rotate e two basic types of magnetic material behaviour (tested in an inhomogeneous eld): diamagnetism and paramagnetism e relativistic aspect of magnetism e schematics, the realization and the operation of a Tesla coil, including spark and corona discharges (© Robert Billon) Li ing a light object – covered with aluminium foil – using high a tension discharge (© Jean-Louis Naudin at http://www.jlnlabs.org) e magnetic eld due to the tides (© Stefan Maus) A unipolar motor e simplest motor (© Stefan Kluge) e correspondence of electronics and water ow e rst of Maxwell’s equations e second of Maxwell’s equations Charged particles a er a collision Vector potentials for selected situations Which one is the original landscape? (NOAA)
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269 586
270 587 271 588
272 593 273 593 274 596
275 597 276 600 277 604 278 604 279 613
631
A plane, monochromatic and linearly polarized electromagnetic wave, with the elds as described by the eld equations of electrodynamics
e rst transmitter (le ) and receiver (right) of electromagnetic (micro-) waves Heinrich Hertz
e primary and secondary rainbow, and the supernumerary bows below the primary bow (© Antonio Martos and Wolfgang Hinz)
e light power transmitted through a slit as function of its width Sugar water bends light
e real image produced by a converging lens and the virtual image produced by a diverging lens Refraction as the basis of the telescope – shown here in the original Dutch design Watching this graphic at higher magni cation shows the dispersion of the lens in the human eye: the letters oat at di erent depths In certain materials, light beams can spiral around each other Masses bend light Re ection at air interfaces is the basis of the Fata Morgana
e last mirror of the solar furnace at Odeillo, in the French Pyrenees (© Gerhard Weinrebe) Levitating a small glass bead with a laser A commercial light mill turns against the light Light can rotate objects Proving that white light is a mixture of colours Umbrellas decompose white light Milk and water simulate the evening sky (© Antonio Martos)
e de nition of important velocities in wave phenomena Positive and negative indices of refraction
e path of light for the dew on grass that is responsible for the aureole A limitation of the eye What is the angle between adjacent horizontal lines?
e Lingelbach lattice: do you see white, grey, or black dots? An example of an infrared photograph, slightly mixed with a colour image (© Serge Augustin) A high quality photograph of a live human retina, including a measured (false colour) indication of the sensitivity of each cone cell (© Austin Roorda)
e recording and the observation of a hologram Sub-wavelength optical microscopy using stimulated emission depletion (© MPI für biophysikalische Chemie/Stefan Hell) Capacitors in series Small neon lamps on a high voltage cable How natural colours (top) change for three types of colour blind: deutan, protan and tritan (© Michael Douma) Cumulonimbus clouds from ground and from space (NASA)
e structure of our planet Trapping a metal sphere using a variable speed drill and a plastic saddle Floating ‘magic’ nowadays available in toy shops Bodies inside a oven at room temperature (le ) and red hot (right) Ludwig Wittgenstein
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319 786
Devices for the de nition of sets (le ) and of relations (right) e surreal numbers in conventional and in bit notation
An example of a quantum system (© Ata Masafumi) An artistic impression of a water molecule Hills are never high enough Leaving enclosures Identical objects with crossing paths Transformation through reaction How do train windows manage to show two superimposed images? A particle and a screen with two nearby slits Illumination by pure-colour light Observation of photons Various types of light An interferometer How to measure photon statistics: with an electronic coincidence counter and the variation by varying the geometrical position of a detector
e kinetic energy of electrons emitted in the photoelectric e ect Light crossing light Interference and the description of light with arrows (at one particular instant of time) Light re ected by a mirror and the corresponding arrows (at one particular instant of time)
e light re ected by a badly placed mirror and by a grating If light were made of little stones, they would move faster inside water Di raction and photons A falling pencil Steps in the ow of electricity in metals Matter di racts and interferes Trying to measure position and momentum On the quantization of angular momentum
e Stern–Gerlach experiment Erwin Schrödinger Climbing a hill
e spectrum of daylight: a stacked section of the rainbow (© Nigel Sharp (NOAO), FTS, NSO, KPNO, AURA, NSF)
e energy levels of hydrogen Paul Dirac Identical objects with crossing paths Two photons and interference Particles as localized excitations An argument showing why rotations by 4π are equivalent to no rotation at all A belt visualizing two spin 1/2 particles Another model for two spin 1/2 particles
e human arm as spin 1/2 model e extended belt trick, modelling a spin 1/2 particle: independently of the number of bands attached, the two situations can be transformed into each other, either by rotating the central object by 4π or by keeping the central object xed and moving the bands around it Some visualizations of spin representations
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Equivalence of exchange and rotation in space-time Belts in space-time: rotation and antiparticles Artist’s impression of a macroscopic superposition Quantum mechanical motion: an electron wave function (actually its module squared) from the moment it passes a slit until it hits a screen Bohm’s Gedanken experiment A system showing probabilistic behaviour
e concepts used in the description of measurements A quantum machine (© Elmar Bartel) Myosin and actin: the building bricks of a simple linear molecular motor Two types of Brownian motors: switching potential (le ) and tilting potential (right) A modern version of the evolutionary tree
e di erent speed of the eye’s colour sensors, the cones, lead to a strange e ect when this picture (in colour version) is shaken right to le in weak light Atoms and dangling bonds An unusual form of the periodic table of the elements Several ways to picture DNA Nuclear magnetic resonance shows that vortices in super uid 3He-B are quantized. An electron hologram Ships in a swell What is the maximum possible value of h/l? Some snow akes (© Furukawa Yoshinori) QED as perturbation theory in space-time
e weakness of gravitation A Gedanken experiment allowing to deduce the existence of black hole radiation A selection of gamma ray bursters observed in the sky Sagittal images of the head and the spine – used with permission from Joseph P. Hornak, e Basics of MRI, http://www.cis.rit.edu/htbooks/mri, Copyright 2003 Henri Becquerel
e origin of human life (© Willibrord Weijmar Schultz) Marie Curie All known nuclides with their lifetimes and main decay modes (data from http://www.nndc.bnl.gov/nudat2) An electroscope (or electrometer) (© Harald Chmela) and its charged (le ) and uncharged state (right) Viktor Heß A Geiger–Müller counter An aurora borealis produced by charged particles in the night sky An aurora australis on Earth seen from space (in the X-ray domain) and one on Saturn
e lava sea in the volcano Erta Ale in Ethiopia (© Marco Fulle) Various nuclear shapes – xed (le ) and oscillating (right), shown realistically as clouds (above) and simpli ed as geometric shapes (below)
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
353 922 Photographs of the Sun at wavelengths of 30.4 nm (in the extreme ultraviolet,
le ) and around 677 nm (visible light, right, at a di erent date), by the SOHO
mission (ESA and NASA)
354 924 A simpli ed drawing of the Joint European Torus in operation at Culham,
showing the large toroidal chamber and the magnets for the plasma con ne-
ment (© EFDA-JET)
355 927 A selection of mesons and baryons and their classi cation as bound states of
quarks
356 929 e spectrum of the excited states of proton and neutron
357 930
e essence of the QCD Lagrangian
358 943 e behaviour of the three coupling constants with energy for the standard
model (le ) and for the minimal supersymmetric model (right) (© Dmitri
Kazakov)
359 961 A simpli ed history of the description of motion in physics, by giving the
limits to motion included in each description
360 970 A ying fruit y, tethered to a string
361 970 Vortices around a butter y wing (© Robert Srygley/Adrian omas)
362 971 Two large wing types
363 972 e wings of a few types of insects smaller than 1 mm (thrips, Encarsia, Ana-
grus, Dicomorpha) (HortNET)
364 973 A swimming scallop (here from the genus Chlamys) (© Dave Colwell)
365 974 Ciliated and agellate motion
366 975 A well-known ability of cats
367 977 A way to turn a sphere inside out, with intermediate steps ordered clockwise
(© John Sullivan)
368 978 e knot diagrams for the simplest prime knots (© Robert Scharein)
369 979 Crossing types in knots
370 979 e Reidemeister moves and the ype
371 979 e diagrams for the simplest links with two and three components (©
Robert Scharein)
372 981 A hag sh tied into a knot
373 981 How to simulate order for long ropes
374 984 e mutually perpendicular tangent e, normal n, torsion w and velocity v of
a vortex in a rotating uid
375 985 Motion of a vortex: the fundamental helical solution and a moving helical
‘wave packet’
376 988 e two pure dislocation types: edge and screw dislocations
377 990 Swimming on a curved surface using two discs
378 990 A large raindrop falling downwards
379 990 Is this possible?
380 999 ‘Tekenen’ by Maurits Escher, 1948 – a metaphor for the way in which
‘particles’ and ‘space-time’ are usually de ned: each with the help of the other
(© M.C. Escher Heirs)
1012 Andrei Sakharov
381 1014 A Gedanken experiment showing that at Planck scales, matter and vacuum
cannot be distinguished
382 1024 Coupling constants and their spread as a function of energy
383 1037 Measurement errors as a function of measurement energy
384 1037 Trees and galaxies
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385 1039 e speed and distance of remote galaxies 386 1064 A Gedanken experiment showing that at Planck scales, matter and vacuum
cannot be distinguished 387 1105 Measurement requires matter and radiation 388 1122 A possible model for a spin 1/2 particle 389 1125 Planck e ects make the energy axis an approximation 390 1164 A precision experiment and its measurement distribution 391 1196 A property of triangles easily provable with complex numbers 393 1199 e hand and the quaternions 392 1200 Combinations of rotations 394 1215 Examples of orientable and non-orientable manifolds of two dimensions: a
disc, a Möbius strip, a sphere and a Klein bottle 395 1215 Compact (le ) and noncompact (right) manifolds of various dimensions 396 1216 Simply connected (le ), multiply connected (centre) and disconnected
(right) manifolds of one (above) and two (below) dimensions 397 1217 Examples of homeomorphic pairs of manifolds 398 1217 e rst four two-dimensional compact connected orientable manifolds: 0-,
1-, 2- and 3-tori 399 1236 A simple way to measure bullet speeds 400 1236 Leaving a parking space – the outer turning radius 401 1237 A simple drawing proving Pythagoras’ theorem 402 1237 e trajectory of the middle point between the two ends of the hands of a
clock 403 1238 e angles de ned by the hands against the sky, when the arms are extended 404 1243 Deducing the expression for the Sagnac e ect 405 1250 e south-pointing carriage 406 1256 A candle on Earth and in microgravity (NASA) 407 1267 Two converging lenses make a microscope 408 1268 e relation between the colour e ect and the lens dispersion
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
P
e mountain photograph on the cover is courtesy and copyright by Dave ompson (http:// www.daveontrek.co.uk). e basilisk running over water, on the back cover and on page , is courtesy and copyright by the Belgian group TERRA vzw and found on their website http://www. terra.vzw.org. e copyrights of the gures on page and on page cannot be traced. e gures on pages , and were made especially for this text and are copyright by Luca Gastaldi.
e famous photograph of the Les Poulains and its lighthouse by Philip Plisson on page is courtesy and copyright by Pechêurs d’Images; see the websites http://www.plisson.com and http:// www.pecheurs-d-images.com. It is also found in Plisson’s magnus opus La Mer, a stunning book of photographs of the sea. e hollow Earth gure on pages and is courtesy of Helmut Diel and was drawn by Isolde Diel. e Moon photographs on page and the analemma on page are courtesy and copyright by Anthony Ayiomamitis; the story of the photographs is told on his website at http://www.perseus.gr. e anticrepuscular photograph on page is courtesy and copyright by Peggy Peterson. e ring caterpillar gure of page is courtesy and copyright of Stanley Caveney. e photograph of the standard kilogram on page is courtesy and copyright by the Bureau International des Poids et Mesures. e measured graph of the walking human on page is courtesy and copyright of Ray McCoy. Figure on page is courtesy and copyright of the Australian Gemini project at http://www.ausgo.unsw.edu.au. e water strider photograph on page is courtesy and copyright by Charles Lewallen. e gure on the triangulation of the
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
meridian of Paris on page is copyright and courtesy of Ken Alder and found on his website http://www.kenalder.com. e geoid of page is courtesy and copyright by the GFZ Potsdam, to be found at http://www.gfz-potsdam.de. e beautiful lm of the lunation on page was calculated from real astronomical data and is copyrighted by Martin Elsässer; it is used here with his permission. It can be found on his website http:://www.mondatlas.de/lunation.html. e moon maps on page are courtesy of the USGS Astrogeology Research Program, http:://astrogeology. usgs.gov, in particular Mark Rosek and Trent Hare. e gure of myosotis on page is courtesy and copyright by Markku Savela. e gure of the soliton in the water canal on page is copyright and courtesy of Dugald Duncan and taken from his website on http://www.ma.hw.ac. uk/solitons/soliton .html. e fractal mountain on page is courtesy and copyright by Paul Martz, who explains on his website http://www.gameprogrammer.com/fractal.html how to program such images. e gure of helium atoms on metal on page is copyright and courtesy of IBM. e oscillon picture on page is courtesy and copyright by Paul Umbanhowar. e drawing of swirled spheres on page is courtesy and copyright by Karsten Kötter. e uid owing over an inclined plate on page is courtesy and copyright by Vakhtang Putkaradze. e photograph of the reconstruction of Fizeau’s experiment is copyright by AG Didaktik und Geschichte der Physik, Universität Oldenburg, and courtesy of Jan Frercks, Peter von Heering and Daniel Osewold. e photograph of a light pulse on page is courtesy and copyright of Tom Mattick.
e data and images of the Michelson–Morely experiment on page are copyright and courtesy of Stephan Schiller. e relativistic images of the travel through the simpli ed Stonehenge on page are copyright of Nicolai Mokros and courtesy of Norbert Dragon. e relativistic views on page and are courtesy and copyright of Daniel Weiskopf. e gures of galaxies on pages , , , , , , , , and are courtesy of NASA. e maps of the universe on page and the Hertzsprung-Russell diagram on page are courtesy and copyright of Richard Powell, and taken from his website http://www.anzwers.org/free/universe. e photograph of lightning on page is copyright Steven Horsburgh (see http://www.horsburgh. com) and used with his permission. e photograph of M. bavaricum on page is copyright by Marianne Hanzlik and is courtesy of Nicolai Petersen. e pictures of the Tesla coil on page
are courtesy and copyright of Robert Billon, and found on his website http://f wm.free.fr. e photograph of a li er on page is courtesy and copyright of Jean-Louis Naudin; more information can be found on his website http://www.jlnlabs.org. e ocean gure on page is courtesy of Stefan Maus, and taken from his http://www.gfz-potsdam.de/pb /pb /SatMag/ ocean_tides.html website. e simple motor photograph on page is courtesy and copyright of Stefan Kluge. e picture of the rainbow on page is from the NOAA website. e secondary rainbow picture on page is courtesy and copyright of Antonio Martos. e supernumerary rainbow picture on page is courtesy and copyright of Wolfgang Hinz and from his website http://www.meteoros.de. e solar furnace photograph on page is courtesy and copyright of Gerhard Weinrebe. e picture of milky water on page was made for this text and is copyright by Antonio Martos. e infrared photograph on page is copyright and courtesy of Serge Augustin. e pictures of retinas on page are courtesy and copyright of Austin Roorda. e microscope picture on page is copyright and courtesy of Stefan Hell. e pictures showing colour blindness on page are courtesy and copyright of Michael Douma, from his splendid website at http://webexhibits.org/causesofcolor/ .html. e cloud photographs on page are courtesy of NASA. e photograph of on page is a present to the author by Ata Masafumi, who owns the copyright. e pictures of snow akes on page are courtesy and copyright of Elmar Bartel. e pictures of snow akes on page are courtesy and copyright of Furukawa Yoshinori. e MRI images of the head and the spine on page are courtesy and copyright of Joseph Hornak and taken from his website http://www.cis.rit.edu/htbooks/mri/. e MRI image on page of humans making love is courtesy and copyright of Willibrord Wejimar Schultz. e
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nuclide chart on page is used with permission of Alejandro Sonzogni of the National Nuclear Data Centre, using data extracted from the NuDat database. e electroscope photograph on page is courtesy and copyright of Harald Chmela and taken from his website http://www. hcrs.at. e gure of the Erta Ale volcano on page is courtesy and copyright of Marco Fulle and part of the photograph collection at http://www.stromboli.net. e drawing of the JET reactor on page is courtesy and copyright of EFDA-JET. e picture of the butter y in a wind tunnel on page is courtesy and copyright of Robert Srygley and Adrian omas. e pictures of feathers insects on page are material used with kind permission of HortNET, a product of
e Horticulture and Food Research Institute of New Zealand. e picture of the scallop on page is courtesy and copyright of Dave Colwell. e picture of the eversion of the sphere on page is courtesy and copyright of John Sullivan from the Optiverse website, on http://new.math.
