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24Groups on null orbits. Plane waves24.1 IntroductionIn classifying space-times according to the group orbits in Chapters 11–22, we postponed the case of null orbits; they will be the subjectof this chapter. All space-times considered here satisfy the conditionR ab k a k b =0.A null surface N m is geometrically characterized by the existence ofa unique null direction k tangent to N m at any point of N m . The nullcongruence k is restricted by the existence of a group of motions actingtransitively in N m .The groups G r , r ≥ 4, on N 3 have at least one subgroup G 3 (Theorems8.5, 8.6 and Petrov (1966), p.179), which may act on N 3 , N 2 or S 2 .(AG 4on N 3 cannot contain G 3 on T 2 since the N 3 contains no T 2 .) For G 3 onS 2 , one obtains special cases of the metric (15.4) admitting either a groupG 3 on N 3 or a null Killing vector (see Barnes (1973a)). For G 3 on N 2 ,the metric also admits a null Killing vector (Petrov 1966, p.154, Barnes1979).Thus we need only consider here the groups G 3 on N 3 (§24.2), G 2 onN 2 (§24.3), and G 1 on N 1 (§24.4). As we study the case of null Killingvectors (G 1 on N 1 ) separately, we can also restrict ourselves to groupsG 3 on N 3 and G 2 on N 2 generated by non-null Killing vectors. It will beshown that in these cases, independent of the group structure, there isalways a non-expanding, non-twisting and shearfree null congruence k.None of the space-times with a G 4 on N 3 is compatible with the typesof energy-momentum tensors considered in this book (see §5.2) (Lautenand Ray 1977), and all space-times admitting a G 3 on N 3 or a G 2 onN 2 (generated by non-null Killing vectors) are algebraically special andbelong to Kundt’s class (Chapter 31).375