Theory of Space, Time and Gravitation (eBook)

Front Cover 
1 
The Theory of Space, Time and Gravitation 4
Copyright Page 5
Table of Contents 6
TRANSLATOR'S PREFACE 10
TRANSLATOR'S PREFACE TO SECOND EDITION 11
PREFACE 12
PREFACE TO SECOND EDITION 12
INTRODUCTION 14
CHAPTER I. THE THEORY OF RELATIVITY 22
1. Coordinates of Space and Time 22
2. The Position of a Body in Space at a given Instant, in a Fixed Reference Frame 22
3. The Law of Propagation of an Electromagnetic Wave Front 24
4. Equations for Rays 27
5. Inertial Frames of Reference 28
6. The Basic Postulates of the Theory of Relativity 29
7. The Galileo Transformations and the Need to Generalize Them 32
8. Proof of the Linearity of the Transformation Linking Two Inertial Frames 33
9. Determination of the Coefficients of the Linear Transformations and of a Scale Factor 37
10. Lorentz Transformations 39
11. Determination of Distances and Synchronization of Clocks within One Inertial Reference Frame 43
12. Time Sequence of Events in Different Reference Frames 46
13. Comparison of Time Differences in Moving Reference Frames. The Doppler Effect 50
14. Comparison of Clock Readings in Moving Reference Frames 53
15. Comparison of Distances and Lengths in Moving Reference Frames 57
16. Relative Velocity 58
17. The Lobachevsky-Einstein Velocity Space 61
CHAPTER II. THE THEORY OF RELATIVITY IN TENSOR FORM 67
18. Some Remarks on the Covariance of Equations 67
19. Definition of a Tensor in Three Dimensions and some Remarks on Covariant Quantities 68
20. Definition of a Four-Dimensional Vector 72
21. Four-Dimensional Tensors 74
22. Pseudo-Tensors 77
23. Infinitesimal Lorentz Transformations 78
24. The Transformation Laws for the Electromagnetic Field and the Covariance of Maxwell's Equations 80
25. The Motion of a Charged Mass-Point in a given External Field 86
26. Approximate Description of a System of Moving Point Charges 90
27. Derivation of the Conservation Laws in the Mechanics of Point Systems 96
28. The Tensor Character of the Integrals of Motion 99
29. A Remark on the Conventional Formulation of the Conservation Laws 102
30. The Vector of Energy Current (Umov's Vector). 104
31. The Mass Tensor 107
32. Examples of the Mass Tensor 114
33. The Energy Tensor of the Electromagnetic Field 119
34. Mass and Energy 123
CHAPTER III: GENERAL TENSOR ANALYSIS 127
35. Permissible Transformations for Space and Time Coordinates 127
36. General Tensor Analysis and Generalized Geometry 132
37. The Definitions of a Vector and of a Tensor. Tensor Algebra 135
38. The Equation of a Geodesic 142
39. Parallel Transport of a Vector 148
40. Covariant Differentiation 152
41. Examples of Covariant Differentiation 155
42. The Transformation Law for Christoffel Symbols and the Locally Geodesic Coordinate System. Conditions for transforming ds2 to a Form with Constant Coefficients 159
43. The Curvature Tensor 163
44. The Basic Properties of the Curvature Tensor 166
CHAPTER IV: A FORMULATION OF RELATIVITY THEORY IN ARBITRARY COORDINATES 171
45. Properties of Space-Time and Choice of Coordinates 171
46. The Equations of Mathematical Physics in Arbitrary Coordinates 174
47. A Variational Principle for the Maxwell-Lorentz System of Equations 178
48. The Variational Principle and the Energy Tensor 183
49. The Integral Form of the Conservation Laws in Arbitrary Coordinates 188
CHAPTER V. THE PRINCIPLES OF THE THEORY OF GRAVITATION 196
50. The Generalization of Galileo's Law 196
51. The Square of the Interval in Newtonian Approximation 197
52. Einstein's Gravitational Equations 202
53. The Characteristics of Einstein's Equations. The Speed of Propagation of Gravitation 205
54. A Comparison with the Statement of the Problem in Newtonian Theory. Boundary Conditions 207
