A linear approximation to black hole evaporation

Abstract

An evaporating Schwarzschild black hole is analysed including back reaction in a linear approximation. The analysis assumes a massless scalar field propagating in a spacetime consisting of two Vaidya metrics corresponding respectively to outgoing radiation and an infalling negative energy flux. For times late relative to the collapse but early relative to the lifetime of the hole, the standard rate is reproduced and has the correct time dependence. The event horizon shrinks at the expected rate. These results are independent of the exact location of the boundary between the regions. The magnitude of the quantum fluxes at various radii suggests that most of the pair production occurs far from the horizon.