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.
This paper describes the electrodynamics of a null and force-free field in completely geometric terms. As was previously established in \cite{Menon_FF20}, solutions to force-free electrodynamics are governed by the existence of certain special types of foliations of spacetime. Here the nature of the foliations in a coordinate-free formalism in the null case is prescribed. All of the general results are illustrated by constructing a null, force-free electrodynamic field in a Friedmann-Lemaitre-Robertson- Walker (FLRW) spacetime.
The Fourier transform method can be applied to obtain electromagnetic knots, which are solutions of Maxwell equations in vacuum with non-trivial topology of the field lines and especial properties. The program followed in this work allows to present the main ideas and the explicit calculations at undergraduate level, so they are not obscured by a more involved formulation. We make use of helicity basis for calculating the electromagnetic helicity and the photon content of the fields.
In null electromagnetic fields the electric and the magnetic field lines evolve like unbreakable elastic filaments in a fluid flow. In particular, their topology is preserved for all time. We prove that for every link $L$ there is such an electromagnetic field that satisfies Maxwell's equations in free space and that has closed electric and magnetic field lines in the shape of $L$ for all time.
We derive a superpotential for null electromagnetic fields in which the field line structure is in the form of an arbitrary torus knot. These fields are shown to correspond to single copies of a class of anti-self-dual Kerr-Schild spacetimes containing the Sparling-Tod metric. This metric is the pure Weyl double copy of the electromagnetic Hopfion, and we show that the Eguchi-Hanson metric is a mixed Weyl double copy of this Hopfion and its conformally inverted state. We formulate two conditions for electromagnetic fields, generalizing torus knotted fields and linked optical vortices, that, vi…
We revisit a newfound construction of rational electromagnetic knots based on the conformal correspondence between Minkowski space and a finite $S^3$-cylinder. We present here a more direct approach for this conformal correspondence based on Carter-Penrose transformation that avoids a detour to de Sitter space. The Maxwell equations can be analytically solved on the cylinder in terms of $S^3$ harmonics $Y_{j;m,n}$, which can then be transformed into Minkowski coordinates using the conformal map. The resultant "knot basis" electromagnetic field configurations have non-trivial topology in that t…
We set up a correspondence between solutions of the Yang-Mills equations on ${\mathbb R}\times S^3$ and in Minkowski spacetime via de Sitter space. Some known Abelian and non-Abelian exact solutions are rederived. For the Maxwell case we present a straightforward algorithm to generate an infinite number of explicit solutions, with fields and potentials in Minkowski coordinates given by rational functions of increasing complexity. We illustrate our method with a nontrivial example.
We find all analytic SU(2) Yang-Mills solutions on de Sitter space by reducing the field equations to Newton's equation for a particle in a particular 3d potential and solving the latter in a special case. In contrast, Maxwell's equations on de Sitter space can be solved in generality, by separating them in hysperspherical coordinates. Employing a well-known conformal map between (half of) de Sitter space and (the future half of) Minkowski space, the Maxwell solutions are mapped to a complete basis of rational electromagnetic knot configurations. We discuss some of their properties and illustr…
In this paper, we present a covariant formalism that connects solutions to force-free electrodynamics in the non-null case and foliations of spacetime. In doing so, we are also able to derive an expression of the general non-null current density vector. Just as in the null case, solutions in the non-null case can give rise to a dual solution, however, as is shown below, this can happen only when the solution describes a vacuum field. All theorems are illustrated with previously known solutions.
We employ a recently developed method for constructing rational electromagnetic field configurations in Minkowski space to investigate several properties of these source-free finite-action Maxwell ("knot") solutions. The construction takes place on the Penrose diagram but uses features of de Sitter space, in particular its isometry group. This admits a classification of all knot solutions in terms of $S^3$ harmonics, labelled by a spin $2j\in{\mathbb N}_0$, which in fact provides a complete "knot basis" of finite-action Maxwell fields. We display a $j{=}1$ example, compute the energy for arbit…
Knotted solutions to electromagnetism are investigated as an independent subsector of the theory. We write down a Lagrangian and a Hamiltonian formulation of Bateman's construction for the knotted electromagnetic solutions. We introduce a general definition of the null condition and generalize the construction of Maxwell's theory to massless free complex scalar, its dual two form field, and to a massless DBI scalar. We set up the framework for quantizing the theory both in a path integral approach, as well as the canonical Dirac method for a constrained system. We make several observations abo…
In this paper, we prove the linear stability to gravitational and electromagnetic perturbations of the Reissner-Nordström family of charged black holes with small charge. Solutions to the linearized Einstein-Maxwell equations around a Reissner-Nordström solution arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to a linearized Kerr-Newman metric. We express the perturbations in geodesic outgoing null foliations, also known as Bondi gauge. To obtain decay of the solution, one must add a residual pure gauge solution which is proved to be itsel…
