Newtonian Gravity and the Bargmann Algebra

TL;DR

This paper demonstrates that Newton-Cartan gravity, the geometric reformulation of Newtonian gravity, can be obtained by gauging the Bargmann (centrally extended Galilean) algebra. The authors first mirror the relativistic construction by gauging the Poincaré algebra and imposing curvature constraints, then adapt this approach to the non-relativistic Bargmann case. By enforcing curvature constraints, two independent Vielbein postulates, and a final set of Ehlers/Trautman-type conditions, they show how the independent fields reduce to the Newton-Cartan data and reproduce the Poisson equation and Newtonian geodesics. The work also outlines extensions to supersymmetric and other non-relativistic algebras with potential relevance to AdS/CFT and non-relativistic holography.

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.