### 6.4 The stationary Einstein–Maxwell system

In the one-dimensional Abelian case, both the off-diagonal Einstein equation (6.17) and the Maxwell
equation (6.18) give rise to scalar potentials, (locally) defined by
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 and , rather than the one-forms and . 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 with norm :
where , etc. The electro-magnetic potentials and and the gravitational scalars
and are obtained from the four-dimensional field strength and the Killing field as
follows (given a two-form , we denote by the one-form with components ):
where . The inner product and the associated “norm” are taken with respect
to the three-metric , which becomes pseudo-Riemannian if is space-like. The additional stationary
symmetry will then imply that the inner products in (6.20) have a fixed sign, despite the fact that is
not a Riemannian metric in this case.
The action (6.20) describes a harmonic mapping into a four-dimensional target space, effectively
coupled to three-dimensional gravity. In terms of the complex Ernst potentials, and
[102, 103], the effective EM action becomes

where . The field equations are obtained from variations with respect to the
three-metric and the Ernst potentials. In particular, the equations for and become
where . 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, , is
, whereas one finds for the stationary EM
equations. If the dimensional reduction is performed with respect to a space-like Killing field, then
. The explicit representation of the coset manifold in terms of the
above Ernst potentials, and , is given by the Hermitian matrix , with components
where is the Kinnersley vector [185], and . It is straightforward to verify that,
in terms of , the effective action (6.23) assumes the invariant form (6.9). The equations of
motion following from this action are the following three-dimensional Einstein equations with sources,
obtained from variations with respect to , and the -model equations, obtained from variations with
respect to :
here all operations are taken with respect to .