uiuc.edu/optiverse/. e knot and link diagrams on pages and are courtesy and copyright of Robert Scharein and taken from his website on http://www.knotplot.com. e copyright of the gure on page belongs to the M.C. Escher Heirs, c/o Cordon Art, Baarn, e Netherlands, who kindly gave permission for its use. All the drawings not explicitly mentioned here are copyright © – by Christoph Schiller.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Dvipsbugw
Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
A
H
LIST OF TABLES
N .P T
1 33 2 38 3 43 4 44 5 46 6 48 7 51 8 52 9 59 10 70 11 79 12 81 13 81 14 85 15 87 16 89 17 89 18 144 19 145 20 147 21 150 22 177 23 184 24 195
25 200
26 204 27 206 28 232 29 233 30 236 31 238 32 244
33 247
Content of books about motion found in a public library Family tree of the basic physical concepts Properties of everyday – or Galilean – velocity Some measured velocity values Selected time measurements Properties of Galilean time Properties of Galilean space Some measured distance values
e exponential notation: how to write small and large numbers Some measured acceleration values Some measured momentum values Some measured mass values Properties of Galilean mass Some measured energy values Some measured power values Some measured rotation frequencies Correspondence between linear and rotational motion An unexplained property of nature: the Titius–Bode rule
e orbital periods known to the Babylonians Some measured force values Selected processes and devices changing the motion of bodies Some action values for changes either observed or imagined Some major aggregates observed in nature Correspondences between the symmetries of an ornament, a ower and nature as a whole
e symmetries of relativity and quantum theory with their properties; also the complete list of logical inductions used in the two elds Some mechanical frequency values found in nature Some wave velocities Steel types, properties and uses Extensive quantities in nature, i.e. quantities that ow and accumulate Some temperature values Some measured entropy values Some typical entropy values per particle at standard temperature and pressure as multiples of the Boltzmann constant Some minimum ow values found in nature
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
H
34 256 Sand patterns in the sea and on land 35 280 Properties of the motion of light 36 402 e expected spectrum of gravitational waves 37 442 Some observations about the universe 38 451 A short history of the universe 39 467 e colour of the stars 40 499 Types of tests of general relativity 41 519 Searches for magnetic monopoles, i.e., for magnetic charges 42 520 Some observed magnetic elds 43 523 Properties of classical electric charge 44 524 Values of electrical charge observed in nature 45 524 Some observed electric elds 46 528 Some observed electric current values 47 538 Voltage values observed in nature 48 564 e electromagnetic spectrum 49 581 Experimental properties of ( at) vacuum and of the ‘aether’ 50 605 Selected matter properties related to electromagnetism, showing among
other things the role it plays in the constitution of matter; at the same time a short overview of atomic, solid state, uid and business physics 51 616 Examples of disastrous motion of possible future importance 52 643 e semantic primitives, following Anna Wierzbicka 53 647 e de ning properties of a set – the ZFC axioms 54 652 Some large numbers 55 662 e ‘scienti c method’ 56 709 Some small systems in motion and the observed action values for their changes 57 782 Particle spin as representation of the rotation group 58 797 Common and less common baths with their main properties 59 819 Motion and motors in living beings 60 823 Approximate number of living species 61 841 e types of rocks and stones 62 843 Signals penetrating mountains and other matter 63 844 Matter at lowest temperatures 64 850 A selection of lamps 65 874 e principles of thermodynamics and those of horizon mechanics 66 901 e main types of radioactivity and rays emitted by matter 67 905 e main types of cosmic radiation 68 916 Some radioactivity measurements 69 917 Human exposure to radioactivity and the corresponding doses 70 920 Natural isotopes used in radiometric dating 71 951 Some comparisons between classical physics, quantum theory and experiment 72 959 Everything quantum eld theory and general relativity do not explain; in other words, a list of the only experimental data and criteria available for tests of the uni ed description of motion 73 962 A selection of the consequences of changing the properties of nature 74 1000 e size, Schwarzschild radius and Compton wavelength of some objects appearing in nature 75 1050 Physical statements about the universe
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76 1052 Properties of nature at maximal, everyday and minimal scales 77 1062 Everything quantum eld theory and general relativity do not explain; in other
words, a list of the only experimental data and criteria available for tests of the uni ed description of motion 78 1063 A small selection of the consequences when changing aspects of nature 79 1107 E ects of various camera shutter times on photographs 80 1144 e ancient and classical Greek alphabets, and the correspondence with Latin and Indian digits 81 1145 e beginning of the Hebrew abjad 81 1156 e derived SI units of measurement 81 1156 e SI unit pre xes 83 1157 Planck’s (uncorrected) natural units 84 1165 Basic physical constants 85 1166 Derived physical constants 86 1167 Astrophysical constants 87 1168 Astronomical constants 88 1174 e elementary particles 89 1175 Elementary particle properties 90 1177 Properties of selected composites 91 1181 e periodic table of the elements known in 2006, with their atomic numbers 92 1181 e elements, with their atomic number, average mass, atomic radius and main properties 93 1220 Properties of the most important real and complex Lie groups 94 1228 e structure of the Arxiv preprint archive 95 1229 Some interesting servers on the world-wide web
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
A
I
NAME INDEX
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A
Page numbers in italic typeface refer to pages where the person is presented in more detail.
A
A
Alber, G.
Anderson, J.D.
Abbott, T.A.
Albert, David Z.
Anderson, J.L.
Abdalla, M.C.B.
Albertus Magnus
Anderson, James A.
Abe, E.
Alcock, Nathaniel W.
Anderson, M.H.
Abe, F. ,
Alcubierre, M.
Anderson, R.
Abramowicz, M.A.
Alder, Ken ,
Andreotti, B.
Abrikosova, I.I.
Aleksandrov, D.V.
Andres, M.R.
Acef, O.
Alexander, R. McN.
Andrews, M.R.
Ackermann, Peter
Alexandrov, P.S. ,
Aniçin, I.V.
Adams, Douglas , ,
Ali, A.
Anton, A.
Alighieri, Dante
Antonini, P. ,
Adelberger, E.
Allen, L.
Aquinas, omas , ,
Adelberger, E.G.
Allen, Les
,
Adenauer, Konrad ,
Allen, Woody
Arago, Dominique-François
Adler, C.G.
Almeida, C. de
,
Aedini, J.
Alpher, R.A.
Arago, François
Aetius , , ,
Alsdorf, D.
Araki, T.
A atooni, K.
Alspector, J.
Archimedes
Aguirregabiria, J.M. ,
Altman, S.L.
Ardenne, Manfred von
Aharonov, Y.
Alvarez Diez, C.
Aripov, Otanazar
Ahlgren, A. ,
Alvarez, E.
Aristarchos ,
Ahluwalia, D.V.
Alväger, T.
Aristarchos of Samos , ,
Ahmad, Q.R.
Amaldi, U. , ,
Ahrens, C. Donald
Amati, D.
Aristarchus of Samos
Aichelburg, P.C.
Amelino–Camelia, G. , Aristotle , , , , ,
Aigler, M.
,,,
,, ,, ,
Aitken, M.J.
Amenomori, M.
,
Ajdari, A.
Ampère, André-Marie , Arlt, J.
Akama, K.
Armstrong, J.T.
Akerboom, F.
Amundsen, Roald ,
Armstrong, Neil
Åkerman, N.
An, K. ,
Arndt, M. ,
Al-Dayeh, M.
Anaxagoras ,
Arnowitt
al-Farisi, Kamal al-Din
Anders, S.
Aronson, Je
al-Hadi al-Hush, Ramadan Anderson
Aronson, Je K.
Anderson, Carl ,
Arseneau, Donald
Alanus de Insulis
Anderson, I.M.
Ascher, Marcia
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Ashburner, J.
Bandler, R.
Bell, J.S.
Ashby, M.F.
Bandler, Richard , , , Bell, John
A
Ashcro , N. Ashcro , Neil
Barber, B.P.
Bellini, Giovanni Benbrook, J.R.
Ashkin, A.
Barberi Gnecco, Bruno
Bender, P.L.
A
Ashkin, Arthur
Barbour, John
Benjamin, J.
Ashtekar, A. , , , Barbour, Julian
Benka, S.
, ,,
Bardeen, J.
Bennet, C.L. ,
Asimov, I. Askin, A.
Bardeen, John , Barnett, S.J. ,
Bennet, S.C. Bennett, C.H. , ,
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Aspect, A.
Barnett, S.M.
Bennett, Charles H.
Aspect, Alain
Barnhill, M.V.
Bentlye, W.A.
Aspelmeyer, M.
Barrow, J.D.
Benzenberg, Johann Friedrich
Aspinwall, P.
Barrow, John D. ,
Asterix
Bartel, Elmar , ,
Berg, E.
Astumian, R.D.
Bartocci, Umberto ,
Berg, H.C.
Ata Masafumi ,
Barton, G.
Bergmann, L.
Audoin, C.
Bartussek, R.
Bergquist, J.
Augel, Barbara and Edgar
Barwise, J.
Bering, E.
Augustin, Serge ,
Basieux, Pierre
Bering, E.A.
Augustine ,
Basri, G.
Berkeley, George
Augustine of Hippo
Bass, omas A.
Berlekamp, E.R.
Ausloos, M.
Batelaan, H.
Berlin, Brent
Autumn, K.
Bateman, H.
Berlin, Isaiah
Avogadro, Amadeo
Batty, R.S.
Bernard, C.
Avron, J.E.
Baudelaire, Charles
Bernoulli, Daniel
Awschalom, D.D.
Bauerecker, S.
Bernoulli, J.
Axel, Richard
Baumann, K.
Bernreuther, W.
Axelrod, R.
Bautista, Ferdinand
Bernstein, Aaron
Ayers, Donald M.
Baylor, D.A.
Berry, M.V. , ,
Ayiomamitis, Anthony
Beale, I.L.
Berry, Michael ,
Beaty, William
Bertotti, B.
B
Beauvoir, B. de
Bertulani, C.A.
Babinet, Jacques
Bechinger, C.
Besnard, P.
Babloyantz, A.
Becker, A.
Bessel ,
Bachem
Becker, J.
Bessel, Friedrich
Bader, Richard F.W.
Becquerel, Henri
Bessel, Friedrich Wilhelm ,
Baez, John
Bedford, D.
Baggett, N.
Beeksma, Herman
Besso, Michele
Bagnoli, Franco
Beenakker, C.W.J. , , Beth, R.
Bailey, J.
Bethe, H.
Bailey, J.M.
Behroozi, C.H. ,
Bethe, Hans
Balachandran, A.P.
Bekenstein, J.D. , , , Bettelheim, Bruno
Balashov, Yuri I.
Bettermann, D.
Balibar, Françoise
Bekenstein, Jakob , , Beutelspacher, Albrecht
Balmer, Johann
Beuys, Joseph , ,
Banach, Stefan , ,
Belfort, François
Bevis, M.
Banavar, J.R.
Belic, D. ,
Beyer, Lothar
Banday, A.J.
Belinfante, F.J.
Bhalotra, S.R.
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Bharucha, C.F.
Bonner, Yelena
Brezger, B.
Bianco, C.L. ,
Bonnor
Briand, J.P. ,
B
Bierce, Ambrose Biggar, Mark
Bonnor, W.B. , Boone, Roggie
Briatore Briatore, L.
Bilaniuk, O.M.
Bordé, Ch.J.
Briggs, F.
B
Bilanuk, O.M.P. Bilger, H.R.
Borelli, G. Borgne, J.-F. Le
Bright, William Brightwell, G.
Billon, Robert ,
Born, M.
Brillouin
Bimonte, G. Biraben, F.
Born, Max , , , Bose, S.N.
Brillouin, L. Brillouin, Louis
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Bird, D.J.
Bose, Satyenra Nath
Brillouin, Léon
Birkho
Boss, Alan P.
Bringhurst, Robert
Bischo , Bernard
Bouchiat, C.C.
Britten, R.J.
Bjorkholm, J.E.
Bouchiat, M.A.
Britten, Roy
Björk, G.
Boughn, S.P.
Broch, Henri
Blaauwgeers, R.
Bour, L.
Brock, J.B.
Bladel, Jean van
Bourbaki, N.
Brody, A.L.
Blagden, Charles
Bousso, R. , , ,
Broglie, L. de
Blair, D.G. ,
Broglie, Louis de ,
Blair, David
Bouwmeester, D.
Bronstein, M.
Blanché, Robert
Bowden, F.P.
Bronstein, Matvey
Blandford, R. ,
Bower, B.
Brooks, Mel
Blandford, R.D.
Bowlby, John
Brookshear, J. Glenn
Blau, Stephen
Boyce, K.R.
Brouwer, Luitzen
Blau, Steven K.
Boyda, E.K.
Brown, B.L.
Bliss, G.W.
Boyer, T.H.
Brown, J.H.
Block, S.M.
Boyer, Timothy
Brown, Peter
Blumensath, Achim
Brackenbury, John
Brown, Robert
Boamfa, M.I.
Bradley
Bruce, Tom
Bocquet, Lydéric
Bradley, C.C.
Brumberg
Bode, Johann Elert
Bradley, James , ,
Brumberg, E.M.
Boer, W. de ,
Braginsky, V.B. , ,
Brune, M.
Boersma, S.L.
Brahe, Tycho ,
Brunetti, M.
Boethius
Brahm, A. de
Brunner, H.
Bohm ,
Brahm, D.E.
Brush, S.
Bohm, D.
Brahmagupta
Bryant, D.A.
Bohm, David
Brandes, John
Bub, J.
Bohr ,
Brandt, E.H.
Buchanan, Mark
Bohr, Aage
Brantjes, R.
Buck, Linda
Bohr, N. , , ,
Brattain, Walter
Buckley, Michael
Bohr, Niels , , , , Brault, J.W.
Buddakian, R.
, , , , , , Braxmeier, C.
Budney, Ryan
,,
Bray, H.L.
Bunn, E.F.
Bollinger, J.J.
Brebner, Douglas
Bunsen, Robert
Boltzmann, Ludwig ,
Brecher, K.
Bunyan, Paul
Bombelli, L. , ,
Brehme, R.W.
Burbidge, G.
Bombelli, Luca ,
Brendel, J.
Burbridge, E. Margaret
Bondi, H.
Brennan, Richard
Burbridge, G.R.
Bondi, Hermann
Brewer, Sydney G.
Buridan, Jean ,
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Burke, D.L.
Caveney, S.
Ciufolini, I. ,
Burša, Milan
Caveney, Stanley ,
Ciufolini, Ignazio ,
B
Busch, Paul Busche, Detlev
Caves, C.M. , Cayley, Arthur , ,
Clairon, A. Clancy, E.P.
Butikov, E.I.
Ceasar, Gaius Julius
Clancy, Tom
B
Butler, Samuel Butoli, André
Cecchini, S. Celsius, Anders
Clark, R. Clarke
Buzek, V.
Chahravarty, S.
Clarke, T.
Bäßler, S. Böhm, A.
Chaineux, J. Chaitin, Gregory J.
Clauser, J.F. Clausius
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Börner, G.
Chan, M.A.
Clausius, Rudolph , ,
Böhncke, Klaus
Chan, W.P.
,,
Chandler, David G.
Clavius, Christophonius
C
Chandler, Seth
Cleobulus
Caianiello, E.R. ,
Chandrasekhar, Subramanyan Clerk Maxwell, James ,
Cajori, Florian
Caldeira, A.O.
Chang, I.
Clery, D.
Calzadilla, A.
Chang, J.
Cli on, R.
Campbell, D.
Chang, P.Z.
Close, F.
Campbell, D.K.
Chantell, M.
Codling, K.
Cantor, Georg
Chaplin, Charlie
Coehoorn, Reinder
Caplan, S.R.
Chapman, M.S.
Cohen, M.H.
Caps, H.
Charlemagne
Cohen, P. J.
Caraway, E.L.
Charpak, G.
Cohen, Paul
Carducci, Giosuè
Chen, B.
Cohen, Paul J.
Carilli, C.L.
Chen, P.
Cohen, Philip
Carlip, S.
Cheseaux, Jean Philippe Loÿs Cohen-Tannoudji, C.
Carlip, Steve , , , , de
Cohen-Tannoudji, G.
Chiao, R.
Cohen-Tannoudji, Gilles
Carlyle, omas
Chiao, R.Y. ,
Colazingari, Elena
Carmona, Humberto
Chiba, D.
Colella, R.
Carneiro, S.
Childs, J.J. ,
Coleman–McGhee, Jerdome
Carnot, Sadi
Chinnapared, R.
Carr, Jim
Chmela, Harald ,
Colladon, Daniel
Carroll, Lewis
Chomsky, Noam ,
Collins, D.
Carroll, Lewis, or Charles
Choquet-Bruhat, Yvonne
Collins, J.J.
Lutwidge Dogson
Christ, N.H.
Columbus
Carruthers, P.
Christiansen, W.A.
Colwell, Dave ,
Carter
Christlieb, N.
Comins, Neil F.
Carter, Brandon
Christodoulou, D.
Commins, E.D.
Cartesius
Christou, A.A.
Compton, A.H. ,
Casati, Roberto
Chu, S. , ,
Compton, Arthur ,
Casimir, H.B.G.
Chuang, I.L.
Conde, J.
Casimir, Hendrik
Chudnovsky, D.V.
