Vaibhav Kalvakota
Describing cosmological entropy using the Newman-Penrose formalism
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ABSTRACT. The gravitational entropy proposal describes the evolution of a cosmology in terms of the Weyl curvature, where the beginning of the universe implies a low entropy state due to the homogeneous and isotropic nature of the universe, implying a per- fect Friedmann-Lemaitre-Robertson-Walker (FLRW) universe. The Weyl curvature of the FLRW solution is zero due to conformal flatness, and as time progresses, gravitational effects play a significant role in determining structure formations, which account for anisotropies. These anisotropies in turn imply a departure from the FLRW metric, and therefore the Weyl curvature increases. By using the Weyl curvature as a quantification of entropy in some sense, Penrose introduced the Weyl curvature hypothesis [6], which considered the Weyl curvature as the description of a gravitational analog of the second law of thermodynamics. In this paper, we will look at gravitational entropy proposals and their ranges, focusing on the Weyl invariant based proposal [1], the Newmann-Penrose formalism [18] approach towards cosmologies as introduced by Clifton, Ellis and Tavakol [11] and the Weyl scalar $\Psi _{2}$ based proposal by Gregoris and Ong [7] (since we will only consider Petrov type D spacetimes in this paper with $\mu =
ho $).