Cardy-Verlinde Formula and AdS Black Holes

TL;DR

This paper tests the Cardy-Verlinde entropy formula across various AdS black holes, confirming its validity in many cases but noting exceptions in Lovelock gravity scenarios.

Contribution

It extends the verification of the Cardy-Verlinde formula to multiple AdS black hole solutions, identifying conditions for its applicability and limitations.

Findings

Formula holds for hyperbolic AdS black holes with a massless ground state.

Valid for charged AdS black holes after subtracting supersymmetric background energy.

Fails for black holes in Lovelock gravity, indicating limitations of the formula.

Abstract

In a recent paper hep-th/0008140 by E. Verlinde, an interesting formula has been put forward, which relates the entropy of a conformal formal field in arbitrary dimensions to its total energy and Casimir energy. This formula has been shown to hold for the conformal field theories that have AdS duals in the cases of AdS Schwarzschild black holes and AdS Kerr black holes. In this paper we further check this formula with various black holes with AdS asymptotics. For the hyperbolic AdS black holes, the Cardy-Verlinde formula is found to hold if we choose the ``massless'' black hole as the ground state, but in this case, the Casimir energy is negative. For the AdS Reissner-Nordstr\"om black holes in arbitrary dimensions and charged black holes in D=5, D=4, and D=7 maximally supersymmetric gauged supergravities, the Cardy-Verlinde formula holds as well, but a proper internal energy which corresponds to the mass of supersymmetric backgrounds must be subtracted from the total energy. It is failed to rewrite the entropy of corresponding conformal field theories in terms of the Cardy-Verlinde formula for the AdS black holes in the Lovelock gravity.