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.Contents 4 Throughout this paper, we use the rather loose notation common in high-energy physics, and refer to any one-dimensional Abelian symmetry group as U (1), regardless of whether or not it is actually compact. Moreover, we use the same notation also for the infinite-dimensional, gauge version of the U (1) symmetry. 5 G ijkℓ can be viewed as a metric on the space of symmetric 2-tensors. As in [1,2], we will denote its inverse by G ijkℓ , to distinguish it from G ijkℓ with all four indices lowered via gij.

We argue that Hořava-Lifshitz (HL) gravity provides the minimal holographic dual for Lifshitz-type field theories with anisotropic scaling and dynamical exponent z. First we show that Lifshitz spacetimes are vacuum solutions of HL gravity, without need for additional matter. Then we perform holographic renormalization of HL gravity, and show how it reproduces the full structure of the z = 2 anisotropic Weyl anomaly in dual field theories in 2 + 1 dimensions, while its minimal relativistic gravity counterpart yields only one of two independent central charges in the anomaly.

We show that holographic renormalization of relativistic gravity in asymptotically Lifshitz spacetimes naturally reproduces the structure of gravity with anisotropic scaling: The holographic counterterms induced near anisotropic infinity take the form of the action for gravity at a Lifshitz point, with the appropriate value of the dynamical critical exponent z. In the particular case of 3 + 1 bulk dimensions and z = 2 asymptotic scaling near infinity, we find a logarithmic counterterm, related to anisotropic Weyl anomaly of the dual CFT, and show that this counterterm reproduces precisely the action of conformal gravity at a z = 2 Lifshitz point in 2 + 1 dimensions, which enjoys anisotropic local Weyl invariance and satisfies the detailed balance condition. We explain how the detailed balance is a consequence of relations among holographic counterterms, and point out that a similar relation holds in the relativistic case of holography in AdS 5 . Upon analytic continuation, analogous to the relativistic case studied recently by Maldacena, the action of conformal gravity at the z = 2 Lifshitz point features in the ground-state wavefunction of a gravitational system with an interesting type of spatial anisotropy.