Arrow of time and gravitational entropy in collapse

Abstract: We investigate the status of the gravitational arrow of time in the case of a spherical collapse of a fluid that conducts heat and radiates energy. In particular, we examine the results obtained by W. B. Bonnor in his 1985 paper where he found that the gravitational arrow of time was opposite to the thermodynamic arrow of time. The measure of gravitational epoch function $P$ used by Bonnor was given by the ratio of the Weyl square to the Ricci square. In this paper, we have assumed the measure of gravitational entropy $P_{1}$ to be given by the ratio of the Weyl scalar to the Kretschmann scalar. Our analysis indicates that Bonnor's result seems to be validated, i.e., the gravitational arrow and the thermodynamic arrow of time point in opposite directions. This strengthens the opinion that the Weyl proposal of gravitational entropy applies only to the universe as a whole (provided that we exclude the white holes).