Galilean first-order formulation for the non-relativistic expansion of general relativity

TL;DR

This paper introduces a new first-order formulation of general relativity that is adapted to Galilean symmetry, facilitating a systematic non-relativistic expansion and clarifying the role of torsion in the Newton-Cartan limit.

Contribution

It develops a Galilean covariant first-order Palatini formulation of GR, enabling a natural non-relativistic expansion and elucidating torsion's origin in the Newton-Cartan framework.

Findings

New Galilean covariant Palatini formulation of GR

Clarifies the role of torsion in non-relativistic limit

Provides a foundation for systematic non-relativistic expansions

Abstract

We reformulate the Palatini action for general relativity (GR) in terms of moving frames that exhibit local Galilean covariance in a large speed of light expansion. For this, we express the action in terms of variables that are adapted to a Galilean subgroup of the $GL(n,\mathbb{R})$ structure group of a general frame bundle. This leads to a novel Palatini-type formulation of GR that provides a natural starting point for a first-order non-relativistic expansion. In doing so, we show how a comparison of Lorentzian and Newton-Cartan metric-compatibility explains the appearance of torsion in the non-relativistic expansion.