Curved non-relativistic spacetimes, Newtonian gravitation and massive matter

TL;DR

This paper constructs a comprehensive geometric framework called Bargmann spacetime that generalizes Newton-Cartan structures, enabling the coupling of massive matter fields in non-relativistic settings and connecting to Newtonian gravity.

Contribution

It provides the first complete construction of Bargmann spacetimes without restrictions, including matter coupling and a non-relativistic limit from Lorentzian spacetimes.

Findings

Bargmann spacetime includes Newton-Cartan structure with an additional gauge field.

The construction allows for arbitrary geometric data, supporting matter response analysis.

It connects non-relativistic limits of Lorentzian spacetimes to Newtonian gravity.

Abstract

There is significant recent work on coupling matter to Newton-Cartan spacetimes with the aim of investigating certain condensed matter phenomena. To this end, one needs to have a completely general spacetime consistent with local non-relativisitic symmetries which supports massive matter fields. In particular, one can not impose a priori restrictions on the geometric data if one wants to analyze matter response to a perturbed geometry. In this paper we construct such a Bargmann spacetime in complete generality without any prior restrictions on the fields specifying the geometry. The resulting spacetime structure includes the familiar Newton-Cartan structure with an additional gauge field which couples to mass. We illustrate the matter coupling with a few examples. The general spacetime we construct also includes as a special case the covariant description of Newtonian gravity, which has been thoroughly investigated in previous works. We also show how our Bargmann spacetimes arise from a suitable non-relativistic limit of Lorentzian spacetimes. In a companion paper [arXiv:1503.02680] we use this Bargmann spacetime structure to investigate the details of matter couplings, including the Noether-Ward identities, and transport phenomena and thermodynamics of non-relativistic fluids.