speed of light and our calculation might be applicable. Since the expanding mass shell itself leads to gravitational particle production, this work might be useful for an estimate of backreaction effects in the final stages of black hole evaporation. We will not study this aspect here, the purpose of this paper being just to calculate the energy spectrum of the particles produced by an expanding mass shell.

There are three main procedures that are used to deal with quantum fields in curved space-time: Hawking’s classical ray-tracing [@Haw], renormalization of the energy-momentum tensor (e.g. by point-splitting) [@Fulling; @Wald], particle production in the reference frame of a uniformly accelerated observer [@Unruh; @GrQED], from which we will choose the first one. A useful overview on computational matters can be found in [@BiDa; @Novi].

The procedure of ray-tracing in a semiclassical limit is appropriate if the energy carried by the quantum field is small with respect to the energy of the source of the gravitational field. Furthermore we have to stay on scales on which possible effects of quantum gravity do not occur. As Wheeler showed in [@Pll] the characteristic length for this to happen is the Planck length $l_{\rm P}=
\sqrt{\frac{G\hbar}{c^3}}\approx 10^{-33}$ cm.

