Abstract
Symmetry compatible time reparametrisations are calculated for orthogonal perfect fluid Hamiltonian Bianchi cosmologies by using an extended phase space containing the time variable itself. As in the vacuum case treated in an earlier article, the scale symmetry and the non-unimodular automorphism can be coupled to yield a constant of the motion which in the perfect fluid case is necessarily explicitly time dependent except for the stiff matter equation of state. To obtain the symmetry compatible lapse functions it is necessary to start from one of three 'source-adapted' time gauges, two of which are the Bogoyavlensky-Novikov and Jacobi time gauges. The latter reduces the field equations to geodesic flows on certain conformally flat Lorentzian geometries. In these Jacobi geometries, the symmetry giving rise to the time-dependent constant of the motion is just a homothetic motion.
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