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376 24 Groups on null orbits. Plane waves24.2 Groups G 3 on N 3In this section we study space-times V 4 for which the orbits of a group G 3are null hypersurfaces N 3 parametrized by u (u being constant in eachN 3 ). The null vector k a = −u ,a is orthogonal to all vectors tangent to N 3 ,and therefore it is orthogonal to the three independent Killing vectors ξ A(A =1,...,3),ξ a Ak a =0=k a k a , ξ A(a;b) =0, k a = −u ,a , k a = c A (x)ξ a A. (24.1)From these relations it follows immediately that the null congruence kand tensors obtained from it by covariant differentiation have zero Liederivatives with respect to ξ A , e.g.Now we use (6.33),k a ;b ξ b A − k b ξ a A;b =0, k c ;caξ a A =0. (24.2)R ab k a k b = k b ;abk a = −k a;b k a;b = −2(σ¯σ +Θ 2 ) ≤ 0, (24.3)where σ and Θ denote the shear and the expansion, respectively. By continuity,the energy conditions (5.18)T ab u a u b ≥ 0, T ab T a cu b u c ≤ 0, u a u a < 0 (24.4)must still be true if we replace the timelike vector u by a null vector k:R ab k a k b ≥ 0, T ab T a ck b k c ≤ 0, k a k a =0. (24.5)Comparison of (24.3) with (24.5) leads to σ =0=Θ,i.e.k a;b =2k (a p b) , p a k a =0, R ab k a k b =0. (24.6)With (24.6), the second energy condition (24.4) readsR ab R a ck b k c = k b ;abk c:a ;c ≤ 0, (24.7)whereas (24.6) leads tok b ;abk c;a ∣;c =2∣R ab m a k b∣ ∣2 ≥ 0. (24.8)Comparison of the two inequalities givesR ab k a k b =0=R ab m a k b ⇔ k [c R a]b k b =0. (24.9)Theorem 24.1 If the energy-momentum tensor of a space-time with aG 3 on N 3 satisfies the energy conditions (24.5), then the non-twisting(and geodesic) null congruence k is non-expandingand shearfree and aneigendirection of the Ricci tensor (Kramer 1980).