Spacelike Singularities and Hidden Symmetries of Gravity

Marc Henneaux
Physique Théorique et Mathématique
Université Libre de Bruxelles & International Solvay Institutes
Boulevard du Triomphe, ULB – C.P. 231
B-1050 Bruxelles, Belgium

'External link'http://www.ulb.ac.be/sciences/ptm/pmif/membres/henneaux.html

Daniel Persson
Physique Théorique et Mathématique
Université Libre de Bruxelles & International Solvay Institutes
Boulevard du Triomphe, ULB – C.P. 231
B-1050 Bruxelles, Belgium

Philippe Spindel
Service de Mécanique et Gravitation
Université de Mons-Hainaut, Académie Wallonie-Bruxelles
Avenue du Champ de Mars 6, B-7000 Mons, Belgium

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 E10, 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.