Overview

Written for researchers focusing on general relativity, supergravity, and cosmology, this is a self-contained exposition of the structure of the cosmological singularity in generic solutions of the Einstein equations, and an up-to-date mathematical derivation of the theory underlying the Belinski–Khalatnikov–Lifshitz (BKL) conjecture on this field. Part I provides a comprehensive review of the theory underlying the BKL conjecture. The generic asymptotic behavior near the cosmological singularity of the gravitational field, and fields describing other kinds of matter, is explained in detail. Part II focuses on the billiard reformulation of the BKL behavior. Taking a general approach, this section does not assume any simplifying symmetry conditions and applies to theories involving a range of matter fields and space-time dimensions, including supergravities. Overall, this book will equip theoretical and mathematical physicists with the theoretical fundamentals of the Big Bang, Big Crunch, Black Hole singularities, the billiard description, and emergent mathematical structures.

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9781107047471

Table of Contents

Reviews

'The present monograph is a carefully developed overview about the mathematical details of the big bang singularity, mainly within (but not restricted to) general relativity theory. Chapter 1 presents the basic structure of the singularity, including the Kasner-like and the oscillatory-like cases. Chapters 2 and 3 deal with the Bianchi models, especially the BLK-cases Bianchi VIII and IX and the chaotic character observed there. In chapter 4, the influence of matter and/or changed space-time dimension are discussed. Chapters 5 and 6 deal with the billiard representation of the dynamical system describing the approach to the singularity by a mathematical equivalence of the system of equations to the motion of a point particle in a region with boundary, where (like in the billiard game), the article is reflected at the boundary. This idea is formalized in chapter 7 by the introduction of the Coxeter group. The appendices are useful for several topics, e.g., the spinor field and the Kac-Moody algebra.' Hans-Jurgen Schmidt, Zentralblatt MATH

Author Information

Vladimir Belinski holds a permanent professor position at the International Center for Relativistic Astrophysics Network (ICRANet), Italy. He is noted for his role in several key developments in theoretical physics, including the Belinski–Khalatnikov–Lifshitz conjecture on the behavior of generic solutions of Einstein equations near a cosmological singularity, the Belinski–Zakharov transform, and the 'Inflationary Attractor'. He is co-author of the book Gravitational Solitons (Cambridge, 2001) and has received the Landau Prize of the Russian Academy of Sciences (1974) and Marcel Grossmann Award (2012). Marc Henneaux is Full Professor at the Université Libre de Bruxelles and Director of the International Solvay Institutes for Physics and Chemistry. He has contributed significantly to understandings of gravity and black holes in three dimensions, and the geometrical and algebraic aspects of theories with a gauge freedom (in particular the Becchi–Rouet–Stora–Tyutin symmetry and its cohomology). He has received the Francqui Prize (2000), the Humboldt Prize (2009), the Bogoliubov Prize (2014), the Fonds National de La Recherche Scientifique (FNRS) Quinquennal Prize in Fundamental Exact Sciences (2015), and two ERC advanced grants (2011–2015 and 2016–2021).

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