Torsional Newton-Cartan geometry from Galilean gauge theory

TL;DR

This paper uses Galilean gauge theory (GGT) to construct torsional Newton–Cartan geometry by systematically gauging the Galilean symmetry and coupling a Galilean-invariant matter theory to Newton–Cartan backgrounds. The authors derive a general torsionful connection whose symmetric part splits into a Dautcourt-like term plus a torsion-driven contortion-like contribution, and they show that torsion is not purely temporal, possessing a spatial component expressible in first-order variables. They also provide an implicit relation tying torsion to metric data and the independent spin connection, with a clear torsionless limit and a recovered NRDI action that concords with the $c\to\infty$ limit of Riemann–Cartan theory. The framework resolves several long-standing ambiguities in NC geometry and offers a covariant, first-order formulation of torsional NC spacetime that parallels the relativistic theory while respecting nonrelativistic diffeomorphism invariance.

Abstract

Using the recently advanced Galilean gauge theory (GGT) we give a comprehensive construction of torsional Newton Cartan geometry. The coupling of a Galilean symmetric model with background NC geometry following GGT is illustrated by a free nonrelativistic scalar field theory. The issue of spatial diffeomorphisn \cite{SW, BMM3} is focussed from a new angle. The expression of the torsionful connection is worked out which is in complete parallel with the relativistic theory. Also smooth transition of the connection to its well known torsionless expression is demonstrated. A complete (implicit) expression of the torsion tensor for the Newton Cartan spacetime is provided where the first order variables occur in a suggestive way. The well known result for the temporal part of torsion is reproduced from our expression.