)
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
, 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 |
