# The Theory of Space, Time and Gravitation

## 2nd Edition

**Authors:**V. Fock

**eBook ISBN:**9781483184906

**Imprint:**Pergamon

**Published Date:**1st January 1964

**Page Count:**460

## Description

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.

## Table of 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 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

## Details

- No. of pages:
- 460

- Language:
- English

- Copyright:
- © Pergamon 1964

- Published:
- 1st January 1964

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483184906