Israel and Wildon [102] have shown that all solutions of this equation fulfill
. In fact, it is not hard to verify that this ansatz solves Eq.
(43
), provided that the complex constants
and
are subject to
. Using asymptotic flatness, and adopting a gauge where the
electro-magnetic potentials and the twist potential vanish in the
asymptotic regime, one has
and
, and thus
It is crucial that this ansatz solves both the equation for
and the one for
: One easily verifies that Eqs. (29
) reduce to the single equation
where
is the three-dimensional
flat
Laplacian.
For
static, purely electric
configurations the twist potential
Y
and the magnetic potential
vanish. The ansatz (44
), together with the definitions of the Ernst potentials,
and
(see Sect.
4.5), yields
Since
, the linear relation between
and the gravitational potential
implies
. By virtue of this, the total mass and the total charge of
every asymptotically flat, static, purely electric Israel-Wilson
solution are equal:
where the integral extends over an asymptotic two-sphere.
The simplest nontrivial solution of the flat Poisson equation (45
),
, corresponds to a linear combination of
n
monopole sources
located at arbitrary points
,
This is the Papapetrou-Majumdar (PM) solution [143], [128], with spacetime metric
and electric potential
. The PM metric describes a
regular
black hole spacetime, where the horizon comprises
n
disconnected components.
In Newtonian terms, the configuration corresponds to
n
arbitrarily located charged mass points with
. The PM solution escapes the uniqueness theorem for the
Reissner-Nordström metric, since the latter applies exclusively
to space-times with
M
> |
Q
|.
Non-static members of the Israel-Wilson class were constructed as well [102], [145]. However, these generalizations of the Papapetrou-Majumdar multi black hole solutions share certain unpleasant properties with NUT spacetime [140] (see also [15], [136]). In fact, the work of Hartle and Hawking [81], and Chrusciel and Nadirashvili [42] strongly suggests that - except the PM solutions - all configurations obtained by the Israel-Wilson technique are either not asymptotically Euclidean or have naked singularities. In order to complete the uniqueness theorem for the PM metric among the static black hole solutions with degenerate horizon, it basically remains to establish the equality M = Q under the assumption that the horizon has some degenerate components. Until now, this has been achieved only by requiring that all components of the horizon have vanishing surface gravity and that all ``horizon charges'' have the same sign [90].
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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 |