motion mountain the adventure of physics

Christoph Schiller

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Motion Mountain
The Adventure of Physics

www.motionmountain.net



Christoph Schiller

Motion Mountain

The Adventure of Physics

available at
www.motionmountain.net


Editio decima octava.
Proprietas scriptoris Christophori Schiller
secundo anno Olympiadis vicesimae sextae
secundo 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.

Eighteenth revision, September .
Copyright ©  –  by Christoph Schiller,
between the second year of the th olympiad
and the second 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 for everybody.
About the cover photograph, see page .


To T.

τῷ ἐµοὶ δαὶµονι


Die Menschen stärken, die Sachen klären.


Contents
Preface
1.

20
An appetizer

First Part :

23
Cl assical Physics – How Do Things and Images Move?

Chapter
II Special R el ativity
6. Maximum speed, observers at rest, and motion of light

249
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Chapter
III Gravitation and R el ativity
7. Maximum force: general relativity in one statement
8. The new ideas on space, time and gravity
9. Motion in general relativity – bent light and wobbling vacuum
10. Why can we see the stars? – Motion in the universe
11. Black holes – falling forever
12. Does space differ from time?
13. General relativity in ten points – a summary for the layman

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344
362
402
440
454
460

Chapter

IV Cl assical Electrodynamics
14. Liquid electricity, invisible fields and maximum speed
15. What is light?
16. Charges are discrete – the limits of classical electrodynamics
17. Electromagnetic effects and challenges
18. Classical physics in a nutshell – one and a half steps out of three

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516
545
547
568

Intermezzo

The Brain, L anguage and the Human Condition

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Second Part :
tions?

Quantum Theory – What Is Matter? What Are Interac-

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Chapter
VI Permu tation of Particles
23. Are particles like gloves?
24. Rotations and statistics – visualizing spin

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Chapter VII Details of Quantum Theory and Electromagnetism
25. Superpositions and probabilities – quantum theory without ideology
26. Applied quantum mechanics – life, pleasure and the means to achieve them
27. Quantum electrodynamics – the origin of virtual reality

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Chapter
V Quanta of Light and Matter
19. Minimum action – quantum theory for poets and lawyers
20. Light – the strange consequences of the quantum of action
21. Motion of matter – beyond classical physics
22. Colours and other interactions between light and matter

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Chapter
I Galilean Motion
2. Why should we care about motion?
3. Galilean physics – motion in everyday life
4. Global descriptions of motion: the simplicity of complexity
5. From the limitations of physics to the limits of motion


28. Quantum mechanics with gravitation – the first approach

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Chapter VIII Inside the Nucleus
29. The structure of the nucleus – the densest clouds
30. The strong nuclear interaction and the birth of matter
31. The weak nuclear interaction and the handedness of nature
32. The standard model of elementary particle physics – as seen on television
33. Grand unification – a simple dream

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Chapter

IX

Advanced Quantum Theory (Not yet Avail able)

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Chapter

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Quantum Physics in a Nu tshell

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Intermezzo

Bacteria, Flies and Knots

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Third Part :
Particles?

Motion Withou t Motion – What Are Space, Time and

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Chapter

Extension and Unification (Not yet Avail able)

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Chapter XIII

The Top of the Mountain (Not yet Avail able)

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Fourth Part :

Appendices

Appendix

A

Notation and Conventions

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Appendix

B

Units, Measurements and Constants

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Appendix

C

Particle Properties

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Appendix

D

Numbers and Spaces

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Appendix

E

Information Sources on Motion

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Appendix

F

Challenge Hints & Solu tions

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Appendix

G

List of Illustrations

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Appendix

H

List of Tables

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Appendix

I

Name Index

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Appendix

J

Subject Index

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Chapter
XI General R el ativity Versus Quantum Mechanics
34. Does matter differ from vacuum?
35. Nature at large scales – is the universe something or nothing?
36. The physics of love – a summary of the first two and a half parts
37. Maximum force and minimum distance: physics in limit statements
38. The shape of points – extension in nature

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Detailed Contents
Preface

20

1. An appetizer

23

First Part : Cl assical Physics
How Do Things and Images Move?

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28

40

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Chapter
I Galilean Motion
2. Why should we care about motion?
Does motion exist? 29 • How should we talk about motion? 31 • What are
the types of motion? 32 • Perception, permanence and change 36 • Does
the world need states? 38 • Curiosities and fun challenges about motion 39
•
3. Galilean physics – motion in everyday life
What is velocity? 41 • What is time? 42 • Why do clocks go clockwise? 47
• Does time flow? 47 • What is space? 48 • Are space and time absolute
or relative? 50 • Size: why area exists, but volume does not 51 • What is
straight? 54 • A hollow Earth? 55 • Curiosities and fun challenges about
everyday space and time 56 •
How to describe motion: kinematics
What is rest? 63 • Objects and point particles 66 • Legs and wheels 69 •
Objects and images
Motion and contact 72 • What is mass? 72 • Is motion eternal? 77 • More
on conservation – energy 79 • Is velocity absolute? – The theory of everyday
relativity 81 • Rotation 83 • Rolling wheels 87 • How do we walk? 87 •
Is the Earth rotating? 89 • How does the Earth rotate? 94 • Does the Earth
move? 96 • Is rotation relative? 99 • Curiosities and fun challenges about
everyday motion 100 • Legs or wheels? – Again 106 •
Dynamics due to gravitation

