# 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

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
, 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.