Conkling, J.A.
Cassini, Givanni
Chudnovsky, G.V.
Conroy, R.S.
Cassius Dio
Chung, K.Y.
Conti, Andrea
Castagnino, M.
Ciafaloni, M.
Conway, J.
Cauchy, Augustin-Louis
Cicero
Conway, J.H.
Cavendish, Henry
Cicero, Marcus Tullius
Conway, John ,
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Conway, John H.
Dalton, John ,
Descartes, R.
Cooper, Heather
Dalton, K.
Descartes, René
C
Cooper, J.N. Cooper, L.N.
Dam, H. van Dam, H. van ,
Descartes, René , , Deser
Cooper, Leon N.
Dambier, G.
Deshpande, V.K.
C
Copernicus, Nicolaus , Copper eld, David ,
Damour Damour, T. ,
Desloge, E.A. Desloge, Edward A. ,
Corballis, M.C.
Damour, ibault
Desmet, S.
Corbin, V. Cordero, N.A.
Danecek, Petr Daniels, Peter T.
Destexhe, A. DeTemple, D.W.
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Coriolis, Gustave-Gaspard , Dante Alighieri
Detweiler, S.
Darius, J.
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Dasari, R.R.
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,,
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Legendre, A.
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San Suu Kyi, Aung
Scholz, Christoph
Rusby, R.L. ,
Sarazin, X.
Schooley, J.F.
Rusby, Richard
Sassen, K.
Schrödinger, Erwin , ,
Ruschewitz, F.
Sastry, G.P.
,
Russel, J.S.
Satake Ichiro
Schramm, Herbert
Russell, Bertrand
Sauer, J.
Schramm, T.
Russell, Betrand
Saupe, D.
Schrie er, J. Robert
Russell, John Scott
Saupe, Dietmar
Schrie er, J.R.
Russo, Lucio
Saussure, Ferdinand de
Schröder, Ulrich E.
Rutherford, Ernest , , Savela, Markku ,
Schröder, Ulrich E.
,
Scarcelli, G.
Schrödinger ,
Rybicki, G.R.
Schadwinkel, H.
Schrödinger, Erwin ,
Rydberg, Johannes
Schaefer, B.E. , ,
Schubert, Max
Rü er, U.
Schaerer, D.
Schucking, E. , ,
Rømer, Ole
Schanuel, Stephen H.
Schulte
Scharein, Robert , ,
Schultz, Charles
S
Schultz, S.
Saa, A.
Scharmann, Arthur
Schunck, N.
Sackett, C.A. ,
Scharnhorst, K.
Schuster, S.C.
Sacks, Oliver
Schata, P.
Schutz, B.F.
Sade, Donatien de ,
Scheer, Elke
Schutz, Bernard
Sagan, Carl
Schelby, R.A.
Schwab, K.
Sagan, Hans
Schi , L.I.
Schwartz, Richard
Sagnac, Georges
Schild, A.
Schwarz, J.H. ,
Sahl, Mort
Schiller, Britta
Schwarzschild
Saitou, T.
Schiller, C. , , , , Schwarzschild, Bertram
Sakar, S.
,,,,
Schwarzschild, Karl
Sakharov, A.D. , ,
Schiller, Christoph ,
Schwenk, Jörg
Schiller, Friedrich
Schwenk, K.
Sakharov, Andrei , , Schiller, Isabella
Schwenkel, D.
,,
Schiller, P. ,
Schwinger, Julian , ,
Salam, A.
Schiller, Peter
Salam, Abdus
Schiller, S.
Schwob, C.
Salamo, G.
Schiller, Stephan
Schäfer, C.
Salditt, T.
Schilthuizen, Menno
Schäfer, G.
Salecker
Schlegel, K.
Schön, M. ,
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Sheldon, E.
Smale, S.
Schörner, E.
Sheldon, Eric
Smale, Stephen
S
Sciama, D.W. Sciama, Dennis ,
Sheng, Z. Shephard, G.C.
Smirnov, B.M. Smith, D.R.
Scott, G.G.
Shepp, L.
Smith, David
S
Scott, Jonathan Scott, W.T.
Sherrill, B.M. Shih, M.
Smith, J.B. Smith, J.F.
Scriven
Shih, Y.
Smith, S.P.
Scriven, L.E. Seaborg, Glenn
Shimony, A. Shinjo, K.
Smith, Steven B. Smolin, L. , , ,
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Shirham, King
Seeger, A. ,
Shockley, William
Smoluchowski, Marian von
Seeger, J.
Short, J.
Segev
Shulman, Polly
Smoluchowski, Marian von
Segev, M.
Shupe, M.A.
Segev, Mordechai
Sierra, Bert
Snider, J.L.
Segrè, Gino
Sigwarth, J.B.
Snow
Seidelmann, P. Kenneth
Silk, J.
Snow, G.A.
Seidl, T.
Sillett, S.C.
Snyder, H.
Seielstad, G.A.
Silverman, M.
Snyder, H.S.
Selig, Carl
Silverman, M.P.
Snyder, Hartland
Semon, Mark D.
Silverman, Mark
Socrates
Sen, A.
Silverman, Mark P. , , So el, M.
Send, W.
,
So el, Michael H.
Send, Wolfgang
Simon, C.
So er, B.H.
Seneca, Lucius Annaeus
Simon, Julia
Soldner ,
Serena, P.A.
Simon, M.D.
Soldner, J. ,
Sessa ben Zahir
Simon, M.I.
Soldner, Johann ,
Sexl, R.U.
Simon, R.S.
Solomatin, Vitaliy
Sexl, Roman ,
Simoni, A.
Solomon, R.
Sexl, Roman U. ,
Simonson, A.J.
Sommerfeld, Arnold , ,
Shackleton, N.J.
Simplicius ,
Shakespeare, William , , Simpson, A.A.
Son, D.T.
Simpson, N.B.
Sonett, C.P.
Shalyt-Margolin, A.E. ,
Singh, C.
Song, K.-Y.
Singh, T.P.
Song, X.D.
Shankland, R.S.
Singhal, R.P.
Sonoda, D.H.
Shapere, A.
Singleton, D.
Sonzogni, Alejandro
Shapiro, A.H.
Singleton, Douglas
Sorabji, R.
Shapiro, Arnold
Sirlin, A.
Sorkin, R.D. , ,
Shapiro, Asher
Sitter, W. de
Sossinsky, Alexei
Shapiro, I.I. ,
Sitter, Willem de ,
Soukoulis, C.M.
Shapiro, Irwin I.
Sivardière, Jean
Sparenberg, Anja
Sharkov, I.
Skrbek, L.
Sparnaay, M.J.
Sharma, Natthi L.
Slabber, André
Sparnaay, Marcus
Shaw, George Bernard , Slater, Elizabeth M.
Specker
Slater, Henry S.
Specker, E.P.
Shaw, R.
Slavnov, A.A.
Spence, J.C.H.
Shea, J.H.
Sloterdijk, Peter
Spencer, R. , ,
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Stromberg, G.
Tamman, Gustav
Spiderman
Stromberg, Gustaf
Tanaka, K.
S
Spieker, H. Spinoza, B.
Strominger, A. Strunk, William
Tangen, K. Tanielian, M.
Spinoza, Baruch
Strunz, W.T.
Tantalos
S
Spiropulu, M. Sreedhar, V.V.
Styer, D. , Su, B.
Tarko, Vlad Tarski, Alfred , ,
Sriramkumar
Su, Y. ,
Tartaglia, A.
Sriramkumar, L. Srygley, R.B.
Subitzky, Edward Suchocki, John
Tartaglia, Niccolò Tauber, G.E.
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Srygley, Robert ,
Sudarshan, E.C.
Taylor, B.N. ,
Stachel, John
Sudarshan, E.C.G. ,
Taylor, Edwin F. , , ,
Starinets, A.O.
Sugamoto, A.
,,,
Stark, Johannes
Sugiyama, S.
Taylor, G.J.
Stasiak, A.
Sullivan, John ,
Taylor, J.H. , ,
Stearns, Stephen C.
Sulloway, Frank J.
Taylor, John R. ,
Stedman, G.E. , ,
Sun, X.L.
Taylor, Joseph ,
Stegeman, George
Sundaram, B.
Taylor, R.E.
Steinberg, A.M. , ,
Supplee, J.M.
Taylor, W.R.
Steiner, Kurt
Surdin, Vladimir ,
Taylor, William
Steinhaus
Surry, D.
Teeter Dobbs, Betty Jo
Stengl, Ingrid
Susskind, L. ,
Tegelaar, Paul
Stephan, S.
Susskind, Leonard
Tegmark, M. , ,
Stephenson, G.
Sussman, G.J.
Telegdi, V.L.
Stephenson, G.J.
Suzuki, M.
Teller, Edward
Stephenson, Richard
Svensmark, H.
Tennekes, Henk
Stern, O.
Svozil, K.
Terence, in full Publius
Stern, Otto
Swackhamer, G.
Terentius Afer ,
Stettbacher, J.K.
Swagten, Henk
Terentius Afer, Publius ,
Steur, P.P.M.
Swenson, C.A.
Stewart, I. , ,
Swi , G.
Terletskii, Y.P.
Stewart, Ian , ,
Swinney, H.L.
Terrell, J.
Stilwell, G.R.
Swope, Martha
Tesla, Nikola
Stocke, J.T.
Synge, J.L.
aler, Jon
Stodolsky, Leo
Szczesny, Gerhard
ales of Miletus
Stoehlker, omas
Szilard, L. , ,
eodoricus Teutonicus de
Stokes, Georges Gabriel
Szilard, Leo , , ,
Vriberg
Stone, Michael
idé, Bo
Stoney, G.J. ,
Szuszkiewicz, E.
ierry, Paul
Stoney, George
ies, Ingo
Stoney, George Johnston
T
irring, H.
Stong, C.L.
Tőrők, Gy.
irring, Hans
Stormer, H.L.
Tabor, D.
istlethwaite, M.
Story, Don
Tait, P.G.
ober, D.S.
Stowe, Timothy
Tajima, T.
omas Aquinas ,
Strassman, Fritz
Takamoto, M.
omas, A.L.R.
Strauch, F.
Takita, A.
omas, Adrian ,
Strauch, S.
Talleyrand
omas, L.
Straumann, N.
Tamm, Igor
omas, Llewellyn
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Trevorrow, Andrew
Unwin, S.C.
ompson, Dave
Trout, Kilgore
Unzicker, A.
T
ompson, R.C. omson (Kelvin), William
Trower, W.P. Truesdell, C.
Upright, Craig Uranus
,
Truesdell, Cli ord
Urban, M.
T
omson, Joseph John
Truesdell, Cli ord A.
Ursin, R.
omson, W.
Tsagas, C.G.
Ustinov, Peter
or
Tsai, W.Y.
orndike, E.M. orne, J.A. Wheeler, K.S.
Tsang, W.W. Tschichold, J.
V Vafa, C.
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Valanju, A.P.
orne, K.
Tsuboi, Chuji
Valanju, P.M.
orne, K.S. ,
Tsubota, M.
Valencia, A.
orne, Kip
Tsui, D.C.
Valentin, Karl ,
orne, Kip S. ,
Tu, L.-C.
Valsiner, Jaan
orp, Edward O.
Tucholski, Kurt
Valsinger, Jaan
urston, William
Tuckermann, R.
van Druten, N.J.
urston, William P.
Tuppen, Lawrence
Vanadis
évenaz, L.
Turatto, M.
Vancea, I.V.
Tian, L.
Turchette, Q.A.
Vandewalle, N.
Tiggelen, Bart van
Turnbull, D.
Vanier, J.
Tillich, Paul
Turner, M.S. ,
Vannoni, Paul
Tino, G.
Twain, Mark , , ,
Van Dam, H. , , ,
Tino, G.M.
Twamley, J.
,
Tipler, Frank J. ,
Twiss, R.Q.
Van Den Broeck, Chris
Tisserand, F.
Twiss, Richard
Vardi, Ilan
Titius, Johann Daniel
Tyler, R.H.
Vavilov
Tittel, Wolfgang
Tyler Bonner, J.
Vavilov, S.I.
Tollett, J.J.
Veer, René van der
Tolman, R.C.
U
Veneziano, G. ,
Tolman, Richard
Ucke, C. ,
Vergilius
Tolman, Richard C.
Udem, T.
Vergilius, Publius
Tomonaga ,
Udem, .
Vermeil, H.
Topper, D.
Ueberholz, B.
Vermeulen, R.
Torge, Wolfgang
U nk, J. ,
Verne, Jules
Torre
U nk, Jos
Vernier, Pierre
Torre, A.C. de la
Uglum, J.
Veselago, V.G. ,
Torre, C.G.
Uguzzoni, Arnaldo
Veselago, Victor
Torrence, R.
Uhlenbeck, G.E.
Vessot
Torricelli, Evangelista
Uhleneck, George
Vessot, R.F.C.
Toschek, P.E. ,
Ulam, Stanislaw
Vestergaard Hau, L. ,
Townsend, P.K. ,
Ulfbeck, Ole
Vico, Giambattista
Tozaki, K.-I.
Ullman, Berthold Louis
Vigotsky, Lev ,
Tran anh Van, J.
Uman, M.A.
Villain, J.
Travis, J.
Umbanhowar, P.B.
Vincent, D.E.
Treder, H.-J.
Umbanhowar, Paul ,
Visser, Matt
Trefethen, L.M.
Unruh, W.G. , , , Viswanath, R.N.
Tregubovich, A.Ya. ,
,
Vitali, Giuseppe
Treille, D.
Unruh, William , ,
Viviani, Vincenzo
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Voigt, Woldemar
Waser, A.
Westerweel, J.
Voit, A.
Washizu, M.
Westphal, V.
V
Volin, Leo Vollmer, G.
Watson, A.A. Wearden, John
Westra, M.T. Weyl, Hermann ,
Vollmer, M.
Webb, R.A.
Wheatstone, Charles
V
Volovik, G.E. Volta, Alessandro
Weber, G. Weber, Gerhard
Wheeler Wheeler, J.A. , , ,
Voltaire , , , ,
Weber, R.L.
,
,, , von Klitzing, K.
Weber, T. Wedin, H.
Wheeler, John , , ,
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Vorobie , P.
Weekes, T.C.
Wheeler, John A. , ,
Vos–Andreae, J.
Weeks, J.
,, , , , ,
Voss, Herbert
Wegener, Alfred ,
,,
Voss, R.F.
Wehinger, S.
Wheeler, John Archibald
Vuorinen, R.T.
Wehner, R.
Whewell, William
Völz, Horst
Weierstall, U.
White, E.B.
Weierstrass, K.
White, Harvey E.
W
Weijmar Schultz, W.C.M.
White, M. ,
Wagner, William G.
Weijmar Schultz, Willibrord Whitehead, Alfred North
Wagon, Steve ,
Whitney, A.R.
Wal, C.H. van der
Weil, André
Whitney, C.A.
Wald
Weinberg, Steven , , Whittaker, Edmund T.
Wald, George
,,,
Widmer-Schnidrig, R.
Wald, R.M. , , , , Weinrebe, Gerhard
Widom, A.
Weis, A.
Wiechert, Johann Emil
Wald, Robert
Weisberg, J.M.
Wiegert, Paul A.
Wald, Robert M.
Weiskopf, Daniel , , Wieman, Carl
Waldhauser, F.
,
Wiemann, C.E.
Wale e, F.
Weiss, M.
Wien, Wilhelm
Walgraef, Daniel
Weiss, Martha
Wiens, R.
Walker, A.G.
Weisskopf, V.F. ,
Wierda, Gerben
Walker, Gabriele
Weisskopf, Victor ,
Wierzbicka, Anna , , ,
Walker, J.
Weisskopf, Victor F.
,,
Walker, Jearl
Weissmüller, J.
Wigner, E.
Walker, R.C.
Weitz, M. ,
Wigner, E.P. , , ,
Wallin, I.
Weitz, Martin
Wallis, J.
Weitzmann, Chaim
Wigner, Eugene , , ,
Wallner, A.
Wejimar Schultz, Willibrord
Walser, R.M.
Wigner, Eugene P.
Walter, H.
Wells, M.
Wijk, Mike van
Walter, Henrik
Weninger, K.
Wijngarden, R.J. ,
Walther, P.
Weninger, K.R.
Wikell, Göran
Wambsganss, J.
Wernecke, M.E.
Wilczek, F. , , ,
Wampler, E. Joseph
Werner, S.A.
Wilczek, Frank , ,
Wang Juntao
Wertheim, Margaret ,
Wilde, Oscar , , ,
Wang, L.J.
,
Wang, Y.
Wesson, John
Wiley, Jack
Wang, Z.J.
Wesson, Paul ,
Wilk, S.R.
Warkentin, John
West, G.B.
Wilkie, Bernard
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W
W
Wilkinson, S.R.
Wollaston, William
Will, C. , ,
Wolpert, Lewis
Will, C.M. ,
Wolsky, A.M.
Willberg, Hans Peter
Wong, S.
William of Occam
Wood, B.
Williams, D.R.
