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476 31 Non-diverging solutions (Kundt’s class)Theorem 31.1 From a known solution (‘background metric’) one cangenerate other solutions with the same P, W ,v and Φ 1 , by choosingnewfunctions W 0 ,G 0 , Φ 0 2 satisfyingthe linear equations (31.24) and by thenchoosinga new function H 0 satisfyingthe linear equation (31.25).To conclude this section we give the tetrad components Ψ 2 and Ψ 3 ofthe Weyl tensor for Einstein–Maxwell (and pure radiation) fields:Ψ 2 = 1 2 P 2 ( W ,vζ − 1 2 W ,vW ,v ),Ψ 3 = P()[ ( ) ]∂ ζ+ 1 2 W 1,v 2 P 2 W ,ζ + W ,ζ− (ln P ) ,u(31.27)−P (P 2 W ) ,ζζ+2P ,ζ (P 2 W ) ,ζ− 1 2 PW ,v(P 2 W ) ,ζ+ 1 2 R 32.For W = 0, the last expression reduces toΨ 3 = 1 2 P [H ,v − (ln P ) ,u ] ,ζ.(31.28)31.5 Vacuum solutions31.5.1 Solutions of types III and NThe vacuum solutions of types III and N in Kundt’s class are completelyknown (Kundt 1961; for the subcase W ,v = 0 see also Pandya and Vaidya1961). The field equation R 12 = 0 and the type III condition Ψ 2 =0give(ln P ) ,ζζ= 0, so that one can use a coordinate transformation (31.10a) tomake P = 1. The field equation R 34 = 0 determines H ,vv asH ,vv = − 1 2 W ,vW ,v . (31.29)The field equation R 11 = 0 and the condition Ψ 2 = 0 lead toW ,v = −2n ,ζ , n = n, (e n ) ,ζζ =0=(e n ) ,ζζ.(31.30)Two cases can occur: either W ,v vanishes, or it can be transformed toW ,v = −2/(ζ + ζ) by the transformations (31.10a) and (31.10c).In the case W ,v = 0, and with P =1,R 31 = 0 gives()R 31 = H ,v + 1 2 W ,ζ − 1 2 W ,ζ =0. (31.31),ζThis equation implies H ,v + 1 2 (W ,ζ − W ,ζ) =f(ζ,u). Because H ,v is real,it follows thatW ,ζ− W ,ζ = f(ζ,u) − f(ζ,u), (31.32)