Horava-Lifshitz Gravity From Dynamical Newton-Cartan Geometry

TL;DR

This paper demonstrates that dynamical torsional Newton-Cartan geometry naturally leads to Horava-Lifshitz gravity, establishing a precise correspondence and providing a geometric foundation for HL theories with implications for their symmetries and effective actions.

Contribution

It establishes a detailed connection between dynamical Newton-Cartan geometry and Horava-Lifshitz gravity, including the origin of U(1) symmetry and effective action construction.

Findings

Dynamical NC geometry yields HL gravity models.

Constructed effective actions in 2+1 dimensions match known HL actions.

Identified U(1) symmetry as arising from Bargmann extension.

Abstract

Recently it has been established that torsional Newton-Cartan (TNC) geometry is the appropriate geometrical framework to which non-relativistic field theories couple. We show that when these geometries are made dynamical they give rise to Horava-Lifshitz (HL) gravity. Projectable HL gravity corresponds to dynamical Newton-Cartan (NC) geometry without torsion and non-projectable HL gravity corresponds to dynamical NC geometry with twistless torsion (hypersurface orthogonal foliation). We build a precise dictionary relating all fields (including the scalar khronon), their transformations and other properties in both HL gravity and dynamical TNC geometry. We use TNC invariance to construct the effective action for dynamical twistless torsional Newton-Cartan geometries in 2+1 dimensions for dynamical exponent 1<z\le 2 and demonstrate that this exactly agrees with the most general forms of the HL actions constructed in the literature. Further, we identify the origin of the U(1) symmetry observed by Horava and Melby-Thompson as coming from the Bargmann extension of the local Galilean algebra that acts on the tangent space to TNC geometries. We argue that TNC geometry, which is manifestly diffeomorphism covariant, is a natural geometrical framework underlying HL gravity and discuss some of its implications.