Vacuum polarization in the Schwarzschild spacetime and dimensional reduction

A massless scalar field minimally coupled to gravity and propagating in Schwarzschild spacetime is considered. After dimensional reduction under spherical symmetry the resulting 2D field theory is canonically quantized and the renormalized expectation values $〈T_{\mathrm{ab}}〉$ of the relevant energy-momentum tensor operator are investigated. Asymptotic behaviors and analytical approximations are given for $〈T_{\mathrm{ab}}〉$ in the Boulware, Unruh and Hartle-Hawking states. Special attention is devoted to the black-hole horizon region where the WKB approximation breaks down.