Revisiting the gravitational "arrow of time"

TL;DR

This work revisits Penrose's Weyl-curvature hypothesis (the gravitational arrow of time) by examining Bonnor's counterexample of shear-free heat-conducting spheres in a Vaidya background. It argues that the apparent contradiction arises from exterior matching; when the same class of solutions is treated as cosmological models with the heat flux interpreted as a peculiar velocity field and without Vaidya matching, Penrose's arrow is recovered. A near-FLRW, positively curved cosmological construction is developed to quantify the Weyl-to-Ricci ratio evolution, yielding a leading-order expression that shows the ratio's behavior is controlled by the cosmological parameters and the expansion history, with Λ driving decay in expanding phases. The paper also juxtaposes this with the CET gravitational entropy, highlighting that different notions of gravitational entropy can exhibit contrasting behavior during expansion versus collapse, underscoring subtlety in connecting curvature-based arrows to entropy notions.

Abstract

We address a long-standing misperception on the gravitational ``arrow of time'', a proposal by Penrose (also known as the ``Weyl-curvature hypothesis") that associates structure formation along timelike directions in which Weyl-curvature scalars become dominant over Ricci scalars. A counterexample of this hypothesis was found by Bonnor on a class of exact solutions describing heat conducting spheres collapsing in a Vaidya background. We show that this result does not hold in the same class of solutions considered as physically viable near FLRW cosmological models, with the heat conduction vector interpreted as a peculiar velocity field. We also discuss the similarities and differences between the gravitational ``arrow of time'' and the gravitational entropy formalism of Clifton, Ellis and Tavakol.

Figures (3)

• Figure 1: The Weyl to Ricci ratio $\mathcal{P}$ as a function of $a(t)$ with parameters $\Delta = 0.1$, $\chi=\pi/2$, with different choices of $\Omega_0^m$ and $\Omega_0^\Lambda$, see figure labels for specific values. See main text for details.
• Figure 2: The Weyl to Ricci ratio $\mathcal{P}$ as a function of $a(t)$ and $y(\chi)$ for the model with $\Delta = 0.1$, $\Omega_0^m = 1.05$, and $\Omega_0^\Lambda = 0.7$. See main text for details.
• Figure 3: The Weyl to Ricci ratio $\mathcal{P}$ as a function of $a(t)$ for the model with $\Delta = 0.1$, $\chi=\pi/2$, $\Omega_0^\Lambda = 0$, and various choices of $\Omega_0^m$. See main text for details.