  We study general relativity with pressureless dust in the canonical
formulation, with the dust field chosen as a matter-time gauge. The resulting
theory has three physical degrees of freedom in the metric field. The
linearized canonical theory reveals two graviton modes and a scalar mode. We
find that the graviton modes remain Lorentz covariant despite the time gauge,
and that the scalar mode is ultralocal. We discuss also a modification of the
theory to include a parameter in the Hamiltonian analogous to that in
Horava-Lifshitz models. In this case the scalar mode is no longer ultralocal,
and acquires a propagation speed dependent on the deformation parameter.
