Symmetries of the Stephani universes

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Abstract
Two classes of solutions for conformally flat perfect fluids exist depending on whether the fluid expansion vanishes or not. The expanding solutions have become known as the Stephani universes and are generalizations of the well known Friedmann-Robertson-Walker solutions; the solutions with zero expansion generalize the interior Schwarzschild solution. The isometry structure of the expansion-free solutions was completely analysed some time ago. For the Stephani universes it is shown that any Killing vector is orthogonal to the fluid flow and so the situation for the expanding case is somewhat simpler than the expansion-free case where `tilted' Killing vectors may exist. The existence of isometries in Stephani universes depends on the dimension r of the linear space spanned by certain functions of time and which appear in the metric. If r is 4 or 5, no Killing vectors exist. If r = 3, the isometry group is one dimensional. If r = 2, the spacetime admits a complete three-dimensional isometry group with two-dimensional orbits. If r = 1, there are six Killing vectors and the spacetime is Friedmann-Robertson-Walker. Not all choices of the metric functions and lead to distinct spacetimes: the ten-dimensional conformal group, which acts on each of the hypersurfaces orthogonal to the fluid flow, preserves the overall form of the metric, but induces a group of transformations on these five metric functions which is locally isomorphic to . A result of the same ilk is derived for the non-expanding solutions.

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