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

TL;DR

This paper calculates the finite, renormalized quantum scalar field entropy for black holes in anti-de Sitter space using the brick wall method, showing that for large black holes, the entropy is mainly captured by the one-loop effective gravitational Lagrangian.

Contribution

It demonstrates how to renormalize black hole entropy in anti-de Sitter space via the brick wall method, accounting for quantum effects and highlighting differences between large and small black holes.

Findings

Renormalized entropy is finite and mainly from the one-loop Lagrangian for large black holes.

Infra-red divergences are absent in anti-de Sitter space, simplifying the analysis.

Non-perturbative effects may be relevant for small black holes.

Abstract

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 infra-red divergences, and all ultra-violet 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-Nordstrom-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.