one easily finds the two equations
where
stands for either of the complex potentials
or
, and where the Laplacian and the inner product refer to the
two-dimensional metric
.
In order to control the boundary conditions for black holes,
it is convenient to introduce prolate spheroidal coordinates
x
and
y, defined in terms of the Weyl coordinates
and
z
by
where
is a constant. The domain of outer communications, that is, the
upper half-plane
, corresponds to the semi-strip
. The boundary
consists of the horizon (x
= 0) and the northern (y
=1) and southern (y
= -1) segments of the rotation axis. In terms of
x
and
y, the Riemannian metric
becomes
, up to a conformal factor which does not enter Eqs. (64
). The Ernst equations finally assume the form (
, etc.)
where
stand for
or
. A particularly simple solution to the Ernst equations is
with real constants
p,
q
and
. The norm
X, the twist potential
Y
and the electro-magnetic potentials
and
(all defined with respect to the axial Killing field) are
obtained from the above solution by using Eqs. (63
) and the expressions
,
,
,
. The off-diagonal element of the metric,
, is obtained by integrating the twist expression (10
), where the twist one-form is given in Eq. (27
).
Eventually, the metric function
h
is obtained from Eqs. (62
) by quadrature.
The solution derived in this way is the ``conjugate'' of the
Kerr-Newman solution [36]. In order to obtain the Kerr-Newman metric itself, one has to
perform a rotation in the
-plane: The spacetime metric is invariant under
,
, if
X,
and
are replaced by
kX,
and
, where
. This additional step in the derivation of the Kerr-Newman
metric is necessary because the Ernst potentials were defined
with respect to the axial Killing field
. If, on the other hand, one uses the stationary Killing field
, then the Ernst equations are singular at the boundary of the
ergo-region.
In terms of Boyer-Lindquist coordinates,
one eventually finds the Kerr-Newman metric in the familiar form:
where the constant
is defined by
. The expressions for
,
and the electro-magnetic vector potential
A
show that the Kerr-Newman solution is characterized by the total
mass
M, the electric charge
Q, and the 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 |