Why is the black hole entropy (almost) linear in the horizon area?

TL;DR

This paper demonstrates that classical thermodynamics alone can derive the linear relationship between black hole entropy and horizon area, emphasizing the quantum nature of any deviations from this proportionality.

Contribution

It shows that classical thermodynamics suffices to derive Bekenstein's entropy formula, highlighting the quantum origin of any non-linear corrections.

Findings

Classical thermodynamics determines the linear entropy-area relation.

Non-linear corrections are inherently quantum mechanical.

The proportionality constant is rooted in quantum gravity.

Abstract

We show that essentially pure classical thermodynamics is sufficient to determine Bekenstein's formula for the black hole's entropy, $S=\eta A$. We base our reasoning on the minimal assumption that since black body radiation is describable by classical thermodynamics, so is the complete black hole-Hawking radiation system. Furthermore, we argue that any non-linear correction to the black hole entropy must be quantum mechanical in nature. The proportionality coefficient, $\eta=1/4\ell_P^2$, must be calculated within a semi-classical or full-fledged quantum mechanical framework.