Localization of the Galilean symmetry and dynamical realization of Newton–Cartan geometry

Newtonian gravity was formulated as a geometrodynamic theory as far back as in 1930s by Elie Cartan, in what is named aptly as Newton–Cartan space time. Though there are several approaches of realizing the algebraic structure of the Newton–Cartan geometry from a contraction of the relativistic results, a dynamical (field theoretic) realization of it is lacking. In this paper we present such a realization from the localization of the Galilean symmetry of nonrelativistic matter field theories.