3.5 Rotating black holes with hair

So far we have addressed the ramifications occurring on the “non-rotating half” of the classification diagram shown in Figure 1View Image: We have argued that non-rotating black holes need not be static, static ones need not be spherically symmetric, and spherically symmetric ones need not be characterized by a set of global charges. The right-hand-side of the classification scheme has been studied less intensively until now. Here, the obvious questions are the following ones: Are all stationary black holes with rotating Killing horizons axisymmetric (rigidity)? Are the stationary and axisymmetric Killing fields hyper-surface orthogonal (circularity)? Are the circular black holes characterized by their mass, angular momentum and global charges (no-hair)?

Let us start with the first issue, concerning the generality of the strong rigidity theorem (SRT). While earlier attempts to proof the theorem were flawed22 and subject to restrictive assumptions concerning the matter fields [84], the recent work of ChruĊ›ciel [4039] 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 [6] 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.23 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.24 Recalling the enormous simplifications of the EM equations arising from the (2+2)-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 [115] 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.25 Using numerical methods, Kleihaus and Kunz have constructed black hole solutions of this kind for both the EYM and the EYM-dilaton system [114].26 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 [72Jump To The Next Citation Point76Jump To The Next Citation Point].27 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 [22Jump To The Next Citation Point]. 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 [19Jump To The Next Citation Point]. For a soliton background the solutions to the perturbation equations describe charged, rotating excitations of the Bartnik–McKinnon solitons [4Jump To The Next Citation Point]. 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,28 whereas the second one consists of neutral black holes with non-vanishing angular momentum.29 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 [19Jump To The Next Citation Point].


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