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Global aspects in gravitation and cosmology. (English) Zbl 0853.53055

International Series of Monographs on Physics. 87. Oxford: Clarendon Press. x, 377 p. (1993).
The purpose of this book is to present basic results and applications of global techniques in Einstein’s general relativity and in cosmology. While the known singularity theorems are the most prominent global results in relativity, the author emphasizes the importance of global methods in other arenas of general relativity as well, pointing out that “the very nature of gravitational force is such that global aspects of space-time inevitably come into the picture whenever we try to understand and interpret this force in detail”. Due to the immensity of the subject, the choice of topics is naturally selective, and some parts of the book provide mere overviews with references to the literature for details.
After an overview of the book in Chapter 1, an introduction to geometry and general relativity is given in Chapter 2 and the most important exact solutions to Einstein’s equations are described in Chapter 3. Chapter 4, “Causality and space-time topology”, discusses the different regularity conditions assumed to avoid causality violations, putting particular emphasis on causal functions. Chapter 5, “Singularities in general relativity”, reviews the development of singularity theorems from the original results of Hawking, Penrose and Geroch to recent generalizations. Proofs are generally omitted in this chapter, the reader is referred to the relevant books that exist on the subject. Consequently, this chapter is rather short compared to the rest of the book. Much room is devoted instead to the issue of gravitational collapse and the related problem of cosmic censorship which are treated in detail in Chapters 6 and 7. Chapter 8, “Global upper limits in cosmology”, develops upper bounds on the age of the universe and on particle masses for globally hyperbolic, not necessarily homogeneous or isotropic spacetime models. In Chapter 9, “Quantum effects near the space-time singularity”, the author approaches the subject by quantizing only a limited degree of freedom of the metric, namely the conformal factor, and applies this approach to several exact solutions and to the evolution of quantum effects in black hole geometry.
The book attempts to be self-contained and should be accessible to graduate students with some basic background in general relativity. References to the original literature abound throughout the text. Apart from some inconsistencies in the notation and occasional mathematical improprieties, the treatment is mathematically sufficiently rigorous to make the book readable also for mathematicians.

MSC:

53Z05 Applications of differential geometry to physics
83-02 Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory
83C75 Space-time singularities, cosmic censorship, etc.
83F05 Relativistic cosmology
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