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Introduction to Tensor Calculus, Relativity and Cosmology

D. F. Lawden
Dover Publications
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List of Constants
Chapter 1 Special Principle of Relativity. Lorentz Transformations
1. Newton's laws of motion
2. Covariance of the laws of motion
3. Special principle of relativity
4. Lorentz transformations. Minkowski space-time
5. The special Lorentz transformation
6. Fitzgerald contraction. Time dilation
7. Spacelike and timelike intervals. Light cone
  Exercises 1
Chapter 2 Orthogonal Transformations. Cartesian Tensors
8. Orthogonal transformations
9. Repeated-index summation convention
10. Rectangular Cartesian tensors
11. Invariants. Gradients. Derivatives of tensors
12. Contraction. Scalar product. Divergence
13. Pseudotensors
14. Vector products. Curl
  Exercises 2
Chapter 3 Special Relativity Mechanics
15. The velocity vector
16. Mass and momentum
17. The force vector. Energy
18. Lorentz transformation equations for force
19. Fundamental particles. Photon and neutrino
20. Lagrange's and Hamilton's equations
21. Energy-momentum tensor
22. Energy-momentum tensor for a fluid
23. Angular momentum
  Exercises 3
Chapter 4 Special Relativity Electrodynamics
24. 4-Current density
25. 4-Vector potential
26. The field tensor
27. Lorentz transformations of electric and magnetic vectors
28. The Lorentz force
29. The engery-momentum tensor for an electromagnetic field
  Exercises 4
Chapter 5 General Tensor Calculus. Riemannian Space
30. Generalized N-dimensional spaces
31. Contravariant and covariant tensors
32. The quotient theorem. Conjugate tensors
33. Covariant derivatives. Parallel displacement. Affine connection
34. Transformation of an affinity
35. Covariant derivatives of tensors
36. The Riemann-Christoffel curvature tensor
37. Metrical connection. Raising and lowering indices
38. Scalar products. Magnitudes of vectors
39. Geodesic frame. Christoffel symbols
40. Bianchi identity
41. The covariant curvature tensor
42. Divergence. The Laplacian. Einstein's tensor
43. Geodesics
  Exercises 5
Chapter 6 General Theory of Relativity
44. Principle of equivalence
45. Metric in a gravitational field
46. Motion of a free particle in a gravitational field
47. Einstein's law of gravitation
48. Acceleration of a particle in a weak gravitational field
49. Newton's law of gravit
50. Freely falling dust cloud
51. Metrics with spherical symmetry
52. Schwarzchild's solution
53. Planetary orbits
54. Gravitational deflection of a light ray
55. Gravitational displacement of spectral lines
56. Maxwell's equations in a gravitational field
57. Black holes
58. Gravitational waves
  Exercises 6
Chapter 7 Cosmology
59. Cosmological principle. Cosmical time
60. Spaces of constant curvature
61. The Robertson-Walker metric
62. Hubble's constant and the deceleration parameter
63. Red shifts of galaxies
64. Luminosity distance
65. Cosmic dynamics
66. Model universes of Einstein and de Sitter
67. Friedmann universes
68. Radiation model
69. Particle and event horizons
  Exercises 7