Torsional Newton-Cartan Geometry from the Noether Procedure

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.

31 pages; v2 references added

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• Fracton and Non-Lorentzian Particle Duality: Gauge Field Couplings and Geometric Implications
• Symmetries and Couplings of Non-Relativistic Electrodynamics
• One-dimensional Carrollian fluids I: Carroll-Galilei duality
• Relativistic Fluids, Hydrodynamic Frames and their Galilean versus Carrollian Avatars