A gravitational entropy proposal

It is commonly assumed that gravitational entropy should be associated with the Weyl degrees of freedom and increase in the course of gravitational clumping.

“As mentioned before, there is no generally accepted notion of gravitational entropy, but it is commonly assumed that such an entropy should be associated with the Weyl degrees of freedom and increase in course of gravitational clumping [42].”

This is coherent with Penrose's argumentation but clashes with the proposal for gravitational entropy in [42] based on a formal analogy with classical thermodynamics.

“This is coherent with the argumentation provided by Penrose [20], but it clashes with the proposal for gravitational entropy, put forward in [42], based on a formal analogy with the classical laws of thermodynamics.”

For a static point-like quantum system of mass m, its gravitational field is described by the Schwarzschild metric, and the prescription of [42] yields dim H_G = e^{4π(m/m_Pl)^2}, in coherence with (4.7).

“It is instructive to analyse a static point-like quantum system of mass m.Its gravitational field is then described by the Schwarzschild metric [75], hence the prescription of [42] yields the formula dim H G = e 4π(m/m Pl ) 2 , in coherence with (4.7).”

This harmonises with the assumption that gravitational degrees of freedom are associated with the Weyl tensor, which does not affect energy-momentum conservation.

“Notably, this harmonises with the assumption that gravitational degrees of freedom shall be associated with the Weyl tensor, which does not affect the energymomentum conservation [42].”