The Theory of Space, Time and Gravitation

The Theory of Space, Time, and Gravitation, 2nd Revised Edition focuses on Relativity Theory and Einstein's Theory of Gravitation and correction of the misinterpretation of the Einsteinian Gravitation Theory.

The book first offers information on the theory of relativity and the theory of relativity in tensor form. Discussions focus on comparison of distances and lengths in moving reference frames; comparison of time differences in moving reference frames; position of a body in space at a given instant in a fixed reference frame; and proof of the linearity of the transformation linking two inertial frames. The text then ponders on general tensor analysis, including permissible transformations for space and time coordinates, parallel transport of a vector, covariant differentiation, and basic properties of the curvature tensor.

The publication examines the formulation of relativity theory in arbitrary coordinates and principles of the theory of gravitation. Topics include equations of mathematical physics in arbitrary coordinates; integral form of the conservation laws in arbitrary coordinates; variational principle and the energy tensor; and comparison with the statement of the problem in Newtonian theory.

The manuscript is a dependable reference for readers interested in the theory of space, time, and gravitation.

Contenu

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
32. A System of Equations for the Components of the Mass Tensor as Functions of the Field
33. Examples of the Mass Tensor
34. The Energy Tensor of the Electromagnetic Field
35. Mass and Energy
    III. General Tensor Analysis
36. Permissible Transformations for Space and Time Coordinates
37. General Tensor Analysis and Generalized Geometry
38. The Definitions of a Vector and of a Tensor. Tensor Algebra
39. The Equation of a Geodesic
40. Parallel Transport of a Vector
41. Covariant Differentiation
42. Examples of Co variant Differentiation
43. The Transformation Law for Christoflel Symbols and the Locally Geodesic Coordinate System. Conditions for Transforming ds2 to a Form with Constant Coefficients
44. The Curvature Tensor
45. The Basic Properties of the Curvature Tensor
    IV. A Formulation of Relativity Theory in Arbitrary Coordinates
46. Properties of Space-Time and Choice of Coordinates
47. The Equations of Mathematical Physics in Arbitrary Coordinates
48. A Variational Principle for the Maxwell-Lorentz System of Equations
49. The Variational Principle and the Energy Tensor
50. The Integral Form of the Conservation Laws in Arbitrary Coordinates
51. Remark on the Relativity Principle and the Covariance of Equations
    V. The Principles of the Theory of Gravitation
52. The Generalization of Galileo's Law
53. The Square of the Interval in Newtonian Approximation
54. Einstein's Gravitational Equations
55. The Characteristics of Einstein's Equations. The Speed of Propagation of Gravitation
56. A Comparison with the Statement of the Problem in Newtonian Theory. Boundary Conditions
57. Solution of Einstein's Gravitational Equations in First Approximation and Determination of the Constant
58. The Gravitational Equations in the Static Case and Conformal Space
59. Rigorous Solution of the Gravitational Equations for a Single Concentrated Mass
60. The Motion of the Perihelion of a Planet
61. The Deflection of a Light Ray Passing Near the Sun
62. A Variational Principle for the Equations of Gravitation
63. On the Local Equivalence of Fields of Acceleration and of Gravitation
64. On the Clock Paradox
    VI. The Law of Gravitation and the Laws of Motion
65. The Equations of Free Motion for a Mass Point and their Connection with the Gravitational Equations
66. General Statement of the Problem of the Motion of a System of Masses
67. The Divergence of the Mass Tensor in Second Approximation
68. The Approximate Form of the Mass Tensor for an Elastic Solid with Inclusion of the Gravitational Field
69. Approximate Expressions for the Christoffel Symbols and Some Other Quantities
70. Approximate Form of the Gravitational Equations
71. The Connection between the Divergence of the Mass Tensor and the Quantities I
72. The Equations of Motion and the Harmonic Conditions
73. The Internal and the External Problems in the Mechanics of Systems of Bodies. Newton's Equations for Translational Motion
74. Newton's Equations for Rotational Motion
75. The Internal Structure of a Body. Liapunov's Equation
76. Evaluation of some Integrals that Characterize the Internal Structure of a Body
77. Transformation of the Integral Form of the Equations of Motion
78. Evaluation of the Momentum in Second Approximation
79. Evaluation of the Force
80. The Equations of Translational Motion in Lagrangian Form
81. The Integrals of the Equations of Motion for Systems of Bodies
82. 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
83. The Problem of Two Bodies of Finite Mass
    VII. Approximate Solutions, Conservation Laws and Some Questions of Principle
84. The Gravitational Potentials for Non-Rotating Bodies (Spatial Components)
85. The Gravitational Potentials for Non-Rotating Bodies (Mixed and Temporal Components)
86. Gravitational Potentials at Large Distances from a System of Bodies (Spatial Components)
87. Gravitational Potentials at Large Distances from a System of Bodies (Mixed and Temporal Components)
88. Solution of the Wave Equation in the Wave Zone
89. The Gravitational Potentials in the Wave Zone
90. Some General Remarks on the Conservation Laws
91. Formulation of the Conservation Laws
92. The Emission of Gravitational Waves and its Role in the Energy Balance
93. The Connection between the Conservation Laws for the Field and the Integrals of Mechani…