# Einstein Spaces

## 1st Edition

**Authors:**A. Z. Petrov

**eBook ISBN:**9781483151847

**Imprint:**Pergamon

**Published Date:**1st January 1969

**Page Count:**426

## Description

Einstein Spaces presents the mathematical basis of the theory of gravitation and discusses the various spaces that form the basis of the theory of relativity. This book examines the contemporary development of the theory of relativity, leading to the study of such problems as gravitational radiation, the interaction of fields, and the behavior of elementary particles in a gravitational field. Organized into nine chapters, this book starts with an overview of the principles of the special theory of relativity, with emphasis on the mathematical aspects. This text then discusses the need for a general classification of all potential gravitational fields, and in particular, Einstein spaces. Other chapters consider the gravitational fields in empty space, such as in a region where the energy-momentum tensor is zero. The final chapter deals with the problem of the limiting conditions in integrating the gravitational field equations. Physicists and mathematicians will find this book useful.

## Table of Contents

Preface to the English Edition

Foreword

Notation

Chapter 1. Basic Tensor Analysis

1. Riemann Manifolds

2. Tensor Algebra

3. Covariant Differentiation

4. Parallel Displacement in a Vn Space

5. Curvature Tensor of a Vn Space

6. Geodesies

7. Special Systems of Coordinates in Vn

8. Riemannian Curvature of Vn. Spaces of Constant Curvature

9. The Principal Axes Theorem for a Tensor

10. Lie Groups in Vn

Chapter 2. Einstein Spaces

11. The Basis of the Special Theory of Relativity. Lorentz Transformations

12. Field Equations in the Relativistic Theory of Gravitation

13. Einstein Spaces

14. Some Solutions of the Gravitational Field Equations

Chapter 3. General Classification of Gravitational Fields

15. Bivector Spaces

16. Classification of Einstein Spaces

17. Principal Curvatures

18. The Classification of Einstein Spaces for n = 4

19. The Canonical Form of the Matrix (Rab) for Ti and Ti Spaces

20. Classification of General Gravitational Fields

21. Complex Representation of Minkowski Space Tensors

22. Basis of the Complete System of Second Order Invariants of a VA Space

Chapter 4. Motions in Empty Space

23. Classification of Ti by Groups of Motions

24. Non-Isomorphic Structures of Groups of Motions Admitted by Empty Spaces

25. Spaces of Maximum Mobility T1, T2 and T8

26. T1 Spaces Admitting Motions

27. T2 and T3 Spaces Admitting Motions

28. Summary of Results. Survey of Well-known Solutions of the Field Equations

Chapter 5. Classification of General Gravitational Fields by Groups of Motions

29. Gravitational Fields Admitting a Gr Group (r ≤ 2)

30. Gravitational Fields Admitting a G3 Group of Motions Acting on a V2 or V2

31. Gravitational Fields Admitting a G3 Group of Motions Acting on a V3 or V3

32. Gravitational Fields Admitting a Simply-Transitive or Intransitive G4 Group of Motions

33. Gravitational Fields Admitting Groups of Motions Gr (r ≥ 5)

Chapter 6. Conformal Mapping of Einstein Spaces

34. Conformal Mapping of Riemann Spaces

35. Conformal Mapping of Riemann Spaces on Einstein Spaces

36. Conformal Mapping of Einstein Spaces on Einstein Spaces; Non-isotropic Case

37. Conformal Mapping of Einstein Spaces; Isotropic Case

Chapter 7. Geodesic Mapping of Gravitational Fields

38. Historical Survey

39. Algebraic Classification of the Possible Cases

40. The Invariant Equations for gij in a Non-Holonomic Orthonormal Tetrad

41. The Canonical Forms of the Metrics of V4 and K4 in a Holonomic Coordinate System

42. The Projective Mapping of Einstein Spaces

Chapter 8. The Cauchy Problem for the Einstein Field Equations

43. The Einstein Field Equations

44. The Exterior Cauchy Problem

45. Freedom Available in Choosing Field Potentials for an Einstein Space

46. Characteristic and Bicharacteristic Manifolds

47. The Energy-Momentum Tensor

48. The Conservation Law for the Energy-Momentum Tensor

49. The Interior Cauchy Problem for the Flow of Matter

50. The Interior Cauchy Problem for a Perfect Fluid

Chapter 9. Special Types of Gravitational Field

51. Reducible and Conformal-Reducible Einstein Spaces

52. Symmetric Gravitational Fields

53. Static Einstein Spaces

54. Centro-Symmetric Gravitational Fields

55. Gravitational Fields with Axial Symmetry

56. Harmonic Gravitational Fields

57. Spaces Admitting Cylindrical Waves

58. Spaces and their Boundary Conditions

References

Index

## Details

- No. of pages:
- 426

- Language:
- English

- Copyright:
- © Pergamon 1969

- Published:
- 1st January 1969

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483151847