Thermodynamics of quantum photon spheres
  Photon spheres, surfaces where massless particles are confined in closed orbits, are expected to be common astrophysical structures surrounding ultracompact objects. In this paper a semiclassical treatment of a photon sphere is proposed. We consider the quantum Maxwell field and derive its energy spectra. A thermodynamic approach for the quantum photon sphere is developed and explored. Within this treatment, an expression for the spectral energy density of the emitted radiation is presented. Our results suggest that photon spheres, when thermalized with their environment, have nonusual thermodynamic properties, which could lead to distinct observational signatures. 
M. C. Baldiotti, Walace S. Elias, C. Molina, Thiago S. Pereira
gr-qc astro-ph.HE hep-th quant-ph