Vol. 11 (2008) > lrr-2008-1

doi: 10.12942/lrr-2008-1
Living Rev. Relativity 11 (2008), 1

Spacelike Singularities and Hidden Symmetries of Gravity

1 Physique Théorique et Mathématique, Université Libre de Bruxelles & International Solvay Institutes, Boulevard du Triomphe, ULB – C.P. 231, B-1050 Bruxelles, Belgium
2 Institut für Theoretische Physik, ETH Zürich, Wolfgang-Pauli-Str. 27, 8093 Zürich, Switzerland
3 Service de Mécanique et Gravitation, Université de Mons-Hainaut, Académie Wallonie-Bruxelles, Avenue du Champ de Mars 6, B-7000 Mons, Belgium

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Article Abstract

We review the intimate connection between (super-)gravity close to a spacelike singularity (the “BKL-limit”) and the theory of Lorentzian Kac-Moody algebras. We show that in this limit the gravitational theory can be reformulated in terms of billiard motion in a region of hyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite) sequence of reflections, which are elements of a Lorentzian Coxeter group. Such Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras, suggesting that these algebras yield symmetries of gravitational theories. Our presentation is aimed to be a self-contained and comprehensive treatment of the subject, with all the relevant mathematical background material introduced and explained in detail. We also review attempts at making the infinite-dimensional symmetries manifest, through the construction of a geodesic sigma model based on a Lorentzian Kac-Moody algebra. An explicit example is provided for the case of the hyperbolic algebra E_10, which is conjectured to be an underlying symmetry of M-theory. Illustrations of this conjecture are also discussed in the context of cosmological solutions to eleven-dimensional supergravity.

Keywords: hidden symmetries, duality, Kac-Moody algebras

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Since a Living Reviews in Relativity article may evolve over time, please cite the access <date>, which uniquely identifies the version of the article you are referring to:

Marc Henneaux and Daniel Persson and Philippe Spindel,
"Spacelike Singularities and Hidden Symmetries of Gravity",
Living Rev. Relativity 11,  (2008),  1. URL (cited on <date>):
http://www.livingreviews.org/lrr-2008-1

Article History

ORIGINAL http://www.livingreviews.org/lrr-2008-1
Title Spacelike Singularities and Hidden Symmetries of Gravity
Author Marc Henneaux / Daniel Persson / Philippe Spindel
Date accepted 12 February 2008, published 24 April 2008
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