Black Hole Evaporation in the Presence of a Short Distance Cutoff

TL;DR

This paper derives a version of the Hawking effect that incorporates a short-distance cutoff, proposing a boundary condition near the horizon that could alter the expected black hole radiation spectrum.

Contribution

It introduces a boundary condition near the horizon in a cutoff theory, challenging the traditional derivation of Hawking radiation and suggesting possible deviations in the radiation spectrum.

Findings

Hawking effect can be derived with a cutoff without high-frequency modes

Boundary conditions near the horizon are crucial for Hawking radiation

Black hole radiation spectrum may differ from Hawking's prediction

Abstract

A derivation of the Hawking effect is given which avoids reference to field modes above some cutoff frequency $\omega_c\gg M^{-1}$ in the free-fall frame of the black hole. To avoid reference to arbitrarily high frequencies, it is necessary to impose a boundary condition on the quantum field in a timelike region near the horizon, rather than on a (spacelike) Cauchy surface either outside the horizon or at early times before the horizon forms. Due to the nature of the horizon as an infinite redshift surface, the correct boundary condition at late times outside the horizon cannot be deduced, within the confines of a theory that applies only below the cutoff, from initial conditions prior to the formation of the hole. A boundary condition is formulated which leads to the Hawking effect in a cutoff theory. It is argued that it is possible the boundary condition is {\it not} satisfied, so that the spectrum of black hole radiation may be significantly different from that predicted by Hawking, even without the back-reaction near the horizon becoming of order unity relative to the curvature.