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 $\lim \left(\frac{\mathrm{dM}}{\mathrm{du}}\right)=0\mathrm{}$ as $M\to 0$, where $u$ is the usual outgoing null coordinate defined on ${\mathcal{I}}^{+}$. This is particularly interesting because $\frac{\mathrm{dM}}{\mathrm{du}}$ is in principle measurable by a distant observer. If $\lim \left(\frac{\mathrm{dM}}{\mathrm{du}}\right)\ne 0$ as $M\to 0$, then the quantized field's outgoing energy flux ($T_{\mathrm{uu}}$) and its integral (the total energy radiated) both diverge as the Cauchy horizon is approached.

• Received 12 December 1980

DOI:https://doi.org/10.1103/PhysRevD.23.2823