# Covariant Poisson's equation in torsional Newton-Cartan gravity

## TL;DR

This paper derives a covariant form of Poisson's equation within torsional Newton-Cartan gravity using a non-relativistic conformal approach, providing new insights into the structure of non-relativistic gravitational theories with torsion.

## Contribution

It introduces a covariant formulation of Poisson's equation in torsional Newton-Cartan gravity via the non-relativistic conformal method, extending previous non-relativistic gravity frameworks.

## Key findings

- Derived covariant Poisson's equation with torsion.
- Fixed all coefficients for the equation in the presence of a cosmological constant.
- Obtained Ehlers conditions and torsion-related equations.

## Abstract

We derive the covariant Poisson's equation of (d+1)-dimensional Newton-Cartan gravity with (twistless) torsion by applying the `non-relativistic conformal method' introduced in arXiv:1512.06277. We apply this method on-shell to a Schr\"odinger field theory on the curved Newton-Hooke background. The covariance of the field equation in the presence of the non-relativistic cosmological constant, entails fixing all coefficients in the covariant Poisson's equation for (twistless) torsional Newton-Cartan gravity. We further derive Ehlers conditions and an equation associated to the torsion in this method.