Torsional Newton-Cartan Geometry from the Noether Procedure

TL;DR

This paper derives torsional Newton-Cartan geometry from the Noether procedure applied to Galilean symmetric theories, revealing new couplings and connections, and applies these findings to Galilean electrodynamics.

Contribution

It demonstrates how the Noether procedure naturally leads to torsional Newton-Cartan geometry and introduces a distinguished affine connection with applications to Galilean electrodynamics.

Findings

Torsional Newton-Cartan geometry arises at the linearized level from the Noether procedure.

The form M_mu couples to conserved mass and topological currents in different theories.

A torsionful affine connection linear in M_mu is naturally obtained.

Abstract

We apply the Noether procedure for gauging space-time symmetries to theories with Galilean symmetries, analyzing both massless and massive (Bargmann) realizations. It is shown that at the linearized level the Noether procedure gives rise to (linearized) torsional Newton-Cartan geometry. In the case of Bargmann theories the Newton-Cartan form $M_\mu$ couples to the conserved mass current. We show that even in the case of theories with massless Galilean symmetries it is necessary to introduce the form $M_\mu$ and that it couples to a topological current. Further, we show that the Noether procedure naturally gives rise to a distinguished affine (Christoffel type) connection that is linear in $M_\mu$ and torsionful. As an application of these techniques we study the coupling of Galilean electrodynamics to TNC geometry at the linearized level.