An Exploration of the Black Hole Entropy via the Weyl Tensor

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.

arXiv comment5 pages

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• Understanding Gravitational Entropy of Black Holes: A New Proposal via Curvature Invariants
• Thermodynamics of Shearing Massless Scalar Field Spacetimes is Inconsistent With the Weyl Curvature Hypothesis
• On Black Hole Thermodynamics, Singularity, and Gravitational Entropy
• Scalar Curvature Invariants in Classical and Quantum Gravity
• An exploration of the black hole entropy in Gauss-Bonnet gravity via the Weyl tensor