### 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 and , rather than the one-forms and . 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 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 (one form) as
follows:
where . The inner product is taken with respect to the three-metric , which
becomes pseudo-Riemannian if 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 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, and
[52, 53], 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 [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, , 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 [106], and . It is straightforward to
verify that, in terms of , the effective action (28) assumes the invariant form
where . The equations of motion following from
the above action are the three-dimensional Einstein equations (obtained from variations with
respect to ) and the -model equations (obtained from variations with respect to ):
By virtue of the Bianchi identity, , and the definition , the -model equations
are the integrability conditions for the three-dimensional Einstein equations.