Localisation of the Galilean symmetry and dynamical realisation of Newton-Cartan geometry

TL;DR

The paper shows how localising Galilean symmetry in nonrelativistic field theories yields a dynamical Newton-Cartan spacetime. By introducing a manifesto set of gauge fields and a vierbein-like structure, it builds degenerate spatial and temporal metrics and a NC-compatible connection without invoking relativistic contraction. The resulting framework provides a direct geometric realisation of Newtonian gravity as a gauge-theoretic, diffeomorphism-invariant theory with explicit metricity and transformation rules. This offers a novel route to nonrelativistic diffeomorphism invariance and potential applications in NR quantum Hall and holographic contexts.

Abstract

Newtonian gravity was formulated as a geometrodynamic theory as far back 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 localisation of the Galilean Symmetry of nonrelativistic matter field theories.