In the minimal formulation of gravity with Lifshitz-type anisotropic scaling, the gauge symmetries of the system are foliation-preserving diffeomorphisms of spacetime. Consequently, compared to general relativity, the spectrum contains an extra scalar graviton polarization. Here we investigate the possibility of extending the gauge group by a local U(1) symmetry to "nonrelativistic general covariance." This extended gauge symmetry eliminates the scalar graviton, and forces the coupling constant $λ$ in the kinetic term of the minimal formulation to take its relativistic value, $λ=1$. The resulting theory exhibits anisotropic scaling at short distances, and reproduces many features of general relativity at long distances.

41 pages; v2: small clarifications, references added