### 5.2 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 [176] or electrovacuum [177] black-hole spacetimes. 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.
In [213], Lee et al. reanalyzed the stability of the Reissner–Nordström (RN) solution in the context of
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 [28, 30, 214, 1].

Motivated by these solutions, Ridgway and Weinberg [277] 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), as discussed in Weinberg’s comprehensive
review on magnetically-charged black holes [317]. In fact, the work of Ridgway and Weinberg shows that
static black holes with magnetic charge need not even be axially symmetric [278]. Axisymmetric, static
black holes without spherical symmetry appear to exist within the pure EYM system and the EYM-dilaton
model [194].

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.