Static black holes without spherical symmetry

3.2 Static black holes without spherical symmetry

The above counterexamples to the generalized no-hair conjecture are static and spherically symmetric. The famous Israel theorem guarantees that spherical symmetry is, in fact, a consequence of staticity, provided that one is dealing with vacuum [99 ] or electrovac [100 ] black hole space-times. The task to extend the Israel theorem to more general self-gravitating matter models is, of course, a difficult one. In fact, the following example proves that spherical symmetry is not a generic property of static black holes.

A few years ago, Lee et al. [125 ] reanalyzed the stability of the Reissner–Nordström (RN) solution in the context of $SU (2)$ EYM–Higgs theory. It turned out that – for sufficiently small horizons – the RN black holes develop an instability against radial perturbations of the Yang–Mills field. This suggested the existence of magnetically charged, spherically symmetric black holes with hair, which were also found by numerical means [12 , 14 , 180 , 1 ].

Motivated by these solutions, Ridgway and Weinberg [149 ] considered the stability of the magnetically charged RN black holes within a related model; the EM system coupled to a charged, massive vector field. Again, the RN solution turned out to be unstable with respect to fluctuations of the massive vector field. However, a perturbation analysis in terms of spherical harmonics revealed that the fluctuations cannot be radial (unless the magnetic charge assumes an integer value).¹⁷ In fact, the work of Ridgway and Weinberg shows that static black holes with magnetic charge need not even be axially symmetric [150 ].¹⁸

This shows that static black holes may have considerably more structure than one might expect from the experience with the EM system: Depending on the matter model, they may allow for nontrivial fields outside the horizon and, moreover, they need not be spherically symmetric. Even more surprisingly, there exist static black holes without any rotational symmetry at all.