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 b roken Yang-Mills gauge theories that have been recently presented as c andidates for an evasion of the scalar-no-hair theorem. Although we sh ow that in principle the conditions for the no-hair theorem do not app ly 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 t he sphaleron sector, which was expected to be unstable for topological reasons, is performed by means of a variational method. As shown, the re exist modes in this sector that are unstable against linear perturb ations. Instabilities exist also in the gravitational sector. A method for counting the gravitational unstable modes, which utilizes a catas trophe-theoretic approach is presented. The role of the catastrophe fu nctional is played by the mass functional of the black hole. The Higgs vacuum expectation value is used as a control parameter, having a cri tical value beyond which instabilities are turned on. The (stable) Sch warzschild solution is then understood from this point of view. The ca tastrophe-theory appproach facilitates enormously a universal stabilit y study of non-Abelian black holes, which goes beyond linearized pertu rbations. Some elementary entropy considerations are also presented th at support the catastrophe theory analysis, in the sense that ''high-e ntropy'' branches of solutions are shown to be relatively more stable than ''low-entropy'' ones. As a partial result of this entropy analysi s, it is also shown that there exist logarithmic divergences in the en tropy of matter (scalar) fields near the horizon, which are up and abo ve the linear divergences, and, unlike them, they cannot be absorbed i n a renormalization of the gravitational coupling constant of the theo ry. The associated part of the entropy violates the classical Bekenste in-Hawking formula which is a proportionality relation between black-h ole 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.