General Relativity for Mathematicians - Rainer Sachs

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Résumé :
This is a book about physics, written for mathematicians. The readers we have in mind can be roughly described as those who: I. are mathematics graduate students with some knowledge of global differential geometry 2. have had the equivalent of freshman physics, and find popular accounts of astrophysics and cosmology interesting 3. appreciate mathematical elarity, but are willing to accept physical motiva? tions for the mathematics in place of mathematical ones 4. are willing to spend time and effort mastering certain technical details, such as those in Section 1. 1. Each book disappoints so me readers. This one will disappoint: 1. physicists who want to use this book as a first course on differential geometry 2. mathematicians who think Lorentzian manifolds are wholly similar to Riemannian ones, or that, given a sufficiently good mathematical back? ground, the essentials of a subject !ike cosmology can be learned without so me hard work on boring detaiis 3. those who believe vague philosophical arguments have more than historical and heuristic significance, that general relativity should somehow be proved, or that axiomatization of this subject is useful 4. those who want an encyclopedic treatment (the books by Hawking-Ellis [1], Penrose [1], Weinberg [1], and Misner-Thorne-Wheeler [I] go further into the subject than we do; see also the survey article, Sachs-Wu [1]). 5. mathematicians who want to learn quantum physics or unified fieId theory (unfortunateIy, quantum physics texts all seem either to be for physicists, or merely concerned with formaI mathematics).

Sommaire:
0 Preliminaries.- 0.0 Review and notation.- 0.1 Physics background.- 0.2 Preview of relativity.- 1 Spacetimes.- 1.0 Review and notation.- 1.1 Causal character.- 1.2 Time orientability.- 1.3 Spacetimes.- 1.4 Examples of spacetimes.- 2 Observers.- 2.0 Mathematical preliminaries.- 2.1 Observers and instantaneous observers.- 2.2 Gyroscope axes.- 2.3 Reference frames.- 3 Electromagnetism and matter.- One: Basic Concepts.- 3.0 Review and notation.- 3.1 Particles.- 3.2 Particle flows.- 3.3 Stress-energy tensors.- 3.4 Electromagnetism.- 3.5 Matter and relativistic models.- Two: Interactions.- 3.6 Some mathematical methods.- 3.7 Maxwell's equations.- 3.8 Particle dynamics.- 3.9 Matter equations: an example.- 3.10 Energy-momentum 'conservation'.- 3.11 Two initial value theorems.- 3.12 Appropriate matter equations.- Three: Other Matter Models.- 3.13 Examples.- 3.14 Normal stress-energy tensors.- 3.15 Perfect fluids.- 4 The Einstein field equation.- 4.0 Review and notation.- 4.1 The Einstein field equation.- 4.2 Ricci flat spacetimes.- 4.3 Gravitational attraction and the phenomenon of collapse.- 5 Photons.- 5.0 Mathematical preliminaries.- 5.1 Photons.- 5.2 Light signals.- 5.3 Synchronizable reference frames.- 5.4 Frequency ratio.- 5.5 Photon distribution functions.- 5.6 Integration on lightcones.- 5.7 A photon gas.- 6 Cosmology.- 6.0 Review, notation and mathematical preliminaries.- 6.1 Data.- 6.2 Cosmological models.- 6.3 The Einstein-de Sitter model.- 6.4 Simple cosmological models.- 6.5 The early universe.- 6.6 Other models.- 6.7 Appendix: Luminosity distance in the Einstein-de Sitter model.- 7 Further applications.- 7.0 Review and notation.- 7.1 Preview.- 7.2 Stationary spacetimes.- 7.3 The geometry of Schwarzschild spacetimes.- 7.4 The solar system.- 7.5 Black holes.- 7.6 Gravitational plane waves.- 8 Optional exercises: relativity.- 8.1 Lorentzian algebra.- 8.2 Differential topology and geometry.- 8.3 Chronology and causality.- 8.4 Isometries and characterizations of gravitational fields.- 8.5 The Einstein field equation.- 8.6 Gases.- 9 Optional exercises: Newtonian analogues.- 9.0 Review and notation.- 9.1 Maxwell's equations.- 9.2 Particles.- 9.3 Gravity.- Glossary of symbols.- Index of basic notations.