Models of evaporating black holes. II. Effects of the outgoing created radiation

The two-dimensional stress-energy tensor of a quantized masssless scalar field is studied on fixed background evaporating-black-hole spacetimes (EBH's). The spherically symmetric EBH backgrounds are constructed by patching together ingoing- and outgoing-null-fluid Vaidya metrics along a timelike hypersurface which is near (but outside) the apparent horizon of the EBH. The timelike hypersurface on which the metrics are joined may be viewed as a shell of pair-creation events. It is found that a necessary (but not sufficient) condition for the quantized field's stress-energy to remain finite on the Cauchy horizon of the EBH is that lim(dM/du)=0 as M→0, where u is the usual outgoing null coordinate defined on I⁺. This is particularly interesting because dM/du is in principle measurable by a distant observer. If lim(dM/du)≠0 as M→0, then the quantized field's outgoing energy flux (Tᵤᵤ) and its integral (the total energy radiated) both diverge as the Cauchy horizon is approached.