Scale Invariance in Newton-Cartan and Ho\v{r}ava-Lifshitz Gravity

TL;DR

This paper analyzes the construction of scale-invariant Hořava-Lifshitz gravity using Newton-Cartan geometry and tensor calculus, highlighting the distinction between scale and Schrödinger invariance and exploring symmetry enhancements.

Contribution

It introduces a method to construct and distinguish scale invariance from Schrödinger invariance in Hořava-Lifshitz gravity using non-relativistic geometric frameworks.

Findings

Constructed $z=2$ and $z eq2$ scale-invariant Hořava-Lifshitz gravity models.

Identified mechanisms to differentiate local scale invariance from Schrödinger invariance.

Discussed symmetry enhancement to the full Schrödinger group.

Abstract

We present a detailed analysis of the construction of $z=2$ and $z\neq2$ scale invariant Ho\v{r}ava-Lifshitz gravity. The construction procedure is based on the realization of Ho\v{r}ava-Lifshitz gravity as the dynamical Newton-Cartan geometry as well as a non-relativistic tensor calculus in the presence of the scale symmetry. An important consequence of this method is that it provides us the necessary mechanism to distinguish the local scale invariance from the local Schr\"odinger invariance. Based on this result we discuss the $z=2$ scale invariant Ho\v{r}ava-Lifshitz gravity and the symmetry enhancement to the full Schr\"odinger group.