Properties of gravitation 111 • Dynamics: how do things move in various dimensions? 115 • Gravitation in the sky 116 • The Moon 117 • Orbits 118
• Tides 120 • Can light fall? 123 • What is mass? – Again 124 • Curiosities and fun challenges about gravitation 125 •
What is classical mechanics?
Should one use force? 134 • Complete states: initial conditions 139 • Do
surprises exist? Is the future determined? 141 • A strange summary about
motion 144 •
Bibliography
4. Global descriptions of motion: the simplicity of complexity
Measuring change with action
The principle of least action 164 • Why is motion so often bounded? 168 •
Curiosities and fun challenges about Lagrangians 170 •
Motion and symmetry
Why can we think and talk? 174 • Viewpoints 174 • Symmetries and
groups 176 • Representations 177 • Symmetries, motion and Galilean


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Chapter
II Special Relativity
6. Maximum speed, observers at rest, and motion of light
Can one play tennis using a laser pulse as the ball and mirrors as rackets? 254
• Special relativity in a few lines 256 • Acceleration of light and the Doppler
effect 257 • The difference between light and sound 260 • Can one shoot
faster than one’s shadow? 261 • The addition of velocities 263 • Observers
and the principle of special relativity 263 • What is space-time? 266 • Can
we travel to the past? – Time and causality 268 •
Curiosities of special relativity
Faster than light: how far can we travel? 269 • Synchronization and aging:
can a mother stay younger than her own daughter? – Time travel to the future 270 • Length contraction 272 • Relativistic movies – aberration and
Doppler effect 275 • Which is the best seat in a bus? 277 • How fast can
one walk? 278 • Is the speed of shadow greater than the speed of light? 278
• Parallel to parallel is not parallel – Thomas rotation 282 • A never-ending
story: temperature and relativity 282 •
Relativistic mechanics
Mass in relativity 283 • Why relativistic snooker is more difficult 284 •
Mass is concentrated energy 285 • Collisions, virtual objects and tachyons 287 • Systems of particles: no centre of mass 289 • Why is most motion so slow? 290 • The history of the mass–energy equivalence formula by
de Pretto and Einstein 290 • Four-vectors 291 • Four-momentum 294

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physics 179 • Reproducibility, conservation and Noether’s theorem 182 •
Curiosities and fun challenges about motion symmetry 187 •
Simple motions of extended bodies – oscillations and waves
Waves and their motion 189 • Why can we talk to each other? – Huygens’
principle 193 • Signals 194 • Solitary waves and solitons 195 • Curiosities
and fun challenges about waves and extended bodies 197 •
Do extended bodies exist?
Mountains and fractals 201 • Can a chocolate bar last forever? 202 • How
high can animals jump? 203 • Felling trees 204 • The sound of silence 205
• Little hard balls 205 • Curiosities and fun challenges about fluids and
solids 208 •
What can move in nature?
Why are objects warm?
Entropy 217 • Flow of entropy 218 • Do isolated systems exist? 219 •
Why do balloons take up space? – The end of continuity 220 • Brownian
motion 221 • Entropy and particles 223 • The minimum entropy of nature:
the quantum of information 224 • Why can’t we remember the future? 226
• Is everything made of particles? 226 • Why stones can be neither smooth
nor fractal, nor made of little hard balls 227 • Curiosities and fun challenges
about heat 228 •
Self-organization and chaos
Curiosities and fun challenges about self-organization 237 •
5. From the limitations of physics to the limits of motion
Research topics in classical dynamics 239 • What is contact? 240 • Precision and accuracy 240 • Can all of nature be described in a book? 241 •
Why is measurement possible? 241 • Is motion unlimited? 242 •

Bibliography


contents
• Four-force 295 • Rotation in relativity 295 • Wave motion 297 • The
action of a free particle – how do things move? 297 • Conformal transformations: Why is the speed of light constant? 298 •
Accelerating observers
Acceleration for inertial observers 301 • Accelerating frames of reference 302 • Event horizons 306 • Acceleration changes colours 307 • Can
light move faster than c? 308 • What is the speed of light? 309 • Limits on
the length of solid bodies 309 •
Special relativity in four sentences
Could the speed of light vary? 311 • What happens near the speed of
light? 311 •
Bibliography

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Chapter
III Gravitation and Relativity
7. Maximum force: general relativity in one statement
The maximum force and power limits 320 • The experimental evidence 322
• Deducing general relativity 323 • Space-time is curved 327 • Conditions of validity of the force and power limits 328 • Gedanken experiments
and paradoxes about the force limit 329 • Gedanken experiments with the
power limit and the mass flow limit 333 • Hide and seek 336 • An intuitive understanding of general relativity 336 • An intuitive understanding of
cosmology 339 • Experimental challenges for the third millennium 339 •
A summary of general relativity 341 • Bibliography 342 •
8. The new ideas on space, time and gravity
Rest and free fall 344 • What is gravity? – A second answer 345 • What
tides tell us about gravity 348 • Bent space and mattresses 349 • Curved
space-time 351 • The speed of light and the constant of gravitation 353 •
Why does a stone thrown into the air fall back to Earth? – Geodesics 354 •
Can light fall? 357 •
Curiosities about gravitation
What is weight? 361 • Why do apples fall? 362 •
9. Motion in general relativity – bent light and wobbling vacuum

