The self-screening Hawking atmosphere

A model is proposed in which the Hawking particles emitted by a black hole are treated as an envelope of matter that obeys an equation of state, and acts as a source in Einstein's equations. This is a crude but interesting way to accommodate for the back reaction. For large black holes, the solution can be given analytically, if the equation of state is $p=\kappa\rho$, with $0<\kappa<1$. The solution exhibits a singularity at the origin. If we assume $N$ free particle types, we can use a Hartree-Fock procedure to compute the contribution of one such field to the entropy, and the result scales as expected as $1/N$. A slight mismatch is found that could be attributed to quantum corrections to Einstein's equations, but can also be made to disappear when $\k$ is set equal to one. The case $\kappa=1$ is further analysed.