An exploration of the black hole entropy via the Weyl tensor

The role of the Weyl tensor C_μνλρ in black hole thermodynamics is explored by looking at the relation between the scalar invariant C_μνλρC^μνλρ 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_μνλρC^μνλρ 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. (orig.)