Some useful operators on differential forms in Galilean and Carrollian spacetimes

TL;DR

This paper explores the potential for defining analogs of the Hodge star operator on Galilean and Carrollian spacetimes, aiming to extend the utility of differential forms in physics beyond Lorentzian geometries.

Contribution

It introduces the concept of intertwining operators as analogs of the Hodge star operator on Galilean and Carrollian spacetimes, facilitating the use of differential forms in these settings.

Findings

Intertwining operators can serve as Hodge star analogs.

Differential forms could become more useful in Galilean and Carrollian physics.

Potential for new mathematical tools in non-Lorentzian spacetimes.

Abstract

Differential forms on Lorentzian spacetimes is a well-established subject. On Galilean and Carrollian spacetimes it does not seem to be quite so. This may be due to the absence of Hodge star operator. There are, however, potentially useful analogs of Hodge star operator also on the last two spacetimes, namely intertwining operators between corresponding representations on forms. Their use could perhaps make differential forms as attractive tool for physics on Galilean and Carrollian spacetimes as forms on Lorentzian spacetimes definitely proved to be.