# Torsional Newton-Cartan gravity from the large $c$ expansion of General Relativity

## TL;DR

This paper explores the large $c$ expansion of General Relativity, revealing that relaxing certain assumptions introduces torsion and extends Newtonian physics to include effects like gravitational time dilation.

## Contribution

It demonstrates that allowing poles in the relativistic connection leads to a torsionful non-relativistic gravity, broadening the understanding of the non-relativistic limit of GR.

## Key findings

- Inclusion of twistless torsion in the non-relativistic limit.
- Extension of Newton-Cartan gravity with gravitational time dilation.
- Relaxation of previous assumptions yields a richer non-relativistic theory.

## Abstract

We revisit the manifestly covariant large $c$ expansion of General Relativity, $c$ being the speed of light. Assuming the relativistic connection has no pole in $c^{-2}$, this expansion is known to reproduce Newton-Cartan gravity and a covariant version of Post-Newtonian corrections to it. We show that relaxing this assumption leads to the inclusion of twistless torsion in the effective non-relativistic theory. We argue that the resulting TTNC theory is an effective description of a non-relativistic regime of General Relativity that extends Newtonian physics by including strong gravitational time dilation.