Entropy of massive fields near a black hole and vacuum polarization: thermodynamics without statistical mechanics

TL;DR

This paper investigates the entropy contribution of quantum massive fields near black holes using vacuum polarization effects, revealing that these contributions can be positive, negative, or zero depending on the field type and conditions, without relying on statistical mechanics.

Contribution

It derives the entropy contribution of massive quantum fields near black holes from vacuum polarization, clarifying its non-statistical origin and behavior across different field types.

Findings

Fermions and scalar fields contribute positively to black hole entropy under certain conditions.

Vector fields can contribute negatively to the entropy.

Massive fields in accelerated spacetimes have zero entropy contribution, similar to massless fields.

Abstract

Starting from the Frolov-Zel'nikov stress-energy tensor of quantum massive fields in the Schwarzschild background, we recover the contribution $S_{q}$ of these field into the entropy of a black hole. For fermions with the spin $%s=1/2$ $S_{q}>0$, for scalar fields $S_{q}>0$ provided the coupling parameter is restricted to some interval, and $S_{q}<0$ for vector fields. The appearance of negative values of $S_{q}$ is attributed to the fact that in the situation under discussion there are no real quanta to contribute to the entropy, so $S_{q}$ is due to vacuum polarization entirely and has nothing to do with the statistical-mechanical entropy. We also consider the spacetime with an acceleration horizon - the Bertotti-Robison spacetime - and show that $S_{q}=0$ for massive fields similarly to what was proved earlier for massless fields.