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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 Lemaître-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. [ABSTRACT FROM AUTHOR]

We consider a formalism of weighed proper volume scalar averages (the "q-average") for the study of quasi-spherical Szekeres models. We show that the q-average of the main fluid flow covariant scalars are spherically symmetric and satisfy FLRW evolution laws, so that fluctuations and perturbations with respect to these averages provide a full description of the deviation of the models from homogeneity and spherical symmetry. The main proper tensors of the models are given in terms of these fluctuations, with the averages of scalar invariant contractions expressed as second order statistical moments of the density and Hubble scalar expansion. We discuss a possible application of this formalism in connection to a gravitational entropy functional in which entropy production is directly related to a negative statistical correlation between density and velocity fluctuations. [ABSTRACT FROM AUTHOR]

An effective acceleration, which mimics the effect of dark energy, may arise in the context of Buchert's scalar averaging formalism. We examine the conditions for such an acceleration to occur in the asymptotic radial range in generic spherically symmetric Lemaître-Tolman-Bondi (LTB) dust models. By looking at the behavior of covariant scalars along space slices orthogonal to the 4-velocity, we show that this effective acceleration occurs in a class of models with negative spatial curvature that are asymptotically convergent to sections of Minkowski spacetime. As a consequence, the boundary conditions that favor LTB models with an effective acceleration are not a void inhomogeneity embedded in a homogeneous FLRW background (Swiss cheese models), but a local void or clump embedded in a large cosmic void region represented by asymptotically Minkowski conditions. [ABSTRACT FROM AUTHOR]

We introduce quasi-local (QL) scalar variables in spherically symmetric LTB models. If the QL scalars are defined as functionals, they become weighed averages that generalize the standard proper volume averages on space slices orthogonal to the 4-velocity. We examine the connection between QL functions and functionals and the “back-reaction” term Q in the context of Buchert’s scalar averaging formalism. With the help of the QL scalars we provide rigorous proof that back-reaction is positive for (i) all LTB models with negative and asymptotically negative spatial curvature, and (ii) models with positive curvature decaying to zero asymptotically in the radial direction. We show by means of qualitative, but robust, arguments that generic LTB models exist, either with clump or void profiles, for which an “effective” acceleration associated with Buchert’s formalism can mimic the effects of dark energy. [ABSTRACT FROM AUTHOR]