Positivity of Entropy in the Semi-Classical 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 $\mathrm{\Delta }S$ by which such fields augment the usual black hole entropy is found to be positive. More precisely, the derivative of $\mathrm{\Delta }S$ 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 $\mathrm{\Delta }S$ has a local minimum there. Explicit calculation shows that indeed $\mathrm{\Delta }S$ 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 $\mathrm{\Delta }S$ as arising from both the radiation fields and their effects on the gravitational field. The back-reaction, no matter how “small”,