Cosmological Billiards

TL;DR

The paper demonstrates that near a spacelike singularity, the complex dynamics of Einstein-dilaton-p-form systems can be effectively modeled as billiard motion in hyperbolic space, revealing a deep geometric and algebraic structure.

Contribution

It provides a detailed Hamiltonian analysis of the near-singularity dynamics, establishing a billiard description in Lobachevskii space for arbitrary spacetime dimensions and linking it to Kac-Moody symmetries.

Findings

Billiard motion accurately describes the asymptotic dynamics near singularities.

Off-diagonal and p-form degrees of freedom become asymptotically frozen.

The approach connects gravitational dynamics to Kac-Moody algebraic structures.

Abstract

It is shown in detail that the dynamics of the Einstein-dilaton-p-form system in the vicinity of a spacelike singularity can be asymptotically described, at a generic spatial point, as a billiard motion in a region of Lobachevskii space (realized as an hyperboloid in the space of logarithmic scale factors). This is done within the Hamiltonian formalism, and for an arbitrary number of spacetime dimensions $D \geq 4$. A key role in the derivation is played by the Iwasawa decomposition of the spatial metric, and by the fact that the off-diagonal degrees of freedom, as well as the p-form degrees of freedom, get ``asymptotically frozen'' in this description. For those models admitting a Kac-Moody theoretic interpretation of the billiard dynamics we outline how to set up an asymptotically equivalent description in terms of a one-dimensional non-linear sigma-model formally invariant under the corresponding Kac-Moody group.