Compressibility of rotating black holes

TL;DR

This paper explores the thermodynamic properties of rotating black holes, focusing on their compressibility and sound speed, revealing how rotation and cosmological constant influence these characteristics.

Contribution

It introduces a detailed analysis of black hole compressibility and sound speed, highlighting the effects of rotation and cosmological constant on their thermodynamic behavior.

Findings

Adiabatic compressibility is zero for non-rotating black holes.

Maximum compressibility occurs in extremal black holes.

Speed of sound decreases with increasing angular momentum.

Abstract

Interpreting the cosmological constant as a pressure, whose thermodynamically conjugate variable is a volume, modifies the first law of black hole thermodynamics. Properties of the resulting thermodynamic volume are investigated: the compressibility and the speed of sound of the black hole are derived in the case of non-positive cosmological constant. The adiabatic compressibility vanishes for a non-rotating black hole and is maximal in the extremal case --- comparable with, but still less than, that of a cold neutron star. A speed of sound $v_s$ is associated with the adiabatic compressibility, which is is equal to $c$ for a non-rotating black hole and decreases as the angular momentum is increased. An extremal black hole has $v_s^2=0.9 \,c^2$ when the cosmological constant vanishes, and more generally $v_s$ is bounded below by $c/ {\sqrt 2}$.