Black Hole Evaporation in the Presence of a Short Distance Cutoff

TL;DR

This work addresses the dependence of Hawking radiation on ultra-high-frequency (trans-Planckian) modes by introducing a short-distance cutoff and a horizon-near boundary condition defined for high-frequency outgoing modes as seen by freely falling observers. By constructing a consistent boundary prescription and analyzing its implications, the paper shows how Hawking radiation can emerge within a cutoff theory and quantifies the resulting precision, which scales as $O\left(\sqrt{\kappa/\omega_c}\right)$ for large black holes. It also discusses the physical plausibility of the boundary condition, the possible deviations from Hawking's spectrum if the boundary condition fails, and the constraints from the stress-energy tensor on back-reaction, highlighting how short-distance physics could influence observable radiation while remaining compatible with small back-reaction under certain scenarios. The results underscore the potential role of horizon redshift as a probe of short-distance physics and motivate further exploration of cutoff models in black hole spacetimes.

Abstract

A derivation of the Hawking effect is given which avoids reference to field modes above some cutoff frequency $ω_c\gg M^{-1}$ 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.