Spherically symmetric black holes with hair

3.1 Spherically symmetric black holes with hair

Requiring spherical symmetry, the task to prove the no-hair theorem for the Einstein–Maxwell (EM) system becomes almost trivial. However, not even this part of the uniqueness proof can be generalized: The first black hole solution demonstrating the failure of the no-hair conjecture was obtained by Gibbons in 1982 [72

] within EM-dilaton theory.¹³ The fact that the Gibbons solution carries no dilatonic charge makes it asymptotically indistinguishable from a Reissner–Nordström black hole with the same mass and electric charge. However, since the latter is not a consistent solution of the EM-dilaton equations, one might expect that – within a given matter model – the stationary black hole solutions are still characterized by a set of global charges (generalized no-hair conjecture). In fact, the Gibbons black hole supports the generalized no-hair conjecture; its uniqueness within EM-dilaton theory was established by Masood-ul-Alam in 1992 [130 ].

However, neither the original nor the generalized no-hair conjecture are correct. For instance, the latter fails to be valid within Einstein–Yang–Mills (EYM) theory: According to the generalized version, 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 [174 , 122 , 9 ] found a family of static black hole solutions with vanishing Yang–Mills charges.¹⁴ Since these solutions are asymptotically indistinguishable from the Schwarzschild solution, and since the latter is a particular solution of the EYM equations, the non-Abelian black holes violate the generalized no-hair conjecture.

As the non-Abelian black holes are not stable [166 , 186 , 178 ],¹⁵ one might adopt the view that they do not present actual threats to the generalized no-hair conjecture. However, during the last years, 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 to the static, spherically symmetric Einstein–Skyrme equations [50 , 92 , 93 , 97 ] and to the EYM equations coupled to a Higgs triplet [12 , 14 , 180 , 1 ].¹⁶ 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, four 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 result is obtained from the positive mass theorem and the Komar expression for the total mass of an asymptotically flat, stationary spacetime; see, e.g. [73 ] or [88 ].)
Second, both Deser’s energy argument [48 ] and Coleman’s scaling method [46 ] show that there exist no pure YM solitons in flat spacetime.
Third, the EM system admits no soliton solutions. (This follows by applying Stokes’ theorem to the static Maxwell equations; see, e.g. [87 ].)
Finally, Deser [49 ] 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 $σ$ -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 4).