As Cold as a Black Hole: Extended Photon Spheres

Abstract

It is widely believed that self-gravitating radiation cannot reach thermal equilibrium with a black hole in asymptotically flat spacetime. We describe a marginally stable continuum exception to the standard instability argument. Our main observation is that the photon sphere controls the Israel junction conditions (IJCs), the Tolman-Oppenheimer-Volkoff (TOV) equation, and finite-radius black hole thermodynamics: the IJCs and TOV equation are equivalent at zero radial pressure, the inverse specific heat at the photon sphere is proportional to $-\Lambda$, and adding shells at fixed total mass \textit{lowers} the asymptotic Hawking temperature if and only if the local specific heat is positive. Using these results, we show how to compute thermodynamic entropies without the Euclidean path integral. The exception described here results from companion work with M.J. Strassler, where we found that a "hillingar black hole" (HBH) mimics an ordinary Schwarzschild black hole of mass $M$, sharing its Hawking temperature, photon ring, and, in equilibrium, its coarse-grained entropy $S = 4 \pi M^2$. Here, we show these features are not tuned; they follow uniquely from joint mechanical and thermodynamic constraints. Conditions for thermodynamic mimicry and for the formation of extended photon spheres are found; for self-similar solutions they coincide. A one-parameter family of self-similar systems -- all of which, excepting the HBH, require massless walls at the edges of their extended photon spheres -- satisfies both conditions. This family includes "stiffest stars" and "frozen stars".