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
, rather than the one-forms
a
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
N
:
where
, etc. The electro-magnetic potentials
and
and the gravitational scalars
N
and
Y
are obtained from the four-dimensional field strength
F
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 Sect.
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 Sect.
4.3
that the coset space,
G
/
H, is
SU
(1,1)/
U
(1), 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
SU
(2,1) 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.
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Stationary Black Holes: Uniqueness and Beyond
Markus Heusler http://www.livingreviews.org/lrr-1998-6 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |