  We attempt to find a function that characterizes gravitational clumping and
that 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 compute $\Omega$ for transverse electromagnetic waves and for
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. We find that $\Omega$, and hence $ln\Omega$ behaves as required.
We also carry out calculations for Bianchi 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 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.
