Holographic Pressure and Extensivity of Rotating Black Holes at Finite Cutoff

TL;DR

This paper explores the thermodynamics of rotating Kerr-AdS black holes with a finite boundary, revealing how angular momentum affects the first law and showing that large black holes behave extensively, supporting holographic duality.

Contribution

It extends the holographic thermodynamics framework to rotating black holes at finite cutoff, deriving explicit thermodynamic relations including angular momentum effects.

Findings

Large black holes exhibit extensive thermodynamic behavior.

Angular momentum introduces a momentum flux term at the boundary.

The holographic interpretation is supported by the recovery of extensivity at large sizes.

Abstract

We investigate the quasi-local thermodynamics of rotating Kerr-AdS black holes enclosed by a finite timelike boundary (cavity). Extending recent work on static systems, we define the holographic pressure and volume via the trace of the Brown-York stress tensor on the cutoff surface. We demonstrate that the inclusion of angular momentum introduces a momentum flux term at the boundary, requiring a generalized first law: $dE = T_{\mathrm{loc}}dS + \Omega_{\mathrm{loc}}dJ - \mathcal{P}d\mathcal{A}$. We derive the explicit expressions for these thermodynamic conjugates and analyze the equation of state. Crucially, we examine the extensivity of the system in the large-size limit. We find that while small rotating black holes exhibit non-extensive behavior typical of self-gravitating systems, large Kerr-AdS black holes recover extensivity, behaving effectively as a thermal fluid on the boundary. This result strengthens the holographic interpretation of the cutoff surface as the domain of a dual field theory.