Self-Screening Hawking Atmosphere in the Presence of a Bulk Viscosity

The recent theory of 't Hooft [ Nucl. Phys. Suppl. {\bf 68}, 174 (1998)] models the black hole as a system endowed with an envelope of matter that obeys an equation of state in the form $ p=(\gamma -1)\rho$, and acts as a source in Einstein's equations. The present paper generalizes the 't Hooft theory so as to take into account a bulk viscosity $\zeta$ in the fluid. It is shown that even a slight positive value of $\zeta$ will suffice to yield complete agreement with the Hawking formula for the entropy of the black hole, if the value of the constant $\gamma$ takes a value that is slightly less than 4/3. The value $\gamma=4/3$ corresponds to a radiation fluid.