Considering two
arbitrary
solutions of the Ernst equations, Robinson was able to construct
an identity [152], the integration of which proved the uniqueness of the Kerr
metric. The complicated nature of the Robinson identity dashed
the hope of finding the corresponding electrovac identity by
trial and error methods.
In fact, the problem was only solved when Mazur [131], [133] and Bunting [24] independently succeeded in deriving the desired divergence
identities by using the distinguished structure of the EM
equations in the presence of a Killing symmetry. Bunting's
approach, applying to a general class of harmonic mappings
between Riemannian manifolds, yields an identity which enables
one to establish the uniqueness of a harmonic map if the target
manifold has negative curvature.
The Mazur identity (34) applies to the relative difference
of two arbitrary hermitian matrices. If the latter are
solutions
of a
-model with symmetric target space of the form
, then the identity implies
where
, and
is the difference between the currents.
The reduction of the EM equations with respect to the axial
Killing field yields the coset
(see Sect.
4.5), which, reduces to the vacuum coset
(see Sect.
4.3). Hence, the above formula applies to both the axisymmetric
vacuum and electrovac field equations, where the Laplacian and
the inner product refer to the pseudo-Riemannian three-metric
defined by Eq. (55
). Now using the existence of the stationary Killing symmetry and
the circularity property, one has
, which reduces Eq. (72
) to an equation on
. Integrating over the semi-strip
and using Stokes' theorem immediately yields
where
and
are the volume form and the Hodge dual with respect to
. The uniqueness of the Kerr-Newman metric follows from the
facts that
The RHS is non-negative because of the following observations:
First, the inner product is definite, and
is a positive volume-form, since
is a Riemannian metric. Second, the factor
is non-negative in
, since
is the image of the upper half-plane,
. Last, the one-forms
and
are space-like, since the matrices
depend only on the coordinates of
.
In order to establish that
on the boundary
of the semi-strip, one needs the asymptotic behavior and the
boundary and regularity conditions of all potentials. A careful
investigation
then shows that
vanishes on the horizon, the axis and at infinity, provided that
the solutions have the same mass, charge and angular
momentum.
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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 |