]. It relates the global concept of the event horizon to the
independently defined - and logically distinct - local notion of
the Killing horizon: Requiring that the fundamental matter fields
obey well behaved hyperbolic equations, and that the
stress-energy tensor satisfies the weak energy condition,
the first part of the SRT asserts that the event horizon of a
stationary
black hole spacetime
is
a Killing horizon.
The latter is called non-rotating if it is generated by the
stationary Killing field, and rotating otherwise. In the rotating
case, the second part of the SRT implies that spacetime is
axisymmetric.
The subdivision provided by the SRT is, unfortunately, not sufficient to apply the uniqueness theorems for the Reissner-Nordström and the Kerr-Newman metric: The latter are based on the stronger requirements that the domain of outer communication (DOC) is either static (non-rotating case) or circular (axisymmetric case). Hence, in both cases one has to establish the Frobenius integrability conditions for the Killing fields beforehand (staticity and circularity theorems).
The
circularity theorem, due to Carter [26], and Kundt and Trümper [118], implies that the metric of a vacuum or electrovac spacetime
can, without loss of generality, be written in the well-known
Papapetrou
-split. The
staticity theorem, implying that the stationary Killing field of a non-rotating,
electrovac black hole spacetime is hyper-surface orthogonal, is
more involved than the circularity problem: First, one has to
establish
strict
stationarity, that is, one needs to exclude ergo-regions. This
problem, first discussed by Hajicek [78], [79], and Hawking and Ellis [84], was solved only recently by Sudarsky and Wald [167], [168], assuming a foliation by maximal slices.
If ergo-regions are excluded, it still remains to prove that the
stationary Killing field satisfies the Frobenius integrability
condition. In the vacuum case, this was achieved by Hawking [82], who was able to extend a theorem due to Lichnerowicz [126] to black hole space-times. In the presence of Maxwell fields
the problem was solved only a couple of years ago [167], [168], by means of a generalized version of the first law of black
hole physics.
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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 |