The Cardy-Verlinde Formula and AdS Black Holes

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, including hyperbolic AdS black holes, AdS Reissner-Nordström black holes, charged black holes in D=5, D=4, and D=7 gauged supergravities, and AdS black holes in higher derivative gravity. We find that for the hyperbolic AdS black holes, the Cardy-Verlinde formula holds 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, the Cardy-Verlinde formula holds as well if we subtract the electric potential energy from the total energy. For other cases, it is failed to rewrite the entropy of corresponding conformal field theories in terms of the Cardy-Verlinde formula. ∗e-mail address: [email protected]