Weak fields
The Thirring effects 363 • Gravitomagnetism 365 • Gravitational
waves 368 • Bending of light and radio waves 375 • Time delay 376
• Effects on orbits 377 • The geodesic effect 379 • Curiosities about weak
fields 379 •
How is curvature measured?
Curvature and space-time 383 • Curvature and motion in general relativity 384 • Universal gravity 385 • The Schwarzschild metric 386 • Curiosities and fun challenges about curvature 386 •
All observers: heavier mathematics
The curvature of space-time 387 • The description of momentum, mass and
energy 388 • Hilbert’s action – how do things fall 390 • The symmetries of
general relativity 391 • Einstein’s field equations 391 • More on the force
limit 394 • Deducing universal gravity 395 • Deducing linearized general
relativity 396 • How to calculate the shape of geodesics 396 • Mass in general relativity 398 • Is gravity an interaction? 398 • The essence of general
relativity 399 • Riemann gymnastics 400 •
10. Why can we see the stars? – Motion in the universe

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Chapter
IV Classical Electrodynamics
14. Liquid electricity, invisible fields and maximum speed
Amber, lodestone and mobile phones
How can one make lightning? 480 • Electric charge and electric fields 483
• Can we detect the inertia of electricity? 487 • Feeling electric fields 489
• Magnets 490 • Can humans feel magnetic fields? 490 • How can one
make a motor? 491 • Magnetic fields 493 • How motors prove relativity
to be right 497 • Curiosities and fun challenges about things electric and
magnetic 498 •
The description of electromagnetic field evolution
Colliding charged particles 505 • The gauge field: the electromagnetic vector potential 506 • Energy, linear and angular momentum of the electromagnetic field 510 • The Lagrangian of electromagnetism 510 • Symmetries:
the energy–momentum tensor 511 • What is a mirror? 512 • What is the
difference between electric and magnetic fields? 514 •
Electrodynamic challenges and curiosities
Could electrodynamics be different? 515 • The toughest challenge for elec-

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Which stars do we see? 402 • What do we see at night? 405 • What is the
universe? 409 • The colour and the motion of the stars 412 • Do stars shine
every night? 414 • A short history of the universe 415 • The history of
space-time 418 • Why is the sky dark at night? 422 • Is the universe open,
closed or marginal? 424 • Why is the universe transparent? 426 • The big
bang and its consequences 427 • Was the big bang a big bang? 427 • Was
the big bang an event? 427 • Was the big bang a beginning? 428 • Does the
big bang imply creation? 429 • Why can we see the Sun? 429 • Why are the
colours of the stars different? 430 • Are there dark stars? 432 • Are all stars
different? – Gravitational lenses 432 • What is the shape of the universe? 434
• What is behind the horizon? 435 • Why are there stars all over the place? –
Inflation 435 • Why are there so few stars? – The energy and entropy content
of the universe 436 • Why is matter lumped? 437 • Why are stars so small
compared with the universe? 437 • Are stars and galaxies moving apart or
is the universe expanding? 437 • Is there more than one universe? 438 •
Why are the stars fixed? – Arms, stars and Mach’s principle 438 • At rest in
the universe 439 • Does light attract light? 440 • Does light decay? 440 •
11. Black holes – falling forever
Why study black holes? 441 • Horizons 441 • Orbits 444 • Hair and entropy 446 • Black holes as energy sources 448 • Paradoxes, curiosities and
challenges 449 • Formation of and search for black holes 451 • Singularities 452 • A quiz: is the universe a black hole? 453 •
12. Does space differ from time?
Can space and time be measured? 455 • Are space and time necessary? 456
• Do closed timelike curves exist? 457 • Is general relativity local? – The
hole argument 457 • Is the Earth hollow? 458 • Are space, time and mass
independent? 459 •
13. General relativity in ten points – a summary for the layman
The accuracy of the description 461 • Research in general relativity and cosmology 461 • Could general relativity be different? 463 • The limits of
general relativity 464 •
Bibliography

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Intermezzo
The Brain, L anguage and the Human Condition

Evolution 585 •
Children and physics
Why a brain? 588 • What is information? 589 • What is memory? 590 •
The capacity of the brain 592 •
What is language?
What is a concept? 597 • What are sets? What are relations? 599 • Infinity 601 • Functions and structures 603 • Numbers 604 • Why use mathematics? 608 • Is mathematics a language? 609 • Curiosities and fun challenges 610 •
Physical concepts, lies and patterns of nature
Are physical concepts discovered or created? 612 • How do we find physical
patterns and rules? 614 • What is a lie? 615 • Is this statement true? 619
• Challenges about lies 620 •
Observations
Have enough observations been recorded? 622 • Are all physical observables
known? 623 • Do observations take time? 625 • Is induction a problem in
physics? 625 •
The quest for precision and its implications
What are interactions? – No emergence 628 • What is existence? 628 •
Do things exist? 630 • Does the void exist? 631 • Is nature infinite? 632
• Is the universe a set? 633 • Does the universe exist? 634 • What is
creation? 635 • Is nature designed? 637 • What is a description? 638 •

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trodynamics 515 •
15. What is light?
The slowness of progress in physics 524 • Does light travel in a straight
line? 524 • The concentration of light 527 • Can one touch light? 528
• War, light and lies 530 • What is colour? 531 • What is the speed of
light? – Again 533 • 200 years too late: negative refraction indices 536 •

