The Theory of Space, Time and Gravitation

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 Co variance 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 Remark on the Conventional Formulation of the Conservation Laws


30. The Vector of Energy-Current (Umov's Vector)


31. The Mass Tensor


31. A System of Equations for the Components of the Mass Tensor as Functions of the Field


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 Christoflel Symbols and the Locally Geodesic Coordinate System. Conditions for Transforming ds2 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 Choice of 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


49. Remark on the Relativity Principle and the Covariance of Equations

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 and Conformal Space


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 Ray 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 I


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. Proof of the Uniqueness of the Energy Momentum Tensor of the Electromagnetic Field

Appendix C. Proof of the Uniqueness of the Hydro-Dynamic Mass Tensor

Appendix D. The Transformations of the Einstein Tensor

Appendix E. The Characteristics of the Generalized D'Alembert Equation

Appendix F. Integration of the Wave Front Equation

Appendix G. Necessary and Sufficient Conditions for the Euclidean Character of Three-Dimensional Space

References

Index