Horava-Lifshitz Gravity From Dynamical Newton-Cartan Geometry

TL;DR

The paper shows that dynamical twistless TTNC geometry provides a natural, diffeomorphism-covariant foundation for Horava-Lifshitz gravity, establishing a detailed dictionary between TTNC fields (including the Khronon and Stückelberg scalar χ) and HL variables. By constructing TTNC-invariant actions in 2+1 dimensions for 1<z≤2 and comparing with HL actions, it demonstrates that projectable HL corresponds to dynamical NC geometry without torsion, while non-projectable HL aligns with TTNC geometry, with the Bargmann extension accounting for U(1) structures. The work further extends to conformal HL gravity via the Schrödinger algebra and discusses the role of constraint equations as alternatives to U(1) invariance, highlighting TTNC geometry as a robust framework for non-relativistic gravity and holography. Overall, it opens avenues for coupling matter to HL gravity, exploring IR/UV behavior, and applying these geometric concepts to condensed-matter systems and non-relativistic holography.

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.