General relativity and the Einstein equations.

*(English)*Zbl 1157.83002
Oxford Mathematical Monographs. Oxford: Oxford University Press (ISBN 978-0-19-923072-3/hbk). xxv, 785 p. (2009).

These almost 800 pages about general relativity represent a fundamental monograph written with mathematical rigour by an author who is working on this topic for more than one half of a century. It covers all necessary ingredients to deduce and solve the Einstein equation and to give its solutions a geometrically oriented meaning. A large portion of the book is dedicated to global questions, where almost all other comparable books concentrate on the local point of view.

The 16 chapters of the main body of the book start with Lorentz geometry, the differential geometrical background necessary for the next chapters. Chapter 2 deals with special relativity, chapter 3 with General relativity and Einstein’s equations. Next chapters are: Schwarzschild spacetime and black holes, Cosmology, local Cauchy problem, constraints, relativistic fluids, relativistic kinetic theory, global hyperbolicity and causality, singularities, and global existence theorems. The seven appendices introduce the following topics: Sobolev spaces on Riemannian manifolds, Cauchy-Kowaleski theorem, Fuchs theorem, conformal methods, Kaluza-Klein theories, and others.

Approximately 100 pages of this book consist of reprints of earlier papers by Y. Choquet-Bruhat, one of them being coauthored with Antonio Greco. The book ends with reference list and index.

The 16 chapters of the main body of the book start with Lorentz geometry, the differential geometrical background necessary for the next chapters. Chapter 2 deals with special relativity, chapter 3 with General relativity and Einstein’s equations. Next chapters are: Schwarzschild spacetime and black holes, Cosmology, local Cauchy problem, constraints, relativistic fluids, relativistic kinetic theory, global hyperbolicity and causality, singularities, and global existence theorems. The seven appendices introduce the following topics: Sobolev spaces on Riemannian manifolds, Cauchy-Kowaleski theorem, Fuchs theorem, conformal methods, Kaluza-Klein theories, and others.

Approximately 100 pages of this book consist of reprints of earlier papers by Y. Choquet-Bruhat, one of them being coauthored with Antonio Greco. The book ends with reference list and index.

Reviewer: Hans-Jürgen Schmidt (Potsdam)

##### MSC:

83-02 | Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory |

83C10 | Equations of motion in general relativity and gravitational theory |

83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |

83A05 | Special relativity |

83C57 | Black holes |

83F05 | Cosmology |

83C75 | Space-time singularities, cosmic censorship, etc. |

53Z05 | Applications of differential geometry to physics |

76E20 | Stability and instability of geophysical and astrophysical flows |

83E15 | Kaluza-Klein and other higher-dimensional theories |