On spacetime foliations and electromagnetic knots

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–Rañada 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.

Finsler geometries from topological electromagnetism

“…Another problem is to study the Finsler geometries corresponding to the polynomial action which implies including the constraints in the fundamental Finsler function. New generalizations of the Rañada fields have been presented recently in the literature, see e. g. [5][6][7][8] for topological solutions in the presence of the gravitational field and [9][10][11][12][13][14] for generalization to the non-linear electrodynamics. It should be interesting to generalize the construction presented in this paper to determine the Finsler geometries associated to these systems.…”

“…The study of the topological solutions to Maxwell's equations in vacuum, firstly proposed by Trautman and Rañada in [1][2][3], has revealed so far a rich interplay between physical systems and mathematical structures which was previously unexpected in the realm of classical electrodynamics and classical field theory [4]. Since then, the subject of the topological electromagnetic fields has gain momentum with very interesting problems investigated recently, such as the existence of topological solutions of the Einstein-Maxwell theory [5][6][7][8] and of the non-linear electrodynamics [9][10][11][12][13][14]. Also, it has been shown that there are interesting mathematical structures that can be associated to the physical systems a e-mail: adina.crisan@mep.utcluj.ro b e-mail: ionvancea@ufrrj.br (corresponding author) with topological electromagnetic fields and play an important role in their dynamics, such as twistors [15], fibrations [16] and rational functions [17,18] (see for recent reviews [19][20][21]).…”

Quasipositive links and electromagnetism

“…We would like to point out that Rañada's construction has also been interpreted using contact geometry [18]. Furthermore, Costa e Silva, Goulart and Ottoni interpreted knotted solutions to Maxwell's equations as foliations of space time [8].…”

Gravitoelectromagnetic Knot Fields

“…This is an active area of research in classical electromagnetism with significant progress made recently in this direction by the discovery and the analysis of a large set of new solutions of the field equations that generalize the electromagnetic Rañada fiels in the linear [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54] and non-linear electrodynamics [55][56][57][58]. From the mathematical point of view, several mathematical structures related to the topologically non-trivial fields have been explored in the recent literature among which are the twistors [42], rational functions [51,54], fibre bundles [59], space-time foliations [60] and generalized Finsler geometries [61]. The interaction of electromagnetic knots with the matter for classical and quantum particles is discussed.…”