Understanding Gravitational Entropy of Black Holes: A New Proposal via Curvature Invariants

TL;DR

This paper proposes a new curvature-invariant based entropy density function for black holes, improving upon previous Weyl curvature proposals, applicable across various static black hole solutions and potentially extendable to higher dimensions and modified gravity theories.

Contribution

It introduces a novel curvature invariant combination to define black hole entropy density, addressing limitations of earlier Weyl curvature-based proposals and enabling broader applicability.

Findings

Works for all static black holes in 4D and 5D general relativity

Potential to extend method to higher dimensions

Suggests different physical effects in modified gravity theories

Abstract

Partly motivated by the arrow of time problem in cosmology and the Weyl curvature hypothesis formulated by Roger Penrose, previous works in the literature have proposed - among other possibilities - the square of the Weyl curvature, as being the underlying entropy density function of black hole entropy, but the proposal suffers from a few drawbacks. In this work, we propose a new entropy density function also based solely on the Weyl curvature, but adopting some other combinations of curvature invariants. As an improvement we find that our method works for all static black hole solutions in four and five dimensional general relativity regardless of whether they are empty space solutions or not. It should also be possible to generalize our method to higher dimensions. This allows us to discuss the physical interpretation of black hole entropy, which remains somewhat mysterious. Extending to modified theories of gravity, our work also suggests that gravitational entropy in some theories is a manifestation of different physical effects since we need to choose different combinations of curvature quantities