Scalar hair on the black hole in asymptotically anti-de Sitter spacetime -art. no. 044007

We examine the no-hair conjecture in asymptotically anti-de Sitter (AdS) sp acetime. First, we consider a real scalar field as the matter field and ass ume static spherically symmetric spacetime. Analysis of the asymptotics sho ws that the scalar field must approach the extremum of its potential. Using this fact, it is proved that there is no regular black hole solution when the scalar field is massless or has a "convex" potential. Surprisingly, whi le the scalar field has a growing mode around the local minimum of the pote ntial. there is no growing mode around the local maximum. This implies that the local maximum is a kind of "attractor" of the asymptotic scalar field. We give two examples of the new black hole solutions with a nontrivial sca lar field configuration numerically in the symmetric or asymmetric double w ell potential models. We study the stability of these solutions by using th e linear perturbation method in order to examine whether or not the scalar hair is physical. In the symmetric double well potential model, we find tha t the potential function of the perturbation equation is positive semidefin ite in some wide parameter range and that the new solution is stable. This implies that the black hole no-hair conjecture is violated in asymptoticall y AdS spacetime.