In the non-rotating case it was Israel who, in his pioneering
work, showed that both static vacuum [99] and electrovac [100
] black hole space-times are spherically symmetric. Israel's
ingenious method, based on differential identities and Stokes'
theorem, triggered a series of investigations devoted to the
static uniqueness problem (see, e.g. [137], [138], [151
], [153]). Later on, Simon [160], Bunting and Masood-ul-Alam [25], and Ruback [154] were able to improve on the original method, taking advantage
of the positive energy theorem.
(The ``latest version'' of the static uniqueness theorem can be
found in [129].)
The key to the uniqueness theorem for rotating black holes
exists in Carter's observation that the stationary and
axisymmetric EM equations reduce to a two-dimensional boundary
value problem [28] (See also [30] and [32].). In the vacuum case, Robinson was able to construct an
amazing identity, by virtue of which the uniqueness of the Kerr
metric followed [152
]. The uniqueness problem with electro-magnetic fields remained
open until Mazur [131
] and, independently, Bunting [24
] were able to obtain a generalization of the Robinson identity
in a systematic way: The Mazur identity (see also [132
], [133
]) is based on the observation that the EM equations in the
presence of a Killing field describe a non-linear
-model with coset space
(provided that the dimensional reduction of the EM action is
performed with respect to the axial Killing field
). Within this approach, the Robinson identity looses its
enigmatic status - it turns out to be the explicit form of the
Mazur identity for the vacuum case,
G
/
H
=
SU
(1,1) /
U
(1).
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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 |