Renormalized black hole entropy in anti-de Sitter space via the "brick wall" method - art. no. 084013

We consider the entropy of a quantum scalar field on a background black hol e geometry in asymptotically anti-de Sitter space-time, using the "brick wa ll" approach. In anti-de Sitter space, the theory has no infrared divergenc es, and all ultraviolet divergences can be absorbed into a renormalization of the coupling constants in the one-loop effective gravitational Lagrangia n. We then calculate the finite renormalized entropy for the Schwarzschild- anti-de Sitter and extremal Reissner-Nordstrom-anti-de Sitter black holes, and show that, at least for large black holes, the entropy is entirely acco unted for by the one-loop Lagrangian, apart possibly from terms proportiona l to the logarithm of the event horizon radius. For small black holes, ther e are indications that non-perturbative quantum gravity effects become impo rtant.