THE FIRST LAW, COUNTERTERMS AND KERR-AdS5 BLACK HOLES

We apply the counterterm subtraction technique to calculate the action and other quantities of the Kerr-AdS black hole in five dimensions using two boundary metrics: the Einstein universe and the rotating Einstein universe with an arbitrary angular velocity. In both cases, the resulting thermodynamic quantities satisfy the first law of thermodynamics. We point out that the reason for the violation of the first law in previous calculations was that the rotating Einstein universe, used as a boundary metric, was rotating with an angular velocity that depended on the black hole rotation parameter. Using a new coordinate system with a boundary metric that has an arbitrary angular velocity, one can show that the resulting physical quantities satisfy the first law.