Signals and predictions 537 • How does the world look when riding on a
light beam? 537 • Does the aether exist? 537 •
Challenges and curiosities about light
How to prove you’re holy 538 • Do we see what exists? 539 • How does one
make pictures of the inside of the eye? 541 • How does one make holograms
and other 3-d images? 542 • Imaging 544 •
16. Charges are discrete – the limits of classical electrodynamics
How fast do charges move? 545 • Challenges and curiosities about charge
discreteness 546 •
17. Electromagnetic effects and challenges
Is lightning a discharge? – Electricity in the atmosphere 551 • Does gravity
make charges radiate? 553 • Research questions 554 • Levitation 555 •
Matter, levitation and electromagnetic effects 558 • Why can we see each
other? 565 • A summary of classical electrodynamics and of its limits 567
•
18. Classical physics in a nutshell – one and a half steps out of three
The future of planet Earth 569 • The essence of classical physics: the infinitely small implies the lack of surprises 571 • Why have we not yet reached
the top of the mountain? 572 •
Bibliography

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contents



Reason, purpose and explanation 639 • Unification and demarcation 640 •
Pigs, apes and the anthropic principle 641 • Does one need cause and effect
in explanations? 643 • Is consciousness required? 644 • Curiosity 644 •

Courage 647 •
Bibliography

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Second Part : Quantum Theory
What Is Matter? What Are Interactions?

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Chapter
VI Permutation of Particles
23. Are particles like gloves?
Why does indistinguishability appear in nature? 721 • Can particles be counted? 721 • What is permutation symmetry? 722 • Indistinguishability and
symmetry 722 • The behaviour of photons 724 • The energy dependence

of permutation symmetry 724 • Indistinguishability in quantum field theory 725 • How accurately is permutation symmetry verified? 726 • Copies,
clones and gloves 726 •
24. Rotations and statistics – visualizing spin

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Chapter
V Quanta of Light and Matter
19. Minimum action – quantum theory for poets and lawyers
Gedanken experiments and challenges 667 •
20. Light – the strange consequences of the quantum of action
What is colour? 668 • What is light? – Again 671 • The size of photons 672
• Are photons countable? – Squeezed light 672 • The position of
photons 675 • Are photons necessary? 676 • How can a wave be made up
of particles? 678 • Can light move faster than light? – Virtual photons 683 •
Indeterminacy of electric fields 684 • Curiosities and fun challenges about
photons 684 •
21. Motion of matter – beyond classical physics
Wine glasses and pencils 685 • Cool gas 686 • No rest 686 • Flows and
the quantization of matter 687 • Quantons 687 • The motion of quantons
– matter as waves 688 • Rotation and the lack of North Poles 690 • Silver,
Stern and Gerlach 692 • The language of quantum theory and its description of motion 693 • The evolution of operators 695 • The state – or wave
function – and its evolution 695 • Why are atoms not flat? Why do shapes
exist? 697 • Rest: spread and the quantum Zeno effect 697 • Tunnelling,
hills and limits on memory 698 • Spin and motion 699 • Relativistic wave
equations 700 • Maximum acceleration 701 • Curiosities and fun challenges about quantum theory 701 •
22. Colours and other interactions between light and matter

What are stars made of? 703 • What determines the colour of atoms? 704
• Relativistic hydrogen 706 • Relativistic wave equations – again 707 •
Antimatter 708 • Virtual particles and QED diagrams 709 • Compositeness 710 • Curiosities and fun challenges about colour 711 • The strength
of electromagnetism 712 •
Bibliography


contents
The belt trick 730 • The Pauli exclusion principle and the hardness of matter 732 • Integer spin 733 • Is spin a rotation about an axis? 734 •
Why is fencing with laser beams impossible? 735 • Rotation requires antiparticles 735 • Limits and open questions of quantum statistics 736 •
Bibliography

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Chapter VII Details of Quantum Theory and Electromagnetism
25. Superpositions and probabilities – quantum theory without ideology
Why are people either dead or alive?
Conclusions on decoherence, life and death 745 •
What is a system? What is an object?
Is quantum theory non-local? – A bit about Einstein-Podolsky-Rosen 747 •
Curiosities 749 •
What is all the fuss about measurements in quantum theory?
Hidden variables 754 •
Conclusions on probabilities and determinism
What is the difference between space and time? 758 • Are we good observers? 758 • What connects information theory, cryptology and quantum
theory? 759 • Does the universe have a wave function? And initial conditions? 760 •
26. Applied quantum mechanics – life, pleasure and the means to achieve them
Biology
Reproduction 762 • Quantum machines 763 • How do we move? – Molecular motors 763 • Curiosities and fun challenges about biology 766 •
The physics of pleasure
The nerves and the brain 771 • Clocks in quantum mechanics 771 • Do
clocks exist? 772 • Living clocks 774 • Metre sticks 775 • Why are predictions so difficult, especially of the future? 775 • Decay and the golden
rule 775 • Zeno and the present in quantum theory 777 • What is motion? 777 • Consciousness: a result of the quantum of action 778 • Why
can we observe motion? 779 • Curiosities and fun challenges about quantum
experience 779 •

