By virtue of Eq. (10), the (locally defined) function
Y
is a potential for the twist one-form,
. In order to write the effective action (9
) in terms of the twist potential
Y, rather than the one-form
a, one considers
as a fundamental field and imposes the constraint
with the Lagrange multiplier
Y
. The variational equation with respect to
f
then yields
, which is used to eliminate
f
in favor of
Y
. One finds
. Thus, the action (9
) becomes
where we recall that
is the inner product with respect to the three-metric
defined in Eq. (8
).
The action (12) describes a harmonic mapping into a two-dimensional target
space, effectively coupled to three-dimensional gravity. In terms
of the complex Ernst potential
[52
], [53
], one has
The stationary vacuum equations are obtained from variations
with respect to the three-metric
[(ij)-equations] and the Ernst potential
[
-equations]. One easily finds
and
, where
is the Laplacian with respect to
.
The target space for stationary vacuum gravity, parametrized
by the Ernst potential
, is a Kähler manifold with metric
(see [60] for details). By virtue of the mapping
the semi-plane where the Killing field is time-like,
, is mapped into the interior of the complex unit disc,
, with standard metric
. By virtue of the stereographic projection,
,
, the unit disc
D
is isometric to the pseudo-sphere,
. As the three-dimensional Lorentz group,
SO
(2,1), acts transitively and isometrically on the pseudo-sphere
with isotropy group
SO
(2), the target space is the coset
. Using the universal covering
SU
(1,1) of
SO
(2,1), one can parametrize
in terms of a positive hermitian matrix
, defined by
Hence, the effective action for stationary vacuum gravity
becomes the standard action for a
-model coupled to three-dimensional gravity [139
],
The simplest nontrivial solution to the vacuum Einstein
equations is obtained in the static, spherically symmetric case:
For
one has
and
. With respect to the general spherically symmetric ansatz
one immediately obtains the equations
and
, the solution of which is the Schwarzschild metric in the usual
parametrization:
,
.
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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 |