Positivity of entropy in the semiclassical theory of black holes and radiation

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 the radius, at a fixed black hole mass, is found to vanish at the horizon for 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 an increasing radius and is positive. (The same conclusions hold for a massless spin-½ field, but the accuracy of the stress-energy tensor we employ has not been confirmed, in contrast with 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," is therefore always significant in describing thermal properties of the spacetime geometries and fields near black holes.