Relativistic quantum gravity at a Lifshitz point

We show that the Horava theory for the completion of General Relativity at UV scales can be interpreted as a gauge fixed Tensor-Vector theory, and it can be extended to an invariant theory under the full group of four-dimensional diffeomorphisms. In this respect, although being fully relativistic, it results to be locally anisotropic in the time-like and space-like directions defined by a family of irrotational observers. We show that this theory propagates generically three degrees of freedom: two of them are related to the four-dimensional diffeomorphism invariant graviton (the metric) and one is related to a propagating scalar mode of the vector. Finally, we note that in the present formulation, matter can be consistently coupled to gravity.