Chemistry – from atoms to DNA
Ribonucleic acid and Deoxyribonucleic acid 782 • Chemical challenges and
curiosities 783 •
Materials science
Why does the floor not fall? 783 • Rocks and stones 784 • How can one
look through matter? 784 • What is necessary to make matter invisible? 785
• How does matter behave at lowest temperatures? 787 • Curiosities and
fun challenges about materials science 787 •
Quantum technology
Motion without friction – superconductivity and superfluidity 789 • Quantized conductivity 791 • The fractional quantum Hall effect 791 • Lasers
and other spin-one vector boson launchers 792 • Can two photons interfere? 794 • Can two electron beams interfere? 795 • Challenges and dreams
about quantum technology 796 •
27. Quantum electrodynamics – the origin of virtual reality
Ships, mirrors and the Casimir effect 796 • The Banach–Tarski paradox for
vacuum 798 • The Lamb shift 799 • The QED Lagrangian 799 • Interactions and virtual particles 799 • Vacuum energy 800 • Moving mirrors 800

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

• Photon hitting photons 800 • Is the vacuum a bath? 801 • Renormalization – why is an electron so light? 801 •
Curiosities and fun challenges of quantum electrodynamics
How can one move on perfect ice? – The ultimate physics test 805 •
Summary of quantum electrodynamics
Open questions in QED 808 •
28. Quantum mechanics with gravitation – the first approach
Corrections to the Schrödinger equation 810 • A rephrased large number hypothesis 810 • Is quantum gravity necessary? 811 • Limits to disorder 811

• Measuring acceleration with a thermometer: Fulling–Davies–Unruh radiation 812 •
Black holes aren’t black
Gamma ray bursts 815 • Material properties of black holes 817 • How do
black holes evaporate? 818 • The information paradox of black holes 818 •
More paradoxes 820 •
Quantum mechanics of gravitation
The gravitational Bohr atom 821 • Decoherence of space-time 821 • Do
gravitons exist? 822 • Space-time foam 822 • No particles 823 • No science fiction 823 • Not cheating any longer 824 •
Bibliography

Advanced Quantum Theory (Not yet Available)

Chapter
X Quantum Physics in a Nutshell
Quantum theory’s essence: the lack of the infinitely small

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Chapter VIII Inside the Nucleus
29. The structure of the nucleus – the densest clouds
A physical wonder: magnetic resonance imaging 836 • The size of nuclei 837
• Nuclei are composed 840 • Nuclei can move alone – cosmic rays 843 •
Nuclei decay 847 • Nuclei can form composites 849 • Nuclei have colours
and shapes 849 • Motion in the nuclear domain – four types of motion 851
• Nuclei react 851 • Bombs and nuclear reactors 852 • The Sun 853 •
Curiosities and fun challenges on radioactivity 855 •
30. The strong nuclear interaction and the birth of matter
Why do the stars shine? 857 • Where do our atoms come from? 860 • The
weak side of the strong interaction 860 • Bound motion, the particle zoo

and the quark model 861 • The mass, shape, and colour of protons 862
• Experimental consequences of the quark model 863 • The Lagrangian
of quantum chromodynamics 864 • The sizes and masses of quarks 866 •
Confinement and the future of the strong interaction 867 • Curiosities about
the strong interactions 867 •
31. The weak nuclear interaction and the handedness of nature
Curiosities about the weak interactions 869 • Mass, the Higgs boson and a
ten thousand million dollar lie 870 • Neutrinium and other curiosities about
the electroweak interaction 871 •
32. The standard model of elementary particle physics – as seen on television
Conclusion and open questions about the standard model 873 •
33. Grand unification – a simple dream
Experimental consequences 874 • The state of grand unification 875 •
Bibliography

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

Achievements in precision 880 • Physical results of quantum theory 882 •
Results of quantum field theory 884 • Is quantum theory magic? 885 • The
dangers of buying a can of beans 886 •
The essence and the limits of quantum theory
What is unexplained by quantum theory and general relativity? 887 • How
to delude oneself that one has reached the top of the Motion Mountain 890 •
What awaits us? 893 •
Bibliography

Intermezzo
Bacteria, Flies and Knots
Bumble-bees and other miniature flying systems 896 • Swimming 899 •
Falling cats and the theory of shape change 903 • Turning a sphere inside
out 903 • Knots, links and braids 904 • Knots in nature and on paper 907 •
Clouds 909 • Fluid space-time 910 • Solid space-time 910 • Swimming
in curved space 912 • Curiosities and fun challenges 913 • Outlook 914
•
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What Are Space, Time and Particles?

