Black hole radiation in the presence of a short distance cutoff

A derivation of the Hawking effect is given which avoids reference to field modes above some cutoff frequency ${\mathrm{\omega }}_{\mathit{c}}$≫${\mathit{M}}^{\mathrm{-}1}$ 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. Because of 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 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.