Minkowski Spacetime: A Hundred Years Later (eBook)

Preface 6
References 9
Contributors 12
Raum und Zeit/Space and Time 15
Part I The Impact of Minkowski Spacetime on the Twentieth Century Physics from a Historical Perspective 44
1 Hermann Minkowski, Relativity and the Axiomatic Approach to Physics 45
1.1 Introduction 45
1.2 The Principle of Relativity 49
1.3 The Basic Equations of Electromagnetic Processes in Moving Bodies 52
1.3.1 Three Meanings of ``Relativity'' 53
1.3.2 Axioms of Electrodynamics 55
1.3.3 Relativity and Mechanics 57
1.3.4 Relativity and Gravitation 59
1.4 Space and Time 64
1.4.1 Groups of Transformations 66
1.4.2 Empirical Considerations 68
1.4.3 Relativity and Existing Physical Theories 70
1.5 Max Born, Relativity, and the Theories of the Electron 72
1.6 Summary and Concluding Remarks 76
References 81
2 Minkowski's Modern World 84
2.1 The Geometry of Physical Space Circa 1907 85
2.2 World-Geometry (1907--1908) 89
2.3 The Reception of ``Raum und Zeit'' 97
2.4 Minkowski's Modern World 100
References 101
Part II Implications of Minkowski Spacetime for Theoretical Physics 103
3 Hermann Minkowski and Special Relativity 104
3.1 Introduction 104
3.2 Newtonian Mechanics 105
3.3 Special Relativity 107
3.4 Minkowski and Special Relativity Geometry and Kinematics
3.5 Minkowski and Special Relativity Electromagnetic Theory 1
3.6 Minkowski and Special Relativity Electromagnetic Theory 2
3.7 Conclusions 119
References 121
4 The Rich Structure of Minkowski Space 122
4.1 General Introduction 122
4.2 Minkowski Space and Its Partial Automorphisms 124
4.2.1 Outline of General Strategy 124
4.2.2 Definition of Minkowski Space and Poincaré Group 125
4.2.3 From Metric to Affine Structures 128
4.2.4 From Causal to Affine Structures 130
4.2.5 The Impact of The Law of Inertia 132
4.2.6 The Impact of Relativity 134
4.2.7 Local Versions 140
4.3 Selected Structures in Minkowski Space 143
4.3.1 Simultaneity 143
4.3.2 The Lattices of Causally and Chronologically Complete Sets 148
4.3.3 Rigid Motion 154
Appendix 1: Sets and Group Actions 164
Appendix 2: Affine Spaces 165
Appendix 3: Affine Maps 166
Appendix 4: Affine Frames, Active and Passive Transformations 168
References 170
5 Minkowski Space-Time and Quantum Mechanics 172
5.1 Time, Space, and Quantum Mechanics 172
5.2 Heisenberg Representation 174
5.3 Relation Between Quantum Field Theories and Quantum Mechanics 175
5.3.1 Particles 176
5.4 Locality 179
5.4.1 Leggett's Extention of Bell's Theorem for Non-Linear Classical System 183
5.5 Conclusion 187
References 187
6 Modern Space-Time and Undecidability 188
6.1 Introduction 188
6.2 Quantum Mechanics with Real Clocks 190
6.3 Fundamental Limits to Realistic Clocks 192
6.4 Possible Experimental Implications 193
6.5 Conceptual Implications 194
6.6 Undecidability and the Laws of Physics 198
6.7 Discussion 199
References 199
7 Quantum Space-Times 201
7.1 Introduction 201
7.2 Quantum Nature of the Big Bang 204
7.2.1 Issue of the Beginning and the End 204
7.2.2 Some Key Questions 205
7.2.3 FRW Models and The WDW Theory 207
7.2.4 Loop Quantum Cosmology: New Quantum Mechanics 211
7.2.5 LQC: Dynamics 213
7.3 Black Holes 217
7.3.1 Horizons 218
7.3.2 Hawking Radiation and Information Loss 221
7.3.3 CGHS Black Holes 223
7.3.4 Quantum Geometry 226
7.4 Discussion 229
References 231
8 Space-Time Extensions in Quantum Gravity 235
8.1 Introduction 235
8.2 Space-Time in Classical Cosmology 237
8.3 Atomic Quantum Space-Time 237
8.3.1 Loop Quantum Cosmology 238
8.3.2 Harmonic Cosmology 239
8.3.2.1 Effective Friedmann Equation 243
8.3.2.2 Cosmic Forgetfulness 244
8.3.3 Interactions: How Quantum Is The Bounce? 246
8.3.4 Black Holes 249
8.4 Effective Equations and Lorentz Invariance 252
8.5 Conclusions 256
References 257
Part III Conceptual and Philosophical Issues of Minkowski Spacetime 261
9 The Adolescence of Relativity: Einstein, Minkowski, and the Philosophy of Space and Time 262
9.1 Introduction 263
9.2 Einstein and the Definition of Space and Time 265
9.3 Minkowski's Analysis of Space and Time 272
9.4 Intermezzo: Space-Time Coordinates as Physical Properties 275
9.5 Simultaneity, Symmetries and Conventionality 278
References 282
10 Hermann Minkowski: From Geometry of Numbers to Physical Geometry 283
10.1 Introduction 283
10.2 Geometry of Numbers 285
10.3 Spacetime Diagrams 287
10.4 Physical Axiomatics 289
10.5 Conclusion 291
References 293
11 The Mystical Formula and The Mystery of Khronos 294
11.1 The Mystical Formula 294
11.2 The Mystery of Kronos 299
11.3 Arrows of Time 306
11.4 Open Questions 309
11.4.1 Are The Arrows of Time Correlated ? 309
11.4.2 Time in Quantum Gravity 311
11.4.3 Closed Time-Like Curves and Time Travel 313
11.4.4 A Cyclic Time ? 314
11.5 Conclusions and Outlook 315
References 316
12 Physical Laws and Worldlines in Minkowski Spacetime 319
12.1 Introduction 319
12.2 Minkowski Spacetime and Reality 321
12.3 Newton's Three Laws and Worldlines 328
12.4 Spacetime and Gravity 331
12.5 Spacetime and Quantum Physics 336
12.6 Conclusions 338
References 339
13 Time as an Illusion 341
13.1 Introduction 341
13.2 Physics and the Flow of Time 343
13.3 Time as a Subjective Ordering Device 345
13.4 Mathematics and Reality 348
13.5 Conclusion 349
References 351
14 Consequences of Minkowski's Unification of Space and Time for a Philosophy of Nature 352
14.1 Philosophy of Nature 352
14.2 What Is Space? 354
14.3 Atom, Monad and Aporon 355
14.4 The Velocity of Communication 356
14.5 Space-Time Instead of Space and Time 357