Black-hole thermodynamics and singular solutions of the Tolman-Oppenheimer-Volkoff equation

Thermodynamic equilibrium of a self-gravitating perfect fluid for a spherically symmetric system containing a black hole of mass $M$ is investigated by means of the Tolman-Oppenheimer-Volkoff (TOV) equation. A singular family of solutions of the TOV equation is described. At $r\gg 2M$ these solutions can be used to represent a perfect fluid (i.e., photon gas) of temperature $T_{\mathrm{BH}}={\left(8\mathrm{}\pi M\right)}^{-1\mathrm{}}$ in equilibrium with a Schwarzschild black hole. The energy density is positive at all $r>\mathrm{}0\mathrm{}$. A singular negative point mass resides at $r=0\mathrm{}$.