General Covariance in Gravity at a Lifshitz Point

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This paper is based on the invited talks delivered by the author at GR 19: the 19th International Conference on General Relativity and Gravitation, Ciudad de México, México, July 2010. In Part 1, we briefly review some of the main features of quantum gravity with anisotropic scaling, and comment on its possible relation to the causal dynamical triangulations (CDT) approach to lattice quantum gravity. Part 2 explains the construction of gravity with anisotropic scaling with an extended gauge symmetry -- essentially a nonrelativistic version of general covariance. This extra symmetry eliminates the scalar graviton polarization, and thus brings the theory closer to general relativity at long distances.