Wood, C.S.
Williams, David
Woodhouse, Nick M.J.
Williams, David R.
Woods, P.M.
Williams, Earle R.
Wootters
Williams, G.E.
Wootters, W.K.
Williams, H.H.
Wouthuysen, S.A. ,
Williams, R.
Wright, B.
Wilson, B.
Wright, E.M.
Wilson, Charles
Wright, K.
Wilson, J.R.
Wright, Steven
Wilson, Robert
Wu, C.
Wilson, Robert W.
Wu, T.T.
Wiltschko, R.
Wul la
Wiltschko, W.
Wussing, H. ,
Wineland, D.J. , ,
Wynands, R.
Winterberg, F.
Würschum, R.
Wirtz
Wirtz, C.
X
Wirtz, Carl
Xavier, A.L.
Wisdom, J. ,
Xu Liangying
Wisdom, Jack
Xue, S.-S. ,
Wise, N.W. ,
Witt, Bryce de
Y
Witte, H.
Yamafuji, K.
Witteborn, F.C.
Yamamoto, H.
Witten, E. ,
Yamamoto, Y.
Witten, Ed
Yamane, T.
Witten, Edward
Yandell, Ben H. ,
Wittgenstein, Ludwig , , Yang Chen Ning
, , , , , , , Yang, C.N.
, , , , , , Yang, J.
, , , , , , Yao, E.
Yazdani, A. ,
Wittke, James P.
Yearian, M.R.
Wodsworth, B.
Yoshida, K.
Woerdman, J.P.
Young, James A.
Woit, Peter
Young, omas ,
Wolf, C.
Yourgray, Wolfgang
Wolf, E.
Yukawa Hideki
Wolf, Emil
Wolf, R.
Z
Wolfendale, A.W.
Zürn, W.
Wolfenstätter, Klaus-Dieter Zabusky, N.J.
Zabusky, Norman
Zaccone, Rick Zakharian, A. Zalm, Peer Zander, Hans Conrad Zanker, J. Zbinden, H. Zedler, Michael Zeeman, Pieter Zeh, H.D. , Zeh, Hans Dieter , , Zeilinger, A. , , ,
Zeller, Eduard Zeno of Elea , , , ,
,, , , , ,, Zensus, J.A. Zermelo, Ernst Zetsche, Frank Zeus Zhang Zhang, J. Zhang, Yuan Zhong Zhao, C. , Zhou, H. Ziegler, G.M. Zimmer, P. Zimmerman, E.J. , , ,, Zimmermann, H.W. Zimmermann, Herbert Zouw, G. van der Zuber, K. Zuckerman, G. Zurek Zurek, W.H. , Zurek, Wojciech Zurek, Wojciech H. , Zuse, Konrad Zwaxh, W. Zweck, Josef , Zweig, G. Zweig, George Zwerger, W. Zwicky, F. Zwicky, Fritz Zybin, K.P.
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A
J
SUBJECT INDEX
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Page numbers in italic typeface refer to pages where the keyword is de ned or presented in detail. e subject index thus contains a glossary.
Symbols n-th harmonic generation ( ) *-algebra + − ë
< =
DNA , EPR experiments TNT energy content
p
Time magazine
∆
∇ -acceleration -coordinates , , -jerk -momentum -vector -velocity
A @ (at sign) α-rays a (year) a (year)
a posteriori a priori a priori concepts abacus Abelian, or commutative aberration , , , abjad Abrikosova absolute value absorption abstraction abugida Academia del Cimento acausal , acausality accelerating frames acceleration , acceleration composition
theorem acceleration, limit to acceleration, maximum acceleration, proper , acceleration, quantum limit
accretion discs accumulability accumulation accuracy , accuracy, maximal
Acinonyx jubatus acoustical thermometry acoustoelectric e ect acoustomagnetic e ect acoustooptic e ect actin actinium actinoids action , , action, limit to action, physical action, principle of least action, quantum of actuators adaptive optics addition , additivity , , , , additivity of area and volume
acceleration, relativistic
adenosine triphosphate ,
behaviour
acceleration, tidal
adjoint representation
acceleration, uniform
adjunct
accelerations, highest
ADM mass ,
accelerator mass spectroscopy adventures, future
Aeneis
accelerometer
aerodynamics
accents, in Greek language
aeroplane, model
aeroplanes
accretion
aether , , ,
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aether and general relativity alphasyllabaries
anthropic principle ,
,
Alps ,
anti-atoms
A
aether models a ne Lie algebra
Altair , aluminium ,
anti-bubbles anti-bunching
AFM, atomic force
aluminium amalgam
anti-gravity devices
microscope
amalgam
anti-Hermitean
Africa collides with Europe Amazonas
anti-the stickers
Amazonas River
anti-unitary
AgBr AgCl ,
amber americium
anticommutator bracket antigravity ,
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age
amoeba
antigravity devices
age of universe ,
ampere
antihydrogen
ageing
amplitude
antiknot
aggregates of matter
Ampère’s ‘law’
antimatter , , ,
AgI
analemma
antimony
agoraphobics
anapole moment
Antiqua
air , ,
AND gate, logical
antisymmetry ,
air cannot ll universe
andalusite
anyons ,
air pressure
Andromeda nebula ,
apes ,
airbag sensors
angel, are you one?
aphelion
akne
angels , , , ,
apogee
Al
angle
Apollo , , ,
albedo
angle, plane
apparatus, classical
Albert Einstein, teenager
angle, solid
apparatus, irreversible
alchemy
angular acceleration
apple as battery
Aldebaran , ,
angular frequency
apple trees
aleph
angular momentum , , apple, standard
algebra ,
,,,,
apples , , ,
algebra, alternative
angular momentum as a
Aquarius
algebra, associated a ne
tensor
Ar
algebra, linear
angular momentum
Arabic numbers
algebraic structure
conservation
Archaeozoicum
algebraic surfaces
angular momentum of light archean
algebraic system
arcs
Alice in Wonderland
angular momentum, extrinsic Arcturus
alkali metals
argon ,
Allen, Woody
angular momentum,
argument
Alluvium
indeterminacy relation
Aries
Alnilam
angular momentum, intrinsic Aristotelian view
Alnitak
Aristotle
Alpha Centauri
angular momentum, limit to arm ,
alpha decay
Armillaria ostoyae ,
alpha ray dating
angular velocity ,
arms, human
alphabet
anholonomic constraints
arrow
alphabet, Greek
animism
arrow of time
alphabet, Hebrew
annihilation ,
arsenic
alphabet, Latin
annihilation operator
artefact , ,
alphabet, phonemic
anode
ash
alphabets, syllabic
Antares ,
Ashtekar variables
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aspects of nature
aurora borealis
Barnett e ect
associative algebra
aurum
barometer light
A
associativity astatine
austenitic steels autism ,
barycentre barycentric coordinate time
asteroid
autistics ,
asteroid hitting the Earth
average , , ,
barycentric dynamical time
asteroid, Trojan
average curvature
asteroids ,
Avogadro’s number ,
baryon number
Astrid, an atom astrology , , ,
awe , axiom of choice ,
baryon number density baryon table
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astronomers, smallest known axioms
base units
axioms of set theory
basic units
astronomical unit
axioms, ZFC of set theory
Basiliscus basiliscus ,
astronomy
axis of the Earth, motion of basilisk
astronomy picture of the day
basis ,
axis, Earth’s
bath, physical ,
astrophysics
axle, impossibility in living bath-tub vortex
asymptotic freedom ,
beings
bathroom scale
at-sign
azoicum
bathroom scales
atmosphere of the Moon
bathtub vortex
atom
B
BaTiO ,
atom formation
β-rays
bats
atom rotation
B+
battery
atomic
Babylonia
atomic force microscope , Babylonians
BCS theory
,
background , , ,
beam, tractor
atomic force microscopes
background radiation , beans
atomic mass unit
,,
Beaufort
atomic number
bacteria , ,
beauty , , ,
atomic radius
bacterium
beauty quark
atoms ,
bacterium lifetime
beauty, origin of
atoms and swimming
badminton
becquerel
atoms are not indivisible
badminton serve, record
beer , ,
atoms are rare
badminton smashes
Beetle
atoms, hollow
bags as antigravity devices
beetle, click
atoms, manipulating single
before the Common Era
bags, plastic
beginning of the universe
atoms, matter is not made of ball lightning
beginning of time ,
balloons
behaviour
ATP , , ,
Banach measure
beings, living
ATP consumption of
Banach–Tarski paradox or
Bekenstein’s entropy bound
molecular motors
theorem , , , ,
ATP synthase
,
Bekenstein-Hawking
atto
banana catching
temperature
Atwood machine
bananas, knotted
belief systems
Au
BaO
beliefs , ,
Auger e ect
barber paradox
beliefs and Ockham
aureole
barium
beliefs in physical concepts
aurora australis
Barlow’s wheel
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
B
B’
Bell’s equation Bell’s inequality Bellatrix bells belt belt trick , belts and spin Bennett–Brassard protocol
BeppoSAX satellite berkelium Bernoulli’s ‘law’ Bernoulli’s principle beryllium Bessel functions Betelgeuse , , , , beth bets, how to win Bi bible , bicycle riding bicycle weight big bang , , big bang was not a singularity
big brother billiards , bimorphs binary pulsars , biographies of
mathematicians biological evolution , biology , bioluminescence biphoton BIPM , , bird appearance bird speed birds , , , birefringence , BiSb BiSeTe bismuth , , , Bi Te bits bits to entropy conversion
black body black body radiation ,
black body radiation constant ,
black hole , , , , ,
black hole collisions black hole entropy black hole halo black hole radiation black hole, entropy of black hole, extremal black hole, Kerr black hole, primordial black hole, rotating black hole, Schwarzschild black hole, stellar black holes , , , ,
black holes do not exist black paint black vortex black-body radiation black-hole temperature blackness blasphemies , blinks block and tackle blood pressure blood supply blue blue colour of the sea blue shi blue whale nerve cell board divers boat , Bode’s rule bodies, rigid , body body, rigid body, solid Bohr radius , bohrium Boltzmann constant , ,
, , ,, , Boltzmann’s constant bomb bombs bones, seeing own books books, information and
entropy in boost , , , boosts, concatenation of boredom as sign of truth boring physics boron Bose–Einstein condensates
,, Bose-Einstein condensate bosonisation bosons , , bottle , bottle, full bottom quark , bottomness boundaries boundary conditions boundary layer boundary of space-time box tightness boxes boxes, limits to bradyons Bragg re ection braid , braid pattern braid symmetry brain , , , , ,
brain and Moon brain and physics paradox brain size of whales brain stem brain’s energy consumption
brain’s interval timer brain, clock in Brans-Dicke ‘theory’ bread , breaking breathings breaths bremsstrahlung brick tower, in nitely high brilliant bromine brooms , brown dwarfs , , ,
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Brownian motion
Cancer
causality and maximum speed
Brownian motors
candela
B
browser brute force approach
candle , , , , canoeing
causality of motion cause
bubble chamber
Canopus ,
cause and e ect
B
bubbles bucket
cans of beans cans of peas
cavitation Cayley algebra
bucket experiment, Newton’s cans of ravioli
Cayley numbers
, bulge
capacitor , Capella
Cd CdS ,
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bullet speed
Capricornus
CeB
bullet speed measurement
capture of light
CeF
Bureau International des
capture, gravitational
celestrocentric system
Poids et Mesures
capture, in universal gravity cell
Burgers vector
cell motility
bus, best seat in
car parking
cell, voltaic
buses
car the
cells
bushbabies
car weight
Celsius temperature scale
butter y
carbohydrates, radicals in
Celtic wiggle stone
butter y e ect
carbon ,
cementite
button, future
Carboniferous
cenozoic
byte
cardinality
censorship, cosmic
cardinals
centi
C
cardinals, inaccessible
centre
C*-algebra
carriage, south-pointing , centre of gravity
c.
centre of mass , ,
C dating
cars
centre, quaternion
cable, eliminating power
cars on highways
centrifugal acceleration ,
CaCO
Cartan algebra
cadmium
Cartan metric tensor
centrifugal e ect
Caenozoicum
Cartan superalgebra
centrifugal force
caesium ,
Cartesian
Cepheids
caesium and parity
Cartesian product
Čerenkov e ect
non-conservation
cartoon physics, ‘laws’ of
Čerenkov radiation
CaF
Casimir e ect , , , cerium
calcite
CERN
calcium
Casimir e ect, dynamical
CERN
calculating prodigies
cat ,
Cetus
calculus of variations
catastrophes
chain
calendar ,
categories
chair as time machine
calendar, modern
category ,
chalkogens
californium
catenary
challenge level ,
calorie
caterpillars
challenge solution number
Calpodes ethlius
cathode
Cambrian
cathode rays ,
challenge, toughest of science
camera
cats
camera obscura
causal
challenges , , , ,
camera, holy
causal connection
–, – , – , ,
Canary islands
causality
–, –, –, ,
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
, –, –,,
, , –, ,
– , –, – ,
, , , , , Christo el symbols of the
C
, –, –, –, – ,
, , , –,
second kind
– , , , , Christophe Clanet
–,, –,
, , , , , chromatic aberrations
–, , – ,
– , , – , chromium
–, , ,
–, , ,
chromosome , ,
–, –,
chandelier
chromosome X and colour
– , ,–,
channel rays
– , – , , , chaos , ,
blindness cilia
Dvipsbugw
,,, , ,
chapter sign
circalunar
, –, –,
characteristic
circle packing
– , , – , charge ,
circularly polarized waves
, –, – ,
charge, amount of
Ciufolini, Ignazio
, , –, ,
charge, central
CL +
–, , ,
charge, radiation due to
class
–, , , ,
acceleration
classical
, – , – , charge, radiation due to
classical apparatus
, , , , , , gravity
classical electron radius
, , – , – , chargino
classical mechanics
–, , –,
charm quark ,
classi cation
–, –,
check
classi cation of concepts
–, –, ,
cheetah ,
classi cations in biology
–, , – ,
chemical elements, origin of classi ers
–, , , ,
heavy
claustrophobics
, –, – ,
chemical mass defect
Clay Mathematics Institute
, , , – , chemistry ,
,
, , , – , , chemoluminescence
cleveite
, – , , , , chess
click beetles
, – , , , , child psychology
clock , ,
, , – , , , childhood ,
clock in brain
, , , – , , chimaera
clock oscillator
– , , , , , Chimborazo, Mount
clock paradox
, , , , , , chiral
clock puzzles ,
, , , – , , chirality
clock synchronisation of ,
, , , , – , chirality in nature
, , , – , chirality of knots in nature
clock, air pressure powered
, ,, , ,
Chlamydomonas
clock, exchange of hands
–, , ,,
Chlamys
clocks , , ,
–, –,
chlorine
clocks and Planck time
– , , –,
chlorophyll
clocks, do not exist
, , , , , , chocolate ,
clockwise rotation
, , , – , , chocolate and the speed of
clone ,
, , –, ,
light
clones, biological
, , , , , chocolate bar
closed system ,
, – , , , chocolate does not last forever clothes, see through ,
, , , –,
clothes, seeing through
– , , , , choice, lack of, at big bang
cloud ,
, , , , , Chomolungma, Mount , cloud chamber
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clouds ,
compact disc
conductivity, electrical
Clunio
compact discs
cones in retina
C
CMOS CNO cycle
compactness comparison with a standard
cones in the retina Conférence Générale des
CO
Poids et Mesures ,
Co
complementarity
con guration
coal
complementarity principle con guration space , ,
coastline length
,
cobalt , CODATA
completeness , , , ,
conformal group conformal invariance , ,
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CODATA
completeness property of sets
coe cient of local
conformal transformations
self-induction
complex conjugate
co ee machines
complex Lie group
conic sections ,
coherence , ,
complex numbers ,
conjectures
coherence length
complexity
connected bodies
coherence time
complexity, ‘in nite’
connected manifold
coherent , ,
composed
consciousness , ,
coil guns
compositeness
conservation , , ,
cold fusion
composition theorem for
conservation of momentum
collapsars
accelerations
collapse , ,
comprehensibility of universe conservative
collision
conservative systems
collisions ,
Compton (wave)length
conserved quantities
colour , , , ,
Compton e ect
constancy of the speed of light
colour blindness
Compton satellite
colour displacement, Wien’s Compton tube
constellations , , ,
Compton wavelength ,
constituent quark mass
colour, world survey
constraints
coloured constellation
Compton wheel
container ,
colours ,
computational high-energy containers
colours in nature
physics
continental motion
comet ,
computer programs
continuity , , , , ,
comet shower
computer science
,
comet, Halley’s
computer science and
continuum
comets
quantum theory
continuum approximation
comic books
computer scientists
continuum hypothesis
Commission Internationale computers
continuum mechanics
des Poids et Mesures
concatenation
continuum physics
communication, faster than concave Earth theory
contraction ,
light
concept , ,
convection
communism
concepts
Convention du Mètre
commutation of Hamiltonian concepts, classi cation of
Conway groups
and momentum operator condensed matter physics
cooking
Cooper pairs
commutation, lack of
condom problem
cooperative structures
commutative ,
conductance
coordinates
commutator ,
conductance quantum
coordinates, fermionic
commute, observables do
conductivity
coordinates, generalised
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
coordinates, Grassmann
equivalence of
curl
Copernicus
countability
current quark mass
C
copper , , , , copy, perfect
counter coupling, minimal
current, electric curvature , , ,
copying machine
coupling, principle of
curvature, Gaussian
copying machines ,
minimal
curvature, intrinsic
core of the Earth
courage
curvature, sectional
Coriolis acceleration ,
covariance, principle of
curvemeter
Coriolis acceleration in atoms general covariant
cutting cycle
Dvipsbugw
Coriolis e ect
covering
cyclotron resonance
cork , ,
cows, ruminating
Cygnus X-
corkscrew
CPT ,
Cyrillic alphabet
cornea ,
Cr
corner gures
crackle
D
corner movie, lower le , crackpots , , , , dx
corner patterns
,
∂
corner pictures
creation , , , ,
D DR
corner, lower le movie ,
,
daemons
corpuscle
creation is a type of motion daisy
corrected Planck units
daleth
correctness
creation is impossible
daltonic
cortex
creation of light
damping , ,
cosmic background radiation creation of motion
dark energy ,
creation operator
dark matter , ,
cosmic censorship , , creation science
dark matter problem
,
Cretaceous
dark stars
cosmic evolution
cricket bowl
dark, speed of the
cosmic mirror
critical magnetic eld ,
darkness
cosmic radiation ,
critical mass density
darmstadtium
cosmic rays , , , , crop circles
dating, radiocarbon
,, ,
cross product ,
day length, past
cosmological constant , crust of the Earth
day, length of
,,, ,
cryptoanalysis
