Modern Special Relativity: A Student's Guide with Discussions and Examples

ISBN-13: 9783030543518 / Angielski / Miękka / 2022 / 466 str.

Johann Rafelski

Spis treści

I Space-Time, Light and the aether 1

1 What is (Special) Relativity? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1 Principle of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Time, a 4th coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Path toward Lorentz coordinate transformations . . . . . . . . . . . . . . 10

1.4 Highlights: How did relativity happen? . . . . . . . . . . . . . . . . . . . 13

2 Light and the aether . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 Measuring space and time: SI unit system . . . . . . . . . . . . . . . . . . 15

2.2 Speed of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Essay: aether and Special Relativity . . . . . . . . . . . . . . . . . . . . . 25

3 Material Bodies in SR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1 The Michelson-Morley Experiment . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Body contraction and time dilation . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Is the Lorentz-FitzGerald body contraction measurable? . . . . . . . . . . 38

3.4 Experiments require understanding of body contraction . . . . . . . . . . 40

3.5 Resolving misunderstandings of SR . . . . . . . . . . . . . . . . . . . . . . 42

II Time Dilation, and Lorentz-Fitzgerald Body Contraction 47

4 Time Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1 Proper time of a traveler . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Relativistic light-clock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3 Talking about time (dilation) . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 The Lorentz-FitzGerald Body Contraction . . . . . . . . . . . . . . . . . . . . . . 61

5.1 Light-clock moving parallel to light path . . . . . . . . . . . . . . . . . . . 61

5.2 Body contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.3 Arbitrary orientation of the light clock . . . . . . . . . . . . . . . . . . . . 66

III The Lorentz Transformation 73

6 Relativistic Coordinate Transformation . . . . . . . . . . . . . . . . . . . . . . . 75

6.1 Derivation of the Lorentz coordinate transformation . . . . . . . . . . . . 75

6.2 Explicit form of the Lorentz transformation . . . . . . . . . . . . . . . . . 79

6.3 The nonrelativistic Galilean limit . . . . . . . . . . . . . . . . . . . . . . . 84

6.4 The inverse Lorentz coordinate transformation . . . . . . . . . . . . . . . 85

7 Some Consequences of Lorentz Transformation . . . . . . . . . . . . . . . . . . . 88

7.1 Invariance of proper time . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.2 Relativistic addition of velocities . . . . . . . . . . . . . . . . . . . . . . . 92

