A Note on Physical Mass and the Thermodynamics of AdS-Kerr Black Holes

TL;DR

This paper clarifies the proper definition of mass for AdS-Kerr black holes, showing that using the physical mass E resolves apparent violations of the second law of thermodynamics related to horizon area changes.

Contribution

It demonstrates that the correct physical mass to use in thermodynamic considerations of AdS-Kerr black holes is E, not M, resolving paradoxes in horizon area behavior.

Findings

Horizon area increases with angular momentum if M is fixed.

Using the physical mass E, the horizon area decreases with angular momentum.

The physical mass E aligns with the First Law of black hole thermodynamics.

Abstract

As with any black hole, asymptotically anti-de Sitter Kerr black holes are described by a small number of parameters, including a "mass parameter" $M$ that reduces to the AdS-Schwarzschild mass in the limit of vanishing angular momentum. In sharp contrast to the asymptotically flat case, the horizon area of such a black hole increases with the angular momentum parameter $a$ if one fixes $M$, this appears to mean that the Penrose process in this case would violate the Second Law of black hole thermodynamics. We show that the correct procedure is to fix not $M$ but rather the "physical" mass $E=M/(1-a^2/L^2)^2$, this is motivated by the First Law. For then the horizon area decreases with $a$. We recommend that $E$ always be used as the mass in physical processes: for example, in attempts to "over-spin" AdS-Kerr black holes.