Frame independence of the inhomogeneous mixmaster chaos via Misner-Chitré-like variables

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 Arnowitt-Deser-Misner reduction of the dynamics to the physical degrees of freedom. The resulting 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 (energylike) constant of the motion allows one to construct the Jacobi metric associated with the geodesic flow and to calculate a nonzero 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-Chitré-like variables which allows one to fix the billiard potential walls.

Figure 1

${\Gamma }_Q(\xi ,\theta )$.

Figure 2

${\Gamma }_Q(u,v)$.