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444 29 Twistingvacuum solutionsTable 29.2.Twisting algebraically special vacuum solutionsP, m +iM and L ,u − (ln P ) ,ζ = G are assumed to be independent of u.∂ ζ I ̸= 0∂ ζ I =0=IL ,u ̸= 0 Equations only: Flat background. All(29.34)–(29.37) solutions are knownand given by (29.53),(29.54), (29.58)L ,u = 0 Equations: (29.41)–(29.44) All solutions areBackground: non-twisting type known and given byIII. Solutions (not exhaustive): (29.60)–(29.61)(29.45)–(29.50), (38.6)The main idea then runs as follows. Assume we are able to solve thedifferential equation (29.30), which is in fact one of the field equations,and have found a solution G. It then turns out that the second fieldequation (29.20b) can be solved in the sense that m +iM is given in termsof G, P and arbitrary functions or constants of integration. The third (andlast) field equation (29.20c) can now be reduced to a linear homogeneousdifferential equation for a real function, or can be solved explicitly. G issometimes called the ‘background’, because m +iM = 0 and L =[G +(ln P ) ,ζ ]u also give algebraically special solutions. Explicit solutions havebeen found for several cases, see Table 29.2 and the following subsections.It is possible to reduce and simplify the field equations in a similar wayeven if m +iM depends on u (so that (29.30) is no longer true). But nosolution is known so far which satisfies (29.29) but not (29.30).A third way of characterizing the known solutions starts from the observationthat the field equation (29.20b), i.e. ∂(m +iM) =3(m +iM)L ,ucan be integrated by introducing a complex function Φ viam +iM =Φ 3 ,u (29.31)(Stephani1983a). In terms of Φ, the field equation (29.20b) then reads∂(Φ ,u )=L ,u Φ ,u , and because of the commutator relation (29.17) it canbe written as∂ u ∂Φ=0. (29.32)This equation is integrated by ∂Φ =ϕ(ζ,ζ), and since (29.31) defines Φonly up to an additive function of ζ and ζ, ϕ can be set zero, and ∂Φ =0can then be read as giving L in terms of Φ:L =Φ ,ζ /Φ ,u . (29.33)