7.3 Two Lorentz coordinate transformations in sequence . . . . . . . . . . . . 99

7.4 Rapidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

IV Measurement 111

8 Time Measurement and Lorentz Transformation . . . . . . . . . . . . . . . . . . 113

8.1 Graphic representation of Lorentz Transformation . . . . . . . . . . . . . 113

8.2 Time dilation and simultaneity . . . . . . . . . . . . . . . . . . . . . . . . 114

9 Methods of Measuring Spatial Separation . . . . . . . . . . . . . . . . . . . . . . 119

9.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

9.2 Determination of spatial separation . . . . . . . . . . . . . . . . . . . . . . 120

9.3 Light illumination emitted in the rest-frame of the observer . . . . . . . . 123

9.4 Train in the tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

10 The Bell Rockets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

10.1 Rockets connected by a thread . . . . . . . . . . . . . . . . . . . . . . . . 130

10.2 The thread breaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

10.3 Lorentz-FitzGerald body contraction measured . . . . . . . . . . . . . . . 133

V Space, Time, Doppler Shift 139

11 The Light-Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

11.1 The future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

11.2 The past . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

12 Space-time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

12.1 Timelike and spacelike event separation . . . . . . . . . . . . . . . . . . . 149

12.2 Time dilation revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

12.3 Essay: Quantum entanglement and causality . . . . . . . . . . . . . . . . 156

13 SR-Doppler Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

13.1 Introducing the nonrelativistic Doppler shift . . . . . . . . . . . . . . . . . 162

13.2 Misunderstanding of the relativistic Doppler eect . . . . . . . . . . . . . 164

13.3 SR-Aberration of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

13.4 SR-Doppler shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

VI Mass, Energy, Momentum 177

14 Mass and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

14.1 Energy of a body at rest . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

14.2 Relativistic energy of a moving body . . . . . . . . . . . . . . . . . . . . . 182

14.3 Mass of a body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

15 Particle Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

15.1 Relation between energy and momentum . . . . . . . . . . . . . . . . . . 186

15.2 Particle rapidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

16 Generalized Mass-Energy Equivalence . . . . . . . . . . . . . . . . . . . . . . . . 201

16.1 Where does energy come from? . . . . . . . . . . . . . . . . . . . . . . . . 201

16.2 Mass equivalence for kinetic energy in a gas . . . . . . . . . . . . . . . . . 202

16.3 Potential energy mass equivalence . . . . . . . . . . . . . . . . . . . . . . 203

16.4 Atomic mass defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

16.5 Rotational energy mass equivalence . . . . . . . . . . . . . . . . . . . . . 206

16.6 Chemical energy mass defect . . . . . . . . . . . . . . . . . . . . . . . . . 207

16.7 Nuclear mass defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

16.8 Origin of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

VII Collisions, Decays 213

17 Preferred Frame of Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

17.1 The center of momentum frame (CM-Frame) . . . . . . . . . . . . . . . . 215

17.2 The Lorentz transformation to the CM-frame . . . . . . . . . . . . . . . . 217

17.3 Particle decay in the CM-frame . . . . . 228 . . . . . . . . . . . . . . . . . . . 220

17.4 Decay energy balance in CM-frame . . . . . . . . . . . . . . . . . . . . . . 222

17.5 Decay of a body in flight . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

18 Particle Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

18.1 Elastic two body reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

18.2 Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

18.3 Elastic bounce from a moving wall . . . . . . . . . . . . . . . . . . . . . . 233

18.4 Inelastic two-body reaction threshold . . . . . . . . . . . . . . . . . . . . . 237

18.5 Energy available in a two body collision . . . . . . . . . . . . . . . . . . . 241

18.6 Inelastic collision and particle production . . . . . . . . . . . . . . . . . . 247

VIII SR-Tests & Open Questions 251

19 Tests of Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

19.1 Overview: Testing SR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

19.2 The Michelson-Morley experiment today . . . . . . . . . . . . . . . . . . . 254

19.3 How constant is the speed of light? . . . . . . . . . . . . . . . . . . . . . . 255

19.4 Tests of SR material body properties . . . . . . . . . . . . . . . . . . . . . 256

19.5 Doppler effect and tests of the Lorentz coordinate transformation . . . . . 258

19.6 Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

20 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

20.1 Accelerated motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

20.2 Can there be acceleration in SR? . . . . . . . . . . . . . . . . . . . . . . . 265

20.3 Evidence for the existence of acceleration . . . . . . . . . . . . . . . . . . 266

20.4 Small and large acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . 268

20.5 Achieving strong acceleration . . . . . . . . . . . . . . . . . . . . . . . . . 269

IX Lorentz Force and Particle Motion 275

21 Acceleration and Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

21.1 Newton's second Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

21.2 Motion in magnetic and electric elds . . . . . . . . . . . . . . . . . . . . 280

21.3 Variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

21.4 Electron Coulomb orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

22 Electrons Riding a Plane Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

22.1 Fields and potentials for a plane wave . . . . . . . . . . . . . . . . . . . . 298

22.2 Role of conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

22.3 Surng the plane wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

X Space Travel 311

23 Spaceship Travel in the Milky Way . . . . . . . . . . . . . . . . . . . . . . . . . . 313

23.1 Space travel with constant acceleration . . . . . . . . . . . . . . . . . . . 313

23.2 The eect of time dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

23.3 How far can we travel? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

23.4 Variable acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

24 Relativistic Rocket equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

24.1 Nonrelativistic rocket equation . . . . . . . . . . . . . . . . . . . . . . . . 322

24.2 Relativistic rocket equation . . . . . . . . . . . . . . . . . . . . . . . . . . 323

24.3 Energy of relativistic rocket . . . . . . . . . . . . . . . . . . . . . . . . . . 325

O autorze

Johann Rafelski is a theoretical physicist working at The University of Arizona in Tucson, USA. Born in Kraków, Poland in 1950, he received his Ph.D. with Walter Greiner at Johann Wolfgang Goethe University, Frankfurt, Germany in 1973. In 1977 Rafelski arrived at CERN-Geneva, where with Rolf Hagedorn he developed the search for quark-gluon plasma in relativistic heavy ion collision as a novel research domain. He invented and developed the strangeness quark flavor as the signature of quark-gluon plasma, advancing the discovery of this new phase of primordial matter. Professor Rafelski teaches Relativity, Quantum, Particle and Nuclear Physics; in addition to CERN and Arizona, he also has held professional appointments at the University of Pennsylvania in Philadelphia, Argonne National Laboratory in Chicago, the University of Frankfurt, the University of Cape Town, the University of Paris-Jussieu, and the Ecole Polytechnique. He has been a DFG Excellence Initiative Professor at Ludwig-Maximillian University Munich. In collaboration with researchers from the Ecole Polytechnique in Paris and ELI-Beamlines in Prague he is using ultra-intense lasers in nuclear and fundamental physics.

Prof. Rafelski is the editor of the open-access book: Melting Hadrons, Boiling Quarks – From Hagedorn Temperature to Ultra-Relativistic Heavy-Ion Collisions at CERN - With a Tribute to Rolf Hagedorn (Springer, 2016) and he has authored the book: Relativity Matters – From Einstein's EMC2 to Laser Particle Acceleration and Quark-Gluon Plasma (Springer, 2017).

Blurb

This book presents Special Relativity in a language accessible to students while avoiding the burdens of geometry, tensor calculus, space-time symmetries, and the introduction of four vectors. The search for clarity in the fundamental questions about Relativity, the discussion of historical developments before and after 1905, the strong connection to current research topics, many solved examples and problems, and illustrations of the material in colloquial discussions are the most significant and original assets of this book. Importantly for first-time students, Special Relativity is presented such that nothing needs to be called paradoxical or apparent; everything is explained.

The content of this volume develops and builds on the book Relativity Matters (Springer, 2017). However, this presentation of Special Relativity does not require 4-vector tools. The relevant material has been extended and reformulated, with additional examples and clarifications.

This introduction of Special Relativity offers conceptual insights reaching well beyond the usual method of teaching relativity. It considers relevant developments after the discovery of General Relativity (which itself is not presented), and advances the reader into contemporary research fields. This presentation of Special Relativity is connected to present day research topics in particle, nuclear, and high intensity pulsed laser physics and is complemented by the current cosmological perspective. The conceptual reach of Special Relativity today extends significantly further compared even to a few decades ago.

As the book progresses, the qualitative and historical introduction turns into a textbook-style presentation with many detailed results derived in an explicit manner. The reader reaching the end of this text needs knowledge of classical mechanics, a good command of elementary algebra, basic knowledge of calculus, and introductory know-how of electromagnetism.