Gravitational entropy of dust sources in General Relativity.

A "gravitational entropy" can be defined for a gravitational field, in the context of General Relativity, as an independent concept from the entropy of its sources (thermal fluids or black holes). We probe two proposals of this gravitational entropy in generic spherically symmetric Lemaitre-Tolman-Bondy (LTB) models with a dust source. The conditions for the growth of gravitational entropy in both proposals is directly governed by a negative correlation of fluctuations of the rest-mass density and the Hubble expansion scalar. These conditions hold throughout the time evolution of the models, except near a non-simultaneous Big Bang where density decaying modes are dominant. Perpetually expanding models reach a stable terminal equilibrium characterized by an inhomogeneous entropy maximum in their late time evolution, while regions with decaying modes and collapsing elliptic models exhibit unstable equilibria associated with an entropy minimum. We examine the convergence of the gravitational entropies near the Big Bang and collapse singularities, as well as in the radial asymptotic range. These entropies behave as intensive variables for models converging radially to a FLRW background and for some of the models radially converging to a Minkowski vacuum. The fact that different independent proposals yield fairly similar conditions for entropy production, time evolution and radial scaling in generic LTB models seems to suggest that their common notion of a gravitational entropy may be a theoretically robust concept applicable to more general spacetimes.