Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Curved non-relativistic spacetimes, Newtonian gravitation and massive matter

Michael Geracie, Kartik Prabhu, Matthew M. Roberts

TL;DR

The paper constructs the most general Bargmann spacetime—Newton–Cartan geometry augmented by a mass gauge field—to couple massive matter while preserving local Galilean invariance, including torsionful connections. It develops an extended coframe and Milne-invariant (boost-invariant) derivative, derives manifestly invariant actions for a massive particle and the Schrödinger field, and shows how the mass gauge field naturally arises as a next-to-leading order Lorentzian datum in the non-relativistic limit. The work clarifies the role of the Newton–Coriolis form, the separation of mass and charge gauge fields, and the interpretation of gravity in this framework, while connecting to Newtonian gravity in torsionless limits and to null reductions of Lorentzian spacetimes. It also demonstrates how these Bargmann backgrounds can be obtained as non-relativistic limits of Lorentzian spacetimes, providing a solid bridge between relativistic and condensed-matter approaches to non-relativistic geometry and transport phenomena.

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.

Curved non-relativistic spacetimes, Newtonian gravitation and massive matter

TL;DR

The paper constructs the most general Bargmann spacetime—Newton–Cartan geometry augmented by a mass gauge field—to couple massive matter while preserving local Galilean invariance, including torsionful connections. It develops an extended coframe and Milne-invariant (boost-invariant) derivative, derives manifestly invariant actions for a massive particle and the Schrödinger field, and shows how the mass gauge field naturally arises as a next-to-leading order Lorentzian datum in the non-relativistic limit. The work clarifies the role of the Newton–Coriolis form, the separation of mass and charge gauge fields, and the interpretation of gravity in this framework, while connecting to Newtonian gravity in torsionless limits and to null reductions of Lorentzian spacetimes. It also demonstrates how these Bargmann backgrounds can be obtained as non-relativistic limits of Lorentzian spacetimes, providing a solid bridge between relativistic and condensed-matter approaches to non-relativistic geometry and transport phenomena.

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.

Paper Structure

This paper contains 14 sections, 95 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Bargmann group and non-relativistic spacetimes
  3. Bargmann group and representations
  4. Bargmann spacetime
  5. Bundle reduction and null compactification
  6. Torsion and restrictions of Bargmann spacetime
  7. Matter actions and currents
  8. Massive particle
  9. Schrödinger field
  10. Electromagnetism
  11. Non-relativistic limit of Lorentzian spacetimes
  12. Massive particle from Lorentzian worldline
  13. Schrödinger from Klein-Gordon
  14. Outlook

Figures (2)

  • Figure 1: Spacetime diagram with spatial hypersurfaces $\Sigma_1$, $\Sigma_2$ and time-like curves $\gamma_1$ and $\gamma_2$. The curves $\Gamma_1$ and $\Gamma_2$ are arbitrary curves lying entirely in the respective spatial hypersurfaces ending on the intersections of the appropriate $\gamma$ and $\Sigma$. The time measured along $\gamma_1$ and $\gamma_2$ differ by the flux of $d\boldsymbol{n}$ through $R$.
  • Figure 2: A summary of Bargmann spacetimes and some invariant restrictions that can be imposed on them.