A Schr\"odinger approach to Newton-Cartan and Ho\v{r}ava-Lifshitz gravities

TL;DR

This paper introduces a non-relativistic conformal method based on Schr"odinger algebra to derive Newton-Cartan and Hořava-Lifshitz gravity equations, providing a new framework for non-relativistic gravitational theories.

Contribution

It develops a Schr"odinger algebra-based non-relativistic conformal method and applies it to derive Newton-Cartan gravity and classify Hořava-Lifshitz scalar theories.

Findings

Derived Newton-Cartan gravity equations with twistless torsion.

Reproduced z=2 Hořava-Lifshitz gravity from scalar field theories.

Introduced a complex scalar field with scale and central charge transformations.

Abstract

We define a `non-relativistic conformal method', based on a Schr\"odinger algebra with critical exponent z = 2, as the non-relativistic version of the relativistic conformal method. An important ingredient of this method is the occurrence of a complex compensating scalar field that transforms under both scale and central charge transformations. We apply this non-relativistic method to derive the curved space Newton-Cartan gravity equations of motion with twistless torsion. Moreover, we reproduce z = 2 Ho\v{r}ava-Lifshitz gravity by classifying all possible Schr\"odinger invariant scalar field theories of a complex scalar up to second order in time derivatives.