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A non-lorentzian primer

Eric Bergshoeff, José Figueroa-O'Farrill, Joaquim Gomis

TL;DR

This work develops a comprehensive framework for non-lorentzian physics by classifying kinematical and aristotelian algebras, constructing homogeneous spacetimes via Klein pairs, and applying nonlinear realisations and coadjoint-orbit methods to derive particle actions across Galilei, Newton–Cartan, and Carroll geometries. It demonstrates how non-relativistic and Carrollian limits of relativistic theories yield consistent dynamics for spins 0, 1/2, and 1, and shows how gravity theories can be obtained by gauging non-lorentzian algebras and performing systematic contractions of general relativity. The treatment connects geometric structures (G-structures, Newton–Cartan, Carroll) to physical models, including harmonic-oscillator NR limits, Schwarzian actions, and SL(2,R) conformal mechanics, offering a unified path from symmetry to dynamics in non-lorentzian spacetimes. The results provide a robust toolkit for exploring non-relativistic holography, fracton physics, and boundary/flat-horizon phenomena, with explicit constructions of matter and gauge sectors in diverse backgrounds. Overall, the paper lays out a cohesive, symmetry-driven programme to understand non-lorentzian gravity and field theories, bridging algebraic classifications with geometric realizations and dynamical actions.

Abstract

We review both the kinematics and dynamics of non-lorentzian theories and their associated geometries. First, we introduce non-lorentzian kinematical spacetimes and their symmetry algebras. Next, we construct actions describing the particle dynamics in some of these kinematical spaces using the method of nonlinear realisations. We explain the relation with the coadjoint orbit method. We continue discussing three types of non-lorentzian gravity theories: Galilei gravity, Newton-Cartan gravity and Carroll gravity. Introducing matter, we discuss electric and magnetic non-lorentzian field theories for three different spins: spin-0, spin-1/2 and spin-1, as limits of relativistic theories.

A non-lorentzian primer

TL;DR

This work develops a comprehensive framework for non-lorentzian physics by classifying kinematical and aristotelian algebras, constructing homogeneous spacetimes via Klein pairs, and applying nonlinear realisations and coadjoint-orbit methods to derive particle actions across Galilei, Newton–Cartan, and Carroll geometries. It demonstrates how non-relativistic and Carrollian limits of relativistic theories yield consistent dynamics for spins 0, 1/2, and 1, and shows how gravity theories can be obtained by gauging non-lorentzian algebras and performing systematic contractions of general relativity. The treatment connects geometric structures (G-structures, Newton–Cartan, Carroll) to physical models, including harmonic-oscillator NR limits, Schwarzian actions, and SL(2,R) conformal mechanics, offering a unified path from symmetry to dynamics in non-lorentzian spacetimes. The results provide a robust toolkit for exploring non-relativistic holography, fracton physics, and boundary/flat-horizon phenomena, with explicit constructions of matter and gauge sectors in diverse backgrounds. Overall, the paper lays out a cohesive, symmetry-driven programme to understand non-lorentzian gravity and field theories, bridging algebraic classifications with geometric realizations and dynamical actions.

Abstract

We review both the kinematics and dynamics of non-lorentzian theories and their associated geometries. First, we introduce non-lorentzian kinematical spacetimes and their symmetry algebras. Next, we construct actions describing the particle dynamics in some of these kinematical spaces using the method of nonlinear realisations. We explain the relation with the coadjoint orbit method. We continue discussing three types of non-lorentzian gravity theories: Galilei gravity, Newton-Cartan gravity and Carroll gravity. Introducing matter, we discuss electric and magnetic non-lorentzian field theories for three different spins: spin-0, spin-1/2 and spin-1, as limits of relativistic theories.
Paper Structure (71 sections, 1 theorem, 355 equations, 1 figure, 5 tables)

This paper contains 71 sections, 1 theorem, 355 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Affine space
  4. Galilei spacetime
  5. Minkowski spacetime
  6. Lie algebra of symmetries
  7. Symmetry
  8. Kinematical Lie algebras
  9. Aristotelian Lie algebras
  10. Central extensions
  11. Geometry
  12. Homogeneous spaces
  13. Group actions on manifolds
  14. Linear isotropy representation and invariant tensors
  15. Klein pairs
  16. ...and 56 more sections

Key Result

Theorem 1

Let $\mathscr{G}$ be a connected Lie group and $(M,\omega)$ a simply-connected homogeneous symplectic manifold of $\mathscr{G}$. Then there exists a covering $\pi: (M,\omega) \to (\mathscr{O},\omega_{\text{KKS}})$, with $\mathscr{O}$ a coadjoint orbit of a one-dimensional central extension of $\math

Figures (1)

  • Figure 1: The clock fibration $\pi : \mathbb{A}^4 \to \mathbb{A}^1$

Theorems & Definitions (3)

  • Definition 1
  • Definition 2
  • Theorem