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18.2 The Ricci tensor on Σ 3 27718.2 The Ricci tensor on Σ 3First we introduce a complex vector Γ on Σ 3 ,Γ a = −2ξ c ξ ∗ c;a = −F ,a +iω a , Γ a ξ a =0, L ξ Γ =0. (18.9)With the aid of (8.22) and the symmetry relations (2.80) of the curvaturetensor it can easily be verified that the equationsξ ∗ a;b ;b = −R ad ξ d , ξ ∗ a;b = −F −1 (ξ [a Γ b] ) ∗ (18.10)hold. Taking the divergence of the complex vector Γ givesΓ a ;a =Γ a ||a + 1 2 F −1 F a Γ a = −F −1 Γ a Γ a +2ξ a ξ b R ab . (18.11)The real and imaginary parts of this equation areF ,a ||a = 1 2 F −1 F ,a F ,a − F −1 ω a ω a − 2ξ a ξ b R ab , (18.12)ω a ||a = 3 2 F −1 F ,a ω a . (18.13)Equation (18.12) expresses the component ξ a ξ b R ab of the Ricci tensor inV 4 in terms of tensors and their covariant derivatives on Σ 3 . In order toderive an analogous formula for the components h a cξ b R ab , we calculate thecurl of ω:ω [b||a] = ε smnr (ξ m ξ n;r ) ;c h c [a hs b] =2ξm ξ d R ∼ dcmsh c [a hs b]= −2ξ d R ∼ d[ab]m ξm = ε abmn ξ m R n d ξ d .(18.14)The result of this short calculation is the formula(−F ) −1/2 ε abc ω c||b =2h a b R b cξ c . (18.15)From this equation we conclude that the vanishing of the componentsh a b Rb cξ c implies that, at least locally, the twist vector ω is a gradient,ω a = ω ,a (see Theorem 2.1).Finally, we can derive the formulaR 3 ab = 1 2 F −1 F ,a||b − 1 4 F −2 F ,a F ,b + 1 2 F −2 (ω a ω b − h ab ω c ω c )+h m a h n b R mn(18.16)by a straightforward calculation in which we insert (18.7) into (18.5),contract (18.5) for the curvature tensor with the projection tensor andapply some of the previous relations of the present section.Equations (18.12), (18.15) and (18.16) express the Ricci tensor of astationary space-time in terms of tensors and their derivatives on the 3-space Σ 3 . All the equations are written in a four-dimensionally covariant