Hořava-Lifshitz gravity from dynamical Newton-Cartan geometry

Jelle Hartong,Niels A. Obers

doi:10.1007/jhep07(2015)155

Abstract

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 Hořava-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 ≤ 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 Hořava 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.

Highlights

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
We argue that torsional Newton-Cartan (TNC) geometry, which is manifestly diffeomorphism covariant, is a natural geometrical framework underlying HL gravity and discuss some of its implications
This includes in particular holography for Lifshitz space-times, for which it was found that the boundary geometry is described by a novel extension of Newton-Cartan (NC) geometry1 with a specific torsion tensor, called torsional Newton-Cartan (TNC) geometry

Summary

Local Galilean transformations

To obtain torsional Newton-Cartan geometry we follow the same logic as in appendix A for the case of the Galilean algebra and its central extension known as the Bargmann algebra. This was first considered in [22] for the case without torsion. Adding torsion to Newton-Cartan geometry can be done by making it locally scale invariant, i.e. gauging the Schrodinger algebra as in [21]. Any tensor redefinition of the connections Γρμν, Ωμa and Ωμab that leaves the covariant derivatives form-invariant leads to an allowed set of connections with the exact same transformation properties.

The affine connection: part 1

Local Bargmann transformations

The affine connection: part 2

Torsion and the Stuckelberg scalar

Curvatures

Horava-Lifshitz actions

11 A constraint equation

12 Conformal HL gravity from the Schrodinger algebra

A Schrodinger covariant derivative Dμχ is given by

13 Discussion

A Gauging Poincare