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12Homogeneous space-times12.1 The possible metricsA 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 wherethe group is or contains a simply-transitive G 4 ; see §8.6 and below. Difficultiesmay arise when there is a multiply-transitive group G r , r>4,not containing a simply-transitive subgroup, and we shall consider suchpossibilities first. In such space-times, there is an isotropy group at eachpoint. From the remarks in §11.2 we see that there are only a limitednumber of cases to consider, and we take each possible isotropy groupin turn.For G r , r ≥ 8, we have only the metrics (8.33) with constant curvatureadmitting an I 6 and a G 10 .If the space-time admits a G 6 or G 7 , and its isotropy group containsthe two-parameter group of null rotations (3.15), but its metric is not ofconstant curvature, then it is either of Petrov type N , in which case wecan find a complex null tetrad such that (4.10) holds, or it is conformallyflat, with a pure radiation energy-momentum tensor, and we can choosea null tetrad such that (5.8) holds with Φ 2 = 1. In either case the tetradis fixed up to null rotations (together with a spatial rotation in the lattercase). The covariant derivative of k in this tetrad must be invariant underthe null rotations, which immediately givesκ = ρ = σ = ε +¯ε = τ +ᾱ + β =0. (12.1)Since τ and σ are invariantly defined for the tetrad described, (7.21p)yieldsτ(τ + β − ᾱ) =0, (12.2)171