On a remarkable electromagnetic field in the Einstein Universe

Abstract

We present a time-dependent solution of the Maxwell equations in the Einstein universe, whose electric and magnetic fields, as seen by the stationary observers, are aligned with the Clifford parallels of the 3-sphere S 3 . The conformal equivalence between Minkowski's spacetime and (a region of) the Einstein cylinder is then exploited in order to obtain a knotted, finite energy, radiating solution of the Maxwell equations in flat spacetime. We also discuss similar electromagnetic fields in expanding closed Friedmann models, and compute the matter content of such configurations.

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]).…”

On the existence of the field line solutions of the Einstein–Maxwell equations

“…Note that, while in the flat space-time, these solutions can be used to construct global electromagnetic knots, the global solutions are not always possible on an arbitrary hyperbolic manifold. Nevertheless, at least one particular example of our general solution is known in the case of spherical leafs [36]. However, it would be interesting to explore further the electromagnetic line configurations to determine other local as well as global solutions of the Maxwell's equations.…”

Gravitoelectromagnetic Knot Fields

“…Here, we focus on the linearized gravity away from sources in the GEM formulation, which originated in the works of Thirring and Lens [62,63]. The existence of the knot fields for gravitational and gravitating electromagnetic fields has been discussed in other research papers [71][72][73][74][75][76][77][78][79]. In particular, the existence of the Hopf-Rañada solutions of the spin 2 field theory and their properties in the GEM formulation in terms of Weyl tensor were analyzed in [71][72][73]78,80].…”