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XI General Relativity Versus Quantum Mechanics
The contradictions 918 •
34. Does matter differ from vacuum?
Planck scales 921 • Farewell to instants of time 923 • Farewell to points in
space 925 • Farewell to the space-time manifold 927 • Farewell to observables and measurements 930 • Can space-time be a lattice? – A glimpse of
quantum geometry 931 • Farewell to particles 932 • Farewell to mass 935
• Curiosities and fun challenges 938 • Farewell to the big bang 940 • The
baggage left behind 941 •
Some experimental predictions
Bibliography
35. Nature at large scales – is the universe something or nothing?
Cosmological scales 952 • Maximum time 953 • Does the universe have
a definite age? 954 • How precisely can ages be measured? 954 • Does
time exist? 955 • What is the error in the measurement of the age of the universe? 956 • Maximum length 959 • Is the universe really a big place? 960
• The boundary of space-time – is the sky a surface? 961 • Does the universe have initial conditions? 962 • Does the universe contain particles and
stars? 962 • Does the universe contain masses and objects? 963 • Do symmetries exist in nature? 965 • Does the universe have a boundary? 965 •
Is the universe a set? 966 • Curiosities and fun challenges 968 • Hilbert’s
sixth problem settled 968 • Does the universe make sense? 969 • A concept
without a set eliminates contradictions 970 • Extremal scales and open questions in physics 971 • Is extremal identity a principle of nature? 971 •
Bibliography
36. The physics of love – a summary of the first two and a half parts

Bibliography


contents



Extension and Unification (Not yet Available)

1001

1009

1016

1019

1025

1027
1030
1033
1035
1036
1037

1042
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XII

993

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985

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37. Maximum force and minimum distance: physics in limit statements
Fundamental limits to all observables
Special relativity in one statement 985 • Quantum theory in one statement 986 • General relativity in one statement 988 • Deducing general
relativity 989 • Deducing universal gravitation 992 • The size of physical
systems in general relativity 992 • A mechanical analogy for the maximum
force 992 •
Units and limit values for all physical observables
Limits to space and time 994 • Mass and energy limits 995 • Virtual
particles – a new definition 996 • Limits in thermodynamics 996 • Electromagnetic limits and units 997 • Vacuum and mass-energy – two sides of
the same coin 998 • Paradoxes and curiosities about Planck limits 999 •
Upper and lower limits to observables
Size and energy dependence 1001 • Angular momentum, action and
speed 1002 • Force, power and luminosity 1003 • Acceleration 1003
• Momentum 1004 • Lifetime, distance and curvature 1004 • Mass
change 1004 • Mass and density 1005 • The strange charm of the entropy
bound 1005 • Temperature 1007 • Electromagnetic observables 1007 •

Paradoxes and challenges around the limits 1008 •
Limits to measurement precision and their challenge to thought
Measurement precision and the existence of sets 1009 • Why are observers
needed? 1011 • A solution to Hilbert’s sixth problem 1011 • Outlook 1012
• Bibliography 1012 •
38. The shape of points – extension in nature
Introduction: vacuum and particles 1017 • How else can we show that matter and vacuum cannot be distinguished? 1018 •
Argument 1: The size and shape of elementary particles
Do boxes exist? 1020 • Can the Greeks help? – The limits of knifes 1020 •
Are cross-sections finite? 1021 • Can one take a photograph of a point? 1021
• What is the shape of an electron? 1023 • Is the shape of an electron
fixed? 1024 •
Argument 2: The shape of points in vacuum
Measuring the void 1026 • What is the maximum number of particles that
fits inside a piece of vacuum? 1027 •
Argument 3: The large, the small and their connection
Small is large? 1028 • Unification and total symmetry 1028 •
Argument 4: Does nature have parts?
Does the universe contain anything? 1032 • An amoeba 1032 •
Argument 5: The entropy of black holes
Argument 6: Exchanging space points or particles at Planck scales
Argument 7: The meaning of spin
Present research
Conceptual checks of extension 1038 • Experimental falsification of models based on extended entities 1039 • Possibilities for confirmation of extension models 1039 • Curiosities and fun challenges 1040 • An intermediate
status report 1041 • Sexual preferences in physics 1041 • A physical aphorism 1041 •
Bibliography


contents



Chapter XIII

The Top of the Mountain (Not yet Available)
Fourth Part :

1047

Appendices

Appendix
A Notation and Conventions
1049
The symbols used in the text 1049 • The Latin alphabet 1051 • The Greek
alphabet 1052 • The Hebrew alphabet and other scripts 1054 • Digits and
numbers 1055 • Calendars 1056 • Abbreviations and eponyms or concepts? 1057 • Bibliography 1058 •
Appendix
B Units, Measurements and Constants
1060
Planck’s natural units 1063 • Other unit systems 1064 • Curiosities 1066
• Precision and accuracy of measurements 1069 • Basic physical constants 1069 • Useful numbers 1074 • Bibliography 1075 •
Appendix
C Particle Properties
Bibliography 1096 •

1078

Appendix
D Numbers and Spaces

Numbers as mathematical structures
Complex numbers 1099 • Quaternions 1101 • Octonions 1105 • Other
types of numbers 1107 • Grassmann numbers 1107 •
Vector spaces
Algebras
Lie algebras 1112 • Classification of Lie algebras 1113 • Lie superalgebras 1114 • The Virasoro algebra 1115 • Kac–Moody algebras 1115 •
Topology – what shapes exist?
Topological spaces 1116 • Manifolds 1117 • Holes, Homotopy and Homology 1119 •
Types and classification of groups
Lie groups 1121 • Connectedness 1122 • Compactness 1122 •
Mathematical curiosities and fun challenges
Bibliography

1098
1098

Appendix

E

Information S ources on Motion

1128

Appendix

F

Challenge Hints & S olutions

1134

1107
1109

1126
1126

1178

Appendix

H

List of Tables

1190

Appendix

I

Name Index

1193

Appendix

J

Subject Index

1216

Copyright © Christoph Schiller November 1997–September 2005

Appendix
G List of Illustrations
Picture credits 1188 •

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1116


Preface
Primum movere, deinde docere.

