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 [310, 208, 24] 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, 329, 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 [27] 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 [28, 30, 214, 1]; 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.

- First, there exist no purely gravitational solitons, that is, the only globally-regular, asymptotically-flat, static vacuum solution to the Einstein equations with finite energy is Minkowski spacetime. This is the Lichnerowicz theorem, which nowadays can be obtained from the positive mass theorem and the Komar expression for the total mass of an asymptotically-flat, stationary spacetime [170]; see, e.g., [127] or [152]. A rather strong version thereof, which does not require asymptotic conditions other than completeness of the space metric, has been established by Anderson [5], see also [4].
- Next, there are no nontrivial static solutions of the EYM equations near Minkowski spacetime [221].
- Further, both Deser’s energy argument [90] and Coleman’s scaling method [87] show that there do not exist pure YM solitons in flat spacetime.
- Moreover, the EM system admits no soliton solutions. (This follows by applying Stokes’ theorem to the static Maxwell equations; see, e.g., [151].)
- Finally, Deser [91] proved that the three-dimensional EYM equations admit no soliton solutions. The argument takes advantage of the fact that the magnetic part of the Yang–Mills field has only one non-vanishing component in 2+1 dimensions.

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 -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).

Living Rev. Relativity 15, (2012), 7
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