Do evaporating black holes form photospheres?

Several authors, most notably Heckler, have claimed that the observable Hawking emission from a microscopic black hole is significantly modified by the formation of a photosphere around the black hole due to QED or QCD interactions between the emitted particles. In this paper we analyze these claims and identify a number of physical and geometrical effects which invalidate these scenarios. We point out two key problems. First, the interacting particles must be causally connected to interact, and this condition is satisfied by only a small fraction of the emitted particles close to the black hole. Second, a scattered particle requires a distance $\sim E/m_e^2$ for completing each bremsstrahlung interaction, with the consequence that it is improbable for there to be more than one complete bremsstrahlung interaction per particle near the black hole. These two effects have not been included in previous analyses. We conclude that the emitted particles do not interact sufficiently to form a QED photosphere. Similar arguments apply in the QCD case and prevent a QCD photosphere (chromosphere) from developing when the black hole temperature is much greater than ${\Lambda }_{\mathrm{QCD}}$, the threshold for QCD particle emission. Additional QCD phenomenological arguments rule out the development of a chromosphere around black hole temperatures of order ${\Lambda }_{\mathrm{QCD}}$. In all cases, the observational signatures of a cosmic or Galactic halo background of primordial black holes or an individual black hole remain essentially those of the standard Hawking model, with little change to the detection probability. We also consider the possibility, as proposed by Belyanin et al. and D. Cline et al., that plasma interactions between the emitted particles form a photosphere, and we conclude that this scenario too is not supported.

Figure 1

The eight Feynman diagrams for electron-electron bremsstrahlung in lowest order perturbation theory. The labels, which are defined in the text, correspond to the 4-momenta of the particles involved in the interaction.

Figure 2

Illustrating how a particle emitted by a black hole of radius $r_{\mathrm{bh}}$ can interact with other particles emitted in a similar radial direction where the angle ${\theta }_{\mathrm{bh}}$ is the deviation from the radial direction due to the finite size of the black hole.

Figure 3

The distribution of background particles for which the formula $\lambda =(n\sigma v{)}^{-1}$ describes the mean-free path of an incident particle of speed $v$. The number density $n$ is assumed to be uniform. In the actual black hole situation the density is not uniform, and all particles are moving radially outwards in the black hole frame.