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32.3 Kerr–Schild Einstein–Maxwell fields 493which is simultaneously a Killing vector of flat space-time. The solutionscan be simplified by performing a Lorentz transformation. One can thusassume that if η ab ξ a ξ b < 0, P √ 2=1+Y Y ;ifη ab ξ a ξ b > 0, P √ 2=1−Y Y ;and if η ab ξ a ξ b =0,P =1.(iv) The points (Pρ −1 ) = 0 are singularities of the Riemannian space.In general Y is a multivalued function with these singularities as branchpoints. The only solution whose singularities are confined to a boundedregion is the Kerr metric (Kerr and Wilson 1979).(v) For a timelike Killing vector ξ, the particular case Φ = −iaY leadsto the Kerr solution (20.34). Its Kerr–Schild form reads (with r given by(x 2 + y 2 )(r 2 + a 2 ) −1 + z 2 r −2 =1)ds 2 =dx 2 +dy 2 +dz 2 − dt 2 ++rar 2 (x dx + y dy) −+ a2 [2mr3r 4 + a 2 z 2 dt + z r dz (32.47)r 2 + a 2 (x dy − y dx) ] 2,the double Debever–Penrose null vector being[ω 3 =2 1/2 rdt + z ]r(x dx + y dy ) a(x dy − y dx)dz +r + z r r 2 + a 2 −r 2 + a 2 = −k i dx i .(32.48)For a different characterization of the Kerr–Schild vacuum solutions see§§29.2.5 and 32.4.1.32.2.2 The case ρ = −(Θ+iω) =0The non-expanding and non-twisting solutions have been treated in Chapter31. The corresponding Kerr–Schild metrics have been considered byTrautman (1962), Urbantke (1972), Debney (1973) and McIntosh (privatecommunication). The result isTheorem 32.6 The Kerr–Schild vacuum fields with a non-expandingand non-twistingnull congruence k are necessarily of Petrov type N. Theyare the subcases W 0 =0of (31.34) and (31.38).32.3 Kerr–Schild Einstein–Maxwell fields32.3.1 The case ρ = −(Θ+iω) ̸= 0Electromagnetic solutions of the Kerr–Schild type were studied by Debneyet al. (1969), again assuming the null vector k to be geodesic. By Theorem32.1, the energy-momentum tensor of the electromagnetic field fulfils theconditionsk a;b k b =0 ↔ T ab k a k b =0 ↔ F ab k a = λk b . (32.49)