Covariant Description of the Inhomogeneous Mixmaster Chaos

TL;DR

This paper demonstrates that the chaotic behavior of the early universe near the singularity is covariant and can be described as a billiard motion on a Lobachevsky plane, independent of gauge choices.

Contribution

It provides a gauge-independent, covariant analysis of Mixmaster chaos, linking the dynamics to a billiard model on a hyperbolic plane and explicitly calculating Lyapunov exponents.

Findings

Chaos is covariant and gauge-independent.

Dynamics is equivalent to a billiard on a Lobachevsky plane.

Explicit Lyapunov exponent calculation confirms chaos.

Abstract

We outline the covariant nature of the chaos characterizing the generic cosmological solution near the initial singularity. Our analysis is based on a "gauge" independent ADM-reduction of the dynamics to the physical degrees of freedom, and shows that the dynamics is isomorphic point by point in space to a billiard on a Lobachevsky plane. The Jacobi metric associated to the geodesic flow is constructed and a non-zero Lyapunov exponent is explicitly calculated. The chaos covariance emerges from the independence of the form of the lapse function and the shift vector.