A short course in general relativity

Suitable for a one-semester course in general relativity for senior undergraduates or beginning graduate students, this text clarifies the mathematical aspects of Einstein's theory of relativity without sacrificing physical understanding.

The text begins with an exposition of those aspects of tensor calculus and differential geometry needed for a proper treatment of the subject. The discussion then turns to the spacetime of general relativity and to geodesic motion. A brief consideration of the field equations is followed by a discussion of physics in the vicinity of massive objects, including an elementary treatment of black holes and rotating objects. The main text concludes with introductory chapters on gravitational radiation and cosmology.

This new third edition has been updated to take account of fresh observational evidence and experiments. It includes new sections on the Kerr solution (in Chapter 4) and cosmological speeds of recession (in Chapter 6). A more mathematical treatment of tensors and manifolds, included in the 1st edition, but omitted in the 2nd edition, has been restored in an appendix. Also included are two additional appendixes - "Special Relativity Review" and "The Chinese Connection" - and outline solutions to all exercises and problems, making it especially suitable for private study.

Author Notes

J.D. Nightingale is Emeritus Professor of Physics at the State University of New York, College at New Paltz. J. Foster is recently retired Senior Lecturer in Mathematics at the University of Sussex. Both have extensive teaching experience in applied mathematics and theoretical physics. Prof. Nightingale's research interests tend towards the physical and cosmological consequences of general relativity, while Prof. Foster's tend towards the more mathematical aspects, such as exact solutions.

Reviews 1

Choice Review

As a short course on general relativity, this book (1st ed., CH, Jul'80; 2nd ed., 1995) joins the large number of other works at about the same level of mathematical sophistication either partially or completely devoted to general relativity, e.g., Introduction to Relativity, by William D. McGlinn (CH, Jun'03, 40-5857); The Tapestry of Modern Astrophysics, by Steven N. Shore (CH, Apr'03, 40-4591). The Foster-Nightingale book seems to display its origin as class notes made into a book. Very compact and concise, the presentation is clear and the choice of subjects standard. However, it is so dominated by the mathematics of the subject that it displays little by way of physical insight and is virtually devoid of any of the considerable data associated with modern cosmology, despite the chapter "Elements of Cosmology." It is presumably aimed at senior undergraduate and first-year graduate students, and would require much mathematical sophistication in students to be suitable at the undergraduate level. Further, the instructor would have to supply much of the "justification" for the math and all of the "romance" of the subject. It could serve very well as a handy resource for the outlines and basic applications of general relativity. ^BSumming Up: Optional. Upper-division undergraduates; graduate students. K. L. Schick emeritus, Union College (NY)

Preface	p. v
Introduction	p. 1
1 Vector and tensor fields	p. 7
1.0 Introduction	p. 7
1.1 Coordinate systems in Euclidean space	p. 7
1.2 Suffix notation	p. 13
1.3 Tangents and gradients	p. 19
1.4 Coordinate transformations in Euclidean space	p. 23
1.5 Tensor fields in Euclidean space	p. 27
1.6 Surfaces in Euclidean space	p. 30
1.7 Manifolds	p. 35
1.8 Tensor fields on manifolds	p. 38
1.9 Metric properties	p. 43
1.10 What and where are the bases?	p. 46
Problems 1 p. 49
2 The spacetime of general relativity and paths of particles	p. 53
2.0 Introduction	p. 53
2.1 Geodesics	p. 56
2.2 Parallel vectors along a curve	p. 64
2.3 Absolute and covariant differentiation	p. 71
2.4 Geodesic coordinates	p. 79
2.5 The spacetime of general relativity	p. 82
2.6 Newton's laws of motion	p. 86
2.7 Gravitational potential and the geodesic	p. 87
2.8 Newton's law of universal gravitation	p. 89
2.9 A rotating reference system	p. 90
Problems 2 p. 94
3 Field equations and curvature	p. 97
3.0 Introduction	p. 97
3.1 The stress tensor and fluid motion	p. 97
3.2 The curvature tensor and related tensors	p. 102
3.3 Curvature and parallel transport	p. 105
3.4 Geodesic deviation	p. 110
3.5 Einstein's field equations	p. 112
3.6 Einstein's equation compared with Poisson's equation	p. 115
3.7 The Schwarzschild solution	p. 116
Problems 3 p. 119
4 Physics in the vicinity of a massive object	p. 123
4.0 Introduction	p. 123
4.1 Length and time	p. 124
4.2 Radar sounding	p. 129
4.3 Spectralshift	p. 131
4.4 General particle motion (including photons)	p. 136
4.5 Perihelion advance	p. 144
4.6 Bending of light	p. 146
4.7 Geodesic effect	p. 149
4.8 Blackholes	p. 152
4.9 Other coordinate systems	p. 157
4.10 Rotating objects; the Kerr solution	p. 158
Problems 4 p. 167
5 Gravitational radiation	p. 169
5.0 Introduction	p. 169
5.1 What wiggles?	p. 170
5.2 Two polarizations	p. 173
5.3 Simple generation and detection	p. 178
Problems 5 p. 182
6 Elements of cosmology	p. 183
6.0 Introduction	p. 183
6.1 Robertson-Walker line element	p. 185
6.2 Field equations	p. 187
6.3 The Friedmann models	p. 189
6.4 Redshift, distance, and speed of recession	p. 192
6.5 Objects with large redshifts	p. 196
6.6 Comment on Einstein's models; inflation	p. 200
6.7 Newtonian dust	p. 203
Problems 6 p. 206
Appendices
A Special relativity review	p. 211
A.0 Introduction	p. 211
A.1 Lorentz transformations	p. 214
A.2 Relativistic addition of velocities	p. 216
A.3 Simultaneity	p. 217
A.4 Time dilation, length contraction	p. 218
A.5 Spacetime diagrams	p. 219
A.6 Some standard 4-vectors	p. 222
A.7 Doppler effect	p. 226
A.8 Electromagnetism	p. 228
Problems A p. 231
B The Chinese connection	p. 233
B.0 Background	p. 233
B.1 Lanchester'stransporteronaplane	p. 234
B.2 Lanchester'stransporteronasurface	p. 237
B.3 A trip at constant latitude	p. 240
C Tensors and Manifolds	p. 241
C.0 Introduction	p. 241
C.1 Vector spaces	p. 241
C.2 Dualspaces	p. 244
C.3 Tensor products	p. 247
C.4 The space T_s^r	p. 250
C.5 From tensors to tensor fields	p. 251
C.6 Manifolds	p. 251
C.7 The tangent space at each point of a manifold	p. 254
C.8 Tensor fields on a manifold	p. 256
Solutions	p. 259
References	p. 283
Index	p. 287

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