The current one-form
is given in terms of the hermitian matrix
, which comprises the norm and the generalized twist potential
of the Killing field, the fundamental scalar fields and the
electric and magnetic potentials arising on the effective level
for each Abelian vector field. If the dimensional reduction is
performed with respect to the axial Killing field
with norm
, then
is Ricci tensor of the pseudo-Riemannian three-metric
, defined by
In the stationary and axisymmetric case under consideration,
there exists, in addition to
m, an asymptotically time-like Killing field
k
. Since
k
and
m
fulfill the Frobenius integrability conditions, the spacetime
metric can be written in the familiar
-split.
Hence, the circularity property implies that
the field equations (54) become a set of partial differential equations on the
two-dimensional Riemannian manifold
:
as is seen from the standard reduction of the Ricci tensor
with respect to the static three-metric
.
The last simplification of the field equations is due to the
circumstance that
can be chosen as one of the coordinates on
. This follows from the facts that
is harmonic (with respect to the Riemannian two-metric
) and non-negative, and that the domain of outer communications
of a stationary black hole spacetime is simply connected [44]. The function
and the conjugate harmonic function
z
are called Weyl coordinates.
With respect to these, the metric
can be chosen to be conformally flat, such that one ends up with
the spacetime metric
the
-model equations
and the remaining Einstein equations
for the function
.
Since Eq. (58
) is conformally invariant, the metric function
does not appear in the
-model equation (61
). Therefore, the stationary and axisymmetric equations reduce to
a boundary value problem for the matrix
on a fixed, two-dimensional background. Once the solution to Eq.
(61
) is known, the remaining metric function
is obtained from Eqs. (62
) by quadrature.
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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 |