The Speciality Index as invariant indicator in the BKL mixmaster dynamics

The long standing difficulty in General Relativity of classifying the dynamics of cosmological models, e.g.\ as chaotic, is directly related to the gauge freedom intrinsic to relativistic spacetime theories: in general the invariance under diffeomorphisms makes any analysis of dynamical evolution dependent from the particular choice of time slicing one uses. We show here that the speciality index, a scalar dimensionless curvature invariant that has been mainly used in numerical relativity as an indicator of the special or non-special Petrov type character of a spacetime, is a time-independent quantity (a pure number) at each Kasner step of the Belinski-Khalatnikov-Lifshitz (BKL) map approximating the Mixmaster cosmology. Thus the BKL dynamics can be characterized in terms of the speciality index, i.e. in terms of curvature invariants directly related to observables. Possible applications for the associated Mixmaster dynamics are discussed.