The theory of space, time and gravitation

Table of contents :
Title page
Date-line
CONTENTS
Translator's Preface
Preface
INTRODUCTION
I. THE THEORY OF RELATIVITY
§ 1. Coordinates of Space and Time
§ 2. The Position of a Body in Space at a given Instant, in a Fixed Reference Frame
§ 3. The Law of Propagation of an Electromagnetic Wave Front
§ 4. Equations for Rays
§ 5. Inertial Frames of Reference
§ 6. The Basic Postulates of the Theory of Relativity
§ 7. The Galileo Transformations and the Need to Generalize them
§ 8. Proof of the Linearity of the Transformation Linking Two Inertial Frames
§ 9. Determination of the Coefficients of the Linear Transformations and of a Scale Factor
§ 10. Lorentz Transformations
§ 11. Determination of Distances and Synchronization of Clocks within One Inertial Reference Frame
§ 12. Time Sequence of Events in Different Reference Frames
§ 13. Comparison of Time Differences in Moving Reference Frames. The Doppler Effect
§ 14. Comparison of Clock Readings in Moving Reference Frames
§ 15. Comparison of Distances and Lengths in Moving Reference Frames
§ 16. Relative Velocity
§ 17. The Lobachevsky-Einstein Velocity Space
II. THE THEORY OF RELATIVITY IN TENSOR FORM
§ 18. Some Remarks on the Covariance of Equations
§ 19. Definition of a Tensor in Three Dimensions and some Remarks on Covariant Quantities
§ 20. Definition of a Four-dimensional Vector
§ 21. Four-dimensional Tensors
§ 22. Pseudo-Tensors
§ 23. Infinitesimal Lorentz Transformations
§ 24. The Transformation Laws for the Electromagnetic Field and the Covariance of Maxwell's Equations
§ 25. The Motion of a Charged Mass-Point in a given External Field
§ 26. Approximate Description of a System of Moving Point Charges
§ 27. Derivation of the Conservation Laws in the Mechanics of Point Systems
§ 28. The Tensor Character of the Integrals of Motion
§ 29. A Eemark on the Conventional Formulation of the Conservation Laws
§ 30. The Vector of Energy-Current (Umov's Vector)
§ 31. The Mass Tensor
§ 32. Examples of the Mass Tensor
§ 33. The Energy Tensor of the Electromagnetic Field
§ 34. Mass and Energy
III. GENERAL TENSOR ANALYSIS
§ 35. Permissible Transformations for Space and Time Coordinates
§ 36. General Tensor Analysis and Generalized Geometry
§ 37. The Definitions of a Vector and of a Tensor. Tensor Algebra
§ 38. The Equation of a Geodesic
§ 39. Parallel Transport of a Vector
§ 40. Covariant Differentiation
§ 41. Examples of Co variant Differentiation
§ 42. The Transformation Law for Christofiel Symbols and the Locally Geodesic Coordinate System. Conditions for Transforming $ds^2$ to a Form with Constant Coefficients
§ 43. The Curvature Tensor
§ 44. The Basic Properties of the Curvature Tensor
IV. A FORMULATION OF RELATIVITY THEORY IN ARBITRARY COORDINATES
§ 45. Properties of Space-Time and Coordinates
§ 46. The Equations of Mathematical Physics in Arbitrary Coordinates
§ 47. A Variational Principle for the Maxwell-Lorentz System of Equations
§ 48. The Variational Principle and the Energy Tensor
§ 49. The Integral Form of the Conservation Laws in Arbitrary Coordinates
V. THE PRINCIPLES OF THE THEORY OF GRAVITATION
§ 50. The Generalization of Galileo's Law
§ 51. The Square of the Interval in Newtonian Approximation
§ 52. Einstein's Gravitational Equations
§ 53. The Characteristics of Einstein's Equations. The Speed of Propagation of Gravitation
§ 54. A Comparison with the Statement of the Problem in Newtonian Theory. Boundary Conditions
