The Weyl curvature conjecture and black hole entropy

The universe today, containing stars, galaxies and black holes, seems to have evolved from a very homogeneous initial state. From this it appears as if the entropy of the universe is decreasing, in violation of the second law of thermodynamics. It has been suggested by Roger Penrose that this inconsistency can be solved if one assigns an entropy to the spacetime geometry. He also pointed out that the Weyl tensor has the properties one would expect to find in a description of a gravitational entropy. In this paper, we make an attempt to use this so-called Weyl curvature conjecture to describe the Hawking–Bekenstein entropy of black holes and the entropy of horizons due to a cosmological constant. Our analysis indicates that in the static spherically symmetric case this is not possible.