A note on the Cardy–Verlinde formula

AbstractWe study static, spherically symmetric, and asymptotically AdS black hole spacetimes with matter distribution satisfying Ttt=Trr. The first law of black hole thermodynamics is checked. We show that the Cardy–Verlinde formula, which is supposed to be an entropy formula of CFTs in any dimensions, is just a rewritten form of Einstein equations on the black hole horizon. Further, the entropy of dual CFT to the black hole solution is found to indeed satisfy the Cardy–Verlinde formula and the energy of matter outside the black hole should be subtracted from the total energy of the dual CFT in the Cardy–Verlinde formula. The implications of our results are discussed