On the quantum width of a black hole horizon

TL;DR

This paper extends Sorkin's argument to relativistic settings, showing that quantum fluctuations give black hole horizons a finite width, effectively regulating the thermal atmosphere's entropy to be negligible compared to the Bekenstein-Hawking entropy.

Contribution

It provides a relativistic, dimension-independent derivation of the horizon width, establishing a geometric mean scale that suppresses the thermal atmosphere's entropy for large black holes.

Findings

Quantum horizon width scales with geometric mean of Planck length and black hole radius.

Thermal atmosphere entropy is parametrically small compared to Bekenstein-Hawking entropy.

Large number of fields and model discrepancies are discussed.

Abstract

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.