  Quantum stress-energy tensors of fields renormalized on a Schwarzschild
background violate the classical energy conditions near the black hole.
Nevertheless, the associated equilibrium thermodynamical entropy $\Delta S$ by
which such fields augment the usual black hole entropy is found to be positive.
More precisely, the derivative of $\Delta S$ with respect to radius, at fixed
black hole mass, is found to vanish at the horizon for {\it all} regular
renormalized stress-energy quantum tensors. For the cases of conformal scalar
fields and U(1) gauge fields, the corresponding second derivative is positive,
indicating that $\Delta S$ has a local minimum there. Explicit calculation
shows that indeed $\Delta S$ increases monotonically for increasing radius and
is positive. (The same conclusions hold for a massless spin 1/2 field, but the
accuracy of the stress-energy tensor we employ has not been confirmed, in
contrast to the scalar and vector cases). None of these results would hold if
the back-reaction of the radiation on the spacetime geometry were ignored;
consequently, one must regard $\Delta S$ as arising from both the radiation
fields and their effects on the gravitational field. The back-reaction, no
matter how "small",
