The Theory of Space, Time and Gravitation – Fock

In this post we will see the book The Theory of Space, Time and Gravitation by V. A. Fock.

About the book

The aim of this book is threefold. Firstly, we intended to give a text-book on Relativity Theory and on Einstein’s Theory of Gravitation. Secondly, we wanted to give an exposition of our own researches on these subjects. Thirdly, our aim was to develop a new, non-local, point of view on the theory and to correct a widespread misinterpretation of the Einsteinian Gravitation Theory as some kind of general relativity.

The second edition differs from the first by some additions and reformulations. The question of the uniqueness of the mass tensor is treated in more detail (Section 31*) and is illustrated by two examples (Appendices B and C). The notion of conformal space is introduced and used as a basis for the treatment of Einsteinian statics (Sections 56 and 57). Greatest care has been applied to the formulation of the basic ideas of the theory and to the elucidation of those points on which the author’s views differ from the traditional (Einsteinian) ones. Thus, in order to discuss the general aspects of the relativity principle Section 49* has been added.

The author’s views on the theory are explicitly formulated in different parts of the book and are implicit in the reasoning throughout the whole text. Their general trend is to lay stress on the Absolute rather than on the Relative.

The basic ideas of Einstein’s Theory of Gravitation are considered to be:
(a) the introduction of a space-time manifold with an indefinite metric,
(b) the hypothesis that the space-time metric is not rigid but can be influenced by physical
processes and
(c) the idea of the unity of metric and gravitation.

On the other hand, the principles of relativity and of equivalence are of limited application and, notwithstanding their heuristic value, they are not unrestrictedly part of Einstein’s Theory of Gravitation as expressed by the gravitational equations.

The book was translated from the Russian by N. Kemmer and was published in 1964.

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Contents

Translator’s Preface ix

Preface xi

INTRODUCTION 1

I. THE THEORY OF RELATIVITY 9

§ 1. Coordinates of Space and Time 9
§ 2. The Position of a Body in Space at a given Instant, in a Fixed Reference Frame 9
§ 3. The Law of Propagation of an Electromagnetic Wave Front 11
§ 4. Equations for Rays 14
§ 5. Inertial Frames of Reference 15
§ 6. The Basic Postulates of the Theory of Relativity 16
§ 7. The Galileo Transformations and the Need to Generalize them 19
§ 8. Proof of the Linearity of the Transformation Linking Two Inertial Frames 20
§ 9. Determination of the Coefficients of the Linear Transformations and of a Scale Factor 24
§ 10. Lorentz Transformations
§ 11. Determination of Distances and Synchronization of Clocks within One Inertial Reference Frame 30
§ 12. Time Sequence of Events in Different Reference Frames 33
§ 13. Comparison of Time Differences in Moving Reference Frames. The Doppler Effect 37
§ 14. Comparison of Clock Readings in Moving Reference Frames 40
§ 15. Comparison of Distances and Lengths in Moving Reference Frames 44
§ 16. Relative Velocity 45
§ 17. The Lobachevsky-Einstein Velocity Space 48

II. THE THEORY OF RELATIVITY IN TENSOR FORM 54

§ 18. Some Remarks on the Covariance of Equations 54
§ 19. Definition of a Tensor in Three Dimensions and some Remarks on Covariant Quantities 55
§ 20. Definition of a Four-dimensional Vector 59
§ 21. Four-dimensional Tensors 61
§ 22. Pseudo-Tensors 64
§ 23. Infinitesimal Lorentz Transformations 65
§ 24. The Transformation Laws for the Electromagnetic Field and the Covariance of Maxwell’s Equations 67
§ 25. The Motion of a Charged Mass-Point in a given External Field 73
§ 26. Approximate Description of a System of Moving Point Charges 77
§ 27. Derivation of the Conservation Laws in the Mechanics of Point Systems 83
§ 28. The Tensor Character of the Integrals of Motion 86
§ 29. A Remark on the Conventional Formulation of the Conservation Laws 89
§ 30. The Vector of Energy-Current (Umov’s Vector) 91
§ 31. The Mass Tensor 94
§ 31*. A System of Equations for the Components of the Mass Tensor as Functions of the Field 98 .
§ 32. Examples of the Mass Tensor 101
§ 33. The Energy Tensor of the Electromagnetic Field 106
§ 34. Mass and Energy 110

III. GENERAL TENSOR ANALYSIS H 4

§ 35. Permissible Transformations for Space and Time Coordinates ] ] 4
§ 36. General Tensor Analysis and Generalized Geometry 449
§ 37. The Definitions of a Vector and of a Tensor. Tensor Algebra 422
§ 38. The Equation of a Geodesic 429
§ 39. Parallel Transport of a Vector 435
§ 40. Covariant Differentiation 439
§ 41. Examples of Covariant Differentiation 442
§ 42. The Transformation Law for Christoffel Symbols and the Locally Geodesic Coordinate System. Conditions for Transforming ds^{2} to a Form with Constant Coefficients 146
§ 43. The Curvature Tensor 150
§ 44. The Basic Properties of the Curvature Tensor 153

