Compressibility of rotating black holes

TL;DR

This work analyzes the extended thermodynamics of rotating black holes by computing the adiabatic compressibility $\beta_S$ and the associated speed of sound $v_s$ under pressure variations while holding entropy $S$ and angular momentum $J$ fixed. It shows $\beta_S$ vanishes for $J=0$ and increases with angular momentum, reaching a maximum in the extremal limit. The derived speed of sound satisfies $0 \le v_s^2 \le c^2$, equals $c$ for non-rotating holes, and for $P=0$ at extremality yields $v_s^2=0.9\,c^2$, with a general lower bound $v_s \ge c/\sqrt{2}$. The analysis discusses the relation between adiabatic and isothermal compressibilities, highlights cosmic censorship subtleties when varying $P$ at fixed $S$ and $J$, and notes potential implications for primordial black holes and the stiff equation of state in the AdS regime.

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 $c/ {\sqrt 2}$.