Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time

TL;DR

This work develops the Carroll group as a dual non-Einsteinian spacetime symmetry, complementary to the Galilean limit, by formulating Carroll structures (degenerate metrics with a distinguished time direction) and embedding them in a unifying Bargmann framework. It shows Carroll arises as a null-hypersurface in Bargmann space alongside Newton–Cartan as a quotient, clarifying the geometric and group-theoretic relations among Bargmann, Galilei, and Carroll. The paper then constructs Carroll electromagnetism in electric- and magnetic-like contractions, presents covariant and contravariant formulations, and connects these Carroll theories to Maxwell theory on Bargmann spaces. Extending further, it discusses non-Einsteinian electrodynamics in media, pre-metric approaches, dualities, and the Chaplygin gas, highlighting potential applications to holography and brane dynamics on null surfaces.

Abstract

The Carroll group was originally introduced by Levy-Leblond [1] by considering the limit of the Poincaré group as $c\to0$. In this paper an alternative definition, based on the geometric properties of a non-Minkowskian, non-Galilean but nevertheless boost-invariant, space-time structure is proposed. A "duality" with the Galilean limit $c\to\infty$ is established. Our theory is illustrated by Carrollian electromagnetism.