Extended dynamics and symmetries in vacuum Bianchi cosmologies

Abstract

Symmetries and time reparametrisations are used in a series of articles to study the Hamiltonian equations for certain vacuum and orthogonal perfect fluid Bianchi cosmologies. Using the framework of an extended phase space containing the time variable itself, one may couple the scale invariance symmetry and the non-unimodular automorphism admitted by all the non-semisimple Bianchi types to obtain a variational symmetry whose associated constant of the motion in general depends explicitly on the time. In the vacuum case this is time independent for certain preferred choices of time gauge, corresponding to preferred choices of lapse functions which may be constructed from symmetry invariants. One such preferred choice of lapse is the Jacobi lapse, for which the field equations are reduced to a geodesic flow on a conformally flat Lorentzian geometry. This article studies such preferred choices of lapse functions and the associated constants of the motion for the vacuum case.