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A First Course In General Relativity
by Bernard F. Schutz 376 pages Level: undergraduate text book Introduction to the theory and mathematics of General Relativity |
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You've already read the popular accounts of relativity. You feel comfortable with
special relativity. You've had enough of the analogies that are supposed to
represent the mathematics of relativity but don't really do so. You're ready
for the hard stuff: Tensors, geodesics and the field equations. The best book
currently available to help you get over the next hurdle is Bernard F. Schutz's
"A First Course in General Relativity".
In the preface Schutz lists the prerequisites as "special relativity, including the Lorentz transformation and relativistic mechanics; Euclidean vector calculus; ordinary and simple partial differential equations; thermodynamics and hydrostatics; Newtonian gravity (simple stellar structure would be useful but not essential); and enough elementary quantum mechanics to know what a photon is. Unlike many other text books on relativity, "A First Course" does not deal with electromagnetism at any great length. Except for the calculus, most of the prior knowledge required from the reader can be obtained from Wheeler and Taylor's "Spacetime Physics". Schutz wisely uses geometrized units throughout. (Geometrized units set certain basic constants such as c the speed of light to one, which simplifies many equations and enables them to more closely reflect physical phenomena without clutter generated by different units.) His notation follows that of MWT's "Gravitation". Most importantly, he presents tensors by the modern index free way, in which they are defined as functions acting on vectors and one-forms. This is not only the technique being used in modern relativity literature, but is easier to understand than the old way in which tensors were some sort of bridge between different coordinate systems. Schutz's explanation of tensor analysis are the clearest that I have seen. While this is one of the "easier" books on the details of General Relativity, it is a serious undergraduate level text book and like all text books that are heavily based on mathematics, it can not be simply read through and understood. There are exercises at the end of each chapter and many of them have either answers or hints. |
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| Review by Ed Ehrlich | |
| Table Of Contents | |
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Preface 1 Special relativity 1.1 Fundamental principles of special relativity theory (SR) 1.2 Definition of an inertial observer in SR 1.3 New units 1.4 Spacetime diagrams 1.5 Construction of the coordinates used by another observer 1.6 Invariance of the interval 1.7 Invariant hyperbolae 1.8 Particularly important results 1.9 The Lorentz transformation 1.10 The velocity-coposition law 1.11 Paradoxes and physical intuition 1.12 Bibliography 1.13 Appendix 1.14 Exercises 2 Vector analysis in special relativity 2.1 Definition of a vector 2.2 Vector algebra 2.3 The four-velocity 2.4 The four-momentum 2.5 Scalar product 2.6 Applications 2.7 Photons 2.8 Bibliography 2.9 Exercises 3 Tensor analysis in special relativity 3.1 The metric tensor 3.2 Definition of tensors 3.3 The (0 1) tensors: one-forms 3.4 The (0 2) tensors 3.5 Metric as a mapping of vectors into one-forms 3.6 finally: (M N) tensors 3.7 Index 'raising' and 'lowering' 3.8 Differentiation of tensors 3.9 Bibliography 3.10 Exercises 4 Perfect fluids in special relativity 4.1 Fluids 4.2 Dust: The number-flux vector N 4.3 One-forms and surfaces 4.4 Dust again; The stress-energy tensor 4.5 General fluids 4.6 Perfect fluids 4.7 Importance for general relativity 4.8 Gauss' law 4.9 Bibliography 4.10 Exercises 5 Preface to curvature 5.1 On the relation of gravitation to curvature 5.2 Tensor algebra in polar coordinates 5.3 Tensor calculus in polar coordinates 5.4 Christoffel Symbols and the metric 5.5 The tensorial nature of Christoffel 5.6 Noncoordinate bases 5.7 Looking ahead 5.8 Bibliography 5.9 Exercises 6 Curved manifolds 6.1 Differentiable manifolds and tensors 6.2 Riemannian manifolds 6.3 covariant differentiation 6.4 Parallel transport, geodesics and curvature 6.5 The curvature tensor 6.6 Bianchi identities; Ricci and Einstein tensors 6.7 Curvature in perspective 6.8 Bibliography 6.9 Exercises 7 Physics in a curved spacetime 7.1 The transition from differential geometry to gravity 7.2 Physics in slightly curved spacetime 7.3 Curved intuition 7.4 Conserved quantities 7.5 Bibliography 7.6 Exercises 8 The Einstein field equations 8.1 Purpose and justification of the field equations 8.2 Einstein's equations 8.3 Einstein's equations for weak gravitational fields 8.4 Newtonian gravitational fields 8.5 Bibliography 8.6 Exercises 9 Gravitational radiation 9.1 The propagation of gravitational waves 9.2 The detection of gravitational waves 9.3 The generation of gravitational waves 9.4 The energy carried away by gravitational waves 9.5 Bibliography 9.6 Exercises 10 Spherical solutions for stars 10.1 Coordinates for spherically symmetric spacetimes 10.2 Static spherically symmetric spacetimes 10.3 Static perfect fluid Einstein equations 10.4 The exterior geometry 10.5 The interior structure of the star 10.6 Exact interior solutions 10.7 Relativistic stars and gravitational collapse 10.8 Bibliography 10.9 Exercises 11 Schwarzschild geometry and black holes 11.1 Trajectories in the Schwarzschild spacetime 11.2 Nature of the surface r = 2M 11.3 More-general black holes 11.4 Quantum mechanical emission of radiation by black holes: The Hawking process 11.5 Bibliography 11.6 Exercises 12 Cosmology 12.1 What is cosmology? 12.2 General-relativistic cosmological models 12.3 Cosmological observations 12.4 Physical cosmology 12.5 Bibliography 12.6 Exercises Appendix A: Summary of linear algebra Appendix B: Hints and solutions to selected exercises References Index |
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