Newton-Cartan Gravity and Torsion

TL;DR

This work analyzes Newton-Cartan (NC) gravity with arbitrary torsion by juxtaposing gauging the Bargmann algebra against null-reduction of General Relativity, and extends the framework to a $z=2$ Schrödinger setting. Off-shell, NC geometry with arbitrary torsion is obtained via gauging, and a null-reduction of GR reproduces the same structure but yields zero torsion on-shell, revealing a tension between these approaches. In three dimensions, the authors overcome this by using a non-relativistic conformal method with two compensating scalars, coupling a Schrödinger field theory to a $z=2$ Schrödinger geometry to produce NC gravity with arbitrary torsion; the resulting equations involve a boost-invariant connection $\Omega_\mu{}^{ab}$ and three independent NC field equations, including a Poisson-type equation for the Newton potential in the torsionless limit. These results clarify the role of torsion in NC and Schrödinger geometries and provide a concrete path to torsionful NC gravity, with potential extensions to supersymmetry and higher dimensions.

Abstract

We compare the gauging of the Bargmann algebra, for the case of arbitrary torsion, with the result that one obtains from a null-reduction of General Relativity. Whereas the two procedures lead to the same result for Newton-Cartan geometry with arbitrary torsion, the null-reduction of the Einstein equations necessarily leads to Newton-Cartan gravity with zero torsion. We show, for three space-time dimensions, how Newton-Cartan gravity with arbitrary torsion can be obtained by starting from a Schroedinger field theory with dynamical exponent z=2 for a complex compensating scalar and next coupling this field theory to a z=2 Schroedinger geometry with arbitrary torsion. The latter theory can be obtained from either a gauging of the Schroedinger algebra, for arbitrary torsion, or from a null-reduction of conformal gravity.