A non-lorentzian primer

TL;DR

This paper reviews non-lorentzian theories, their geometries, and dynamics, introducing new formulations of particle actions, gravity theories, and matter fields as limits of relativistic models.

Contribution

It provides a comprehensive overview of non-lorentzian kinematics, geometries, and gravity theories, and constructs new particle and matter actions using nonlinear realizations.

Findings

Introduces non-lorentzian kinematical spacetimes and symmetry algebras.

Constructs particle actions in non-lorentzian spaces via nonlinear realizations.

Discusses non-lorentzian gravity theories and matter fields as relativistic limits.

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.