6.4 The stationary Einstein–Maxwell system

In the one-dimensional Abelian case, both the off-diagonal Einstein equation (6.17View Equation) and the Maxwell equation (6.18View Equation) give rise to scalar potentials, (locally) defined by
dψ := V ¯∗(F¯ + ϕf ), dY := − V2¯∗f + 2ϕdψ − 2ψd ϕ. (6.19 )
Similarly to the vacuum case, this enables one to apply the Lagrange multiplier method to express the effective action in terms of the scalar fields Y and ψ, rather than the one-forms a and A¯. It turns out that in the stationary-axisymmetric case, to which we return in Section 8, we will also be interested in the dimensional reduction of the EM system with respect to a space-like Killing field. Therefore, 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 − 2N 2 , (6.20 )
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 as follows (given a two-form β, we denote by iξβ the one-form with components ξμβμν):
dϕ = − iξF , dψ = iξ ∗ F , (6.21 )
N = g(ξ,ξ), dY = 2 (ω + ϕdψ − ψd ϕ), (6.22 )
where 2ω := ∗(ξ ∧ dξ). The inner product ⟨⋅,⋅⟩ and the associated “norm” | ⋅ | are taken with respect to the three-metric g¯, which becomes pseudo-Riemannian if ξ is space-like. The additional stationary symmetry will then imply that the inner products in (6.20View Equation) have a fixed sign, despite the fact that g¯ is not a Riemannian metric in this case.

The action (6.20View 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 [102, 103], the effective EM action becomes

∫ ( 2 ¯ 2) Seff = ¯∗ ¯R − 2 | dΛ-|-− 1-| dE-+-2-ΛdΛ-|- , (6.23 ) N 2 N 2
where 2 --- |d Λ| := ⟨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
⟨dE, dE + 2Λ¯d Λ ⟩ ⟨d Λ,dE + 2¯Λd Λ⟩ ¯ΔE = − -----------------, ¯Δ Λ = − ----------------, (6.24 ) N N
where − N = Λ ¯Λ + 12(E + ¯E). The isometries of the target manifold are obtained by solving the respective Killing equations [250] (see also [186, 187, 189, 188]). 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 and a timelike Killing vector we have seen in Section 6.2 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
ΦAB = ηAB + 2sig(N )¯vAvB, (v0,v1,v2) := -∘-1---(E − 1,E + 1, 2Λ), (6.25 ) 2 |N |
where vA is the Kinnersley vector [185], and η := diag(− 1,+1, +1 ). It is straightforward to verify that, in terms of Φ, the effective action (6.23View Equation) assumes the SU (2,1 ) invariant form (6.9View Equation). The equations of motion following from this action are the following three-dimensional Einstein equations with sources, obtained from variations with respect to ¯g, and the σ-model equations, obtained from variations with respect to Φ:
1 ¯Rij = 4Trace (𝒥i𝒥j) , d¯∗𝒥 = 0; (6.26 )
here all operations are taken with respect to g¯.

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