First Law, Counterterms and Kerr-AdS_5 Black Holes

Understand

We apply the counterterm subtraction technique to calculate the action and other quantities for the Kerr--AdS black hole in five dimensions using two boundary metrics; the Einstein universe and rotating Einstein universe with 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 is that the rotating Einstein universe, used as a boundary metric, was rotating with an angular velocity that depends 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.