  We propose a thermodynamically motivated measure of gravitational entropy
based on the Bel-Robinson tensor, which has a natural interpretation as the
effective super-energy-momentum tensor of free gravitational fields. The
specific form of this measure differs depending on whether the gravitational
field is Coulomb-like or wave-like, and reduces to the Bekenstein-Hawking value
when integrated over the interior of a Schwarzschild black hole. For scalar
perturbations of a Robertson-Walker geometry we find that the entropy goes like
the Hubble weighted anisotropy of the gravitational field, and therefore
increases as structure formation occurs. This is in keeping with our
expectations for the behaviour of gravitational entropy in cosmology, and
provides a thermodynamically motivated arrow of time for cosmological solutions
of Einstein's field equations. It is also in keeping with Penrose's Weyl
curvature hypothesis.