55. Solution of Einstein's Gravitational Equations in First Approximation and Determination of the Constant 210
56. The Gravitational Equations in the Static Case and Conformal Space 216
57. Rigorous Solution of the Gravitational Equations for a Single Concentrated Mass 222
58. The Motion of the Perihelion of a Planet 228
59. The Deflection of a Light Ray Passing Near the Sun 234
60. A Variational Principle for the Equations of Gravitation 237
61. On the Local Equivalence of Fields of Acceleration and of Gravitation 241
62. On the Clock Paradox 247
CHAPTER VI. THE LAW OF GRAVITATION AND THE LAWS OF MOTION 251
63. The Equations of Free Motion for a Mass Point and their Connection with the Gravitational Equations 251
64. General Statement of the Problem of the Motion of a System of Masses 254
65. The Divergence of the Mass Tensor in Second Approximation 257
66. The Approximate Form of the Mass Tensor for an Elastic Solid with Inclusion of the Gravitational Field 260
67. Approximate Expressions for the Christoffel Svmbols and Some Other Quantities 262
68. Approximate Form of the Gravitational Equations 267
69. The Connection Between the Divergence of the Mass Tensor and the Quantities 272
70. The Equations of Motion and the Harmonic Conditions 276
71. The Internal and the External Problems in the Mechanics of Systems of Bodies. Newton's Equations for Translational Motion 280
72. Newton's Equations for Rotational Motion 285
73. The Internal Structure of a Body. Liapunov's Equation 290
74. Evaluation of Some Integrals that Characterize the Internal Structure of a Body 293
75. Transformation of the Integral Form of the Equations of Motion 296
76. Evaluation of the Momentum in Second Approximation 300
77. Evaluation of the Force 304
78. The Equations of Translational Motion in Lagrangian Form 310
7 9 . The Integrals of the Equations of Motion for Systems of Bodies 313
80. Additional Remarks on the Problem of the Motion of a System of Bodies. The Explicit Form of the Integrals of Motion for the Case of Non-Rotating Masses 320
CHAPTER VII. APPROXIMATE SOLUTIONS, CONSERVATION LAWS AND SOME QUESTIONS OF PRINCIPLE 331
82. The Gravitational Potentials for Non-Rotating Bodies (Spatial Components) 331
83. The Gravitational Potentials for Non-Rotating Bodies (Mixed and Temporal Components) 337
84. Gravitational Potentials at Large Distances from a System of Bodies (Spatial Components) 343
85. Gravitational Potentials at Large Distances from a System of Bodies (Mixed and Temporal Components) 347
86. Solution of the Wave Equation in the Wave Zone 353
87. The Gravitational Potentials in the Wave Zone 355
88. Some General Remarks on the Conservation Laws 362
89. Formulation of the Conservation Laws 363
90. The Emission of Gravitational Waves and its Role in the Energy Balance 370
91. The Connection between the Conservation Laws for the Field and the Integrals of Mechanics 373
92. The Uniqueness Theorem for the Wave Equation 378
9 3 . On the Uniqueness of the Harmonic Coordinate System 382
94. Friedmann-Lobachevsky Space 388
9 5 . Theory of the Red Shift 396
96. The Development of the Theory of Gravitation and of the Motion of Masses (A Critical Survey) 405
CONCLUSION 413
APPENDIX A: ON THE DERIVATION OF THE LORENTZ TRANSFORMATIONS 416
APPENDIX B: PROOF OF THE UNIQUENESS OF THE ENERGY MOMENTUM TENSOR OF THE ELECTROMAGNETIC FIELD 424
APPENDIX C: PROOF OF THE UNIQUENESS OF THE HYDRODYNAMIC MASS TENSOR 430
APPENDIX D: THE TRANSFORMATION OF THE EINSTEIN TENSOR 435
APPENDIX E: THE CHARACTERISTICS OF THE GENERALIZED D'ALEMBERT EQUATION 444
APPENDIX F: INTEGRATION OF THE WAVE FRONT EQUATION 447
APPENDIX G: NECESSARY AND SUFFICIENT CONDITIONS FOR THE EUCLIDEAN CHARACTER OF THREE-DIMENSIONAL SPACE 451
REFERENCES 454
INDEX 456