The stationary Einstein–Maxwell system

4.5 The stationary Einstein–Maxwell system

In the Abelian case, both the off-diagonal Einstein equation (22

) and the Maxwell equation (23

) give rise to scalar potentials, (locally) defined by

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 $N$ :

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

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 (25

) have a fixed sign, despite the fact that $¯g$ is not a Riemannian metric in this case.

The action (25 ) 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 $¯ E ≡ − N − Λ Λ + iY$ [52 , 53 ], the effective EM action becomes

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

where $¯ 1 ¯ − N = ΛΛ + 2(E + E)$ . The isometries of the target manifold are obtained by solving the respective Killing equations [139 ] (see also [107 , 108 , 109 , 110 ]). 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

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 (28

) assumes the $SU (2,1)$ invariant form

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 $Φ$ ):

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.