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.
Highlights
In recent years the study of nonrelativistic approximations to general relativity (GR) has gained renewed interest
We focus on the 1=c2 expansion of GR, which has been shown to lead to a modification of the original notion of Newton-Cartan (NC) geometry, known as type II torsional Newton-Cartan (TNC) geometry [2,3], as the correct framework for a covariant action of nonrelativistic gravity
We provide a new perspective on the geometric interpretation of the PNR parametrization using moving frames, which clarifies the appearance of torsion and the local Galilean symmetries associated to NewtonCartan geometry
Summary
In recent years the study of nonrelativistic approximations to general relativity (GR) has gained renewed interest. The PNR parametrization (1) is a convenient way of recasting the Lorentzian metric variables of GR in such a way that the appropriate nonrelativistic geometry appears naturally in the expansion. We provide a new perspective on the geometric interpretation of the PNR parametrization using moving frames, which clarifies the appearance of torsion and the local Galilean symmetries associated to NewtonCartan geometry. By embedding both the local Lorentz symmetry of GR and the local Galilean symmetry of NC geometry inside the GLðn; RÞ frame bundle and its associated general (linear) affine connections, we can translate between the corresponding notions of torsion and metric-compatibility. This affine perspective naturally connects to the “triality” between the usual metric formulation of GR and its equivalent formulations in terms of torsion or nonmetricity [21,22]
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