day, sidereal
cosmological constant
cryptography
day, time unit
paradox
cryptography, quantum
de Broglie wavelength
cosmological constant
cryptology
dead alone are legal
problem
crystallization dating
dead water
cosmological principle
Cs
death , , , , , ,
cosmonauts , , , , CsNO
,, , ,,
,
Cu
death sentence
cosmonauts, lifetime of
cube, magic
deca
cosmos ,
cubes, magic
decay , ,
Cotton–Mouton e ect
cucumber
decay of photons
coulomb ,
cumulonimbus , ,
deceleration parameter
Coulomb explosion
cuprum
deci
Coulomb force
curiosities
decoherence ,
Coulomb gauge
curiosity , ,
decoherence process
Coulomb’s and Gauss’s ‘laws’, curium
decoherence time ,
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
deep sea sh
di erent
distinguish
deer
di erential
distinguishability , , ,
D
degenerate matter degree Celsius
di erential manifold
,
di raction , , , , distribution, Gaussian normal
degree, angle unit
deity
di raction and photons
distribution, normal ,
delta function
di raction limit ,
divergence
demarcation
di raction of matter by light divine surprises
Demodex brevis demon, Maxwell’s
di raction pattern
division division algebra ,
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denseness ,
di usion , ,
DNA
density functional
digamma
DNA , , , , ,
density matrix
digital versatile discs, or DVD
density perturbations
DNA (human)
density, proper
dihedral angles
DNA molecules
deoxyribonucleic acid
dilations ,
DNS
dependence on r
Diluvium
doctrine
derivative
dimension ,
Dolittle
derivative at a point
dimension, fourth
dolphins
Derjaguin
dimensionality ,
domain of de nition
description ,
dimensionality, spatial
door sensors
descriptions, accuracy of
dimensionless
dopamine
design
dimensions
Doppler e ect ,
design, intelligent
dimensions, three spatial
Doppler red-shi
Desoxyribonukleinsäure
dinosaurs
double cover
details
diopter
double numbers
details and pleasure
dipole strength
doublets
details of nature
Dirac equation and chemisty doubly special relativity
detector
Dove prisms
detectors of motion
direction
down quark ,
determinism , , ,
disappearance of motion
Draconis, Gamma
,
disasters, future
drag
deutan
discovery of physical concepts drag coe cient ,
deuterium
dragging of vector potential
devils ,
discrete
by currents
Devonian ,
disentanglement
dreams
diagonal argument
disentanglement process
dri speed
diamagnetism ,
disinformation
drop
diameter
dislocation
Drosophila bifurca
diamond , , ,
dislocations ,
Drosophila melanogaster
diamond, breaking
dispersion ,
Drosophila melanogaster
diamonds
dispersion relation ,
dual eld tensor
dichroism ,
dispersion, anomalous
duality between large and
dielectricity
dissection of volumes
small ,
dielectrics
dissipative
duality in string theory
di eomorphism invariance dissipative systems , , duality of coupling constants
,
di erences are approximate distance
duality, electromagnetic
distance, rod
duality, space-time
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
dubnium
Earth, age of
electric e ects
Duckburg
Earth, at
electric eld ,
D
ducks, swimming Duration
Earth, attened Earth, hollow ,
electric eld, limit to electric polarizability
dust ,
Earth, length contraction
electric potential
duvet
Earth, mass of
electric signal speed
DVD
Earth, ring around
electrical conductance
dwarfs
Earth, shape of
electrical resistance
dx dyadic numbers
Earth-alkali metals earthquake ,
electricity electricity, start of
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dyadic product ,
earthquake, triggered by
electri cation ,
dyadic rational numbers
humans
electro-optical activity
dyadosphere
earthquakes , , ,
electro-osmosis
DyI
EBK
electroactive polymers
dynabee
EBK quantization
electrochromicity
dynamic friction
eccentricity ,
electrode
dynamical Casimir e ect
eccentricity of Earth’s axis
electrodynamics
dynamics
eccentrics
electrokinetic e ect
dynamos
echo ,
electroluminescence
dysprosium
ecliptic
electrolyte
eddies
electrolytes
E
edge dislocations
electrolytic activity
ε
e ect
electromagnetic coupling
E. coli
e ect, sleeping beauty
constant
e.g.
e ective width
electromagnetic eld , ,
ear , , , , ,
e ort
,
ear problems
EHF, extremely high
electromagnetic eld,
Earnshaw’s theorem
frequency
linearity
Earth ,
Ehrenfest’s paradox
electromagnetic smog
Earth core solidi cation
eigenvalue , ,
electromagnetic unit system
Earth dissection
eigenvalues , ,
,
Earth formation
eigenvector ,
electromagnetic waves,
Earth from space
eight-squares theorem
spherical
Earth rotation and
Einstein algebra
electromagnetism
super uidity
Einstein tensor
electromagnetism as proof of
Earth rotation slowing
Einstein, Albert
special relativity
Earth speed
Einstein–Cartan theory
electromotive eld
Earth stops rotating
Einstein–de Haas e ect
electron ,
Earth’s age ,
Einstein–Podolsky–Rosen
electron radius , ,
Earth’s av. density
paradox
electron speed ,
Earth’s axis
einsteinium
electron, weak charge
Earth’s gravitational length Ekert protocol
electrons
Ekman layer
electronvolt
Earth’s radius
elasticity ,
electroscope, capacitor
Earth’s rotation
elders
electrostatic machines
Earth’s rotation change of
Elea ,
electrostatic unit system ,
Earth’s shadow
electrets
Earth’s speed through the
electric charge
electrostriction
universe
electric charge is discrete
electrowetting
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
element of set , , , , energy velocity
ephemeris time
,
energy width
epistemology
E
element, adjoint elementary particle ,
energy, free energy, observer
eponyms EPR ,
elementary particle, shape of
independence
equilibrium , ,
,,
energy, potential
equipotential lines
elementary particle, size of energy, relativistic kinetic , equivalence principle ,
erasable
elementary particles
energy, relativistic potential
elementary particles, electric
erasing memory erbium
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polarizability
energy, scalar generalisation ergosphere ,
elementary particles, size of
of
Erta Ale
energy, total
ESA
elements ,
energy, undiscovered
escape velocity ,
elephants , ,
energy–momentum -vector Escherichia coli ,
Eliza
essence of universe
ellipse , ,
energy–momentum tensor et al.
Ellis
,
Eta Carinae
elongation
energy–momentum tensor of eth ,
elves ,
the electromagnetic eld ethel
ELW
ether, also called luminiferous
email
engines, maximum power of
ether
embryo
ethics
emergence , , ,
English language
Ettinghausen e ect
emergence of properties
English language, size of
Ettinghausen–Nernst e ect
emergent properties
enlightenment
emission, spontaneous
ensemble
Eucalyptus regnans
emissivity ,
ensembles
Euclidean space ,
emit waves
entanglement , , , Euclidean vector space , ,
empirical
empty space ,
entanglement, degree of
europium
EMS
entities
EUV
Encyclopédie
entropic forces
evanescent waves
end of applied physics
entropy , , , ,
evaporation ,
end of fundamental physics
Evarcha arcuata
entropy ow
evenings, lack of quietness of
end of science
entropy of a black hole
energy , , , , , entropy of black hole
event ,
,
entropy, limit to
event horizon
energy conservation , , entropy, quantum of
event symmetry ,
,
entropy, smallest in nature
events
energy consumption in First
Everest, Mount ,
World countries
entropy, state of highest , everlasting life
energy ows
eversion
energy ux
environment , , , , evolution , ,
energy is bounded
evolution equation, rst order
energy of a wave
enzymes
energy of the universe , Eocene
evolution equations ,
Epargyreus clarus ,
evolution, marginal
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
evolutionary biologists
F
eld emission
ex nihilo
fact ,
eld evaporation
E
Exa excess radius
faeces Falco peregrinus
eld ionization eld lines
exclamation mark
fall ,
eld of scalars
exclusion principle
fall and ight are independent eld theory
existence of mathematical
eld, mathematical
concepts
fall is not vertical
eld, physical
existence of the universe existence, physical
fall is parabolic fall of Moon
Fields medal elds, morphogenetic
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existence, psychological
fall, permanent
gure-eight
expansion
false
gures in corners
expansion of the universe
fame, way to reach
lament
expansions
familiarity
ne structure constant ,
experience ,
fan-out e ect
,,,, ,
Experience Island ,
farad
,,
experimental high-energy
Faraday cage
ne tuning
physics
Faraday e ect
ngers
experimental nuclear physics Faraday rotation
ngers prove the wave
farting for communication
properties of light
experimental physicists
nite , ,
experimentalists
faster than light
nite number
experiments
faster than light motion
re , ,
explanation ,
observed in an accelerated re y
exploratory drive
frame
rst law of black hole
explosion
faster than light motion, in
mechanics
explosion of volcano
collisions
rst law of horizon mechanics
explosion of Yellowstone
fate
exponential notation
Fe ,
rst property of quantum
exposition, SI unit
feathers
measurements
extended bodies, non-rigid feelings and lies
Fischer groups
feet
sh in water ,
extension
feldspar ,
sh’s eyes
extension, tests of
femto
sh, weakly-electric
extensive quantities ,
femtosecond laser
agella
extrasolar planets ,
fence
agella, prokaryote ,
extraterrestrials
fencing
agellar motor
extremal identity
Fermi coupling constant
agellum
extremum
Fermi problems
ame , , ,
extrinsic curvature
fermionic coordinates ,
atness
eye , ,
atness, asymptotic
eye and the detection of
fermions ,
attening of the Earth
photons
fermium
eas
eye motion
ferritic steels
ies
eye sensitivity ,
ferroelectricity
ight simulation
eye, human ,
ferromagnetism ,
ip movie
eyelid
ferrum
ip movie, explanation of
eyes of birds
Fiat Cinquecento
ow
eyes of sh
Fibonacci series
ow of time ,
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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frames of reference
gadolinium ,
uctuations
francium ,
Gaia
F
uctuations, zero-point uid mechanics
frass fraud
gait galaxy ,
uorescence
Fraunhofer lines , ,
galaxy centre
uorine
galaxy cluster
ute
free fall, permanent
galaxy formation
ux
free fall, speed of
galaxy group
ux, electric y, common ,
free will Freederichsz e ect
galaxy supercluster Galilean physics , ,
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ying saucers
frequency mixing
Galilean space
ying systems
friction , , , , , Galilean time
foam
Galilean velocity
focal point
Friction between planets and gallium
focus
the Sun
gallium arsenide
force , , ,
friction produced by tides
galvanometer
force limit
friction, importance of
gamma ray bursts , ,
force, central
frog legs
,,,,
force, de nition of
froghopper
gamma-ray bursts , ,
force, maximum
front velocity
force, minimum in nature
Froude number, critical
garlic-smelling
force, perfect
fruit y ,
semiconductor
force, physical
fuel consumption
gas
force, use of
full width at half maximum gas discharge lamps
Ford and precision
gas planets
forerunner velocity
Fulling–Davies–Unruh e ect gases ,
forest
,
gasoline
forgery
Fulling–Davies–Unruh
gate, logical AND
forgetting
radiation ,
gauge change
form, mathematical
function
gauge invariance
formal sciences
function collapse, wave
gauge symmetries
formula of life
function, mathematical
gauge symmetry
formulae, ISO
fundamental group
gauge theory
formulae, liking them
funnel ,
gauge transformation
Foucault pendulum ,
fusion
gauge transformations ,
four-squares theorem
fusion reactors
Fourier components
Futhark
gauge, Coulomb
Fourier transformation
futhorc ,
Gauss’s ‘law’
fourth dimension
future and present
Gauss’s theorem
fourth dimension to special future button
Gaussian curvature
future light cone
Gaussian distribution ,
fractals , ,
future, xed
fractals do not appear in
future, remembering
Gaussian integers
nature
Gaussian primes
fractional quantum Hall e ect G
Gaussian unit system ,
γ-rays ,
Gd
frame dragging ,
g-factor
GdSiGe
frame of reference
G-parity
gecko
frame-dragging
GaAs ,
Gedanken experiment
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
G
G
–M
Geiger–Müller counters Gell-Mann matrices Gemini General Motors general physics general relativity , general relativity and
quantum cosmology general relativity in one
paragraph general relativity in one
statement general relativity in ten points
general relativity, accuracy of
general relativity, rst half general relativity, second half
general relativity, statements of
generalized coordinates generalized indeterminacy
principle generators , , genes , genetic di erence between
man and chimpanzee genius , , , geocentric system geodesic deviation geodesic e ect , geodesic, lightlike geodesic, timelike geodesics geodynamo geoid geometrodynamic clock geostationary satellites germanium Germany, illegality of life Gerridae ghosts , , , , giant tsunami from Canary
islands Giant’s Causeway giants Gibbs’ paradox Giga
gimel girl-watching glass , , , global warming globular clusters glory glossary glove problem gloves gloves, di erence with
quantum systems glow of eyes of a cat glow-worm glueball gluino gluon gnomonics goblin , god , god’s existence goddess , , , gods , , , , , ,
,, , , , , , , , ,, ,, Goethe, Johann Wolfgang von ,, goggles, night gold , , golden rule golf balls Gondwana Gondwanaland gorilla test for random numbers Gothic alphabet Gothic letters
GPS
GPS, global positioning system
grace , , , gradient grammatical grampus , grand uni cation , graphics, three-dimensional
graphite , , grass
grasshopper Grassmann coordinates Grassmann numbers grating gravitation , gravitation and
measurements gravitation as braking
mechanism gravitational acceleration,
standard gravitational and inertial
mass identity gravitational Bohr radius gravitational constant gravitational constant is
constant gravitational coupling
constant gravitational Doppler e ect
gravitational energy , gravitational eld gravitational lensing , gravitational mass gravitational radiation gravitational red-shi ,
gravitational waves gravitational waves, speed of
, gravitodynamics gravitoelectric eld gravitoluminescence gravitomagnetic eld gravitomagnetic vector
potential gravitomagnetism graviton , gravity , gravity Faraday cages gravity inside matter shells
gravity measurement with a thermometer
Gravity Probe B gravity wave detectors and
quantum gravity gravity wave emission delay
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hag sh
heliocentric system
gravity waves , ,
hair growth
helioseismologists
G
gravity waves, spin of gravity, centre of
hair whirl hair, diameter
helium , , , , , , ,,,,
gravity, essence of
hair, gray
,,,, ,
gravity, sideways action of
hairs
gray ,
half-life
helium burning
Great Wall
Hall e ect ,
helium, discovery of
Great Wall in China Greek alphabet
Hall e ect, fractional quantum
helium, super uid helix
Dvipsbugw
Greek number system
halo , , ,
hell
Greeks
halogens
Helmholtz
green ,
Hamilton’s principle
henry
green ash
hammer drills
Hentig, Harmut von
green ideas
Hanbury Brown–Twiss e ect Hering lattices
green ray
Hermann lattice
green star
hand
Hermitean
greenhouse e ect
hand in vacuum
Hermitean vector space
Gregorian calendar
hand, for quaternion
herring, farting
Gregorian calendar reform
visualization
hertz
grey hair
handedness
Hertzsprung–Russell diagram
group
hands of clock
group of components
Hanle e ect
Hg
group velocity
hard discs, friction in
hidden variables
group velocity can be in nite harmonic functions
Higgs
harmonic motion
higgsino
group, conformal
hassium
Hilbert space ,
group, mathematical
Hausdor space
Hilbert’s problems ,
group, monster
He–Ne
Hilbert’s sixth problem ,
group, simple
heart beats
growth , ,
heartbeat
Hilversum
growth of deep sea
heat ,
Himalaya age
manganese crust
heat & pressure
hips
Gulf Stream
heat capacity of metals
Hiroshima
Gulliver’s travels
heat radiation
Hitachi
guns and the Coriolis e ect Heaviside formula
hodograph , ,
GUT epoch
Heaviside–Lorentz unit
HoI
gymnasts
system ,
hole argument
gyromagnetic ratio ,
heavy metals
hole paradox
gyromagnetic ratio of the
Hebrew alphabet
holes
electron
hecto
holes in manifolds
gyroscope
Heiligenschein
hollow Earth hypothesis
Göttingen
Heisenberg picture ,
hollow Earth theory ,
Gödel’s theorem
Heisenberg’s indeterminacy Hollywood
relations
Hollywood movies , ,
H
helicity ,
,, ,
HO
helicity of stairs
holmium
hadrons
helicopter ,
Holocene ,
hafnium
helicopters
hologram
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
holograms with electron
human Y chromosome
image ,
beams
humour
image, real
H
holograms, moving holography ,
humour, physical hurry
image, virtual images , ,
holonomic
hydrargyrum
images, and focussing devices
holonomic systems
hydrodynamics
holonomic–rheonomic
hydrogen , ,
imaginary
holonomic–scleronomic
hydrogen atoms
imaginary mass
homeomorphism Homo sapiens appears
hydrogen atoms, existence of imaginary number imagination
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Homo sapiens sapiens
hydrogen fusion
imagine
homogeneity ,
hyperbola ,
immediate
homogeneous
hyperbolas
immovable property
homomorphism
hyperbolic
impact
honey
hyperbolic cosine
impact parameter
honey bees ,
hyperbolic secant
impact parameters
Hooke
hyperbolic sine
impenetrability ,
hoop conjecture
hyperbolic tangent
impenetrability of matter ,
Hopi
hypernovae
,,
hops
hyperreals
in all directions
Horace, in full Quintus
hypersurfaces
InAs:Mn
Horatius Flaccus
hypotheses
incandescence , , ,
horizon , , , ,
,
,
I
incandescent lamps
horizon and acceleration
i.e.
