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29.1 Twistingvacuum solutions – the field equations 439withΓ 213 = iIm[PρW |1 + Hρ +(lnρ) |3 ], Γ 433 = H |4 , (29.9)Γ 321 = −iIm[PρW |1 + Hρ]+Re[(lnP ρ) |3 ]. (29.10)We start with R 34 = 0 (which is in fact a consequence of the mainequations, including R 12 = 0) and integrate it byΓ 213 +Γ 433 = −(ln P ) ,u +(m +iM)ρ 2 , (m +iM) |4 =0, (29.11)m and M being real functions of integration. The real part of (29.11)yieldsH = −(r+r 0 )(ln P ) ,u +Re [(m +iM)ρ]+r 0 ,u +K/2, K |4 =0. (29.12)Inserting both results into R 12 = 0 and making repeated use of (27.24) and(29.6), we obtain K in terms of P and L. The result can be summarizedas follows:Theorem 29.1 A space-time admits a geodesic, shearfree and divergingnull congruence k and satisfies R 44 = R 14 = R 11 = R 12 = R 34 =0exactlyif the metric can be written asds 2 =2ω 1 ω 2 − 2ω 3 ω 4 , ω 1 = −dζ/Pρ = ω 2 ,ω 3 =du + Ldζ + Ldζ, ω 4 =dr + W dζ + W dζ + Hω 3 ,the metric functions satisfying(29.13a)ρ −1 = −(r + r 0 +iΣ), 2iΣ =P 2 (∂L − ∂L), (29.13b)W = L ,u /ρ + ∂(r 0 +iΣ), ∂ ≡ ∂ ζ − L∂ u , (29.13c)H = 1 2 K − (r + r0 )(ln P ) ,u − m(r + r0 )+MΣ(r + r 0 ) 2 +Σ 2 + r,u, 0 (29.13d)[]K =2P 2 Re ∂(∂ ln P − L ,u ) , (29.13e)[]M =ΣK + P 2 Re ∂∂Σ − 2L ,u ∂Σ − Σ∂ u ∂L(29.13f)(Kerr 1963a, Debney et al. 1969, Robinson et al. 1969a, Trim and Wainwright1974).The line element (29.13) shows a remarkably simple r-dependence. Furthermore,all remaining functions of ζ,ζ and u are given in terms of thecomplex function L and the real functions r 0 ,P and m. Asr 0 and P canbe removed by coordinate transformations (see §29.1.4), L (complex) andm (real) can be considered to represent the metric.