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**The large scale structure of space-time.**
*(English)*
Zbl 0265.53054

Cambridge Monographs on Mathematical Physics 1. London: Cambridge University Press. xi, 391 p. £10.00 net (1973).

The subject of the book under review is the structure of space-time from the radius of an elementary particle up to the radius of the universe. The main purpose is the discussion of singularities and the treatment of the resulting predictions about the universe: the collapse of massive stars to black holes and the existence of an initial singularity in the universe. Most of the results described in this book are independent of the detailed nature of the physical laws. They merely depend upon the description of space-time by pseudo-Riemannian geometry and some additional properties. This has the advantage that many results also are valid for modifications of Einstein’s General Relativity such as the Brans-Dicke theory. The study of singularities of space-time needs a lot of information about the local and global structure of Lorentz manifolds. Therefore the major parts of this book are devoted to considerations of mathematical problems, essentially in geometry, topology and analysis. Nevertheless, the authors have succeeded in combining a detailed and exact presentation of the theory under consideration with an excellent motivation by physical and cosmological theories and problems. Furthermore they give an almost complete and up to date survey of the research articles concerning the problems discussed in this book. (The bibliography contains about 180 items.) This altogether makes the book highly interesting for mathematicians as well as for physicists. The following description of the detailed content will emphasize the mathematical aspects.

The first chapter gives a survey of the topics discussed in this book and is followed by an introduction to differential geometry. The material (differentiable manifolds, Lie derivative, connections, curvature tensors, metrics, hypersurfaces etc.) is presented in a modern manner. Only indications of proofs are given. In the third chapter the theory of General Relativity is introduced as a number of postulates about a mathematical model for space-time: local causality, local energy conservation and the validity of the field equations. Under the title “Physical significance of curvature” the fourth chapter discusses the vorticity, shear and expansion of families of non-space like curves, the energy conditions (implying inequalities for the Ricci tensor) and the structure of geodesics (conjugate points etc.) of a Lorentz manifold under such energy conditions. Chapter 5 is devoted to a detailed study of the geometry of the most important cosmological models: Minkowski space-time, de Sitter and anti-de Sitter space times, Robertson-Walker spaces, Bianchi I spaces, the Schwarzschild and the Reissner-Nordström solutions, the Kerr solution, Gödel’s universe, Taub-NUT space etc. Chapter 6 deals with properties of causal relationships of space-time. The causal and the topological structure are compared, the existence of cosmological times is established and the concepts of Cauchy developments, global hyperbolicity and causal boundary are introduced and studied. In Chapter 7 the authors give an outline of the Cauchy problem in General Relativity. The main topic of this book, the singularities of space-time, are treated in the eighth chapter. After some preparations of the final definition of a singularity the singularity theorems of Hawking and Penrose are demonstrated. Furthermore the character and the description of singularities are discussed. The astrophysical and cosmological implications of these results are considered in the last two chapters.

The first chapter gives a survey of the topics discussed in this book and is followed by an introduction to differential geometry. The material (differentiable manifolds, Lie derivative, connections, curvature tensors, metrics, hypersurfaces etc.) is presented in a modern manner. Only indications of proofs are given. In the third chapter the theory of General Relativity is introduced as a number of postulates about a mathematical model for space-time: local causality, local energy conservation and the validity of the field equations. Under the title “Physical significance of curvature” the fourth chapter discusses the vorticity, shear and expansion of families of non-space like curves, the energy conditions (implying inequalities for the Ricci tensor) and the structure of geodesics (conjugate points etc.) of a Lorentz manifold under such energy conditions. Chapter 5 is devoted to a detailed study of the geometry of the most important cosmological models: Minkowski space-time, de Sitter and anti-de Sitter space times, Robertson-Walker spaces, Bianchi I spaces, the Schwarzschild and the Reissner-Nordström solutions, the Kerr solution, Gödel’s universe, Taub-NUT space etc. Chapter 6 deals with properties of causal relationships of space-time. The causal and the topological structure are compared, the existence of cosmological times is established and the concepts of Cauchy developments, global hyperbolicity and causal boundary are introduced and studied. In Chapter 7 the authors give an outline of the Cauchy problem in General Relativity. The main topic of this book, the singularities of space-time, are treated in the eighth chapter. After some preparations of the final definition of a singularity the singularity theorems of Hawking and Penrose are demonstrated. Furthermore the character and the description of singularities are discussed. The astrophysical and cosmological implications of these results are considered in the last two chapters.

Reviewer: Bernd Wegner (Berlin)

### MSC:

53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |

53B30 | Local differential geometry of Lorentz metrics, indefinite metrics |

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

83-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to relativity and gravitational theory |

83C60 | Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism |

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

57N99 | Topological manifolds |

35L99 | Hyperbolic equations and hyperbolic systems |