Galilean and Carrollian Hodge star operators

TL;DR

This paper introduces analogs of the Hodge star operator for Galilean and Carrollian spacetimes, enabling the formulation of physics equations like electrodynamics in these non-metric geometries.

Contribution

It proposes new Hodge star operators tailored for Galilean and Carrollian spacetimes, overcoming the lack of a canonical metric tensor.

Findings

Defined Hodge star analogs for Galilean spacetimes

Defined Hodge star analogs for Carrollian spacetimes

Facilitated formulation of physics equations in non-metric geometries

Abstract

The standard Hodge star operator is naturally associated with metric tensor (and orientation). It is routinely used to concisely write down physics equations on, say, Lorentzian spacetimes. On Galilean (Carrollian) spacetimes, there is no canonical (nonsingular) metric tensor available. So, the usual construction of the Hodge star does not work. Here we propose analogs of the Hodge star operator on Galilean (Carrollian) spacetimes. They may be used to write down important physics equations, e.g. equations of Galilean (Carrollian) electrodynamics.