  The propagator of a scalar field on a stationary slowly varying in space
gravitational background is derived retaining only the second derivatives of
the metric. The corresponding one-loop effective action is constructed. The
propagator and the effective action turn out to depend nontrivially on the
Killing vector defining the vacuum state and the Hamiltonian of a scalar field.
The Hawking particle production is described in the quasiclassical
approximation and the quasiclassical formula for the Hawking temperature is
derived. The behaviour of the Unruh detector on a curved background is
considered and the quasiclassical formula for the Unruh acceleration
determining the Unruh temperature is derived. The radiation reaction problem on
a curved background is discussed in view of the new approximate expression for
the propagator. The correction to the mass squared of a scalar particle on a
stationary gravitational background is obtained. This correction is analogous
to the quantum correction to the particle mass in a strong electromagnetic
field. For a vacuum solution to the Einstein equations, it is equal to minus
one-fourth of the free fall acceleration squared.
