Newtonian Gravity and the Bargmann Algebra

TL;DR

This paper derives Newtonian gravity from the Bargmann algebra via gauging, connecting algebraic structures to geometric formulations and discussing implications for AdS-CFT correspondence.

Contribution

It presents a novel derivation of Newton-Cartan gravity from the Bargmann algebra through a gauging procedure, clarifying the geometric and algebraic foundations.

Findings

Gauging the Bargmann algebra yields Newton-Cartan gravity.

Imposing curvature constraints converts symmetries into coordinate transformations.

The approach links algebraic structures to geometric gravity theories.

Abstract

We show how the Newton-Cartan formulation of Newtonian gravity can be obtained from gauging the Bargmann algebra, i.e., the centrally extended Galilean algebra. In this gauging procedure several curvature constraints are imposed. These convert the spatial (time) translational symmetries of the algebra into spatial (time) general coordinate transformations, and make the spin connection gauge fields dependent. In addition we require two independent Vielbein postulates for the temporal and spatial directions. In the final step we impose an additional curvature constraint to establish the connection with (on-shell) Newton-Cartan theory. We discuss a few extensions of our work that are relevant in the context of the AdS-CFT correspondence.