§ 55. Solution of Einstein's Gravitational Equations in First Approximation and Determination of the Constant
§ 56. The Gravitational Equations in the Static Case
§ 57. Rigorous Solution of the Gravitational Equations for a Single Concentrated Mass
§ 58. The Motion of the Perihelion of a Planet
§ 59. The Deflection of a Light Eay Passing Near the Sun
§ 60. A Variational Principle for the Equations of Gravitation
§ 61. On the Local Equivalence of Fields of Acceleration and of Gravitation
§ 62. On the Clock Paradox
VI. THE LAW OF GRAVITATION AND THE LAWS OF MOTION
§ 63. The Equations of Free Motion for a Mass Point and their Connection with the Gravitational Equations
§ 64. General Statement of the Problem of the Motion of a System of Masses
§ 65. The Divergence of the Mass Tensor in Second Approximation
§ 66. The Approximate Form of the Mass Tensor for an Elastic Solid with Inclusion of the Gravitational Field
§ 67. Approximate Expressions for the Christoffel Symbols and Some Other Quantities
§ 68. Approximate Form of the Gravitational Equations
§ 69. The Connection between the Divergence of the Mass Tensor and the Quantities $\Gamma^\nu$
§ 70. The Equations of Motion and the Harmonic Conditions
§ 71. The Internal and the External Problems in the Mechanics of Systems of Bodies. Newton's Equations for Translational Motion
§ 72. Newton's Equations for Rotational Motion
§ 73. The Internal Structure of a Body. Liapunov's Equation
§ 74. Evaluation of some Integrals that Characterize the Internal Structure of a Body
§ 75. Transformation of the Integral Form of the Equations of Motion
§ 76. Evaluation of the Momentum in Second Approximation
§ 77. Evaluation of the Force
§ 78. The Equations of Translational Motion in Lagrangian Form
§ 79. The Integrals of the Equations of Motion for Systems of Bodies
§ 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
§ 81. The Problem of Two Bodies of Finite Mass
VII. APPROXIMATE SOLUTIONS, CONSERVATION LAWS AND SOME QUESTIONS OF PRINCIPLE
§ 82. The Gravitational Potentials for Non-Rotating Bodies (Spatial Components)
§ 83. The Gravitational Potentials for Non-Rotating Bodies (Mixed and Temporal Components)
§ 84. Gravitational Potentials at Large Distances from a System of Bodies (Spatial Components)
§ 85. Gravitational Potentials at Large Distances from a System of Bodies (Mixed and Temporal Components)
§ 86. Solution of the Wave Equation in the Wave Zone
§ 87. The Gravitational Potentials in the Wave Zone
§ 88. Some General Remarks on the Conservation Laws
§ 89. Formulation of the Conservation Laws
§ 90. The Emission of Gravitational Waves and its Role in the Energy Balance
§ 91. The Connection between the Conservation Laws for the Field and the Integrals of Mechanics
§ 92. The Uniqueness Theorem for the Wave Equation
§ 93. On the Uniqueness of the Harmonic Coordinate System
§ 94. Friedmann-Lobachevsky Space
§ 95. Theory of the Red Shift
§ 96. The Development of the Theory of Gravitation and of the Motion of Masses (A Critical Survey)
CONCLUSION
APPENDIX A. ON THE DERIVATION OF THE LORENTZ TRANSFORMATIONS
APPENDIX B. THE TRANSFORMATION OF THE CURVATURE TENSOR
(a) Transformation of the Second Rank Curvature Tensor
(b) Transformation of the Invariant
(c) Transformation of the Einstein Tensor
APPENDIX C. THE CHARACTERISTICS OF THE GENERALIZED D'ALEMBERT EQUATION
APPENDIX D. INTEGRATION OF THE WAVE FRONT EQUATION
APPENDIX E. NECESSARY AND SUFFICIENT CONDITIONS FOR THE EUCLIDEAN CHARACTER OF THREE-DIMENSIONAL SPACE
REFERENCES
INDEX