On Black Holes Surrounded by Radiation II: Thermodynamics

In a companion paper we considered a Schwarzschild black hole of mass $m$ enveloped by a thick "ocean'' of massless particles that extends the black hole's photon sphere into a region of finite depth. There we showed that this "hillingar black hole'', of ADM mass $M$, optically mimics an ordinary black hole of the same mass. Here we find it also mimics the black hole thermodynamically: the formal assumption of thermal equilibrium implies the system has the same temperature and entropy as an ordinary black hole of mass $M$. We check this result carefully using multiple methods; a further method and indications of metastability are given by one of us in a companion paper. In AdS space, the mimicry does not hold, and the solutions have a richer structure. While it is far from clear that these systems are models for more realistic ones, we note possible connections with black hole evolution. In particular, assuming thermal equilibrium can be established and maintained, an HBH in a cavity of radius $\geq 3M$ can evaporate, potentially posing the information puzzle in a small finite volume.