  The method of Hawking to obtain black hole evaporation through Bogoljubov
transformation between asymptotic modes (in and out) is generalized. The
construction is local in that the in modes (of say positive frequency) are
decomposed by Bogoljubov transformation into positive and negative frequency
local inertial modes (i.e. those which are solutions of the d'Alembertian in
terms of the local normal coordinates). From this follows an interesting
reexpression of the local energy momentum tensor, more particularly of the
outgoing energy flux. One finds that even in the local description there exists
a partial thermal character parametrized by a local temperature. There exists
quantum interference effects as well. These become negligible at large distance
from the black hole.
