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534 34 Application of generation techniques to general relativity(for the details of these calculations see the original papers of Kinnersleyand Chitre or Hoenselaers 1984).Now we have cast the relevant equations into the form (10.58) and wealso know that the solution of the linearized equations can be written inthe form (10.63). Before we proceed to integrate (10.64) we shall digressfor historical reasons and examine (10.67) and (10.68) in more detail.Here, the matrix function G(λ, µ) as defined in (10.65) satisfies∇G(λ, µ) =ɛF + (λ)ɛ∇F (µ). (34.59)For static solutions, i.e. solutions of Weyl’s class, cp. §20.2, the matrixF (λ) can be calculated without much ado. Indeed, if f is given by(f = A −1 RA, A = diag (e U , e −U 0 W2 )), R = , (34.60)1 0thenF (λ) =A −1 Y (λ)B(λ), B(λ) = diag (e β(λ) , e −β(λ) ),(Y (λ) = 1 −i(1 − 2λV + S(λ)) −2+2λV + S(λ)2 S−1 (λ)2λ−2i), (34.61)∇(W ∇U) =0, S(λ)∇β(λ) =(1− 2λV )∇U − 2λW ˜∇U, β(0) = U.34.3.2 Infinitesimal transformations and transformations preservingMinkowski spaceIn this subsection we shall investigate (10.67). We recall that the matricesN nm are defined as the expansion coefficients ofG(λ, µ) = 1∞∑λ − µ [−λ1 + µF −1 (λ)F (µ)] = N nm λ n µ m (34.62)m,n=0and transform ask∑Ṅ nm = α k N n+k,m − N n,m+k α k − N ns α k N k−s,m . (34.63)s=1Note that, due to (34.56), G(0,µ)=iF (µ). The lower left element of H,H 21 , is the Ernst potential and from the explicit expression (34.61) withU = β = 0 one can verify by direct calculation thatα k,21 : E→1 − iɛ(2r) k+1 P k+1 (cos ϕ),r 2 = W 2 + V 2 ,α k,11 : E→1 − ɛ(2r) k P k (cos ϕ), (34.64)tan ϕ = V/W,α k,12 : E→1 − iɛ(2r) k−1 P k−1 (cos ϕ),