Introduction to Tensor Calculus, Relativity and Cosmology

Introduction to Tensor Calculus, Relativity and Cosmology

by D. F. Lawden


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This elementary introduction pays special attention to aspects of tensor calculus and relativity that students tend to find most difficult. Its use of relatively unsophisticated mathematics in the early chapters allows readers to develop their confidence within the framework of Cartesian coordinates before undertaking the theory of tensors in curved spaces and its application to general relativity theory.
Topics include the special principle of relativity and Lorentz transformations; orthogonal transformations and Cartesian tensors; special relativity mechanics and electrodynamics; general tensor calculus and Riemannian space; and the general theory of relativity, including a focus on black holes and gravitational waves. The text concludes with a chapter offering a sound background in applying the principles of general relativity to cosmology. Numerous exercises advance the theoretical developments of the main text, thus enhancing this volume’s appeal to students of applied mathematics and physics at both undergraduate and postgraduate levels. Preface. List of Constants. References. Bibliography.

Product Details

ISBN-13: 9780486425405
Publisher: Dover Publications
Publication date: 01/27/2003
Series: Dover Books on Physics Series
Edition description: 3RD
Pages: 224
Sales rank: 370,874
Product dimensions: 6.14(w) x 9.21(h) x (d)

Table of Contents

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 gravitation
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

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