An exploration of the black hole entropy via the Weyl tensor

The role of the Weyl tensor $$C_{\mu \nu \lambda \rho }$$ in black hole thermodynamics is explored by looking at the relation between the scalar invariant $$C_{\mu \nu \lambda \rho }C^{\mu \nu \lambda \rho }$$ and the entropy of n-dimensional static black holes. It is found that this invariant can be identified as the entropy density of the gravitational fields for classical 5-dimensional black holes. We calculate the proper volume integrals of $$C_{\mu \nu \lambda \rho }C^{\mu \nu \lambda \rho }$$ for the Schwarzschild and Schwarzschild–anti-de Sitter black holes and show that these integrals correctly lead to the Bekenstein–Hawking entropy formulas, only up to some coefficients.