Phase space approach to the gravitational arrow of time

As the universe evolves, it becomes more inhomogeneous due to gravitational clumping. We attempt to find a function that characterizes this behavior and increases monotonically as inhomogeneity increases. We choose $S=\ln \Omega$ as the candidate “gravitational entropy” function, where $\Omega$ is the phase-space volume below the Hamiltonian $H$ of the system under consideration. We perform a direct calculation of $\Omega$ for transverse electromagnetic waves and gravitational wave, radiation, and density perturbations in an expanding FLRW universe. These calculations are carried out in the linear regime under the assumption that the phases of the oscillators comprising the system are random. Entropy is thus attributed to the lack of knowledge of the exact field configuration. The time dependence of $H$ leads to a time-dependent $\Omega$. We find that $\Omega$, and, hence, $\ln \Omega$ behaves as required. We also carry out calculations for Bianchi type IX cosmological models and find that, even in this homogeneous case, the function can be interpreted sensibly. We compare our results with Penrose’s $C^2$ hypothesis. Because $S$ is defined to resemble the fundamental statistical mechanics definition of entropy, we are able to recover the entropy in a variety of familiar circumstances including, evidently, black-hole entropy. The results point to the utility of the relativistic Arnowitt-Deser-Misner (ADM) Hamiltonian formalism in establishing a connection between general relativity and statistical mechanics, although fully nonlinear calculations will need to be performed to remove any doubt.