Renormalized black hole entropy in anti-de Sitter space via the "brick wall" method

We consider the entropy of a quantum scalar field on a background black hole geometry in asymptotically anti–de Sitter space-time, using the “brick wall” approach. In anti–de Sitter space, the theory has no infrared divergences, and all ultraviolet divergences can be absorbed into a renormalization of the coupling constants in the one-loop effective gravitational Lagrangian. We then calculate the finite renormalized entropy for the Schwarzschild–anti-de Sitter and extremal Reissner–Nordström–anti-de Sitter black holes, and show that, at least for large black holes, the entropy is entirely accounted for by the one-loop Lagrangian, apart possibly from terms proportional to the logarithm of the event horizon radius. For small black holes, there are indications that non-perturbative quantum gravity effects become important.