  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 $\lambda$ in the kinetic term of the minimal
formulation to take its relativistic value, $\lambda=1$. The resulting theory
exhibits anisotropic scaling at short distances, and reproduces many features
of general relativity at long distances.
