On the quantum width of a black hole horizon

The many low energy modes near a black hole horizon give the thermal atmosphere a divergent entropy which becomes of order $A/4G$ with a Planck scale cut-off. However, Sorkin has given a Newtonian argument for 3+1 Schwarzschild black holes to the effect that fluctuations of such modes provide the horizon with a non-zero quantum mechanical width. This width then effectively enforces a cut-off at much larger distances so that the entropy of the thermal atmosphere is negligible in comparison with $A/4G$ for large black holes. We generalize and improve this result by giving a relativistic argument valid for any spherical black hole in any dimension. The result is again a cut-off $L_c$ at a geometric mean of the Planck scale and the black hole radius; in particular, $L_c^d \sim \frac{R}{T_H} \ell_p^{d-2}$. With this cut-off, the entropy of the thermal atmosphere is again parametrically small in comparison with the Bekenstein-Hawking entropy of the black hole. The effect of a large number $N$ of fundamental fields and the discrepancies from naive predictions of a stretched horizon model are also discussed.