Energy-momentum tensor of the evaporating black hole and local Bogoljubov transformations

Hawking's method of obtaining black hole evaporation through Bogoljubov transformations 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 transformations 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 re-expression 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 exist quantum interference effects as well. These become negligible at large distances from the black hole.