Cardy-Verlinde Formula and entropy bounds in Kerr-Newman-AdS$_4$/dS$_4$ black holes backgrounds

Abstract

The Cardy-Verlinde formula is further verified by using the Kerr-Newman-AdS$_4$ and Kerr-Newman-dS$_4$ black holes. In the Kerr-Newman-AdS$_4$ spacetime, we find that, for strongly coupled CFTs with AdS duals, to cast the entropy of the CFT into the Cardy-Verlinde formula the Casimir energy must contains the terms $ -n ({\mathcal{J}} \Omega_H+ \frac{Q\Phi}{2}+ \frac{Q\Phi_0}{2})$, which associate with rotational and electric potential energies, and the extensive energy includes the term $-Q \Phi_0$. For the Kerr-Newman-dS$_4$ black hole, we note that the Casimir energy is negative but the extensive energy is positive on the cosmological horizon; while the Casimir energy is positive but the extensive energy is negative on the event horizon (the definitions for the two energies possess the same forms as the corresponding quantities of the Kerr-Newman-AdS$_4$ black hole). Thus we have to take the absolute value of the Casimir (extensive) energy in the Cardy-Verlinde formula for the cosmological (event) horizon. The result for the Kerr-Newman-dS$_4$ spacetime provides support of the dS/CFT correspondence. Furthermore, we also obtain the Bekenstein-Verlinde-like entropy bound for the Kerr-Newman-AdS$_4$ black hole and the D-bound on the entropy of matter system in Kerr-Newman-dS$_4$ spacetime. We find that both the bounds are tightened by the electric charge.