4.5 The stationary Einstein–Maxwell system

In the Abelian case, both the off-diagonal Einstein equation (22View Equation) and the Maxwell equation (23View Equation) give rise to scalar potentials, (locally) defined by
( ) d ψ ≡ σ ¯∗ F¯+ ϕf , dY ≡ − σ2 ¯∗f + 2ϕd ψ − 2ψd ϕ. (24 )
Like for the vacuum system, this enables one to apply the Lagrange multiplier method in order to express the effective action in terms of the scalar fields Y and ψ, rather than the one-forms a and A¯. As one is often interested in the dimensional reduction of the EM system with respect to a space-like Killing field, we give here the general result for an arbitrary Killing field ξ with norm N:
∫ ( ) ¯ |d-ϕ|2-+-|dψ-|2- |dN-|2 +-|dY--−-2ϕd-ψ-+-2ψd-ϕ|2 Seff = ¯∗ R − 2 N − 2 N 2 , (25 )
where ¯∗|d ϕ|2 ≡ dϕ ∧ ¯∗dϕ, etc. The electro-magnetic potentials ϕ and ψ and the gravitational scalars N and Y are obtained from the four-dimensional field strength F and the Killing field (one form) as follows:45
dϕ = − iξF , dψ = iξ ∗ F, (26 )
N = (ξ, ξ ) , dY = 2 (ω + ϕd ψ − ψd ϕ), (27 )
where 2ω ≡ ∗(ξ ∧ dξ). The inner product ⟨ , ⟩ is taken with respect to the three-metric ¯g, which becomes pseudo-Riemannian if ξ is space-like. In the stationary and axisymmetric case, to be considered in Section 6, the Kaluza–Klein reduction will be performed with respect to the space-like Killing field. The additional stationary symmetry will then imply that the inner products in (25View Equation) have a fixed sign, despite the fact that ¯g is not a Riemannian metric in this case.

The action (25View Equation) describes a harmonic mapping into a four-dimensional target space, effectively coupled to three-dimensional gravity. In terms of the complex Ernst potentials, Λ ≡ − ϕ + iψ and ¯ E ≡ − N − Λ Λ + iY [52Jump To The Next Citation Point53Jump To The Next Citation Point], the effective EM action becomes

∫ ( | dΛ |2 1 | dE + 2 ¯ΛdΛ |2) Seff = ¯∗ R¯ − 2-------− ---------2------- , (28 ) N 2 N
where |d Λ|2 ≡ ⟨dΛ , dΛ-⟩. The field equations are obtained from variations with respect to the three-metric ¯g and the Ernst potentials. In particular, the equations for E and Λ become
¯ ¯ ¯ΔE = − ⟨dE-, dE-+-2Λd-Λ⟩-, ¯Δ Λ = − ⟨d-Λ-, dE-+-2ΛdΛ-⟩, (29 ) N (E,Λ ) N (E, Λ)
where ¯ 1 ¯ − N = ΛΛ + 2(E + E). The isometries of the target manifold are obtained by solving the respective Killing equations [139] (see also [107108109110]). This reveals the coset structure of the target space and provides a parametrization of the latter in terms of the Ernst potentials. For vacuum gravity we have seen in Section 4.3 that the coset space, G∕H, is SU (1, 1)∕U (1), whereas one finds G ∕H = SU (2,1)∕S (U(1,1) × U (1)) for the stationary EM equations. If the dimensional reduction is performed with respect to a space-like Killing field, then G ∕H = SU (2,1 )∕S (U (2) × U (1)). The explicit representation of the coset manifold in terms of the above Ernst potentials, E and Λ, is given by the hermitian matrix Φ, with components
Φ = η + 2sig(N )¯v v , (v ,v ,v ) ≡ -∘1---(E − 1,E + 1,2Λ ), (30 ) AB AB A B 0 1 2 2 |N |
where vA is the Kinnersley vector [106], and η ≡ diag(− 1,+1, +1 ). It is straightforward to verify that, in terms of Φ, the effective action (28View Equation) assumes the SU (2,1) invariant form
∫ ( 1 ) −1 𝒮eff = ¯∗ ¯R − 4-Trace⟨𝒥 , 𝒥 ⟩ , with 𝒥 ≡ Φ dΦ , (31 )
where A B ij A B Trace⟨𝒥 , 𝒥 ⟩ ≡ ⟨𝒥 B , 𝒥 A⟩ ≡ ¯g (𝒥i )B(𝒥j )A. The equations of motion following from the above action are the three-dimensional Einstein equations (obtained from variations with respect to ¯g) and the σ-model equations (obtained from variations with respect to Φ):
¯ 1- Rij = 4 Trace {𝒥i𝒥j }, d ¯∗𝒥 = 0. (32 )
By virtue of the Bianchi identity, ¯ ¯ij ∇j G = 0, and the definition −1¯ 𝒥i ≡ Φ ∇iΦ, the σ-model equations are the integrability conditions for the three-dimensional Einstein equations.


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