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504 32 Kerr–Schild metrics(Martín and Senovilla 1986, Senovilla and Sopuerta 1994). The resultingperfect fluid solutions are of Petrov type D and admit a Killingvector ∂ u (and a second Killing vector ∂ y if b = 0,h = 1, leadingto a diagonal non-separable metric). The four-velocity ũ a , ũ a dx a =−dt [2M(t)+2S(x, y)] −1/2 ,doesnot lie in the plane spanned by the twonull eigenvectors of the Weyl tensor. If k is non-shearing and non-twisting(σ =0=ρ − ρ), several classes have been constructed by Martín andSenovilla (1986, 1988). Among them are the type D solutions (33.11) andseveral static spherically symmetric perfect fluids. Sopuerta (1998b) studiedthe case of a non-shearing but expanding k, with a rigidly rotatingperfect fluid, and found a class of new solutions (of Petrov types D andII).All solutions (33.9) with H 0 ̸= 0 are of the generalized Kerr–Schildform (32.84).All static spherically-symmetric space-times can be written in the generalizedKerr–Schild formd˜s 2 =e 2U(r) ds 2 0 − 2S(r)(dt ± dr) 2 , (32.100)where ds 2 0 is a Minkowski space (Mitskievic and Horský 1996), but thefirst part is not necessarily a perfect fluid metric.Starting with a perfect fluid LRS metric[ds 2 = −dt 2 + A 2 (t)dx 2 + B 2 (t) dy 2 +Σ 2 (y, k)dz 2] , (32.101)see §13.1, one can make the transformation (Senovilla 1987a)d˜s 2 =ds 2 + A 2 (t)f(t)(dx ± dt/A) 2 , f(t) =( ) B 2 [ ∫ ] Ac 1A B 4 dt + c 2 .(32.102)The new metric is again of the LRS class, with pressure and mass densitybeing related by˜µ + ˜p =(µ + p)(1 + f). (32.103)Patel and Vaidya (1983) showed that a generalized Kerr–Schild transformationd˜s 2 = a 2[ dx 2 +dy 2 + 1 2 e2x dz 2 − (dt +e x dz) 2] −2S(y)(dy−e x dz−dt) 2 ,(32.104)S(y) =c sin(y √ 2), c= const, κ 0 p =(a 2 − 2S)/2a 4 ,κ 0 (µ +3p) =2/a 2 ,