On the quantum width of a black hole horizon

Donald Marolf

paper

On the quantum width of a black hole horizon

arXiv:hep-th/0312059

Abstract

The many low energy modes near a black hole horizon give the thermal atmosphere a divergent entropy which becomes of order $A /4 G$ 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 $A /4 G$ 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, $L_{c}^{d} \sim \frac{R}{T _{H}} ℓ_{p}^{d - 2}$ . 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.

14 pages, JHEP style, for the Adriatic 2003 conference, minor changes

On the quantum width of a black hole horizon

Abstract

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