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30.4 Charged vacuum metrics 459where the terms of higher order in ρ occurring in Ψ 3 or Ψ 4 vanish identicallyif Ψ 2 or Ψ 2 and Ψ 3 , respectively, vanish.30.4 Charged vacuum metricsBy inspecting the field equations (30.19a)–(30.19c) one immediately seesthat they reduce to the vacuum case if no free radiation field is present,i.e. ifΦ 0 2 =0, Φ 0 1 = Q(ζ,ζ,u)/2 (30.22)holds. In that case, Maxwell’s equations (30.18) yieldand (in the gauge P ,u =0)Q = q(ζ,ζ)P 2 , (30.23)[∂ ζ − 2(L ,u − ∂ ζ ln P )] q =0. (30.24)As q and P do not depend on u, the same is true for L ,u − ∂ ζ ln P , i.e. asa consequence of Maxwell’s equations we obtainL ,u − ∂ ln P = G(ζ,ζ), P ,u =0. (30.25)These conditions coincides with the assumption (29.29) we made in thevacuum case to get special classes of solutions, see §29.2.1. We thushave proved the following generalization of Theorem 28.5 (on Robinson–Trautman Einstein–Maxwell fields):Theorem 30.2 All Einstein–Maxwell fields (aligned case) admittingadiverging, geodesic and shearfree null congruence with a non-radiative(Φ 0 2 =0)Maxwell field are given byds 2 =ds 2 0 − 1 2 κ 0QQρρ(du + Ldζ + Ldζ) 2 ,(Q = α(ζ)P 2 exp 2 ∫ )G(ζ,ζ)dζ , Φ 1 = Qρ 2 /2,(30.26)Φ 2 = 1 2 Pρ2 (2L ,u − ∂ ζ)Q +iPρ 3 Q(ΣL ,u − ∂Σ),where ds 2 0 is an algebraically special vacuum metric subject to (30.25), andα(ζ) is a disposable function (Robinson et al. 1969b, Trim and Wainwright1974).In §29.2 we gave a survey of all explicitly known vacuum solutionswhich satisfy the above conditions (and, moreover, (m +iM) ,u =0). For