6.2 The coset structure of vacuum gravity
For many applications, in particular for the black-hole uniqueness theorems, it is convenient to replace
the one-form
by a function, namely the twist potential. We have already pointed out that
, parameterizing the non-static part of the metric, enters the effective action (6.2) only
via the field strength,
. For this reason, the variational equation for
(that is, the
off-diagonal Einstein equation) takes in vacuum the form of a source-free Maxwell equation:
By virtue of Eq. (6.3), the (locally-defined) function
is a potential for the twist one-form,
.
In order to write the effective action (6.2) in terms of the twist potential
, rather than the one-form
, one considers
as a fundamental field and imposes the constraint
with the Lagrange
multiplier
. The variational equation with respect to
then yields
, which is used
to eliminate
in favor of
. One finds
. Thus, the action (6.2)
becomes
where we recall that
is the inner product with respect to the three-metric
defined in
Eq. (6.1).
The action (6.5) describes a harmonic map into a two-dimensional target space, effectively coupled to
three-dimensional gravity. In terms of the complex Ernst potential
[102
, 103
], one has
The stationary vacuum equations are obtained from variations with respect to the three-metric
[
-equations] and the Ernst potential
[
-equations]. One easily finds
and
, where
is the Laplacian with respect to
.
The target space for stationary vacuum gravity, parameterized by the Ernst potential
, is a Kähler
manifold with metric
(see [115] 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
is isometric to
the pseudo-sphere,
. As the three-dimensional
Lorentz group,
, acts transitively and isometrically on the pseudo-sphere with isotropy group
, the target space is the coset
(see, e.g., [196
] or [26
] for the general
theory of symmetric spaces). Using the universal covering
of
, one can
parameterize
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 [250
],
where
and the currents
are defined as
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:
,
.