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428 28 Robinson–Trautman solutionsWe can now attack the remaining field equationsR 12 = R 34 =2κ 0 Φ 1 Φ 1 , (28.34)R 13 =2κ 0 Φ 1 Φ 2 , R 33 =2κ 0 Φ 2 Φ 2 , (28.35)cp. (7.10)–(7.15) and (7.29). The expressions for R 12 ,R 13 ,R 33 and R 34in terms of the metric function can be taken from (28.2)–(28.5).Equation (28.34) gives the r-dependence of the function H as2H =∆lnP − 2r(ln P ) ,u − 2m(ζ,ζ,u)/r + κ 0 QQ/2r 2 , (28.36)with ∆ ≡ 2P 2 ∂ ζ ∂ ζ, and (28.35) yields the differential equations givenbelow in (28.37d)–(28.37e).Theorem 28.3 If an Einstein–Maxwell field admits a geodesic, shearfree,diverging but normal null vector field k which is an eigenvector of theMaxwell and Weyl tensors, then the metric is algebraically special andcan be written in the formds 2 =2r 2 P −2 (ζ,ζ,u)dζ dζ − 2du dr(28.37a)[− ∆lnP − 2r(ln P ) ,u − 2 r m(ζ,ζ,u)+ κ ]02r 2 Q(ζ,u)Q(ζ,u) du 2 ,and the electromagnetic field is given byΦ 1 = Q/2r 2 , Φ 2 = −P Q ,ζ/2r 2 + Ph(ζ,ζ,u)/r. (28.37b)P and m are real functions, h is complex and Q analytic in ζ. The fourfunctions have to obey∆∆ ln P +12m(ln P ) ,u − 4m ,u =4κ 0 P 2 hh,(28.37c)QQ ,u − QQ ,u =2P 2 (h Q ,ζ− hQ ,ζ ), (28.37d)h ,ζ =(Q/2P 2 ) ,u , m ,ζ = κ 0 h Q. (28.37e)(Equation (28.37d) is in fact a consequence of (28.37e) and the reality ofm.) Like all metrics of the Robinson–Trautman class (27.37), the line elementis preserved under the transformations (28.9) with the appropriatechange Q ′ = F,u −2 Q, h ′ = F,u −1 f ′−1 h. In addition to that, the Einstein andMaxwell field equations are invariant with respect to Q ′ =e ia Q, h ′ =e −ia h, a = const.To have a rough idea of the physical meaning of the quantities appearingin (28.37a)–(28.37b) one may think of m and Q as representing,