12 - Homogeneous space-times

Summary

The possible metrics

A homogeneous space-time is one which admits a transitive group of motions. It is quite easy to write down all possible metrics for the case where the group is or contains a simply-transitive G₄; see §8.6 and below. Difficulties may arise when there is a multiply-transitive group G_r, r > 4, not containing a simply-transitive subgroup, and we shall consider such possibilities first. In such space-times, there is an isotropy group at each point. From the remarks in §11.2 we see that there are only a limited number of cases to consider, and we take each possible isotropy group in turn.

For G_r, r ≥ 8, we have only the metrics (8.33) with constant curvature admitting an I₆ and a G₁₀.

If the space-time admits a G₆ or G₇, and its isotropy group contains the two-parameter group of null rotations (3.15), but its metric is not of constant curvature, then it is either of Petrov type N, in which case we can find a complex null tetrad such that (4.10) holds, or it is conformally flat, with a pure radiation energy-momentum tensor, and we can choose a null tetrad such that (5.8) holds with Φ² = 1. In either case the tetrad is fixed up to null rotations (together with a spatial rotation in the latter case).