Gravitational entropy: Beyond the black hole

We attempt to establish the existence of gravitational entropy for a thin spherical shell of matter as a function of radius by examining the modifications to the thermodynamics of a black hole at the center of such a shell. We show that the shell has the effect of depressing the temperature of the hole, but that small exchanges of energy between the hole and its environment at the same temperature remain isentropic in the presence of the shell. The model is generalized to include de Sitter horizons too, and we find that the shell can be positioned so that a ‘‘back flow’’ of radiation takes place from the de Sitter horizon into the hole, thus enabling the hole-plus-shell system to be used as a device for ‘‘mining the Universe.’’ In all cases our results are consistent with simple arguments indicating that there is no gravitational entropy associated with spherical shells which have not collapsed into black holes.