Cosmological Singularities, Einstein Billiards and Lorentzian Kac-Moody Algebras

TL;DR

This paper explores the behavior of solutions near cosmological singularities, revealing they can be modeled as billiard motions in Lorentzian space, with connections to Kac-Moody algebras and string theory.

Contribution

It demonstrates that near singularities, Einstein-matter solutions can be described by billiard dynamics related to Lorentzian Kac-Moody algebras, providing a unifying algebraic framework.

Findings

Billiard motion describes asymptotic behavior near singularities.

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

Evidence of a link between supergravity solutions and E(10)/K(E(10)) coset 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).