The First Law of Thermodynamics for Kerr-Anti-de Sitter Black Holes

TL;DR

This work derives consistent expressions for the mass and angular momenta of Kerr–AdS black holes across dimensions, showing they satisfy the first law with entropy $S=\frac{A}{4}$ and that the thermodynamic potential relates to the Euclidean action via $\beta\Phi=I_D$. It demonstrates that the bulk mass $E$ coincides with the Ashtekar–Magnon–Das conformal mass when evaluated with the non-rotating infinity frame, resolving ambiguities tied to boundary frames. By contrasting with prior literature, it explains discrepancies arising from frame choices and conformal boundary representatives and establishes a coherent AdS/CFT-compatible thermodynamic framework. The results extend to $D\ge 6$, providing unified expressions for masses, angular momenta, and actions, and solidify the link between bulk black hole thermodynamics and conformal boundary data.

Abstract

We obtain expressions for the mass and angular momenta of rotating black holes in anti-de Sitter backgrounds in four, five and higher dimensions. We verify explicitly that our expressions satisfy the first law of thermodynamics, thus allowing an unambiguous identification of the entropy of these black holes with $\ft14$ of the area. We find that the associated thermodynamic potential equals the background-subtracted Euclidean action multiplied by the temperature. Our expressions differ from many given in the literature. We find that in more than four dimensions, only our expressions satisfy the first law of thermodynamics. Moreover, in all dimensions we show that our expression for the mass coincides with that given by the conformal conserved charge introduced by Ashtekar, Magnon and Das. We indicate the relevance of these results to the AdS/CFT correspondence.