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

Torsional Newton-Cartan Geometry from the Noether Procedure