  The effective action associated with the trace anomaly provides a general
algorithm for approximating the expectation value of the stress tensor of
conformal matter fields in arbitrary curved spacetimes. In static, spherically
symmetric spacetimes, the algorithm involves solving a fourth order linear
differential equation in the radial coordinate r for the two scalar auxiliary
fields appearing in the anomaly action, and its corresponding stress tensor. By
appropriate choice of the homogeneous solutions of the auxiliary field
equations, we show that it is possible to obtain finite stress tensors on all
Reissner-Nordstrom event horizons, including the extreme Q=M case. We compare
these finite results to previous analytic approximation methods, which yield
invariably an infinite stress-energy on charged black hole horizons, as well as
with detailed numerical calculations that indicate the contrary. The
approximation scheme based on the auxiliary field effective action reproduces
all physically allowed behaviors of the quantum stress tensor, in a variety of
quantum states, for fields of any spin, in the vicinity of the entire family (0
le Q le M) of RN horizons.
