By Vladimir Belinski and Marc Henneaux
Cambridge University Press
This monograph discusses at length the structure of the general solution of the Einstein equations with a cosmological singularity in Einstein-matter systems in four and higher space–time dimensions, starting from the fundamental work of Belinski (the book’s lead author), Khalatnikov and Lifshitz (BKL) – published in 1969.
The text is organised in two parts. The first, comprising chapters one to four, is dedicated to an exhaustive presentation of the BKL analysis. The authors begin deriving the oscillatory, chaotic behaviour of the general solution for pure Einstein gravity in four space–time dimensions by following the original approach of BKL. In chapters two and three, homogeneous cosmological models and the nature of the chaotic behaviour near the cosmological singularity are discussed. In these three chapters, the properties of the general solution of the Einstein equation are studied in the case of empty space in four space–time dimensions. The fourth chapter instead deals with different systems: perfect fluids in four space–time dimensions; gauge fields of the Yang–Mills and electromagnetic types and scalar fields, also in four space–time dimensions; and pure gravity in higher dimensions.
The second part of the book (chapters five to seven) is devoted to a model in which the chaotic oscillations discovered by BKL can be described in terms of a “cosmological billiard” system. In chapter five, the billiard description is provided for pure Einstein gravity in four dimensions, without any simplifying symmetry assumption, while the following chapter extends this analysis to arbitrary higher space–time dimensions and to general systems containing gravity coupled to matter fields. Finally, chapter seven covers the intriguing connection between the BKL asymptotic regime and Coxeter groups of reflections in hyperbolic space. Four appendices complete the treatment.
Quite technical and advanced, this book is meant for theoretical and mathematical physicists working on general relativity, supergravity and cosmology.
