Cardy-Verlinde formula and AdS black holes

Abstract

In a recent paper by E. Verlinde, hep-th/0008140, 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 anti–de Sitter (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-Nordström black holes in arbitrary dimensions and charged black holes in $D=5,\hspace{0.5em}D=4,$ and $D=7\mathrm{}$ 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. We fail to rewrite the entropy of corresponding conformal field theories in terms of the Cardy-Verlinde formula for the AdS black holes in Lovelock gravity.