Motion Mountain
www.motionmountain.net
Copyright © Christoph Schiller November 1997–September 2005

The 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 dying out with the end of school education, were
allowed to thrive in an environment without bounds, reaching from the atoms to the

stars? Probably each adolescent would know more about nature than most senior physics
teachers today. This text tries to provide this possibility to the reader. It acts as a guide
in such an exploration, free of all limitations, of the world of motion. The 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 and their understanding, while reducing the mathematics to the necessary minimum. Learning the concepts of physics is
given precedence over using formulae in calculations. All topics are within the reach of
an undergraduate. For the main domains of physics, the simplest summaries possible are
presented. It is shown that physics describes motion in three steps. First there is everyday physics, or classical continuum physics. In the second step each domain of physics is
based on an inequality for the main observable. Indeed, statistical thermodynamics limits
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 electrodynamics
limits change of charge by ∆q e. By basing these domains of physics on limit principles,
a simple, rapid and intuitive introduction is achieved. It is shown that the equations of
each domain follow from the corresponding limit. The third step of physics is the unification of all these limits in a single description of motion. This way to learn physics should
reward the curiosity of every reader – whether student or professional.
In order to be up-to-date, the text includes quantum gravity, string theory and M theory. But also the standard topics – mechanics, electricity, light, quantum theory, particle
physics and general relativity – are greatly enriched by many gems and research results
that are found scattered throughout the scientific literature.
In order to be vivid, a text wants to challenge, to question and to dare. This text tries
to startle the reader as much as possible. Reading a book on general physics should be
similar to a visit to a magic show. We watch, we are astonished, we do not believe our eyes,
we think and finally – maybe – we understand the trick. When we look at nature, we often
have the same experience. The text tries to intensify this by following a simple rule: on
each page, there is at least one surprise or one provocation to think about. Numerous
challenges are proposed. All are as interesting as possible. Hints or answers are given in
the appendix.
A surprise has the strongest effect whenever it questions everyday observations. In
this text most surprises are taken from daily life, in particular, from the experiences one
makes when climbing a mountain. Observations about trees, stones, the Moon, the sky

and people are used wherever possible; complex laboratory experiments are mentioned
only where necessary. All surprises are organized to lead in a natural way to the most
extreme conclusion of all, namely that continuous space and time do not exist. These
concepts, useful as they may be in everyday life, are only approximations that are not


preface



valid in the general case. Time and space turn out to be mental crutches that hinder the
complete exploration of the world.
Enjoying curiosity to full intensity and achieving freedom of thought leads to a strong
and dependable character. Indeed, exploring a limit requires courage. Courage is also
needed to drop space and time as tools for the description of the world. Changing thinking habits produces fear; but nothing is more intense and satisfying than overcoming
one’s own 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.
Eindhoven and other places,  September 

In exchange for getting this text for free, please send a short email on the following issues:
What was unclear?
What should be improved?
What did you miss?

C. Schiller


Copyright © Christoph Schiller November 1997–September 2005

Material on the specific points listed on the />html web page is most welcome of all. Thank you in advance for your input, also in the

name of all other readers. For a particularly useful contribution you will be mentioned
in the acknowledgements, receive a reward, or both. But above all, enjoy the reading.

www.motionmountain.net

Challenge 1 ny

Motion Mountain

A request


Acknowledgements

Motion Mountain
www.motionmountain.net
Copyright © Christoph Schiller November 1997–September 2005

Page 1188

Many people who have kept their gift 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. Also 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 provided valuable advice and encouragement.
The project and the collection of material owes to many. 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 Thaler, 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, Norbert Dragon, 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,
Jeff Aronson, Roggie Boone, Lawrence Tuppen, Quentin David Jones, Arnaldo Uguzzoni,
Frans van Nieuwpoort, Alan Mahoney, Britta Schiller, Petr Danecek, Ingo Thies, Vitaliy Solomatin, Carl Offner, 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, Thomas 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 Dahlmann and all those who wanted to remain
unnamed.
The software tools were refined with the extended 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 that shape this text were made available by the copyright holders.
A warm thank you to all of them; they are mentioned in the dedicated credit section in
Appendix E. In particular, Luca Gastaldi and Antonio Martos produced specific images
for this text. Both the book and the website owe most to the suggestions and support of
my wife Britta.


1. An appetizer
Die Lösung des Rätsels des Lebens in Raum und Zeit

liegt außerhalb von Raum und Zeit.*
Ludwig Wittgenstein, Tractatus, .

