Hawking radiation from an evaporating black hole via Bogoliubov transformations

We study Hawking radiation on a Vaidya space-time with a gravitational collapse followed by evaporation. The collapsing body is a null thin-shell and the evaporation is induced by a negative energy collapsing null-shell. This mimics the back-reaction to the Hawking radiation. Using Hawking's original method of Bogoliubov transformations we characterize the radiated spectrum as dominated by a thermal emission with an increasing effective temperature. We compute this time dependent temperature and find numerical agreement with results obtained by other techniques. The known divergences at the evaporation time are explained by the divergent nature of the effective temperature. As a consistency check, we re-derived the results from a zero mass limit of a remnant BH scenario.