Einstein Spaces (eBook)

Front Cover 1
Einstein Spaces 4
Copyright Page 5
Table of Contents 6
Preface to the English Edition 10
Foreword 12
Notation 14
Chapter 1. Basic Tensor Analysis 16
1. Riemann Manifolds 16
2. Tensor Algebra 21
3. Covariant Differentiation 27
4. Parallel Displacement in a Vn Space 31
5. Curvature Tensor of a Vn Space 35
6. Geodesies 41
7. Special Systems of Coordinates in Vn 45
8. Riemannian Curvature of Vn. Spaces of Constant Curvature 60
9. The Principal Axes Theorem for a Tensor 64
10. Lie Groups in Vn 71
Chapter 2. Einstein Spaces 82
11. The Basis of the Special Theory of Relativity. Lorentz Transformations 82
12. Field Equations in the Relativistic Theory of Gravitation 88
13. Einstein Spaces 90
14. Some Solutions of the Gravitational Field Equations 93
Chapter 3. General Classification of Gravitational Fields 103
15. Bivector Spaces 103
16. Classification of Einstein Spaces 106
17. Principal Curvatures 108
18. The Classification of Einstein Spaces for n = 4 110
19. The Canonical Form of the Matrix (Rab) for Ti and .i Spaces 117
20. Classification of General Gravitational Fields 127
21. Complex Representation of Minkowski Space Tensors 136
22. Basis of the Complete System of Second Order Invariants of a V4 Space 141
Chapter 4. Motions in Empty Space 147
23. Classification of Ti by Groups of Motions 147
24. Non-isomorphic Structures of Groups of Motions Admitted by Empty Spaces 155
25. Spaces of Maximum Mobility T1, T2 and T3 164
26. T1 Spaces Admitting Motions 181
27. T2 and T3 Spaces Admitting Motions 201
28. Summary of Results. Survey of Well-known Solutions of the Field Equations 209
Chapter 5. Classification of General Gravitational Fields by Groups of Motions 213
29. Gravitational Fields Admitting a Gr Group (r = 2) 214
30. Gravitational Fields Admitting a G3 Group of Motions Acting on a V2 or V2 220
31. Gravitational Fields Admitting a G3 Group of Motions Acting on a V3 or V3 226
32. Gravitational Fields Admitting a Simply-transitive or Intransitive G4 Group of Motions 242
33. Gravitational Fields Admitting Groups of Motions Gr (r = 5) 256
Chapter 6. Conformal Mapping of Einstein Spaces 272
34. Conformal Mapping of Riemann Spaces 272
35. Conformal Mapping of Riemann Spaces on Einstein Spaces 275
36. Conformal Mapping of Einstein Spaces on Einstein Spaces Non-isotropic Case
37. Conformal Mapping of Einstein Spaces Isotropie Case
Chapter 7. Geodesic Mapping of Gravitational Fields 291
38. Historical Survey 292
39. Algebraic Classification of the Possible Cases 293
40. The Invariant Equations for gii in a Non-holonomic Orthonormal Tetrad 297
41. The Canonical Forms of the Metrics of V4 and V4 in a Holonomic Coordinate System 301
42. The Projective Mapping of Einstein Spaces 312
Chapter 8. The Cauchy Problem for the Einstein Field Equations 318
43. The Einstein Field Equations 318
44. The Exterior Cauchy Problem 322
45. Freedom Available in Choosing Field Potentials for an Einstein Space 327
46. Characteristic and Bicharacteristic Manifolds 335
47. The Energy-momentum Tensor 338
48. The Conservation Law for the Energy-momentum Tensor 347
49. The Interior Cauchy Problem for the Flow of Matter 350
50. The Interior Cauchy Problem for a Perfect Fluid 352
Chapter 9. Special Types of Gravitational Field 357
51. Reducible and Conformal-reducible Einstein Spaces 357
52. Symmetric Gravitational Fields 365
53. Static Einstein Spaces 369
54. Centro-symmetric Gravitational Fields 372
55. Gravitational Fields with Axial Symmetry 378
56. Harmonic Gravitational Fields 385
57. Spaces Admitting Cylindrical Waves 390
58. Spaces and their Boundary Conditions 394
References 400
Index 422