Abstract
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, via the zero rest mass equation for spin 1 and spin 2, defines solutions of linearized Einstein’s equation possessing a Hopf fibration as the curves along which no stretching, compression or precession will occur. We report on numerical findings relating the stability of the linked and knotted zeros of the Weyl tensor and their relation to linked optical vortices.
Highlights
Knotted electromagnetic fieldsThe electromagnetic Hopfion is intimately related to the Robinson congruence in Minkowski space M4
JHEP07(2019)004 the formalism developed by Plebanski [25] and Tod [26]
These fields are shown to correspond to single copies of a class of anti-self-dual Kerr-Schild spacetimes containing the SparlingTod 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
Summary
The electromagnetic Hopfion is intimately related to the Robinson congruence in Minkowski space M4. In which f(xa) is an arbitrary scalar function that does not affect the integral curves of the corresponding vector field in M4. This vector field is obtained with the help of soldering forms (Infeld van der Waerden symbols) and is geodesic and shear free, since ΠRob satisfies ΠAΠB∇BB ΠA = 0 [29]. Α, β are used to construct null electromagnetic fields F = ∇α×∇β, with F ≡ E + iB the Riemann-Silberstein vector. = pqαp−1βq−1∇α × ∇β, which is the Riemann-Silberstein vector for a null electromagnetic field with field lines in the form of (p, q) torus knots [27, 38]. Since F ∝ ∇α × ∇β and we used the Hopfion Bateman variables, we again have the same Poynting vector structure as before
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