Reversible-process black hole thermodynamics

Bekenstein, Hawking, Gibbons and Perry have discussed the case of placing a Schwarzschild black hole inside an ideal, reflecting box, thereby setting up an equilibrium state. In this paper, we discuss more generally the thennodynamical properties of such a system. Starting from adiabatic expansion of the system, we are naturally led to the definition of total entropy and of black hole entropy. We next point out the two conditions for stable equilibrium between the black hole and the radiation, V < V E and E r < M/4 are not equivalent: only the former is necessary and sufficient. Lastly, we examine three quasi-static processes of evaporation of the black hole, expansion at constant energy, energy release at constant volume and adiabatic expansion.