Aspects of hairy black holes in spontaneously broken Einstein-Yang-Mills systems: Stability analysis and entropy considerations

We analyze (3+1)-dimensional black-hole space-times in spontaneously broken Yang-Mills gauge theories that have been recently presented as candidates for an evasion of the scalar-no-hair theorem. Although we show that in principle the conditions for the no-hair theorem do not apply to this case, we, however, prove that the "spirit" of the theorem is not violated, in the sense that there exist instabilities in both the sphaleron and gravitational sectors. The instability analysis of the sphaleron sector, which was expected to be unstable for topological reasons, is performed by means of a variational method. As shown, there exist modes in this sector that are unstable against linear perturbations. Instabilities exist also in the gravitational sector. A method for counting the gravitational unstable modes, which utilizes a catastrophe-theoretic approach is presented. The role of the catastrophe functional is played by the mass functional of the black hole. The Higgs vacuum expectation value is used as a control parameter, having a critical value beyond which instabilities are turned on. The (stable) Schwarzschild solution is then understood from this point of view. The catastrophe-theory appproach facilitates enormously a universal stability study of non-Abelian black holes, which goes beyond linearized perturbations. Some elementary entropy considerations are also presented that support the catastrophe theory analysis, in the sense that "high-entropy" branches of solutions are shown to be relatively more stable than "low-entropy" ones. As a partial result of this entropy analysis, it is also shown that there exist logarithmic divergences in the entropy of matter (scalar) fields near the horizon, which are up and above the linear divergences, and, unlike them, they cannot be absorbed in a renormalization of the gravitational coupling constant of the theory. The associated part of the entropy violates the classical Bekenstein-Hawking formula which is a proportionality relation between black-hole entropy and the horizon area. Such logarithmic divergences, which are associated with the presence of non-Abelian gauge and Higgs fields, persist in the "extreme case," where linear divergences disappear.