Spacelike Singularities and Hidden Symmetries of Gravity

TL;DR

This paper explores the connection between spacelike singularities in gravity, billiard dynamics in hyperbolic space, and the role of Lorentzian Kac-Moody algebras as potential symmetries of gravitational theories, especially in M-theory.

Contribution

It provides a comprehensive review of how infinite-dimensional Kac-Moody algebras relate to gravitational dynamics near singularities and constructs models to make these symmetries explicit.

Findings

Billiard motion in hyperbolic space models near singularities.

Coxeter groups correspond to Weyl groups of Kac-Moody algebras.

E10 algebra may underlie M-theory symmetries.

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.