inclination of Earth’s axis
horizon force
ibid.
incubation
horizon, moving faster than Icarus ,
independently
light
ice age ,
indeterminacy relation ,
horizons
ice ages , ,
horizons as limit systems
iceberg
indeterminacy relation for
horror vacui
icicles
angular momentum
horse power
icon
indeterminacy relation of
horsepowers, maximum value icosane
thermodynamics
of
idea, platonic
indeterminacy relations ,
HortNET
ideal ,
hot air balloons ,
ideal gas ,
index
hour
ideal gas constant ,
index nger
Hubble constant
ideas, green
index of refraction
Hubble parameter
idem
index, negative refraction
Hubble space telescope
identity, extremal
Indian numbers ,
Hubble time
if
indigo, violet
human
igneous rocks
indistinguishable ,
human body, light emission of ignition
indium
ill-tempered gaseous
individuality ,
human eye
vertebrates
inductions
human growth
illuminance
inertia
human language ,
illumination
inertial
human observer
illusion, motion as
inertial frame
human sciences
illusions, optical
inertial frame of reference
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
inertial mass
intention
involution
inf.
interaction , , ,
Io ,
I
in nite in nite coastlines
interaction, is gravity an interaction, reciprocity of
iodine ion
in nite number of SI pre xes interactions
ionic radii
interference , ,
ionization ,
in nite-dimensional
interference and photons
ionosphere , ,
in nitesimals
interference fringes
ionosphere, shadow of
in nities in nity , , , , ,
interferometer , , ,
ions , ,
IPA
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in nity in physics
interferometers , , , IRA or near infrared
in ation , , ,
IRB or medium infrared
in aton eld ,
intermediate black holes
IRC or far infrared
information
intermediate bosons
iridium
information in the universe internal
iron , ,
internal photoelectric e ect irradiance
information science and
irreducible
quantum theory
International Earth Rotation irreducible mass
information table
Service , ,
irreducible radius
information, erasing of
International Geodesic Union irreversibility of motion
information, quantum of
irreversible ,
infrared
International Latitude Service Island, Experience
infrared light
,
ISO
infrared rays
International Phonetic
isolated system
infrasound ,
Alphabet
isomorphism
inhomogeneous Lorentz
internet ,
isotomeograph
group
interpretation of quantum
isotopes , , ,
initial conditions , , , mechanics ,
isotropic
interstellar gas cloud ,
Istiophorus platypterus
injective ,
intersubjectivity
italic typeface ,
ink sh
interval
IUPAC
inner product
intrinsic
IUPAC ,
inner product spaces
intrinsic angular momentum IUPAP
inner world theory
IUPAP
InP
intrinsic properties
InSb ,
invariance , ,
J
insects , , ,
invariance, conformal
Jacobi identity
inside
invariant
Jacobi identity, super
instability of solar system
invariant property
Janko groups
instant ,
invariant, link
jerk ,
instant, human
invariant, topological
Jesus
instant, single
invariants of curvature tensor JET, Joint European Torus
instruments
jets ,
insulation
inverse element
jewel textbook
insulation power
inversion , ,
Joint European Torus
insulators
inversion symmetry
Josephson constant
integers ,
inverted commas
Josephson e ect ,
integration ,
invisibility of objects
Josephson frequency ratio
intelligent design ,
invisible loudspeaker
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Joule
knowledge, humorous
laser activity
joule
de nition of
laser and glass beads
J
Journal of Irreproducible Results
knuckle angles koppa
laser and Moon laser beam
Julian calendar
Korteweg–de Vries equation laser beam, tubular
J
jump jump, long
KPO
laser cavities laser distance measurement of
Jupiter ,
krypton
Moon
Jupiter’s mass Jupiter’s moons
Kuiper belt ,
laser levitation laser loudspeaker
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Jurassic
L
laser sword ,
ladder, sliding
laser, helim-neon
K
lady’s dress
lasers
k-calculus ,
LAGEOS
lateral force microscopes
Kac–Moody algebra ,
LAGEOS satellites
lateralization ,
KAlSi O
Lagrange’s equations of
latex
kaleidoscope
motion
Latin
kaon ,
Lagrangian
Latin alphabet
kaons and quantum gravity Lagrangian (function)
Laurasia
Lagrangian is not unique
lava
katal
Lagrangian libration points lava, radioactivity of
ke r
law
Kelvin
Lagrangian of the
law of cosmic laziness
kelvin
electromagnetic eld
law of nature, rst
Kepler’s relation
Lagrangian operator
lawrencium
Kerr e ect
LaH
laws are laziness
ketchup motion ,
Lamb shi , ,
laws of nature ,
Killing form
lamp
Lawson criterium
kilo
lamp, ideal
lawyers ,
kilogram
lamps, sodium
laziness of physics
kilogram, prototype
Landau levels
laziness, cosmic, principle of
kilotonne ,
Landolt–Börnstein series
,
kinematics ,
language
lead ,
kinesin
language, human
leaf, falling
kinesiology
language, spoken
leap day
kinetic energy ,
language, written
learning ,
Kirlian e ect
languages on Earth
learning mechanics
Klein bottles
languages spoken by one
least e ort
Klitzing, von – constant ,
person
lecture scripts
lanthanoids
le -handed material
knife limits
lanthanum
le -handers
knocking
Laplace acceleration
leg
knot sh
large
leg performance
knot problem, simplest
Large Electron Positron ring legends, urban
knot theory
Lego ,
knot, mathematical
large number hypothesis , lego
KnotPlot
legs
knotted protein
larger
Leibniz
knowledge
laser ,
Leidenfrost e ect
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
length , , , ,
light de ection
lightning, colour of
length contraction , , light de ection near masses lightning, zigzag shape of
L
length scale
Lilliput light dispersion and quantum limbic system
lens
gravity
limit concept
Leo
light does not move
limit on resolution
LEP
light emitting diodes
limit size of physical system
lepton number
light is electromagnetic
levitation , , , levitation, human
light microscope light mill ,
limit statements and physics
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levitation, neutron
light onion
limit values for measurements
Levitron
light polarization
lexical universals
light pressure
limit, de nition of
liar’s paradox
light pulses, circling each
limits to cutting
Libra
other
limits to motion
library
light source
limits to observables,
librations
light speed measurement
additional
lichen growth
light speed observer is
LiNbO ,
Lie algebra ,
impossible
line
Lie algebra,
light speed, nite
linear spaces
nite-dimensional
light swords
linear vector spaces
Lie algebras
light tower
linearity of electromagnetic
Lie group
light tunnelling
eld
Lie group, compactness of
light year
linearity of quantum
light, angular momentum of
mechanics
Lie group, connectedness of
lines, high voltage
light, constant speed in
linguists
Lie group, linear
electromagnetism
link
Lie multiplication
light, detection of oscillations Linux
Lie superalgebra
liquid
lies , ,
light, faster than
liquid crystal e ect
lies, general
light, longitudinal
liquid crystals
life ,
polarization
liquids
life appearance
light, macroscopic
Listing’s ‘law’
life time
light, made of bosons
lithium , ,
life’s basic processes
light, massive
litre
life’s chemical formula
light, slow group velocity
living beings
life, sense of
light, the unstoppable
living thing, heaviest ,
life, shortest
light, weighing of
living thing, largest
lifetime
lightbulb temperature
lizard
lifetime, atomic
lighthouse
lobbyists
Lifshitz
lightlike
local
li into space
lightlike geodesics
local time
li ers
lightning , , , , localisation
light , , , ,
,
locality ,
light acceleration
lightning emits X-rays
localization (weak, Anderson)
light beams, twisting
lightning rod
light bulbs
lightning rods, laser
locusts
light can hit light
lightning speed
logarithms
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
logicians
Magellanic clouds
Majorana e ect
long jump , ,
magentization of rocks
mammals ,
L
Lorentz acceleration Lorentz boosts
magic , , magic cubes and squares
mammals, appearance of man, wise old ,
Lorentz gauge
magic moment
man-years
Lorentz group
magic moments
manganese
Lorentz relation
magmatites
Manhattan as copper mine
Lorentz symmetry breaking Magna Graecia
and quantum gravity Lorentz transformations of
magnesium magnet
manifold , manifold, analytic
Dvipsbugw
space and time
magnetar ,
manifold, connected
Loschmidt’s number
magnetars
manifolds
loudspeaker
magnetic circular dichroism manta
loudspeaker with laser
mantle of the Earth
loudspeaker, invisible
magnetic e ects
many types
love ,
magnetic eld , ,
many worlds interpretation
love, making ,
magnetic eld, limit to
low-temperature physics
magnetic ux
many-body problem
lower frequency limit
magnetic ux density
mapping
lowest temperature
magnetic ux quantum
marble, oil covered
Lucretius Carus
magnetic induction
marriage
lumen
magnetic monopole
Mars , ,
luminary movement
magnetic monopoles
mars trip
luminescence
magnetic pole in a mirror
martensitic steels
luminous bodies
magnetic resonance
maser ,
luminous density
magnetic resonance force
masers
luminous pressure
microscope
Maslov index
Lunakhod , , ,
magnetic resonance imaging mass , , , , ,
lunar calendar
,,
mass change, limit to
lutetium
magnetic thermometry
mass change, maximum
lux ,
magnetic vector potential
mass conservation implies
LW, long waves
magnetism
momentum conservation
Lyapounov exponent
magneto–Seebeck e ect
Lydéric Bocquet
magneto-optical activity
mass defect, measurement
Lyman-α
magnetoacoustic e ect
mass density, limit to
Lyman-alpha line
Magnetobacterium bavaricum mass is conserved
mass measurement error
M
magnetocaloric e ect
mass, centre of ,
µ
magnetoelastic e ect
mass, equality of inertial and
M theory
magnetoencephalography
gravitational
M
magneton
mass, Galilean
M
magnetoresistance
mass, gravitational ,
M
magnetorheologic e ect
mass, identity of gravitational
M , galaxy
magnetostriction
and inertial
mach
magnets ,
mass, imaginary
Mach’s principle , ,
magnifying glass
mass, inertial , ,
machine ,
magnitudes
mass, limit to
macroscopic system
main result of modern science mass, negative , , ,
macroscopically distinct
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
M
mass, total, in general relativity
mass–energy equivalence match boxes and universe
material systems material, le -handed materials science materials, dense optically math forum mathematical physics mathematicians , , ,
mathematics , , ,
mathematics is applied physics
Mathieu groups maths problem of the week
matrix, adjoint matrix, antisymmetric matrix, complex conjugate matrix, orthogonal matrix, real matrix, symmetric matrix, transposed matter , matter and antimatter,
indistinguishability matter and vacuum, mix-up
matter domination matter is not made of atoms
matter shell, gravity inside matter transformation matter, composite matter, impenetrability of matter, metastable mattress , , , ,
mattress analogy of vacuum
Mauna Kea Maurice Maeterlink maximal ageing maximal ideal Maxwell’s demon ,
meaning measurability , , , ,
measure measured measurement , , measurement apparatus measurement limits measurement requires matter
and radiation measurements measurements and
gravitation measurements disturb mechanics mechanics, classical mechanics, quantum medicines Mega megaparsec Megatonne megatonne Megrez Meissner e ect meitnerium membrane theory memories memorize memory , , , ,
, memory erasing memory, write once Mendel’s ‘laws’ of heredity mendelevium menstrual cycle , Mercalli scale Mercury mercury , , , mercury lamps meridian meson table Mesopotamia mesoscopic mesozoic Messier object listing metacentric height metal alloys metal halogenide lamps metal multilayers ,
metallic shine metallurgy metals , metals, transition metamorphic rocks metamorphites metaphor meteorites metre metre bars metre rule metre sticks metric , metric connection metric space , , metricity , , , , Mg micro micronystagmus microscope , microscope, atomic force ,
microscopes, light microscopic microscopic motion microscopic system microscopy, stimulated
emission depletion microwave background
temperature microwaves Mie scattering migration mile milk , , , , Milky Way Milky Way’s age Milky Way’s mass Milky Way’s size milli million dollars mind mind change mind-reading minerals mini comets minimum minimum electrical charge
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M
minimum force in nature Minkowski space-time Mintaka minute , Miocene miracles , , mirror , , , , mirror molecules mirror, concave mirror, moving mirrors , mixed state mixture of light Mn MnO Mo mobile phone mole , molecular motors molecule molecule size molecules molecules, mirror molybdenum momenergy moment moment of inertia , , ,
moment of inertia, extrinsic
moment of inertia, intrinsic
momentum , , , , ,
momentum as a liquid momentum change momentum conservation
follows from mass conservation momentum ows momentum of a wave momentum, angular momentum, ow of momentum, limit to momentum, relativistic monade money, humorous de nition of monism
monochromatic monocycle monopole, magnetic , monopoles Monster group monster group month Moon , , , , Moon and brain Moon and laser Moon calculation Moon formation Moon path around Sun Moon phase Moon size illusion Moon size, angular Moon size, apparent Moon’s atmosphere Moon’s mass Moon’s mean distance Moon’s radius Moon, dangers of Moon, fall of Moon, laser distance
measurement moons morals mornings, quietness of motion , , , motion and measurement
units motion as an illusion motion as illusion , , motion backwards in time motion detectors motion does not exist , motion in living beings motion is based on friction
motion is change of position with time
motion is due to particles motion is relative motion is the change of state
of objects Motion Mountain motion of continents motion of images motion reversibility
motion, dri -balanced motion, faster than light motion, harmonic motion, hyperbolic motion, limits to motion, manifestations motion, non-Fourier motion, passive , motion, simplest motion, superluminal motion, volitional motion, voluntary motor motor, electrostatic motor, linear motor, unipolar motorbike motors mountain movable property movement movie mozzarella MRI mu-metal multiple universes multiplet , multiplication , multiplicity muon , muon neutrino muons , ,
Musca domestica muscles , music record MW, middle waves mycoplasmas myosin myosotis mysteries mystery of motion myths
Myxine glutinosa Möbius strip Mössbauer e ect Mößbauer e ect
N n-Ge
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
n-Si
neutrino ux on Earth
non-stationary
Na
neutrino, Earth
non-unitarity
N
nabla NaCl
neutrino, electron neutrino, fossil
nonholonomic constraints nonius
NaI
neutrino, man-made
nonlinear sciences
-S
naked singularities ,
neutrinos
nonsense
nano
neutrinos, atmospheric
nonstandard analysis
nanoscopic
neutrinos, solar
Nordtvedt e ect ,
NASA NASA ,
neutron , neutron levitation
norm , , normal distribution
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natural
neutron star ,
normality
natural numbers
neutron stars ,
normality of π
natural sciences
neutron trap
North Pole , , , ,
natural units ,
neutrons
,
nature , ,
new age
north poles ,
nature and computer science New Galactic Catalogue
nose
newspaper
notion
nature, sense of
Newton ,
nova ,
Navier–Stokes equations
newton
novae
Nb-Oxide-Nb
Newtonian physics ,
novelty seeking
Ne
NGC
nuclear explosions
necessities, science of
NGC
nuclear ssion
symbolic
Ni
nuclear magnetic resonance
necklace of pearls
Niagara
,
negative ,
Niagara Falls
nuclear magneton
negative group velocity
nickel ,
nuclear material accident or
negative mass
nigh goggles
weapon use
neighbourhood ,
niobium ,
nuclear motion, bound
neocortex
Nit
nuclear reaction
neodymium
nitrogen ,
nuclear reactor, natural
Neogene
NMR
nuclear warhead
neon
no
nuclei ,
Neptune ,
no-cloning theorem ,
nucleon
neptunium
NOAA
nucleosynthesis
Nernst e ect
Nobel Prizes, scientists with nucleus ,
nerve signal speed
two
nuclides ,
nerve signals
nobelium
null
nerves
noble gases
null geodesics
network
node
null vectors ,
neural networks
Noether charge
number of particle
neurological science
Noether’s theorem
number of stars
neurologists
noise ,
number theory
neurons , ,
noise thermometry
numbers ,
neutral bodies
noise, (physical)
numbers, hypercomplex ,
neutral element
non-Cantorian
neutralino
non-classical
numbers, trans nite
neutrinium
non-classical light ,
nutation
neutrino , , , , non-local
nutshell, general relativity in a
, , ,, ,
non-singular matrix
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
nymphs
operation, (binary)
oxygen depletion
operator
oxygen, appearance in
N
O
Ophiuchus opposite ,
atmosphere ozone shield reduction
object , , , ,
optical activity ,
object are made of particles optical black holes
P
optical Kerr e ect
π
object, full list of properties optical nonlinear e ects
π and gravity
object, levitation
optical radiation thermometry
π, normality of p-Ge
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objects ,
optically induced anisotropy packing of spheres
objects, real
paerlite
objects, virtual
optoacoustic a ect
paint, black
obliquity
optogalvanic e ect
pair creation ,
observable limits,
orange
Paleocene
system-dependent
orbifold
Paleogene
observables , , ,
orbits
paleozoic
observables, discrete
order , , , ,
palladium
observation , ,
order parameter
Pangaea
observations ,
order structures
paper aeroplanes
observer, comoving
order, partial
paper boat contest
Occam’s razor ,
order, total
parabola , , , , ,
ocean oors
ordered pair
ocean levels ,
ordinal numbers
parabolic
octaves
ordinals
parachutes
octet
Ordovician
paradox of incomplete
octonions ,
ore-formers
description
odometer ,
organelles
paradox of overcomplete
ohm
orientation change needs no
description
oil , ,
background
paradox, EPR
oil lm experiment
origin, human
paradox, liar’s
oil tankers
original Planck constant
paraelectricity
Oklo
origins of the heavy chemical parallax ,
Olber’s paradox
elements
parallelepiped
Olbers
Orion , ,
paramagnetism ,
Olbers’ paradox
orthogonality
Paramecium
Oligocene
oscillation
parameter
Olympic year counting
oscillation, harmonic or
parametric ampli cation
Olympus mons
linear
parenthesis
omega
oscillator
parity
one million dollar prize
oscillons
parity invariance
one-body problem
osmium
parity non-conservation
onset
osmosis ,
parity violation in electrons
onto
otoacoustic emissions
ontological reach
outer product
parking challenge
Oort cloud , ,
oven
Parmenides, Plato’s
op. cit.
