Einstein billiards and spatially homogeneous cosmological models

TL;DR

This paper investigates the chaotic behavior of spatially homogeneous cosmological models using billiard dynamics, identifying conditions for chaos and linking finite volume billiards to hyperbolic Kac-Moody algebras.

Contribution

It provides a comprehensive classification of models exhibiting chaos and connects billiard dynamics to algebraic structures in higher-dimensional spacetimes.

Findings

Chaos occurs in D=5 when off-diagonal metric elements are present.

Finite volume billiards correspond to fundamental Weyl chambers of hyperbolic Kac-Moody algebras.

Certain initial conditions lead to different algebraic structures.

Abstract

In this paper, we analyse the Einstein and Einstein-Maxwell billiards for all spatially homogeneous cosmological models corresponding to 3 and 4 dimensional real unimodular Lie algebras and provide the list of those models which are chaotic in the Belinskii, Khalatnikov and Lifschitz (BKL) limit. Through the billiard picture, we confirm that, in D=5 spacetime dimensions, chaos is present if off-diagonal metric elements are kept: the finite volume billiards can be identified with the fundamental Weyl chambers of hyperbolic Kac-Moody algebras. The most generic cases bring in the same algebras as in the inhomogeneous case, but other algebras appear through special initial conditions.