Criticality and Surface Tension in Rotating Horizon Thermodynamics

We study a modified horizon thermodynamics and the associated criticality for rotating black hole spacetimes. Namely, we show that under a virtual displacement of the black hole horizon accompanied by an independent variation of the rotation parameter, the radial Einstein equation takes a form of a "cohomogeneity two" horizon first law, $dE=TdS+ΩdJ-σdA$, where $E$ and $J$ are the horizon energy (an analogue of the Misner-Sharp mass) and the horizon angular momentum, $Ω$ is the horizon angular velocity, $A$ is the horizon area, and $σ$ is the surface tension induced by the matter fields. For fixed angular momentum, the above equation simplifies and the more familiar (cohomogeneity one) horizon first law $dE=TdS-PdV$ is obtained, where $P$ is the pressure of matter fields and $V$ is the horizon volume. A universal equation of state is obtained in each case and the corresponding critical behavior is studied.

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