Compressibility of rotating black holes

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 nonpositive cosmological constant. The adiabatic compressibility vanishes for a nonrotating 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 equal to $c$ for a nonrotating black hole and decreases as the angular momentum is increased. An extremal black hole has ${v}_{s}^{2}=0.9\text{ }\text{ }{c}^{2}$ when the cosmological constant vanishes, and more generally ${v}_{s}$ is bounded below by $c/\sqrt{2}$.

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