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32.5 Generalizations of the Kerr–Schild ansatz 501(iii) If Ṽ4 and V 4 are vacuum space-times, and V 4 is algebraically specialwith k as a repeated null eigenvector, then the same is true for Ṽ4(Thompson 1966).(iv) If k is geodesic and if V 4 is algebraically special with k as a repeatednull eigenvector (κ =Ψ 0 =Ψ 1 = 0), then Ṽ4 has the same properties(Bilge and Gürses 1983).(v) If k is a principal null direction of both Weyl tensors, then it is geodesic(Bilge and Gürses 1983).(vi) If the tetrad (m, m, l, k) is parallelly propagated along k in V 4 , thenthe same holds in Ṽ4 (Bilge and Gürses 1983).When using the generalized Kerr–Schild transformation (32.84) in thesearch for solutions, both metrics – g ab and ˜g ab – have to be restricted.We shall discuss now the different choices investigated so far.32.5.2 Non-flat vacuum to vacuumAs shown above, the null congruence k is necessarily geodesic, but itsshear may or may not vanish. Gergely and Perjés (1993, 1994a, 1994b,1994c) showed that the vacuum solutions with non-vanishingshear (σ ̸=0) are the Kóta–Perjés metrics (18.48)–(18.50) generated from a type Nvacuum metric. In general, they have non-vanishing twist (ω ̸= 0), butthe metrics with ω = 0 are contained as a limiting case (Kupeli 1988b,Gergely and Perjés 1994c): they are the Kasner metrics (13.53)–(13.51),obtained from a plane wave V 4 (Kóta and Perjés 1972).For a general metric with vanishingshear (which is algebraically special)it may happen that the non-flat background metric g ab is already ofthe Kerr–Schild type,˜g ab = g ab − 2S 2 k a k b = η ab − 2S 1 k a k b − 2S 2 k a k b . (32.91)An example for this is the Schwarzschild metric, for which S 1 introducesa mass into Minkowski space, and S 2 only changes the mass parameter.These cases will be considered trivial.To get non-trivial solutions with non-zero expansion (ρ ̸= 0), one canstart from the line element (29.4), see §29.1 for further details. Because ofits definition, a generalized Kerr–Schild transformation is a transformationwhich adds a function S to H such that P, L and W are not changedand that ˜H = H + S is again a solution of the field equations. An inspectionof (29.13) reveals that the only admissible change of H is theaddition of a mass term, ˜m = m + m S ,S= m S Re ρ = m S r/(r 2 +Σ 2 ), cp.also Dozmorov (1971b) and Talbot (1969). The field equations then imply(P −3 m S ) ,u =0, ∂m S = m S L ,u . (32.92)