  We investigate the effects of quantum backreaction on the Schwarzschild
geometry in the semiclassical approximation. The renormalized stress-energy
tensor (RSET) of a scalar field is modelled via an order reduction of the
analytical approximation derived by Anderson, Hiscock and Samuel (AHS). As the
resulting AHS semiclassical Einstein equations are of fourth-derivative order
in the metric, we follow a reduction of order prescription to shrink the space
of solutions. Motivated by this prescription, we develop a method that allows
to obtain a novel analytic approximation for the RSET that exhibits all the
desired properties for a well-posed RSET: conservation, regularity, and correct
estimation of vacuum-state contributions. We derive a set of semiclassical
equations sourced by the order-reduced AHS-RSET in the Boulware state. We
classify the self-consistent solutions to this set of field equations, discuss
their main features and address how well they resemble the solutions of the
higher-order semiclassical theory. Finally, we establish a comparison with
previous results in the literature obtained through the Polyakov approximation
for minimally coupled scalar fields.
