An Action for Extended String Newton-Cartan Gravity

TL;DR

This paper develops a four-dimensional action for extended string Newton-Cartan gravity, extending previous models and connecting to nonrelativistic string theory through a nonrelativistic limit of Einstein-Hilbert gravity.

Contribution

It introduces a new four-dimensional extended string Newton-Cartan gravity action based on a central extension of the algebra, derived from a nonrelativistic limit of Einstein-Hilbert gravity.

Findings

Constructed a four-dimensional action for extended string Newton-Cartan gravity.

Showed the relation between 3D Chern-Simons and 4D actions via reduction and truncation.

Connected the model to extended p-brane Newton-Cartan gravity in higher dimensions.

Abstract

We construct an action for four-dimensional extended string Newton-Cartan gravity which is an extension of the string Newton-Cartan gravity that underlies nonrelativistic string theory. The action can be obtained as a nonrelativistic limit of the Einstein-Hilbert action in General Relativity augmented with a term that contains an auxiliary two-form and one-form gauge field that both have zero flux on-shell. The four-dimensional extended string Newton-Cartan gravity is based on a central extension of the algebra that underlies string Newton-Cartan gravity. The construction is similar to the earlier construction of a three-dimensional Chern-Simons action for extended Newton-Cartan gravity, which is based on a central extension of the algebra that underlies Newton-Cartan gravity. We show that this three-dimensional action is naturally obtained from the four-dimensional action by a reduction over the spatial isometry direction longitudinal to the string followed by a truncation of the extended string Newton-Cartan gravity fields. Our construction can be seen as a special case of the construction of an action for extended p-brane Newton-Cartan gravity in p+3 dimensions.