Moses Fayngold

Special Relativity and How it Works

WILEYVCH WI LEY-VCH Verlag GmbH & Co. KGaA

IX

Contents

Preface XIII 1

Introduction 1

1.1 1.2 1.3

The Meaning of Relativity 1 Weirdness of Light 10 A Steamer in the Stream 12 Problems 15

2

Light and Relativity 17

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

The Michelson Experiment 17 The Speed of Light and the Principle of Relativity 22 "Obvious" Does Not Always Mean "True!" 24 Light Determines Simultaneity 25 Light, Times, and Distances 29 The Lorentz Transformations 32 The Relativity of Simultaneity 38 A Proper Length and a Proper Time 41 Problems 43

3

3.1 3.2 3.3 3.4 3.4.1 3.4.2

47 The Addition of Collinear Velocities 47 The Addition of Arbitrarily Directed Velocities 50 The Velocities' Play 53 Observable Manifestations 56 The Fizeau Experiment 56 Aberration of Light 60 Problems 62

4

Space-Time

4.1 4.2

The Velocities' Play

65 Minkowski's World 65 Rotations in Space-Time (Representation in Eudidean Plane) 73

Special Relativity and How it Works. Moses Fayngold Copyright © 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40607-4

X! Contents

4.3 4.4 4.5 4.6 4.7

Lorentz Transformations as Rotations 81 What is Horizontal and What is Vertical? 87 General Lorentz Transformations 99 The Path Integrals 103 4-Vectors and Tensors 106 Problems 120

5

Relativity at Work 1: Mechanics 125

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

4-Velocity 125 4-Acceleration 130 4-Momentum 132 Relativistic Angular Momentum 137 Force and Motion 141 Mass and Energy 151 Mechanics From the Least Action 170 Relativistic Motion in Coulomb's Field 176 Problems 185

6

Relativity at Work 11: Electromagnetism and Optics 191

6.1

Electric Field and Something Else (The Origin of Magnetic Field) 191 Magnetic Field and Something Else ... 205 Is Electric Charge Invariant? (The Subtleties of "Neutrality") 209 Maxwell's Equations 218 Lenz's Law 225 4-Potential, Gauge Transformations, and Gauge Invariance 228 Electrodynamics and The Principle of the Least Action 233 Electrodynamics in Tensor Notations 239 Running Waves 244 The Relativistic Doppler Effect I 247 The Origin of Light 248 Aberration of Light - 2 252 Why Do Droplets Sparkle? Or Can We Trap the Light in a Droplet? 256 Problems 263

6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13

7

7.1 7.2 7.3 7.4 7.5

Relativistic Paradoxes 267

Seeing and Observing (on the Appearance of Fast-Moving Objects) 267 Can We See the Relativistic Length Contraction? 273 The Lorentz Contraction Paradox 280 Predicaments of Relativistic Train 283 The Dynamics of Relativistic Length Contraction 298

I

Contents XI

7.6 7.7 7.8 7.9

The Conveyor Belt Paradox 317 What is Rigid in Relativity? (What Constitutes Deformation?) 323 The Twin Paradox 334 The Three Friends' Paradox 344 Problems 348

8

Miracles of a Spinning World 351

8.1

8.2 8.3 8.4 8.5

The Ehrenfest Paradox 351 Circumnavigations with the Atomic Clocks 365 Surprises of the Rotland 375 Photons' Races in a Centrifuge 376 The Thomas Precession 387 Problems 394

9

Theory of Relativity is the Theory of Absoluteness

9.1 9.2 9.3 9.4 9.5

What Is Relative and What Is Absolute 397 The Speed of Light and the Invariant Speed 398 Space-Time Intervals 403 Relativistic Doppler Effect II 407 Relative Is Real 413 Problems 422

10

Relativity at Work III: Quantum Mechanics 425

10.1

Basic Ideas of Quantum Mechanics 425 Relativistic QM Indeterminacy 454 Relativistic Wave Equations 461 The Klein—Gordon Equation 461 Dirac's Equation 469 Relativistic Doppler Effect III: A Child of Relativity and Quantum Mechanics 477 Field—Particle Interactions: The Compton Effect 482 Problems 493

10.2 10.3 10.3.1 10.3.2 10.4 10.5

397

11

Relativity and Causality 497

11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9

Space-Time and Causality 497 Tachyons and Tardyons 500 The Tolman Paradox 509 Tachyons and Causality 516 Wave Packet Propagation and Position Measurements 523 Superluminal Quantum Tunneling 527 Wave Front Propagation 537 What Constitutes a Signal? 541 Direct Signaling Into the Past 550

XIII Contents

11.11

The Mystery of Quantum Telecommunication 558 The "No Cloning" Theorem 564 Problems 569

12

Applied Relativity 571

12.1 12.2

Relativistic Jet Propulsion: A "Star Trek" Cruiser 571 Principles of Design and Functioning of Particle Accelerators 580 Problems 593

13

A Bit of General Relativity 595

13.1 13.2

Basic Ideas of GR 595 Kinematics of GR 603

11.10

Appendix A State Function and the Continuity Equation 615 Appendix B Representation of Observables by Operators 617 Appendix C The Pauli Matrices 619 Appendix D Dimensionality of Dirac's Matrices 621 Appendix E Optical Barriers for a Photon 623 Appendix F Cause and Effect 627 Appendix G Permittivity and Refractive Index References 643 Index 647

637