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

TL;DR

This paper investigates the stability of hairy black holes in spontaneously broken Einstein-Yang-Mills theories, revealing instabilities in both sphaleron and gravitational sectors and introducing a catastrophe-theoretic approach for stability analysis.

Contribution

It introduces a novel catastrophe-theoretic method to analyze the stability of non-Abelian black holes, extending beyond traditional linear perturbation techniques.

Findings

Existence of unstable modes in sphaleron sector.

Presence of instabilities in the gravitational sector.

Critical Higgs v.e.v. triggers stability changes.

Abstract

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, however we 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 r\^ole of the catastrophe functional is played by the mass functional of the black hole. The Higgs vacuum expectation value (v.e.v.) 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...