overdescription
parsec ,
Opel
Oxford
particle , ,
open questions in physics
oxygen ,
Particle Data Group
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
particle data group
theorems ,
Philips
particle exchange
pentaquarks , ,
philosophers
P
particle number , particle, ultrarelativistic
people perceptions
philosophers of science phosphorescence
particle, virtual
perfect copy
phosphorus
particles
performers
photino
particles, massive
periastron
photoacoustic e ect , ,
particles, virtual , ,
periastron shi
,
partons
perigee perihelion , ,
photoconductivity photoe ect
Dvipsbugw
parts , ,
perihelion shi , ,
photoelectricity
parts in nature
periodic table
photoelectromagnetic e ect
parts, sum of
periodic table of the elements
pascal
,
photography ,
Paschen–Back e ect
permanence , ,
photography, limits of
passim
permanence of nature
photoluminescence
passive motion
permanent free fall
photon
past light cone
permeability
photon as elementary particle
path ,
permeability, vacuum
paths of motion
Permian
photon cloning
patterns
permittivity
photon decay
patterns of nature
permittivity of free space
photon drag e ect
Paul traps
permutation symmetry , photon number density
Pauli exclusion principle ,
photon sphere
,,,,
perpetual motion machine photon, mass of
Pauli pressure
perpetuum mobile
photon, position of
Pauli spin matrices ,
perpetuum mobile, rst and photon-photon scattering
Pb
second kind
photonic Hall e ect
PbLaZrTi
person
photons , ,
PbSe
perturbation calculations
photons and interference
PbTe
perverted
photons and naked eye
pea dissection
Peta
photons as arrows
pearl necklace paradox
phanerophyte, monopodal photons, entangled
pearls
photons, eye detection of
peas
phase ,
single
pee research
phase conjugated mirror
photons, spin of
peers
activity
photons, virtual
Peirce’s puzzle
phase factor
photorefractive materials ,
Peltier e ect
phase space , ,
pencil, invention of
phase space diagram
photostriction
pendulum
phase velocity
physical aphorism
penetrability of matter
pheasants
physical concepts, discovery
penetration
phenomena, supernatural
of
penguins
phenomena, unnatural
physical explanation
Penning e ect
phenomenological
physical observations
Penning traps ,
high-energy physics
physical system , ,
Penrose inequality ,
phenomenon
physicists
Penrose process
Philaenus spumarius
physics , ,
Penrose–Hawking singularity Philippine healers
physics as limit statements
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
physics papers
planet–Sun friction
polonium
physics problems
planetoids ,
polymer ,
P
physics, boring physics, etymology of
plankton , plants
polymers, electroactive pool, game of
physics, everyday
plants appear
pop
physics, open questions in
plasma ,
porcine principle ,
plasmas , ,
Pororoca
physics, outdated de nition plate tectonics
posets
physiology
plates platinum
position positional system
Dvipsbugw
piccolissimi quanti
platonic ideas
positions
pico
platonism
positive ,
piezoelectricity ,
platypus
positive or negative
pigeons ,
play ,
positivity
pigs ,
Pleiades star cluster
positron
pile
Pleistocene
positron charge
pinch e ect
Pliocene
positron tomography
ping command, in UNIX, to plumb-line
possibility of knots
measure light speed
plumbum
post-Newtonian formalism
ping-pong ball
Pluto
pion ,
plutonium
potassium ,
Pioneer anomaly
Pockels e ect
potato as battery
Pioneer satellites
Poincaré algebra
potential energy
Pisces
point exchange
potential energy in relativity
Planck acceleration
point mass
Planck area, corrected
point particle
potential, gravitational
Planck density
point particles, size of
potential, spherical
Planck force
point, mathematical
power ,
Planck length ,
point-like for the naked eye power emission, limit to
Planck length,
pointer
power lines
Planck limit
points , ,
power set ,
Planck limits, corrected
points, shape of
power supply noise
Planck mass
points, size of –
power, humorous de nition
Planck stroll
poise
of
Planck time ,
poisons
power, maximum
Planck time.
Poisson equation
power, maximum in nature
Planck units, corrected
Poisson’s point
Planck values
polar motion
power, physical ,
Planck’s (reduced) constant Polaris
Poynting vector ,
polarizability
PPN, parametrized
Planck’s (unreduced) constant polarization , , ,
post-Newtonian
formalism
Planck’s constant ,
polarization of light
praeseodymium
Planck’s natural length unit polarization, electrical
pralines
polders
praying e ects
Planck’s natural units
pole, magnetic, in a mirror precession ,
plane gravity wave
precession for a pendulum
planet
poles
precession, equinoctial
planet formation
police
precision , , , , ,
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
,, ,
probability
public
precision does not increase probability amplitude
pulley
P
with energy precision, maximal
probability distribution problems, physical
pullovers pulsar
precision, maximum
Proca Lagrangian
pulsar period
predicates
process, chemical
pulsars ,
pre xes ,
process, sudden
pulse
pre xes, SI
processes
pulsed impulse kill laser
prejudice preon models
Procyon , prodigies
pure pure blue
Dvipsbugw
preprints ,
product set
pure green
present ,
product, dyadic
pure red
present, Zeno and the absence product, outer
pure state
of the
projective planes
pure truth
presocratics
promethium
purpose
pressure , ,
pronunciation, Erasmian
pyramid
pressure of light
proof
pyroelectricity
primal scream
propeller
Pythagoras’ theorem
primates
propellers in nature
primates, appearance of
proper distance
Q
prime knots ,
proper length
Q-factor
principal quantum number proper time , ,
q-numbers
proper velocity ,
Q+
Principe, island of
properties for the elementary QED
principle of equivalence
particles
QED diagrams
principle of gauge invariance properties of nature
QED in one statement
properties, emergent ,
qoppa
principle of general
protactinium
quadrupoles
covariance
protan
quality factor
principle of general relativity proterozoic
quanta
proton , ,
quantitative biology
principle of least action , proton age
quantities ,
proton lifetime
quantities, conserved
principle of minimal coupling proton mass
quantities, extensive
proton radius
quantity of matter ,
principle of relativity ,
proton shape, variation of
quantization
principle of the straightest
proton stability or decay
quantization, EBK
path
proton–electron mass ratio quantons ,
principle, anthropic
quantum action principle
principle, correspondence
protonvolt
quantum computers
principle, equivalence
prototype kilogram
quantum computing ,
principle, extremal
pseudo-science
quantum electrodynamics
principle, physical
pseudovector ,
quantum uctuations
principle, simian
PSR +
quantum geometry , ,
principle, variational
PSR + ,
, ,,
printed words
PSR B +
quantum Hall e ect
prism
PSR J -
quantum lattices
prison
psychokinesis
quantum mechanical system
prize, one million dollar
psychological existence
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
quantum mechanics
radiation, observers made of
quantum mechanics applied
ravioli
Q
to single events quantum numbers ,
radiation, thermal radiative decay
ray days ray form
quantum of action , , radicals
rayon vert
radio eld
rays
quantum of change
radio interference
reaction rate
quantum of entropy
radio speed
reactions
quantum of information quantum particles
radio waves , radioactivity , ,
reactions, nuclear reactor, natural nuclear
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quantum physics
radioactivity, arti cial
real numbers , , , ,
quantum principle
radiocarbon dating
quantum ratchets
radiocarbon method
reality
quantum theory ,
radiocativity, of human body reals
quantum theory and
reason
computer science
radiometer
recognition ,
quantum theory in one
radiometer e ect
recognize
statement
radiometric dating , , recoil
quark con nement ,
recombination ,
quark mass
radium
record ,
quark stars
radius, covalent
rectilinear
quarks , , ,
radius, ionic
red ,
quartets
radon
red-shi ,
quartz , ,
rail guns
red-shi mechanisms
quartz, transparency of
rain drops ,
red-shi number
quasar ,
rain speed
red-shi tests
quasars ,
rainbow , , ,
red-shi values
quasistatic elds
rainbow due to gravity
reduced Planck constant
Quaternary
rainbow, explanation
reducible
quaternion, conjugate
rainbows, supernumerary
reductionism
quaternions , ,
raindrops
reel
quietness of mornings
Raleigh scattering
re ection ,
quotation marks
RAM
re ection of waves
quotations, mathematical
Raman e ect
re ectivity ,
random
refraction , , , ,
random access memory
,, , ,
R
random errors
refraction index, negative
R-complex
random pattern
refraction of matter waves
Rad
randomness
refraction, vacuum index of
radar
randomness, experimental
radian
refractive index
radiation , , , , range
Regulus ,
,,
rank
Reinhard Krüger
radiation composite
Ranque–Hilsch vortex tube Reissner–Nordström black
radiation exposure
holes
radiation thermometry
rapidity
relation
radiation, black body
ratchet
relation, binary
radiation, observer
rational coordinates
relations , ,
dependence of
rational numbers , , relativistic acceleration
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relativistic contraction
riddle
rotation of the Earth
relativistic correction
Riemann curvature tensor
rotation rate
R
relativistic jerk relativistic kinematics
Riemann tensor Riemann-Christo el
rotation sense in the athletic stadium
relativistic mass
curvature tensor
rotation speed
relativistic velocity
Riemannian manifold
rotation, absolute or relative
relativity, doubly special
Riga
relativity, Galileo’s principle of Rigel
rotational energy
relaxation
Righi–Leduc e ect right-hand rule
rotational motors roulette and Galilean
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religion
right-handers
mechanics
reluctance to turn
rigid bodies
rubidium
rem
rigid bodies do not exist in Rubik’s Cube ,
representation , , , nature
ruby ,
rigid coordinate system
rugby
representation of the
rigidity
rules
observables
ring
rules of nature
representations
ring interferometers
runaway breakdown
reproducibility
rings, astronomical, and tides Runic script
reproduction
running on water
research
Roadrunner
running reduces weight
research fraud
Robertson–Walker solutions ruthenium
reservoir
rutherfordium
reset mechanism
robot
Rydberg atoms
resin
robotics
Rydberg constant , ,
resinous
rock cycle
resistance of single atoms
rock magnetization
Röntgen, unit
resistivity, Joule e ect
rock types
resolution
rocket
S
resources
rocket launch sites
S-duality
rest , , ,
rocket motor
Sackur–Tetrode formula
rest energy
rod distance
Sagarmatha, Mount ,
rest mass
rods in retina , ,
Sagittarius ,
rest, Galilean
roentgenium
Sagnac e ect ,
retina
rolling
Sahara
retrore ecting paint
rolling puzzle
Sahara, Hz signal in
reversal of Earth’s magnetic rolling wheels
middle of
eld
Roman number system
sail sh
reversibility of motion
rope attempt
sailing ,
reversible ,
rose
Saiph
Reynolds number , , rosetta
salamander
rosetta paths
Salmonella ,
rhenium ,
Rostock, University of
salt ,
rhodium
Roswell incident
salt-formers
rhythmites, tidal
rotation , ,
Salticidae
Ricci scalar , ,
rotation and arms
samarium
Ricci tensor ,
rotation axis
sampi
Richardson e ect
rotation change of Earth
san
Richter magnitude
rotation of atoms
sand , ,
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
S
S
–S
Sasaki–Shibuya e ect satellite satellite phone and light speed
satellite, arti cial satellites , , satellites, observable saturable absorption saturation Saturn , scalar , , scalar multiplication scalar part of a quaternion
scalar product , , scale , scale factor , scale invariance scale symmetry scallop theorem scallops scandium scanning for imaging
Scarabeus scattering , scattering experiment Schadt–Helfrichs e ect Schottky e ect Schrödinger’s equation of
motion Schrödinger equation,
complex numbers in Schrödinger equation, for
extended entities Schrödinger picture Schrödinger’s cat Schrödinger’s equation Schumann resonances Schwarzschild black holes Schwarzschild metric , Schwarzschild radius , Schwarzschild radius as
length unit ScI science science ction , science ction, not in
quantum gravity science of symbolic
necessities , , science, end of scienti c method scientism scientist scissor trick , scissors Scorpius Scotch tape screw dislocations scripts, complex Se sea wave energy sea waves, highest seaborgium search engines searchlight e ect season second , , second harmonic generation
second principle of thermodynamics ,
second property of quantum measurements:
secret service sedenions sedimentary rocks sedimentites Sedna see through clothes Seebeck e ect seeing selectron selenium self-acceleration self-adjoint self-organization self-referential self-similarity semi-ring , , semiconductivity semiconductor,
garlic-smelling semiconductors semimajor axis semisimple sensations sense of life
sense of nature senses sensors, animal separability , separability of the universe
separable sequence , , , , Sequoiadendron giganteum serious sesquilinear set , sets in nature sets, connected seven sages sex , , , , sexism in physics , sextant sha shadow shadow of Great Wall in
China , shadow of ionosphere shadow of the Earth shadow of the Earth during
an eclipse shadow with halo or aureole
shadow with hole shadows , , , , shadows and attraction of
bodies shadows and radiation shadows not parallel shadows of cables shadows of sundials shadows, colour of shadows, speed of , ,
shape , , , , shape deformation and
motion shape of points shape of the Earth shape, optimal sharks , sharp s shear stress, theoretical sheep, Greek
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
Shell study
mechanics
sodium
shell, gravity inside matter
singlets
sodium lamps
S
SHF, super high frequency
singular singular point
so gamma repeater solar cells
ship ,
singularities , , ,
solar constant, variation
S
ships and relativity ships, critical speed
singularities, naked
solar data solar sail e ect
shit ,
singularity, dressed
solar system
shock waves shoe laces
singularity, naked sink vortex
solar system formation solar system simulator
Dvipsbugw
shoelaces ,
Sirius , , ,
solar system, future of
shore birds
situation
solid bodies
short
size
solid body, acceleration and
short pendulum
size limit
length limit
shortest measured time
skatole
solid state lamps
shot noise
skew eld
solid state physics, in society
shot, small
skew-symmetric
shoulders
skin , ,
solidity ,
showers, cosmic ray
skin e ect
solitary waves
shroud, Turin
skipper
soliton
Shubnikov–de Haas e ect
sky , ,
solitons ,
shutter
sky, moving
solvable
shutter time
sleeping beauty e ect
sonoluminescence ,
shutter times
slide rule
Sophie Germain primes
shuttlecocks
slide rules
soul
Si
slime eel
souls
SI pre xes
slingshot e ect
sound
SI system
Sloan Digital Sky Survey , sound channel
SI units ,
sound speed
SI units, supplementary
sloth
sound waves
siemens
slow motion
source
sievert ,
smallest experimentally
sources ,
sign
probed distance
South Pole
signal
Smekal–Raman e ect
south poles ,
signal distribution
smiley
south-pointing carriage ,
signal, physical
smoke
silent holes
smuon
soviets
silicon , ,
snake constellation
space
Silurian
snakes
space is necessary
silver ,
snap
space li
simian principle ,
sneutrino
space of life
similar
snooker
space points
Simon, Julia
snow akes
space travel
simple
snowboard
space, absolute , ,
simply connected
snowboarder, relativistic
space, absoluteness of
sine curve
snow ake speed
space, mathematical
single atom , , , , snow akes ,
space, physical
,
soap bubbles
space, relative or absolute
single events in quantum
Sobral, island of
space, white
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space-time , , space-time diagram space-time duality space-time elasticity space-time foam space-time interval space-time lattices space-time, uid space-time, relative or
absolute space-time, solid space-time, swimming
through curved spacelike , spacelike convention spanned spanners, optical spark spark chambers sparks spatial inversion special conformal
transformations special orthogonal group special relativity special relativity before the
