Embolic aspects of black hole entropy

TL;DR

This paper explores a mesoscopic approach to understanding black hole entropy in four-dimensional spacetimes, emphasizing the topological and geometric properties of the horizon surface to explain entropy origin.

Contribution

It introduces a novel mesoscopic framework linking horizon topology and geometry to black hole entropy, utilizing injectivity radius and topological entropy as key concepts.

Findings

Topological entropy correlates with black hole entropy.

Injectivity radius encodes elementary cell dimensions.

Isoembolic inequalities relate geometry to entropy bounds.

Abstract

We attempt to provide a mesoscopic treatment of the origin of black hole entropy in (3+1)-dimensional spacetimes. We treat the case of horizons having space-like sections $\Sigma$ which are topological spheres, following Hawking's and the Topological Censorship theorems. We use the injectivity radius of the induced metric on $\Sigma$ to encode the linear dimensions of the elementary cells giving rise to such entropy. We use the topological entropy of $\Sigma$ as the fundamental quantity expressing the complexity of $\Sigma$ on which its entropy depends. We point out the significance, in this context, of the Berger and Croke isoembolic inequalities.