Let us start with the first issue, concerning the generality of the strong rigidity theorem (SRT). While earlier attempts to proof the theorem were flawed and subject to restrictive assumptions concerning the matter fields [84], the recent work of Chrusciel [37], [39] has shown that the SRT is basically a geometric feature of stationary space-times. It is, therefore, conceivable to suppose that both parts of the theorem - that is, the existence of a Killing horizon and the existence of an axial symmetry in the rotating case - are generic features of stationary black hole space-times. (See also [40] for the classification of asymptotically flat space-times.)
The counterpart to the staticity problem is the
circularity
problem: As the non-rotating black holes are, in general, not
static, one expects that the axisymmetric ones need not
necessarily be circular. This is, indeed, the case: While
circularity is a consequence of the EM equations and the symmetry
properties of the electro-magnetic field, the same is not true
for the EYM system.
Hence, the familiar Papapetrou ansatz for a stationary and
axisymmetric metric is too restrictive to take care of all
stationary and axisymmetric degrees of freedom of the EYM system.
Recalling the enormous simplifications of the EM equations
arising from the
-split of the metric in the Abelian case, an investigation of
the non-circular EYM equations will be rather awkward. As
rotating black holes with hair are most likely to occur already
in the circular sector (see the next paragraph), a systematic
investigation of the EYM equations with circular constraints is
needed as well.
The
static
subclass of the circular sector was investigated in recent
studies by Kleihaus and Kunz (see [111] for a compilation of the results). Since, in general, staticity
does not imply spherical symmetry, there is a possibility for a
static branch of axisymmetric black holes without spherical
symmetry.
Using numerical methods, Kleihaus and Kunz have constructed
black hole solutions of this kind for both the EYM and the
EYM-dilaton system [115].
The new configurations are purely magnetic and parametrized by
their winding number and the node number of the relevant gauge
field amplitude. In the formal limit of infinite node number, the
EYM black holes approach the Reissner-Nordström solution, while
the EYM-dilaton black holes tend to the Gibbons-Maeda black hole
[73
], [76
].
Both the soliton and the black hole solutions of Kleihaus and
Kunz are unstable and may, therefore, be regarded as gravitating
sphalerons and black holes inside sphalerons, respectively.
Slowly rotating
regular and black hole solutions to the EYM equations were
recently established in [21]. Using the reduction of the EYM action in the presence of a
stationary symmetry reveals that the perturbations giving rise to
non-vanishing angular momentum are governed by a self-adjoint
system of equations for a set of gauge invariant fluctuations [18
]. For a soliton background the solutions to the perturbation
equations describe charged, rotating excitations of the
Bartnik-McKinnon solitons [4
]. In the black hole case the excitations are combinations of two
branches of stationary perturbations: The first branch comprises
charged black holes with vanishing angular momentum,
whereas the second one consists of neutral black holes with
non-vanishing angular momentum.
In the presence of bosonic matter, such as Higgs fields, the
slowly rotating solitons cease to exist, and the two branches of
black hole excitations merge to a single one with a prescribed
relation between charge and angular momentum [18
].
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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 |