Stability of topological black holes

Abstract

We explore the classical stability of topological black holes in $d$-dimensional anti-de Sitter spacetime, where the horizon is an Einstein manifold of negative curvature. According to the gauge invariant formalism of Ishibashi and Kodama, gravitational perturbations are classified as being of scalar, vector, or tensor type, depending on their transformation properties with respect to the horizon manifold. For the massless black hole, we show that the perturbation equations for all modes can be reduced to a simple scalar field equation. This equation is exactly solvable in terms of hypergeometric functions, thus allowing an exact analytic determination of potential gravitational instabilities. We establish a necessary and sufficient condition for stability, in terms of the eigenvalues $\ensuremath{\lambda}$ of the Lichnerowicz operator on the horizon manifold, namely $\ensuremath{\lambda}\ensuremath{\ge}\ensuremath{-}4(d\ensuremath{-}2)$. For the case of negative mass black holes, we show that a sufficient condition for stability is given by $\ensuremath{\lambda}\ensuremath{\ge}\ensuremath{-}2(d\ensuremath{-}3)$.