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19.5 Various forms of the field equations for vacuum fields 301ξ satisfies the differential equation(ξ ¯ξ − 1)W −1 (Wξ ,M ) ;M =2¯ξξ ,M ξ ,M . (19.47)This last version of the field equations has proved to be especially usefulin constructing new solutions (see §20.5).(iv) The function k in the metric (19.21) satisfies a differential equationof the fourth order, which can be derived from the fact that the right-handside of (19.31) (with Φ = 0) multiplied by (−2)dx M dx N is the metric ofa space of constant negative curvature. From the left-hand side of (19.31)we infer that the line element ρ −1 (ρk ,A ) ,A (dρ 2 +dz 2 ) − 2k ,A dx A dρ/ρ isassociated with this space of constant curvature so that we obtain a differentialequation for k alone:2D(A ,zz + C ,ρρ − 2B ,ρz ) − D ,z A ,z − D ,ρ C ,ρ − BA ,ρ C ,z + BA ,z C ,ρ+2CA ,ρ B ,z +2AC ,z B ,ρ − 4BB ,ρ B ,z =4D 2 , (19.48)A ≡ k ,ρ /ρ − 2k ,AA , C ≡−k ,ρ /ρ − 2k ,AA , B ≡ 2k ,z /ρ, D ≡ AC − B 2 .From a given (non-constant) solution k of (19.48), the potential ξ canbe constructed up to a phase factor, ξ → e iα ξ (ignoring pure coordinatetransformations). Independently, Cosgrove (1978a), Herlt (1978a)and Cox and Kinnersley (1979) have also reduced the field equationsto a fourth-order differential equation for a real superpotential, see alsoTomimatsu (1981) and Lorencz and Sebestyén (1986).(v) Perjés (1985a), see also Theorems 18.5 and 18.6, introduced theErnst potential and its complex conjugate as new space coordinates. Otherforms of the field equations are given, e.g., in Chandrasekhar (1978) andKramer and Neugebauer (1968b).Comparison of (19.40) with (19.43), and with the very similar equations(19.37) for electrostatic fields, leads toTheorem 19.3 Given a stationary axisymmetric vacuum solution(U, ω), the substitutionS ′ = −U ′ + 1 2 ln W = U, A′ =iω, (19.49)yields another vacuum solution (U ′ ,A ′ ) (Kramer and Neugebauer 1968b),and the substitutionU ′ =2U, χ =iω, k ′ =4k (19.50)generates from a stationary vacuum solution (U, ω, k) a static solution(U ′ ,χ,k ′ ) of the Einstein–Maxwell equations (Bonnor 1961).