5.1 Spherically symmetric black holes with hair

One can find in the literature the naive expectation that – within a given matter model – the stationary black-hole solutions are uniquely characterized by a set of global charges; this will be referred to as the generalized no-hair conjecture. A model in which this might possibly be correct is provided by the static sector of the EM-dilaton theory, discussed at the beginning of Section 4.6.

The failure of this generalized no-hair conjecture is demonstrated by the Einstein–Yang–Mills (EYM) theory: According to the conjecture, any static solution of the EYM equations should either coincide with the Schwarzschild metric or have some non-vanishing Yang–Mills charges. This turned out not to be the case when, in 1989, various authors [310Jump To The Next Citation Point, 208Jump To The Next Citation Point, 24Jump To The Next Citation Point] found a family of static black-hole solutions with vanishing global Yang–Mills charges (as defined, e.g., in [74]); these were originally constructed by numerical means and rigorous existence proofs were given later in [299, 297, 298, 29, 227]; for a review see [311]. These solutions violate the generalized no-hair conjecture.

As the non-Abelian black holes are unstable [301, 329Jump To The Next Citation Point, 315], one might adopt the view that they do not present actual threats to the generalized no-hair conjecture. (The reader is referred to [37] for the general structure of the pulsation equations, [309, 40], to [27Jump To The Next Citation Point] for the sphaleron instabilities of the particle-like solutions, and to [292] for a review on sphalerons.) However, various authors have found stable black holes, which are not characterized by a set of asymptotic flux integrals. For instance, there exist stable black-hole solutions with hair of the static, spherically-symmetric Einstein–Skyrme equations [94, 156, 157, 161, 241] and to the EYM equations coupled to a Higgs triplet [28Jump To The Next Citation Point, 30Jump To The Next Citation Point, 214Jump To The Next Citation Point, 1Jump To The Next Citation Point]; it should be noted that the solutions of the EYM–Higgs equations with a Higgs doublet are unstable [27, 324]. Hence, the restriction of the generalized no-hair conjecture to stable configurations is not correct either.

One of the reasons why it was not until 1989 that black-hole solutions with self-gravitating gauge fields were discovered was the widespread belief that the EYM equations admit no soliton solutions. There were, at least, five reasons in support of this hypothesis.

All this shows that it was conceivable to conjecture a nonexistence theorem for soliton solutions of the EYM equations in 3+1 dimensions, and a no-hair theorem for the corresponding black hole configurations. On the other hand, none of the above examples takes care of the full nonlinear EYM system, which bears the possibility to balance the gravitational and the gauge field interactions. In fact, a closer look at the structure of the EYM action in the presence of a Killing symmetry dashes the hope to generalize the uniqueness proof along the lines used in the Abelian case: The Mazur identity owes its existence to the σ-model formulation of the EM equations. The latter is, in turn, based on scalar magnetic potentials, the existence of which is a peculiarity of Abelian gauge fields (see Section 6).

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