Analytical approximations for of a quantized scalar field in
ultrastatic asymptotically flat spacetimes are obtained. The field is assumed
to be both massive and massless, with an arbitrary coupling $\xi$ to the scalar
curvature, and in a zero or nonzero temperature vacuum state. The expression
for is divided into low- and high-frequency parts. The expansion
for the high-frequency contribution to this quantity is obtained. This
expansion is analogous to the DeWitt-Schwinger one. As an example, the
low-frequency contribution to is calculated on the background of
the small perturbed flat spacetime in a quantum state corresponding to the
Minkowski vacuum at the asymptotic. The limits of the applicability of these
approximations are discussed.Comment: revtex4, 9 page
