A First Course in General Relativity

by Bernard Schutz

• ISBN13: 9780521887052
• ISBN10: 0521887054
• Edition: 2nd
• Format: Hardcover
• Copyright: 2009-06-22
• Publisher: Cambridge University Press

Summary

Clarity, readability and rigor combine in the second edition of this widely-used textbook to provide the first step into general relativity for undergraduate students with a minimal background in mathematics. Topics within relativity that fascinate astrophysical researchers and students alike are covered with Schutz's characteristic ease and authority - from black holes to gravitational lenses, from pulsars to the study of the Universe as a whole. This edition now contains discoveries by astronomers that require general relativity for their explanation; a revised chapter on relativistic stars, including new information on pulsars; an entirely rewritten chapter on cosmology; and an extended, comprehensive treatment of modern detectors and expected sources. Over 300 exercises, many new to this edition, give students the confidence to work with general relativity and the necessary mathematics, whilst the informal writing style makes the subject matter easily accessible. Password protected solutions for instructors are available at www.cambridge.org/9780521887052.

Table of Contents

Preface to the second edition	p. xi
Preface to the first edition	p. xiii
Special relativity	p. 1
Fundamental principles of special relativity (SR) theory	p. 1
Definition of an inertial observer in SR	p. 3
New units	p. 4
Spacetime diagrams	p. 5
Construction of the coordinates used by another observer	p. 6
Invariance of the interval	p. 9
Invariant hyperbolae	p. 14
Particularly important results	p. 17
The Lorentz transformation	p. 21
The velocity-composition law	p. 22
Paradoxes and physical intuition	p. 23
Further reading	p. 24
Appendix: The twin 'paradox' dissected	p. 25
Exercises	p. 28
Vector analysis in special relativity	p. 33
Definition of a vector	p. 33
Vector algebra	p. 36
The four-velocity	p. 41
The four-momentum	p. 42
Scalar product	p. 44
Applications	p. 46
Photons	p. 49
Further reading	p. 50
Exercises	p. 50
Tensor analysis in special relativity	p. 56
The metric tensor	p. 56
Definition of tensors	p. 56
The (01) tensors: one-forms	p. 58
The (02) tensors	p. 66
Metric as a mapping of vectors into one-forms	p. 68
Finally: (MN) tensors	p. 72
Index 'raising' and 'lowering'	p. 74
Differentiation of tensors	p. 76
Further reading	p. 77
Exercises	p. 77
Perfect fluids in special relativity	p. 84
Fluids	p. 84
Dust: the number-flux vector N	p. 85
One-forms and surfaces	p. 88
Dust again: the stress-energy tensor	p. 91
General fluids	p. 93
Perfect fluids	p. 100
Importance for general relativity	p. 104
Gauss' law	p. 105
Further reading	p. 106
Exercises	p. 107
Preface to curvature	p. 111
On the relation of gravitation to curvature	p. 111
Tensor algebra in polar coordinates	p. 118
Tensor calculus in polar coordinates	p. 125
Christoffel symbols and the metric	p. 131
Noncoordinate bases	p. 135
Looking ahead	p. 138
Further reading	p. 139
Exercises	p. 139
Curved manifolds	p. 142
Differentiable manifolds and tensors	p. 142
Riemannian manifolds	p. 144
Covariant differentiation	p. 150
Parallel-transport, geodesics, and curvature	p. 153
The curvature tensor	p. 157
Bianchi identities: Ricci and Einstein tensors	p. 163
Curvature in perspective	p. 165
Further reading	p. 166
Exercises	p. 166
Physics in a curved spacetime	p. 171
The transition from differential geometry to gravity	p. 171
Physics in slightly curved spacetimes	p. 175
Curved intuition	p. 177
Conserved quantities	p. 178
Further reading	p. 181
Exercises	p. 181
The Einstein field equations	p. 184
Purpose and justification of the field equations	p. 184
Einstein's equations	p. 187
Einstein's equations for weak gravitational fields	p. 189
Newtonian gravitational fields	p. 194
Further reading	p. 197
Exercises	p. 198
Gravitational radiation	p. 203
The propagation of gravitational waves	p. 203
The detection of gravitational waves	p. 213
The generation of gravitational waves	p. 227
The energy carried away by gravitational waves	p. 234
Astrophysical sources of gravitational waves	p. 242
Further reading	p. 247
Exercises	p. 248
Spherical solutions for stars	p. 256
Coordinates for spherically symmetric spacetimes	p. 256
Static spherically symmetric spacetimes	p. 258
Static perfect fluid Einstein equations	p. 260
The exterior geometry	p. 262
The interior structure of the star	p. 263
Exact interior solutions	p. 266
Realistic stars and gravitational collapse	p. 269
Further reading	p. 276
Exercises	p. 277
Schwarzschild geometry and black holes	p. 281
Trajectories in the Schwarzschild spacetime	p. 281
Nature of the surface r = 2M	p. 298
General black holes	p. 304
Real black holes in astronomy	p. 318
Quantum mechanical emission of radiation by black holes: the Hawking process	p. 323
Further reading	p. 327
Exercises	p. 328
Cosmology	p. 335
What is cosmology?	p. 335
Cosmological kinematics: observing the expanding universe	p. 337
Cosmological dynamics: understanding the expanding universe	p. 353
Physical cosmology: the evolution of the universe we observe	p. 361
Further reading	p. 369
Exercises	p. 370
Summary of linear algebra	p. 374
References	p. 378
Index	p. 386
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