age of four special relativity in one
statement speci c mechanical energy
spectrometers spectrum spectrum, photoacoustic speculations speculative speed speed of dark speed of darkness speed of gravitational waves
, speed of light speed of light inside the Sun
speed of light, nite speed of light, one-way speed of light, theories with
variable speed of light, two-way
speed of shadows speed of sound , speed of sound, values speed of the tip of a lightning
bolt speed, electron dri speed, highest speed, in nite speed, limit to speed, lowest speed, perfect , sperm , sperm motion sphere packing sphere, hairy Spica spiders , spin , , spin particles spin / and quaternions spin particles spin and classical wave
properties spin and extension spin foams spin myth spin of a wave spin of gravity waves spin valve e ect spin, electron spin, importance of spin–orbit coupling spin–spin coupling spin-statistics theorem spinor spinors , , spirits spirituality , split personality spoon spring spring constant spring tides sprites , square, magic squark , squeezed light , SrAlO St. Louis arch
stadia stainless steel staircase formula stairs stalactite stalagmites , standard apple standard cyan standard deviation , standard kilogram standard orange standard quantum limit for
clocks standard yellow standing wave stannum star age star algebra star approach star classes , , star speed measurement star, green stardust Stark e ect stars , , , stars, neutron stars, number of start of physics state , , , , state allows one to distinguish
state function state of an mass point,
complete state of motion state of universe state space diagram state, physical statements statements, boring statements, empirical statements, undecidable states , static friction , static limit stationary , statistical mechanics stauon steel
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steel, stainless
,,
surface, physical ,
Stefan–Boltzmann black body Sun dogs
surfaces of genus n
S
radiation constant , Stefan–Boltzmann constant
Sun size, angular
sur ng
Sun will indeed rise tomorrow surjective
,
surprises ,
Steiner’s parallel axis theorem Sun’s age
surprises in nature
Sun’s heat emission ,
surreal
stellar black hole
Sun’s luminosity , ,
surreal numbers
steradian stibium
Sun’s mass
surreals suspenders
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sticking friction
Sun’s motion around galaxy SW, short waves
stigma
swimming
stilts
Sun, stopping
swimming and atoms
stimulated emission
Sun–planet friction
swimming through curved
Stirling’s formula
sundials ,
space-time
stokes
sun ower
swimming, olympic
stone formation
sunny day
Swiss cheese
stones , , , , , , , sunset
switch, electrical
, , , , , , superalgebras
switch, inverter
, , , , , , superbracket
switchable magnetism
, , , , , , supercommutator
switchable mirror
,, , ,
superconductivity ,
switching o the lights
Stoney units
superconductors
swords in science ction
stopping time, minimum
super uidity , ,
syllabary
straightness , , , ,
supergiants
symbol, mathematical
strain
superlubrication
symbolic necessities, science
strange quark ,
superluminal
of
streets
superluminal motion
symbols ,
Streptococcus mitis
superluminal speed
symmetric
stretch factor
supermarket
symmetries
strike with a eld
supermassive black holes
symmetry ,
string theory ,
supernatural phenomena , symmetry of the whole
strong coupling constant
Lagrangian
strong eld e ects
supernovae , , ,
symmetry operations
strong nuclear interaction
supernumerary rainbows
symmetry, event
strontium
superposition ,
symmetry, external
structure constants ,
superposition, coherent
symmetry, low
stun gun
superposition, incoherent
synapses
subalgebra
superradiation
synchronisation of clocks ,
subgroup
superstition
subjects
superstring theory
syntactic
submarine, relativistic
supersymmetry ,
syrup
submarines
suprises, divine
Système International
subscripts
surface ,
d’Unités (SI)
sugar , ,
surface gravity of black hole system ,
sulphur
system, cloning of
sulphuric acid
surface tension
macroscopic
sum of parts
surface tension waves
system, geocentric
Sun , , , , , , surface, compact
system, heliocentric
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
system, isolated
tensor , ,
thermodynamic degree of
system, macroscopic
tensor of curvature
freedom
S
systematic errors systems, conservative
tensor order tensor product ,
thermodynamic equilibrium
systems, dissipative
tensor trace
thermodynamic limit
systems, simple
tensor, antisymmetric
thermodynamics
tensor, energy–momentum thermodynamics in one
T
statement
T-duality table as antigravity device
tensors Tera
thermodynamics, rst ‘law’ of Dvipsbugw
tachyon , ,
terabyte
thermodynamics, rst
tachyon mass
terahertz waves , , ,
principle
tachyons , ,
,
thermodynamics,
tankers, sinking
terbium
indeterminacy relation of
tantalum , , ,
terms
Tarantula nebula
terrestrial dynamical time
thermodynamics, second ‘law’
taste
Tertiary
of
tau
tesla ,
thermodynamics, second
tau neutrino
Tesla coil
principle
Taurus
Tesla coils
thermodynamics, second
tax collection
test
principle of ,
TB
testicle
thermodynamics, third
TbCl
testimony
principle
teapot ,
tetra-neutrons
thermodynamics, zeroth
tear shape
tetrahedron ,
principle
technetium
tetraquark
thermoelectric e ects
technology
tetraquarks
thermoluminescence
tectonic activity
thallium ,
thermomagnetic e ects
tectonics
ames
thermometer
teeth , ,
theoretical high-energy
thermometry
telecommunication
physics
thermostatics
telecopes in nature
theoretical nuclear physics
theses
telekinesis ,
thin lens formula
telephathy
theoretical physicists
irring e ect
telephone speed
theoreticians
irring–Lense e ect , ,
teleportation , , , , theory
,
theory of everything
omas precession ,
telescope ,
theory of evolution
omson e ect
television , ,
theory of motion
thorium ,
tellurium
theory of relativity
thorn , ,
temperature ,
theory, physical
three-body problem
temperature scale
thermal de Broglie
thriller
temperature, absolute
wavelength
thrips
temperature, human
thermal emission
throw
temperature, lower limit to thermal energy
throwing speed, record ,
thermal equilibrium
throwing, importance of
temperature, lowest in
thermal radiation , , , thulium
universe
thumb
temperature, relativistic
thermoacoustic engines
thunderclouds are batteries
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tokamak ,
tree growth ,
thunderstorms
TOM
tree height, limits to
T
Ti:sapphire laser tidal acceleration
tomography tongue ,
tree leaves and Earth rotation
tidal e ects , , ,
tonne, or ton
tree, family
tides , , , ,
tooth decay ,
trees
tides and friction
toothbrush ,
trees and electricity
tie knots
toothpaste
trees and pumping of water
time , , , , , time average
toothpick top quark , ,
trees appear
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time delay
topness
trefoil knot
time dilation factor
topoisomerases
Triassic
time does not exist
topological invariant
triboelectricity
time independence of G
topological space ,
triboluminescence
time intervals
topological structures
triboscopes
time is deduced by comparing topology , ,
tripod
motions
torque , ,
tritan
time is necessary
torsion
tritium
time is what we read from a torus, n-
trivial knot
clock
total momentum
Trojan asteroids
time machine
total symmetry
tropical year
time machines
touch
trousers
time measurement, ideal
touch sensors
trout
time of collapse
touching
true
time translation
tourmaline
true velocity of light
time travel ,
toys, physical
truth ,
time travel to the future
trace ,
truth quark
time, absolute , ,
tractor beam
tsunami
time, absoluteness of
train windows
tubular laser beam
time, arrow of
trains
tu
time, beginning of
trajectory
Tullio Levi-Civita
time, de nition of
trans nite number
tumbleweed
time, ow of
trans nite numbers
tungsten , , ,
time, limits to
transformation of matter
tunnel
time, proper, end of
transformation relations
tunnel through the Earth
time, relatove or absolute
transformation, conformal tunnelling
time-bandwidth product
tunnelling e ect ,
timelike ,
transformations ,
tunnelling of light
timelike convention
transformations, linear
tunnelling rate
tin
transforms
turbulence
titanium
transition radiation
Turin shroud
Titius’s rule
translation
Turing machines
TmI
translation invariance ,
tv
TNT
transparency
tv tube
Tocharian
transport
tweezers, optical
TOE
transporter of energy
twin paradox
tog
transsubstantiation
twins as father(s)
toilet brushes
travel into the past
two-body problem
toilet research
tree , , , ,
two-dimensional universe
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
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two-squares theorem Tyrannosaurus rex
U U-duality udeko Udekta UFOs , UHF, ultra high frequency
ultrarelativistic particle ultrasound , ultrasound imaging ultrasound motor ultraviolet ultraviolet light ultraviolet rays umbrella umbrellas unboundedness , , , unboundedness, openness uncertainty relation of
thermodynamics uncertainty relations , uncertainty, total uncountability understand understanding quantum
theory undisturbed motion Unicode uni cation , uni cation is possible uniqueness , unit , , , unit systems unitarity , unitary unitary vector space units units, astronomical units, natural units, non-SI units, Stoney units, true natural unity universal ‘law’ of gravitation
universal ‘law’ of gravity
universal gravity as consequence of maximum force
universal gravity, deviation from
universal semantic primitives
universal time coordinate , ,
universality of gravity universals universe , , , , ,
,, universe is comprehensible
universe not a physical system
universe not a set universe recollapse universe’ luminosity limit
universe’s initial conditions do not exist
universe, age of , , universe, believed universe, description by
universal gravitation universe, energy of universe, essence universe, existence of universe, lled with water or
air universe, full universe, only one universe, oscillating universe, size of universe, state of universe, two-dimensional universe, visible universe, volume of universe, wave function of universes, other UNIX
UNIX
unnatural phenomena unpredictable unstoppable motion, i.e. light
ununbium
ununhexium ununoctium ununpentium ununquadium ununseptium ununtrium up quark , uranium , , urban legends , URL Usenet UTC ,
UVA
UVB
UVC
V vacuum , , , ,
, vacuum curvature vacuum elasticity vacuum energy density ,
vacuum permeability vacuum permittivity vacuum polarisation vacuum polarization vacuum state vacuum wave resistance vacuum, hand in vacuum, human exposure to
vacuum, swimming through
vacuum, unstable vacuum–particle
indistinguishability value vampire vanadium vanilla ice cream vanishing , variability variable , variance variation variational principle Varuna vector , , ,
Dvipsbugw
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
vector part of a quaternion virtual photons
virus
watt
V
vector potential, dragging by virus crystallisation
charges
vis viva
wave wave angular momentum
vector potential, magnetic
viscosity
wave equation
vector product ,
viscosity, kinematic ,
wave function ,
vector space ,
viscosity,dynamic
wave function, symmetry of
vector space, Euclidean , vitamin B
vector space, Hermitean
vitamin C vitreous
wave impedance/resistance
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vector space, unitary
vocabulary ,
wave interference
Vega
void ,
wave re ection
Vega at the North pole
Voigt e ect
wave vector
Vela satellites
volcano, giant
wave, evanescent
velars
volcanoes
wave, harmonic
velocimeters
Volkswagen
wave, linear
velocity ,
volt
wave–particle duality
velocity as derivative
volume
wavelength ,
velocity composition formula volume of the universe
wavelet transformation
voluntary motion ,
waves in relativity
velocity is not Galilean
von Neumann equation
waves, circularly polarised
velocity measurements
vortex
velocity of an electron
vortex in bath tub or sink
waves, deep
velocity of light, one-way
vortex lines
waves, electromagnetic
velocity of light, two-way
vortex tube
waves, evanescent
velocity, escape
vortex, black
waves, long
velocity, faster than light
vortex, circular
waves, longitudinal
velocity, perfect
Voyager satellites
waves, shallow
velocity, proper ,
Vulcan
waves, short
velocity, relative
Vulcanoids
waves, terahertz , , ,
velocity, relative - unde ned
,
W
waves, transverse
vendeko
W,
waves, water
Vendekta
W boson
waw
Venus
Waals, van der, forces at feet weak energy condition
verb
of Geckos and spiders
weak mixing angle
veri cation
wafers, consecrated
weapons with light
vernier
walking , ,
weber
VHF, very high frequency
walking on two legs
week
Viagra
walking speed
week days, order of
video recorder
walking, Olympic
Weib
viewpoint-independence
warp drive situations
Weigert e ect
viewpoints
water , , , ,
weighing light
Virasoro algebra ,
water cannot ll universe
weight , ,
Virgo
water density
weight of the Moon
virtual image
water drops and droplets
weko
virtual particle
water taps
Wekta
virtual particles , , , water waves ,
whale brain
,
water waves, group velocity of whale brain size
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Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.net Copyright © Christoph Schiller November 1997–May 2006
W
whales , , , , ,
wheel axle, e ect of Wheeler–DeWitt equation
, wheels and propellers wheels in living beings wheels in nature whip cracking whip, speed of whirl whirl, hair whiskey white white colour does not last white dwarf white dwarfs , whole, the wholeness, Bohm’s unbroken
wholeness, unbroken Wien displacement law
constant Wien’s colour displacement
‘law’ Wien displacement constant
wife Wikipedia wind wind resistance window frame windows in trains wine , , wine arcs wine bottle , wine, dating of Wino Wirbelrohr
wise old man ,
WMAP
wolframates women , , , women and physics women, dangers of looking
a er words words heard words spoken work , , work, physical world , , world colour survey World Geodetic System ,
world question center world, chaos or system world-line world-wide web worm holes wound healing wristwatch time , writhe writing , written texts wrong , , wyn
X X bosons X-ray lasers X-rays , , X-rays, hard X-rays, so xenno xenon , Xenta
Y year, number of days yellow yo-yo yocto yogh yot Yotta youth e ect youth, gaining ytterbium yttrium Yucatan impact
Z Z boson Z boson mass Z-graded Lie algebras Zeeman e ect Zener e ect zenith zenith angle Zeno e ect, quantum zepto zero , zero-body problem zero-point uctuations ,
Zetta ZFC ZFC axioms zinc zino zippo zirconium ZnS ZnSb zodiac
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...
ma la religione di voi è qui
e passa
di generazione in generazione
ammonendo
che S
L
.
Giosuè Carducci*
* ‘... but the religion of you all is here and passes from generation to generation, admonishing that .’ Giosuè Carducci (1835–1907), important Italian poet and scholar, received the Nobel Prize
for literature in 1906. e citation is from Carducci’s inscription in the entry hall of the University of Bologna, the oldest university of the world.
MOTION MOUNTAIN
The Adventure of Physics
Why do change and motion exist? How does a rainbow form? What is the most fantastic voyage possible? Is ‘empty space’ really empty? How can one levitate things? At what distance between two points does it become
impossible to nd room for a third one in between? What does ‘quantum’ mean? Which problems in physics are unsolved?
Answering these and other questions on motion, the book gives an entertaining and mind-twisting introduction into modern physics – one that is surprising and challenging on every page.
Starting from everyday life, the adventure provides an overview of the recent results in mechanics, thermodynamics, electrodynamics, relativity, quantum theory, quantum gravity and uni cation. It is written for undergraduate students and for anybody interested in physics.
Christoph Schiller, PhD Université Libre de Bruxelles, is a physicist with more than years of experience in the presentation of physical topics.
available free of charge at www.motionmountain.net