Pressure and volume in the first law of black hole thermodynamics

The mass of a black hole is interpreted, in terms of thermodynamic potentials,
\nas being the enthalpy, with the pressure given by the cosmological constant.
\nThe volume is then defined as being the Legendre transform of the pressure
\nand the resulting relation between volume and pressure is explored in the case
\nof positive pressure. A virial expansion is developed and a van der Waals
\nlike critical point determined. The first law of black hole thermodynamics
\nincludes a PdV term which modifies the maximal efficiency of a Penrose
\nprocess. It is shown that, in four dimensional space-time with a negative
\ncosmological constant, an extremal charged rotating black hole can…