Einstein Spaces
- 1st Edition - October 30, 2013
- Author: A. Z. Petrov
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 1 - 1 9 4 2 - 7
- eBook ISBN:9 7 8 - 1 - 4 8 3 1 - 5 1 8 4 - 7
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… Read more
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.
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
- Edition: 1
- Published: October 30, 2013
- Language: English
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