The possible metrics

This chapter is concerned with metrics admitting a group of motionstransitive on S3 or T3. Some solutions, such as the well-known Taub– NUT (Newman, Unti, Tamburino) metrics (13.49), cover regions of both types, joined across a null hypersurface which is a special group orbit (metrics admitting a Gr whose general orbits are N3 are considered in Chapter 24). As in the case of the homogeneous space-times (Chapter 12) we first consider the cases with multiply-transitive groups. From Theorems 8.10 and 8.17 we see that only G6 and G4 are possible.

Metrics with a G6 on V3

From §12.1, the space-times with a G6 on S3 have the metric (12.9); this always admits G3 transitive on hypersurfaces t = const and the various cases are thus included in (13.1)–(13.3) and (13.20) below. The relevant G3 types are V and VIIh if k = -1, I and VII0 if k = 0, and IX if k = 1.

Of the energy-momentum tensors considered in this book, the spacetimes with a G6 on T3 permit only vacuum and Λ-term Ricci tensors (see Chapter 5). Thus they will give only the spaces of constant curvature, with a complete G10, which also arise with G6 on S3 and those energymomentum types. Metrics with maximal G6 on S3 are non-empty and have an energy-momentum of perfect fluid type: see §14.2.