Frame-independence of the Inhomogeneous Mixmaster Chaos via Misner-Chitre'-like variables

TL;DR

This paper demonstrates that the chaotic behavior of the inhomogeneous Mixmaster universe near the singularity is frame-independent, using a covariant, gauge-independent approach with Misner-Chitre-like variables to analyze the dynamics as a billiard on a Lobachevsky plane.

Contribution

It introduces a covariant, gauge-independent method to analyze Mixmaster chaos, showing its frame-independence and mapping the dynamics to a billiard on a Lobachevsky plane using Misner-Chitre-like variables.

Findings

Chaos is covariant and independent of the choice of reference frame.

The Mixmaster dynamics can be represented as a billiard on a Lobachevsky plane.

A non-zero Lyapunov exponent confirms the presence of chaos at each spatial point.

Abstract

We outline the covariant nature,with respect to the choice of a reference frame, of the chaos characterizing the generic cosmological solution near the initial singularity, i.e. the so-called inhomogeneous Mixmaster model. Our analysis is based on a "gauge" independent ADM-reduction of the dynamics to the physical degrees of freedom. The out coming picture shows how the inhomogeneous Mixmaster model is isomorphic point by point in space to a billiard on a Lobachevsky plane. Indeed, the existence of an asymptotic (energy-like) constant of the motion allows to construct the Jacobi metric associated to the geodesic flow and to calculate a non-zero Lyapunov exponent in each space point. The chaos covariance emerges from the independence of our scheme with respect to the form of the lapse function and the shift vector; the origin of this result relies on the dynamical decoupling of the space-points which takes place near the singularity, due to the asymptotic approach of the potential term to infinite walls. At the ground of the obtained dynamical scheme is the choice of Misner-Chitre' like variables which allows to fix the billiard potential walls.