General Covariance in Quantum Gravity at a Lifshitz Point

TL;DR

The paper develops a generally covariant framework for gravity with Lifshitz-type anisotropic scaling by extending the gauge symmetry from foliation-preserving diffeomorphisms to a U(1)_Σ semidirect Diff(M, F) group, which eliminates the extra scalar graviton and fixes the kinetic coupling λ to 1. It analyzes both linearized and nonlinear regimes, identifies obstructions to gauging U(1)_Σ in the nonlinear theory, and then remedies them by introducing a Newton prepotential ν and a Newton-like deformation Ω, enabling a consistent gauging in general dimensions. In 2+1 dimensions the extended symmetry yields a topologically constrained theory with no local gravitons, while in higher dimensions the spectrum contains only tensor modes in the IR and Lifshitz UV behavior is preserved. The framework yields Schwarzschild-type static solutions, discusses Lorentz-symmetry aspects, and outlines cosmological solutions, highlighting the potential to reproduce GR-like phenomenology at long distances while maintaining Lifshitz scaling at short distances.

Abstract

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.