  The expectation value of the energy-momentum tensor and the Hawking flux of a
scalar field on a Schwarzschild spacetime is calculated using the zeta-function
regularisation of the heat kernel. In particular, massless particles are
considered in a spherically reduced dilaton model. The effective action is
thereby obtained by the covariant perturbation theory and the boundary
conditions are fixed by means of the energy-momentum conservation equation. In
contrast to previous approaches the expectation values are calculated directly
from the effective action which itself is derived in a straightforward manner.
