Revisiting the gravitational "arrow of time"

Abstract: We address a long-standing misperception on the gravitational arrow of time'', a proposal by Penrose (also known as theWeyl-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.