Let us start with the first issue, concerning the generality of the strong rigidity theorem (SRT). The existence of a second Killing vector field to the future of a bifurcation surface can be established by solving a characteristic Cauchy problem [107], which makes it clear that axial symmetry will hold for a large class of matter models satisfying the, say, dominant energy condition.
The counterpart to the staticity problem is the circularity problem: As general non-rotating black holes
are not static, one expects that the axisymmetric ones need not 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. In the Abelian case, the proof rests on the fact that the field
tensor satisfies ,
and
being the stationary and the axial
Killing field, respectively; for Yang–Mills fields these conditions do no longer follow from the
field equations and their invariance properties (see Section 8.1 for details). 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. However, there are other
matter models for which the Papapetrou metric is sufficiently general: the proof of the circularity
theorem for self-gravitating scalar fields is, for instance, straightforward [150
]. Recalling the key
simplifications of the EM equations arising from the (2+2)-splitting of the metric in the Abelian
case, an investigation of non-circular EYM equations is expected to 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 studies by Kleihaus and Kunz (see [194] 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 [192]. The related axisymmetric soliton solutions without spherical symmetry were
previously obtained by the same authors [190, 191]; see also [193] for more details. The new
configurations are purely magnetic and parameterized 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 [126, 131
]. The solutions themselves are neutral and
not spherically symmetric; however, their limiting configurations are charged and spherically
symmetric. 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.
Existence of slowly rotating regular black-hole solutions to the EYM equations was established in [38].
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 [35
]. With a soliton background, the solutions to
the perturbation equations describe charged, rotating excitations of the Bartnik–McKinnon
solitons [14
]. 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,8
whereas the second one consists of neutral black holes with non-vanishing angular momentum. (A particular
combination of the charged and the rotating branch was found in [312
].) 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 [35
]. More information about the EYM–Higgs system can be found
in [209, 254].
http://www.livingreviews.org/lrr-2012-7 |
Living Rev. Relativity 15, (2012), 7
![]() This work is licensed under a Creative Commons License. E-mail us: |