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 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 vs 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 = 0.9 c 2 when the cosmological constant vanishes, and more generally vs is bounded below by c/ √ 2.