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 c(2) when the cosmological constant vanishes, and more generally v(s) is bounded below by c/root 2.