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34.1 Methods usingharmonic maps 519and use the methods outlined in §10.8. For a null Killing vector see Juliaand Nicolai (1995). To recapitulate the essential results from Chapter 18,it is convenient to introduce a three-dimensional metric γ ab defined byγ ab = |F |(g ab − F −1 ξ a ξ b ), γ = det(γ ab ) (34.2)(Neugebauer 1969). The field equations assume the formˆR ab = 1 2 F −2 (E ,(a +2ΦΦ ,(a )(E ,b) 2ΦΦ ,b) )+2F −1 Φ ,(a Φ ,b) ,(34.3a)0=F E ;a,a + γ ab E ,a (E ,b +2ΦΦ ,b ), (34.3b)0=F Φ ;a,a + γ ab Φ ,a (E ,b +2ΦΦ ,b ), F = −Re E−ΦΦ. (34.3c)By inspection we have the following:Theorem 34.1 For sourcefree Einstein–Maxwell fields admittinga nonnullKillingvector ξ, there exists a set {Φ, E,γ ab } such that the Einstein–Maxwell equations (34.3) follow from a variational principle with the LagrangianL = √ γ[ ˆR + 1 2 F −2 γ ab (E ,a + ΦΦ ,a )(E ,b +ΦΦ ,b )+2F −1 γ ab Φ ,a Φ ,b ] (34.4)( ˆR denotes the curvature scalar with respect to γ ab ), i.e. (34.3) areδL/δγ ab =0, δL/δΦ =0, δL/δE =0. (34.5)The second term of this Lagrangian is precisely of the form (10.8) andthe methods discussed there can be applied. The curvature scalar in theLagrangian relates the two complex – or equivalently four real – fields Φand E to the three-dimensional metric γ ab . For the definitions of the twocomplex scalar potentials Φ and E the reader is referred to (18.31) and(18.35).From a given solution (Φ, E,γ ab ) of the Einstein–Maxwell equations(34.3), the quantities F, ξ a , the space-time metric and the Maxwell tensorcan be reconstructed – in that order – via (cp. §18.2)−F = 1 2 (E + E∗ )+ΦΦ, K ∗ ab =2F −1 (ξ [a E ,b] ) ∗ ,g ab = |F | −1 γ ab + F −1 ξ a ξ b , √ κ 0 /2F ∗ ab =2F −1 (ξ [a Φ ,b ) ∗ .(34.6)Taking Φ and E, respectively their real and imaginary parts, as coordinatesin a four-dimensional potential space, its metric G AB from §10.8has according to (34.4) the formdS 2 = 1 2 F −2 |dE +2ΦdΦ| 2 +2F −1 dΦ dΦ, (34.7)