A massless scalar field minimally coupled to gravity and propagating in Sch
warzschild spacetime is considered. After dimensional reduction under spher
ical symmetry the resulting 2D field theory is canonically quantized and th
e renormalized expectation values [T-ab] Of the relevant energy-momentum te
nsor operator are investigated. Asymptotic behaviors and analytical approxi
mations are given for [T-ab] in the Boulware, Unruh and Hartle-Hawking stat
es. Special attention is devoted to the black-hole horizon region where the
WKB approximation breaks down.