Formation of a Hawking-radiation photosphere around microscopic black holes

We show that once a black hole surpasses some critical temperature $T_{\mathrm{crit}},$ the emitted Hawking radiation interacts with itself and forms a nearly thermal photosphere. Using QED, we show that the dominant interactions are bremsstrahlung and electron-photon pair production, and we estimate $T_{\mathrm{crit}}\sim m_e/{\alpha }^{5/2}$, which when calculated more precisely is found to be $T_{\mathrm{crit}}\approx$45 GeV. The formation of the photosphere is purely a particle physics effect, and not a general relativistic effect, since the photosphere forms roughly ${\alpha }^{-4}$ Schwarzschild radii away from the black hole. The temperature $T$ of the photosphere decreases with distance from the black hole, and the outer surface is determined by the constraint $T\sim m_e$ (for the QED case), since this is the point at which electrons and positrons annihilate, and the remaining photons free stream to infinity. Observational consequences are discussed, and it is found that, although the QED photosphere will not affect the Page-Hawking limits on primordial black holes, which is most important for 100 MeV black holes, the inclusion of QCD interactions may significantly effect this limit, since for QCD we estimate $T_{\mathrm{crit}}\sim {\Lambda }_{\mathrm{QCD}}.$ The photosphere greatly reduces the possibility of observing individual black holes with temperatures greater than $T_{\mathrm{crit}},$ since the high energy particles emitted from the black hole are processed through the photosphere to a lower energy, where the $\gamma$-ray background is much higher. The temperature of the plasma in the photosphere can be extremely high, and this offers interesting possibilities for processes such as symmetry restoration.