Positivity of Entropy in the Semi-Classical Theory of Black Holes and Radiation

TL;DR

This paper demonstrates that the thermodynamic entropy increase due to quantum fields around a Schwarzschild black hole remains positive and increases monotonically with radius, considering back-reaction effects.

Contribution

It provides a detailed analysis showing the positivity and monotonic increase of entropy from quantum stress-energy tensors near black holes, including back-reaction effects.

Findings

Entropy increase is positive and monotonic with radius.

Derivative of entropy with respect to radius vanishes at the horizon.

Second derivative of entropy at the horizon is positive, indicating a local minimum.

Abstract

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",