Cosmological Singularities, Billiards and Lorentzian Kac-Moody Algebras

TL;DR

This paper explores the asymptotic behavior of solutions near cosmological singularities, revealing a billiard dynamics described by Lorentzian Kac-Moody algebras, especially highlighting the role of E_{10} in supergravity.

Contribution

It demonstrates that the near-singularity dynamics of Einstein-matter systems can be modeled as billiard motions in Lorentzian spaces linked to Kac-Moody algebras, establishing a deep algebraic structure in cosmological solutions.

Findings

Billiard tables correspond to Weyl chambers of Lorentzian Kac-Moody algebras.

Pure Einstein gravity in any dimension exhibits billiard dynamics related to these algebras.

Evidence suggests a link between supergravity solutions and geodesics in E_{10}/K(E_{10}) space.

Abstract

The structure of the general, inhomogeneous solution of (bosonic) Einstein-matter systems in the vicinity of a cosmological singularity is considered. We review the proof (based on ideas of Belinskii-Khalatnikov-Lifshitz and technically simplified by the use of the Arnowitt-Deser-Misner Hamiltonian formalism) that the asymptotic behaviour, as one approaches the singularity, of the general solution is describable, at each (generic) spatial point, as a billiard motion in an auxiliary Lorentzian space. For certain Einstein-matter systems, notably for pure Einstein gravity in any spacetime dimension D and for the particular Einstein-matter systems arising in String theory, the billiard tables describing asymptotic cosmological behaviour are found to be identical to the Weyl chambers of some Lorentzian Kac-Moody algebras. In the case of the bosonic sector of supergravity in 11 dimensional spacetime the underlying Lorentzian algebra is that of the hyperbolic Kac-Moody group E_{10}, and there exists some evidence of a correspondence between the general solution of the Einstein-three-form system and a null geodesic in the infinite dimensional coset space E_{10} / K (E_{10}), where K (E_{10}) is the maximal compact subgroup of E_{10}.