  In terms of the Painlev{\'e}-Gullstrand-Lema{\^\i}tre coordinates a rather
general scenario for the gravitational collapse of an object and the subsequent
formation of a horizon is described by a manifestly $C^\infty$-metric. For a
1+1 dimensional model of the collapse the leading contributions to the
Bogoliubov coefficients are calculated explicitely and the Hawking temperature
is recovered. But depending on the particular dynamics of the collapse the
final state represents either evaporation or anti-evaporation. The
generalization of the calculation to 3+1 dimensions is outlined and possible
implications are addressed. PACS-numbers: 04.70.Dy, 04.70.-s, 04.62.+v.
