Linearized gravity with matter time

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.