The automorphism groups for Bianchi universe models and computer-aided invariant classification of metrics

Abstract

An implementation of the time-dependent automorphism formalism for Bianchi cosmologies, in the computer algebra system SHEEP, is described. Its application to the particular case of Bianchi cosmologies admitting vacuum plane-wave solutions is given in relation to the equivalence problem. Using this approach, it is shown that the invariant characterisations of these metrics allow many properties of these spacetimes to be obtained in a straightforward manner. In particular, the invariant characterisations are used to determine the subclasses of spatially homogeneous vacuum plane-wave solutions which simultaneously admit distinct foliations with different Bianchi symmetries. The subclasses of the general vacuum plane-wave solution which admit a Bianchi group on a spacelike hypersurface are evaluated. These results are found to be expressible graphically, in the form of a phase plane which is physically interpretable as describing the amplitude and phase of polarisation of the plane wave. As a final application, it is shown that the Lifshitz-Khalatnikov Bianchi VIh solution is the only vacuum plane-wave solution for which the gravitational wave is linearly polarised.