Introduction to tensor calculus, relativity and cosmology

Cover......Page 1
Contents......Page 4
Preface......Page 7
List of Constants......Page 11
1. Newton's laws of motion......Page 12
2. Covariance of the laws of motion......Page 14
3. Special principle of relativity......Page 15
4. Lorentz transformations. Minkowski space-time......Page 17
S. The special Lorentz transformation......Page 20
6. Fitzgerald contraction. Time dilation......Page 23
7. Spacelike and timelike intervals. Light cone......Page 25
Exercises 1......Page 28
8. Orthogonal transformations......Page 32
9. Repeated-index summation convention......Page 34
10. Rectangular Cartesian tensors......Page 35
11. Invariants. Gradients. Derivatives of tensors......Page 38
12. Contraction. Scalar product. Divergence......Page 39
13. Pseudotensors......Page 40
14. Vector products. Curl......Page 41
Exercises 2......Page 42
15. The velocity vector......Page 50
16. Mass and momentum......Page 52
17. The force vector. Energy......Page 55
18. Lorentz transformation equations for force......Page 57
19. Fundamental particles. Photon and neutrino......Page 58
20. Lagrange's and Hamilton's equations......Page 59
21. Energy-momentum tensor......Page 61
22. Energy-momentum tensor for a fluid......Page 64
23. Angular momentum......Page 68
Exercises 3......Page 70
24. 4-Current density......Page 84
25. 4-Vector potential......Page 85
26. The field tensor......Page 86
27. Lorentz transformations of electric and magnetic vectors......Page 88
29. The energy-momentum tensor for an electromagnetic field......Page 90
Exercises 4......Page 93
30. Generalized N-dimensional spaces......Page 97
31. Contravariant and covariant tensors......Page 100
32. The quotient theorem. Conjugate tensors......Page 105
33. Covariant derivatives. Parallel displacement. Affine connection......Page 106
34. Transformation of an affinity......Page 109
35. Covariant derivatives of tensors......Page 111
36. The Riemann-Christoffel curvature tensor......Page 113
37. Metrical connection. Raising and lowering indices......Page 116
38. Scalar products. Magnitudes of vectors......Page 118
39. Geodesic frame. Christoffel symbols......Page 120
41. The covariant curvature tensor......Page 122
42. Divergence. The Laplacian. Einstein's tensor......Page 123
43. Geodesics......Page 125
Exercises 5......Page 128
44. Principle of equivalence......Page 138
45. Metric in a gravitational field......Page 141
46. Motion of a free particle in a gravitational field......Page 144
47. Einstein's law of gravitation......Page 146
48. Acceleration of a particle in a weak gravitational field......Page 148
49. Newton's law of gravitation......Page 150
50. Freely falling dust cloud......Page 151
51. Metrics with spherical symmetry......Page 153
52. Schwarzschild's solution......Page 156
53. Planetary orbits......Page 158
54. Gravitational deflection of a light ray......Page 161
55. Gravitational displacement of spectral lines......Page 163
56. Maxwell's equations in a gravitational field......Page 165
57. Black holes......Page 166
58. Gravitational waves......Page 170
Exercises 6......Page 174
59. Cosmological principle. Cosmical time......Page 185
60. Spaces of constant curvature......Page 187
61. The Robertson-Walker metric......Page 191
62. Hubble's constant and the deceleration parameter......Page 192
63. Red shift of galaxies......Page 193
64. Luminosity distance......Page 194
65. Cosmic dynamics......Page 196
66. Model universes of Einstein and de Sitter......Page 199
67. Friedmann universes......Page 200
68. Radiation model......Page 204
69. Particle and event horizons......Page 206
Exercises 7......Page 208
References......Page 210
Bibliography......Page 211
Index......Page 212