Torsional Newton-Cartan geometry from Galilean gauge theory

TL;DR

This paper constructs torsional Newton-Cartan geometry using Galilean gauge theory, illustrating its coupling with scalar fields and analyzing the torsionful connection and its transition to torsionless form.

Contribution

It provides a comprehensive construction of torsional Newton-Cartan geometry from Galilean gauge theory, including explicit expressions for torsion and connections, and explores their physical implications.

Findings

Explicit torsionful connection expression derived

Smooth transition to torsionless connection demonstrated

Complete implicit torsion tensor expression provided

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.