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Exact space-times in Einstein’s general relativity. (English) Zbl 1184.83003
Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press (ISBN 978-0-521-88927-8/hbk). xvii, 525 p. (2009).
The present monograph represents a detailed study of the geometric and physical properties of some of the key solutions to the Einstein field equation. It covers the following: Chapter 3: Minkowski space-time, Chapter 4: de Sitter space-time, Chapter 5: Anti-de Sitter space-time, Chapter 6: Friedmann space-times, Chapter 7: Bertotti-Robinson solution, Chapter 8: Schwarzschild space-time, …Chapter 10: Weyl metric, Chapter 11: Rotating black holes, Chapter 12: Taub-NUT space-time, …Chapter 17: Plane and pp-waves, Chapter 18: Kundt solutions, Chapter 19: Robinson-Trautman solutions, Chapter 20: Impulsive waves, Chapter 21: Colliding plane waves.
The book starts with some introduction to the whole topic and to several differential geometric preliminaries, and then, in each chapter, deals with several forms of the metric and curvature of the space-times under discussion. Both local and global questions are covered, so a non-standard global structure for the Minkowski space-time gives rise to a model for a cosmic string.
Several illuminating figures help understanding the structure of the solutions. Two appendices present several forms of the spaces of constant curvature in 2 and 3 dimensions resp., accompanied by the concrete formulas of the coordinate transformations necessary to relate these forms to each other. The very useful reference list covers 42 pages.

83-02 Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C57 Black holes
83C15 Exact solutions to problems in general relativity and gravitational theory
83C22 Einstein-Maxwell equations
83F05 Relativistic cosmology
83C75 Space-time singularities, cosmic censorship, etc.
83C35 Gravitational waves
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53Z05 Applications of differential geometry to physics
83E30 String and superstring theories in gravitational theory
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
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