Introduction

In classifying space-times according to the group orbits in Chapters 11–22, we postponed the case of null orbits; they will be the subject of this chapter. All space-times considered here satisfy the condition Rabkakb = 0.

A null surface Nm is geometrically characterized by the existence of a unique null direction k tangent to Nm at any point of Nm. The null congruence k is restricted by the existence of a group of motions acting transitively in Nm.

The groups Gr, r ≥ 4, on N3 have at least one subgroup G3 (Theorems 8.5, 8.6 and Petrov (1966), p.179), which may act on N3, N2 or S2. (A G4 on N3 cannot contain G3 on T2 since the N3 contains no T2.) For G3 on S2, one obtains special cases of the metric (15.4) admitting either a group G3 on N3 or a null Killing vector (see Barnes (1973a)). For G3 on N2, the metric also admits a null Killing vector (Petrov 1966, p.154, Barnes 1979).

Thus we need only consider here the groups G3 on N3 (§24.2), G2 on N2 (§24.3), and G1 on N1 (§24.4). As we study the case of null Killing vectors (G1 on N1) separately, we can also restrict ourselves to groups G3 on N3 and G2 on N2 generated by non-null Killing vectors. It will be shown that in these cases, independent of the group structure, there is always a non-expanding, non-twisting and shearfree null congruence k.