Newton-Cartan Gravity and Torsion

TL;DR

This paper explores the relationship between Newton-Cartan gravity with torsion and various geometric and algebraic constructions, revealing how different approaches yield consistent or distinct torsion properties, especially in three dimensions.

Contribution

It demonstrates the equivalence of gauging the Bargmann algebra and null-reduction for arbitrary torsion, and constructs Newton-Cartan gravity with torsion from Schrödinger field theory and conformal gravity.

Findings

Gauging Bargmann algebra matches null-reduction results for torsion.

Null-reduction of Einstein equations yields torsion-free Newton-Cartan gravity.

Construction of torsionful Newton-Cartan gravity from Schrödinger field theory in 3D.

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.