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Newton-Cartan, Galileo-Maxwell and Kaluza-Klein

Dieter Van den Bleeken, Cagin Yunus

TL;DR

The paper analyzes Kaluza-Klein reduction within Newton-Cartan gravity and shows that dimensional reduction and the nonrelativistic limit commute, producing a NC–Maxwell–dilaton theory in which Galilean electromagnetism back-reacts on the spatial geometry. It furnishes explicit reduction equations and demonstrates how the Maxwell fields source spatial curvature, extending NC gravity beyond the Newtonian potential. A concrete magnetic monopole–type solution based on Taub-NUT/Gibbons–Hawking metrics illustrates curvature sourcing and scalar hair in the nonrelativistic setup. The work connects to holographic and condensed matter contexts, highlighting nontrivial nonrelativistic geometries that arise beyond traditional post-Newtonian approximations.

Abstract

We study Kaluza-Klein reduction in Newton-Cartan gravity. In particular we show that dimensional reduction and the nonrelativistic limit commute. The resulting theory contains Galilean electromagnetism and a nonrelativistic scalar. It provides the first example of back-reacted couplings of scalar and vector matter to Newton-Cartan gravity. This back-reaction is interesting as it sources the spatial Ricci curvature, providing an example where nonrelativistic gravity is more than just a Newtonian potential.

Newton-Cartan, Galileo-Maxwell and Kaluza-Klein

TL;DR

The paper analyzes Kaluza-Klein reduction within Newton-Cartan gravity and shows that dimensional reduction and the nonrelativistic limit commute, producing a NC–Maxwell–dilaton theory in which Galilean electromagnetism back-reacts on the spatial geometry. It furnishes explicit reduction equations and demonstrates how the Maxwell fields source spatial curvature, extending NC gravity beyond the Newtonian potential. A concrete magnetic monopole–type solution based on Taub-NUT/Gibbons–Hawking metrics illustrates curvature sourcing and scalar hair in the nonrelativistic setup. The work connects to holographic and condensed matter contexts, highlighting nontrivial nonrelativistic geometries that arise beyond traditional post-Newtonian approximations.

Abstract

We study Kaluza-Klein reduction in Newton-Cartan gravity. In particular we show that dimensional reduction and the nonrelativistic limit commute. The resulting theory contains Galilean electromagnetism and a nonrelativistic scalar. It provides the first example of back-reacted couplings of scalar and vector matter to Newton-Cartan gravity. This back-reaction is interesting as it sources the spatial Ricci curvature, providing an example where nonrelativistic gravity is more than just a Newtonian potential.

Paper Structure

This paper contains 21 sections, 69 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview
  3. Newton-Cartan gravity
  4. Galileo-Maxwell electromagnetism
  5. Kaluza-Klein reduction
  6. Non-relativistic limit
  7. Example
  8. Notation and conventions
  9. Space and time
  10. Time derivative
  11. Lie derivative
  12. Time reparametrization invariant formulations
  13. Metric formulation of Newton-Cartan gravity
  14. Setup
  15. Exhibiting the degrees of freedom
  16. ...and 6 more sections

Figures (1)

  • Figure 1: This commutative diagram shows the two ways one can go from general relativity (GR) to Newton-Cartan-Maxwell dilaton theory (NCMD). The first option is to pass via Newton-Cartan gravity (NC) by first taking the nonrelativistic, i.e. $c\rightarrow\infty$, limit and then performing Kaluza-Klein reduction (KK). Or one could go via Einstein-Maxwell dilaton theory (EMD) by first performing Kaluza-Klein reduction and then taking the nonrelativistic limit.