hat is the most daring, amazing and exciting journey we can make in a lifetime?
hat is the most interesting place to visit? We can travel to places that are as remote
as possible, like explorers or cosmonauts, we can look into places as far away as we can
imagine, like astronomers, we can visit the past, like historians or archaeologists, or we
can delve as deeply as possible into the human soul, like artists or psychologists. All these
voyages lead either to other places or to other times (or nowadays, to other servers on the
internet). However, we can do better.
The most daring trip is not the one leading to the most inaccessible place, but the
trip 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 all restrictions, but also of any security
of thought. There, discoveries are still to be made and adventures to be fought. Almost
nobody has ever been there; humanity has so far taken  years for the trip and still
has not completely achieved it.
To venture into this part of nature, we need to be curious about the essence of travel
itself, and in particular about its details and its limits. The essence of any travel is motion.
By exploring motion we will be lead to the most fascinating adventures in the universe.
The 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 apparent,
this text tells the story of the quest as the ascent of a mountain. Every step towards the top
corresponds to a step towards higher precision in the description of motion. In addition,
each step will increase the pleasure and the encountered delights. At the top of the mountain we shall arrive in the domain we were looking for, where ‘space’ and ‘time’ are words
that have lost all meaning and where the sight of the world’s beauty is overwhelming and
unforgettable.
Thinking without time or space is difficult but fascinating. In order to get a taste of
the issues involved, try to answer the following questions without ever referring to either
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 ‘left’ mean, or what a mirror is?
Have you ever tried to make a telephone appointment with a friend without using any
time or position term, such as clock, hour, place, where, when, at, near, before, after, near,
upon, under, above, below?
Can you describe the fall of a stone without using space or time?

W

www.motionmountain.net
Copyright © Christoph Schiller November 1997–September 2005

* The solution of the riddle of life in space and time lies outside space and time.
** Solution to challenges are either given on page 1134 or later on in the text. Challenges are classified as
research level (r), difficult (d), normal student level (n) and easy (e). Challenges with no solution yet are
marked (ny).

Motion Mountain

Challenge 2 n




. an appetizer

Motion Mountain
www.motionmountain.net

Copyright © Christoph Schiller November 1997–September 2005

Do you know of any observation at all that you can describe without concepts from the
domains ‘space’, ‘time’ or ‘object’?
Can you explain what time is? And what clocks are?
Can you imagine a finite history of the universe, but without a ‘first 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?
This book tells 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 shoestrings remain tied? They do so because space has three dimensions. Why not another number? Finding the answer has required the combined effort of
researchers over thousands of years. The answer was only found by studying motion up
to its smallest details and by exploring each of its limits.
Why do the colours of objects differ? 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 fire is not? Why is
the universe so big? Why can birds fly but men can’t? Why is lightning not straight? Why
are atoms neither square, nor the size of cherries? These questions seem to have little in
common; but that impression is wrong. They are all about motion – about its details and
its limitations. Indeed, they all appear and are answered in what follows. 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 find that there are more
cells in the brain than stars in the galaxy; people almost literally 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 find that in contrast to personal experience, motion never stops. We will find out why the floor cannot fall. We will understand
why the speed of computers cannot be made arbitrary high. 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
literally does not exist. We will find out 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 not different from what it is.

Understanding motion, together with all its details and all its limits, implies asking
and answering three specific questions.
How do things move? The usual answer states that motion is an object changing position over time. This seemingly boring statement encompasses general relativity, one of
the most amazing descriptions of nature ever imagined. We find that space is warped, that
light does not usually travel in a straight line and that time is not the same for everybody.
We discover that there is a maximum force of gravity and that, nevertheless, gravity is
not an interaction, but rather the change of time with position. We understand that the
blackness of the sky at night proves that the universe has a finite age. We also discover that
there is a smallest entropy in nature, which prevents us from knowing everything about
a physical system. In addition, we discover the smallest electrical charge. These and other
strange properties of motion are summarized in the first part of this text, whose topic is
classical physics. It directly leads to the next question.
What are things? Things are composites of a few types of particles. In addition, all
interactions and forces – those of the muscles, those that make the Sun burn, those


. an appetizer

Page 886

that make the Earth turn, those that determine the differences between attraction, repulsion, indifference, friction, creation and annihilation – are made of particles as well.
The growth of trees, the colours of the sky, the burning of fire, the warmth of a human
body, the waves of the sea and the mood changes of people are all variations of motion of
particles. This story is told in more detail in the second part of the text, that on quantum
mechanics. Here we will learn that there is a smallest change in nature. This minimum
value forces everything to keep constantly changing. In particular, we will learn that it is
impossible to completely fill a glass of wine, that eternal life is impossible, and that light
can be transformed into matter. If this is still boring, read about the substantial dangers
you incur when buying a can of beans.
The first two parts of this text can be summarized with the help of a few limit principles:

statistical thermodynamics limits entropy:
special relativity limits speed:
general relativity limits force:
quantum theory limits action:
quantum electrodynamics limits charge:

S
v
F
L
∆q

k 
c
c  G
ħ 
e.

(1)

www.motionmountain.net
Copyright © Christoph Schiller November 1997–September 2005

We will see that each of the constants of nature k , c, c  G, ħ  and e that appears on
the right side is also a limit value. We will discover that the equations of the corresponding
domain of physics follow from this limit property. After these results, the path is prepared
for the final theme of the mountain climb.
What are particles, position and time? The recent results of an age-long search are
making it possible to start answering this question. One just needs to find a description
which explains all limit principles at the same time. This third part is not complete yet, because the final research results are not yet available. Nevertheless, the intermediate results

are challenging:
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, in contrast
to what is often said.
It seems that position, time and every particle are aspects of a complex, extended entity
that is incessantly varying in shape.
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.
Research is presently discovering that motion is an intrinsic property of matter and
radiation and that, as soon as we introduce these two concepts in the description of nature,
motion appears automatically. On the other hand, 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.
Research is also presently uncovering that the final description of nature, with complete
precision, does not use any form of infinity. We find, step by step, that all infinities appearing in the human description of nature, both the infinitely large as well as the infinitely

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