An Exploration of the Black Hole Entropy via the Weyl Tensor

TL;DR

This paper investigates the relationship between the Weyl tensor's scalar invariant and black hole entropy, proposing it as an entropy density in higher-dimensional spacetimes, and confirms its validity through volume integrals for specific black holes.

Contribution

It introduces a novel interpretation of the Weyl scalar invariant as an entropy density in 5D black holes, linking geometric quantities to thermodynamical properties.

Findings

Weyl scalar invariant is proportional to black hole entropy in 5D.

Volume integrals of the invariant reproduce entropy formulas.

Interpretation is dimension-dependent, valid in 5D spacetime.

Abstract

The Weyl tensor is the trace-free part of the Riemann tensor. Therefore, it is independent of the energy-momentum tensor and is thus not linked to the dynamics of gravitational fields. In this paper, we explore its possible thermodynamical property (i.e. its relationship with the black hole entropy). For a Schwarzschild black hole, the Weyl scalar invariant, $C_{\mu\nu\lambda\rho}C^{\mu\nu\lambda\rho}$, is proportional to its Bekenstein--Hawking entropy. This observation inspires us to interpret $C_{\mu\nu\lambda\rho}C^{\mu\nu\lambda\rho}$ as the entropy density of the gravitational fields of black holes. A dimensional analysis indicates that this interpretation is only valid in 5-dimensional space-time. We calculate the volume integrals of $C_{\mu\nu\lambda\rho}C^{\mu\nu\lambda\rho}$ for the 5-dimensional Schwarzschild and Schwarzschild--anti-de Sitter black holes, and find that these integrals indeed lead to the correct entropy formulae, only up to some coefficients.