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Galilean and Carrollian Hodge star operators

Marián Fecko

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.

Galilean and Carrollian Hodge star operators

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.
Paper Structure (19 sections, 92 equations)

This paper contains 19 sections, 92 equations.

Table of Contents

  1. Introduction - standard Hodge star operator
  2. Galilean and Carrollian Hodge stars
  3. Explicit formulae and some properties
  4. General explicit formulae
  5. Basic properties
  6. Behaviour with respect to pull-back
  7. An application: Electrodynamics
  8. Standard electrodynamics
  9. Galilei and Carroll electrodynamics
  10. Summary and conclusions
  11. Appendices
  12. Galilean and Carrollian manifolds
  13. Galilean manifold
  14. Carrollian manifold
  15. Top-degree forms and polyvectors
  16. ...and 4 more sections