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378 24 Groups on null orbits. Plane wavessupposition, the two spacelike Killing vectors ξ and η span a null surfaceN 2 , i.e.ξ [a η b] ξ a η b =0. (24.14)At any point of N 2 we have a unique null direction tangent to N 2 :k = ξ − Ωη, Ω ≡ (η c η c ) −1 ξ b η b (24.15)(cp. the similar expression after (19.19) in the case of stationary axisymmetricfields). The null vector field k is orthogonal to the two Killingvectors,k a ξ a =0=k a η a , (24.16)and has the propertiesk a ;abk b =0, k a;b k b =0, ω =ik [a;b] m a m b =0,√2m a =(η c η c ) −1/2 η a +iq a , q a η a =0=q a k a , q a q a =1,(24.17)which can be verified with the aid of the Killing equations, the commutatorrelation for ξ and η, and the formula (24.14). Equations (24.17) holdfor both the group structures G 2 I and G 2 II (§8.2).We insert the information (24.17) on k into (6.33) and conclude, againfrom (24.3) together with (24.5), that the null vector field (24.15) is nonexpandingand shearfree, Θ = σ = 0. (The function Ω in (24.15) obeysthe relations Ω ,a k a =0=Ω ,a m a .) From conditions (24.5) it follows thatk is a Ricci eigenvector. So we can conclude again from Theorem 7.1 thatif a vacuum, Einstein–Maxwell or pure radiation field admits a group ofmotions G 2 on N 2 , then it is algebraically special.Theorem 24.3 In vacuum, Einstein–Maxwell or pure radiation fieldsadmittinga group G 2 on N 2 there is a non-expanding, non-twisting, andshearfree null congruence and the metric can be transformed into (cp.§31.2)ds 2 =2P −2 dζd¯ζ − 2du(dv + W dζ + W d¯ζ + Hdu),P, H real, W complex, P ,v =0, W ,vv =0.(24.18)The gauge transformationu ′ = h(u), v ′ = v/h ,u + g(ζ, ¯ζ,u), ζ ′ = ζ ′ (ζ,u) (24.19)preserving the form of the line element (24.18) can be used to bring theKilling vectors ξ and η satisfying (24.14) into the simple formη = ∂ x , ξ = u∂ x + S(u, y)∂ v ,√2ζ = x +iy. (24.20)