Relativistic Quantum Gravity at a Lifshitz Point

TL;DR

The paper reconceptualizes Horava gravity as a covariant tensor–vector theory by introducing a dynamical, irrotational vector field $n_{\alpha}$ that defines a spacelike foliation, thereby restoring full four-dimensional diffeomorphism invariance while preserving Lifshitz-like UV behavior. The resulting action contains a covariant kinetic term built from the extrinsic curvature $K_{\alpha\beta}$ and higher-spatial-derivative potentials, with the UV dynamics driven by $z=3$ scaling and a Cotton-tensor–controlled ultraviolet structure. The theory generically propagates three degrees of freedom (two from the metric and one from $n_{\alpha}$), but can be formulated so that all physical content is captured by $h_{\alpha\beta}$, yielding a power-counting renormalizable UV completion that reduces to GR in the IR; matter can be coupled consistently in this fully covariant setting. The framework preserves Lorentz invariance in flat backgrounds and provides a platform to study cosmological perturbations, with the scalar sector showing a single propagating mode under certain gauges and matter configurations. Overall, the work offers a covariant path to UV-complete gravity with controlled higher-spatial-derivative terms and a clear route to coupling to matter and recovering GR at large scales.

Abstract

We show that the Horava theory for the completion of General Relativity at UV scales can be interpreted as a gauge fixed 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. Finally, we note that in the present formulation, matter can be consistently coupled to gravity.