Gravitational entropy in Szekeres class I models

Abstract

Developing a self-consistent notion of gravitational entropy in the context of cosmological structure formation has been so far an elusive task. 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 ΛCDM 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.

Arrow of time and gravitational entropy in collapse

“…As a result, in this limit we find that P 1 → 0. Here t s is the constant of integration which comes from the second integral of the equation (21). Without any hesitation we can conclude that the GE given by the Rudjord proposal decreases with time in this scenario.…”

“…Another important work was carried out on the analysis of gravitational entropies associated with exotic wormholes and galactic halos by Lima et al [20]. Recently a detailed study have been conducted on structure formation using the CET proposal in Szekeres class I models by Pizaña et al [21].…”

“…We can now use the expression of Y from (36) and substitute ˙f, f using ( 22), (24) to get the Kretschmann scalar in terms of f. Now taking the limit t = −∞ and also using (21), the Kretschmann scalar in this limit becomes…”

On Black Hole Thermodynamics, Singularity, and Gravitational Entropy

“…12 The Clifton-Ellis-Tavakol gravitational entropy was also shown to be increasing during structure formation in Szekeres Class I models [99]. Interestingly, in the presence of shear, the Clifton-Ellis-Tavakol gravitational entropy can increase even though the Weyl curvature decreases.…”