IV. A FORMULATION OF RELATIVITY THEORY IN ARBITRARY COORDINATES 158

§ 45. Properties of Space-Time and Choice of Coordinates 158
§ 46. The Equations of Mathematical Physics in Arbitrary Coordinates 161
§ 47. A Variational Principle for the Maxwell-Lorentz System of Equations 165
§ 48. The Variational Principle and the Energy Tensor 170
§ 49. The Integral Form of the Conservation Laws in Arbitrary Coordinates 175
§ 49*. Remark on the Relativity Principle and the Covariance of Equations 178

V. THE PRINCIPLES OF THE THEORY OF GRAVITATION 183

§ 50. The Generalization of Galileo’s Law 183
§ 51. The Square of the Interval in Newtonian Approximation 184
§ 52. Einstein’s Gravitational Equations 189
§ 53. The Characteristics of Einstein’s Equations. The Speed of Propagation of Gravitation 192
§ 54. A Comparison with the Statement of the Problem in Newtonian Theory. Boundary Conditions 194
§ 55. Solution of Einstein’s Gravitational Equations in First Approximation and Determination of the Constant 197
§ 56. The Gravitational Equations in the Static Case and Conformal Space 203
§ 57. Rigorous Solution of the Gravitational Equations for a Single Concentrated Mass 209
§ 58. The Motion of the Perihelion of a Planet 215
§ 59. The Deflection of a Light Ray Passing Near the Sun 221
§ 60. A Variational Principle for the Equations of Gravitation 224
§ 61. On the Local Equivalence of Fields of Acceleration and of Gravitation 228
§ 62. On the Clock Paradox 234

VI. THE LAW OF GRAVITATION AND THE LAWS OF MOTION 238

§ 63. The Equations of Free Motion for a Mass Point and their Connection with the Gravitational Equations 238
§ 64. General Statement of the Problem of the Motion of a System of Masses 241
§ 65. The Divergence of the Mass Tensor in Second Approximation 244
§ 66. The Approximate Form of the Mass Tensor for an Elastic Solid with Inclusion of the Gravitational Field 247
§ 67. Approximate Expressions for the Christoffel Symbols and Some Other Quantities 249
§ 68. Approximate Form of the Gravitational Equations 254
§ 69. The Connection between the Divergence of the Mass Tensor and the Quantities 𝚪^{𝛎} 259
§ 70. The Equations of Motion and the Harmonic Conditions 263
§ 71. The Internal and the External Problems in the Mechanics of Systems of Bodies. Newton’s Equations for Translational Motion 267
§ 72. Newton’s Equations for Rotational Motion 272
§ 73. The Internal Structure of a Body. Liapunov’s Equation 277
§ 74. Evaluation of some Integrals that Characterize the Internal Structure of a Body 280
§ 75. Transformation of the Integral Form of the Equations of .Motion 283
§ 76. Evaluation of the Momentum in Second Approximation 287
§ 77. Evaluation of the Force 291
§ 78. The Equations of Translational Motion in Lagrangian Form 297
§ 79. The Integrals of the Equations of Motion for Systems of Bodies 300
§ 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 307
§ 81. The Problem of Two Bodies of Finite Mass 311

VII. APPROXIMATE SOLUTIONS, CONSERVATION LAWS AND SOME QUESTIONS OF PRINCIPLE 318

§82. The Gravitational Potentials for Non-Rotating Bodies (Spatial Components) 318
§ 83. The Gravitational Potentials for Non-Rotating Bodies (Mixed and Temporal Components) 324
§ 84. Gravitational Potentials at Large Distances from a System of Bodies (Spatial Components) 330
§ 85. Gravitational Potentials at Large Distances from a System of Bodies (Mixed and Temporal Components) 334
§ 86. Solution of the Wave Equation in the Wave Zone 340
§ 87. The Gravitational Potentials in the Wave Zone 342
§ 88. Some General Remarks on the Conservation Laws 349
§ 89. Formulation of the Conservation Laws 350
§ 90. The Emission of Gravitational Waves and its Role in the Energy Balance 357
§ 91. The Connection between the Conservation Laws for the Field and the Integrals of Mechanics 360
§ 92. The Uniqueness Theorem for the Wave Equation 365
§ 93. On the Uniqueness of the Harmonic Coordinate System 366
§ 94. Friedmann-Lobachevsky Space 375
§ 95. Theory of the Red Shift 383
§ 96. The Development of the Theory of Gravitation and of the Motion of Masses (A Critical Survey) 392

CONCLUSION 400

APPENDIX A. ON THE DERIVATION OF THE LORENTZ TRANSFORMATIONS 403
APPENDIX B. PROOF OF THE UNIQUENESS OF THE ENERGY MOMENTUM TENSOR OF THE ELECTROMAGNETIC FIELD 411
APPENDIX C. PROOF OF THE UNIQUENESS OF THE HYDRODYNAMIC MASS TENSOR 417
APPENDIX D. THE TRANSFORMATION OF THE EINSTEIN TENSOR 422
APPENDIX E. THE CHARACTERISTICS OF THE GENERALIZED D’ALEMBERT EQUATION 431
APPENDIX F. INTEGRATION OF THE WAVE FRONT EQUATION 434
APPENDIX G. NECESSARY AND SUFFICIENT CONDITIONS FOR THE EUCLIDEAN CHARACTER OF THREE-DIMENSIONAL SPACE 438

REFERENCES 441

INDEX 443

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