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 ∆S by which such fields augment the usual black hole entropy is found to be positive. More precisely, the derivative of ∆S with respect to radius, at 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 ∆S has a local minimum there. Explicit calculation shows that indeed ∆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 ∆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.