On spacetime foliations and electromagnetic knots

TL;DR

This paper explores electromagnetic knots through foliation theory, revealing intrinsic topological properties and offering new insights into their structure and potential applications in electromagnetic field analysis.

Contribution

It introduces a foliation-based framework for understanding electromagnetic knots, emphasizing intrinsic leaf properties over observer-dependent quantities.

Findings

Foliation theory describes electromagnetic knots as intersecting codimension-2 foliations.

Topological invariants of knots emerge naturally from the foliation structure.

The approach offers new perspectives on the Hopf-Ra dnada solution and potential future developments.

Abstract

The present paper has a number of distinct purposes. First is to give a description of a class of electromagnetic knots from the perspective of foliation theory. Knotted solutions are then interpreted in terms of two codimension-2 foliations whose knotted leaves intersect orthogonally everywhere in spacetime. Secondly, we show how the foliations give rise to field lines and how the topological invariants emerge. The machinery used here emphasizes intrinsic properties of the leaves instead of observer dependent quantities - such as a time function, a local rest frame or a Cauchy hypersurface. Finally, we discuss the celebrated Hopf-Ra\~nada solution in details and stress how the foliation approach may help in future developments of the theory of electromagnetic knots. We conclude with several possible applications, extensions and generalizations.