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29.2 Some general classes of solutions 449information needed to construct the full metric according to (29.13). As Pcan be made equal to 1 by a coordinate transformation (29.52), the abovevacuum solutions contain three disposable analytic functions. In generalthey will not admit any Killing vector. Physically the class of algebraicallyspecial diverging vacuum solutions that satisfy the conditions (29.51) canbe characterized (Trim and Wainwright 1974) as the only solutions whichare non-radiative in the sense that the Weyl tensor asymptotically (forlarge r) behavesasC abcd = II abcd /r 3 + O(r 4 ). (29.59)29.2.5 The case I =0=L ,uIf we add L ,u = 0 to the conditions (29.51), then the metric becomesindependent of u; ξ = ∂ u is a Killing vector. It can be shown (Trimand Wainwright 1974) that instead of L ,u = 0 the equivalent conditionΣ ,u = 0 may be imposed. Like I =0, the condition L ,u = 0 is not invariant(strictly speaking, the condition is that L ,u can be made zero). So thecoordinate transformations are now restricted to (29.52) with constant k.Because of (29.51), L ,u = 0 yields G = −(ln P ) ,ζ , which is compatiblewith I = G 2 − G ,ζ = 0 only if P ,ζζ =0, i.e. with real P only ifP = αζζ + βζ + β ζ + δ. (29.60)Equation (29.60) shows that the 2-space 2dζdζ/P 2 has constant curvatureK =2(αδ − ββ).Instead of (29.53), (29.54) and (29.58) the constitutive parts of themetric are now given by (29.60) andm +iM = Z(ζ), L = − 12P 2 ∫Z(ζ)dζ(αζ + β) + l 1(ζ)2 P 2 , (29.61)with Z = −A/(αζ + β) 3 (note that we have chosen a gauge with G =−(αζ + β)/P ̸= 0, so that in the case K = 0 we have to stick to a Pwith α or β ̸= 0). The solutions contain the disposable functions Z(ζ)and l 1 (ζ). The value of K can be transformed to 0, ±1, and L may besimplified by a transformation u ′ = u + h(ζ,ζ), L ′ = L − h ,ζ , e.g. so thatL does not contain terms in m.A subcase of these solutions is the class m +iM = const. It containssome of the well-known type D solutions such as Kerr and NUT (seebelow in §29.5) as well as the Kerr and Debney (1970)/Demiański(1972)four-parameter (m, M, a, c) solution, which corresponds toL = −P −2 [2iM/ζ +iζ(M + a)+ 1 4 icζ ln(ζ/√ 2)]. (29.62)