Gravitational Entropy in Szekeres Class I Models
  Gravitational entropy is an elusive concept. Various theoretical proposals
have been presented, initially based on Penrose's Weyl Curvature Hypothesis,
and variations of it. A more recent proposal by Clifton, Ellis, and Tavakol
(CET) considered a novel approach by defining such entropy from a Gibbs
equation constructed from an effective stress-energy tensor that emerges from
the 'square root' algebraic decomposition of the Bel-Robinson tensor, the
simplest divergence-less tensor related to the Weyl tensor. Since, so far all
gravitational entropy proposals have been applied to highly restrictive and
symmetric spacetimes, we probe in this paper the CET proposal for a class of
much less idealized spactimes (the Szekeres class I models) capable of
describing the joint evolution of arrays of arbitrary number of structures:
overdensities and voids, all placed on selected spatial locations in an
asymptotic $\Lambda$CDM backgound. By using suitable covariant variables and
their fluctuations, we find the necessary and sufficient conditions for a
positive CET entropy production to be a negative sign of the product of the
density and Hubble expansion fluctuations. To examine the viability of this
theoretical result we examine numerically the CET entropy production for two
elongated over dense regions surrounding a central spheroidal void, all
evolving jointly from initial linear perturbations at the last scattering era
into present day Mpc-size CDM structures. We show that CET entropy production
is positive for all times after last scattering at the precise spatial
locations where structure growth occurs and where the exact density growing
mode is dominant. The present paper provides the least idealized (and most
physically robust) probe of a gravitational entropy proposal in the context of
structure formation.
