arXiv subjects

Jan Willem Dalhuisen

Publications and source records attributed to Jan Willem Dalhuisen.

3 recordsLinked to original sources

arXiv2019

Anti-Self-Dual Spacetimes, Gravitational Instantons and Knotted Zeros of the Weyl Tensor

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.

hep-th↗

arXiv2017

Knotted optical vortices in exact solutions to Maxwell's equations

We construct a family of exact solutions to Maxwell's equations in which the points of zero intensity form knotted lines topologically equivalent to a given but arbitrary algebraic link. These lines of zero intensity, more commonly referred to as optical vortices, and their topology are preserved as time evolves and the fields have finite energy. To derive explicit expressions for these new electromagnetic fields that satisfy the nullness property, we make use of the Bateman variables for the Hopf field as well as complex polynomials in two variables whose zero sets give rise to algebraic links. The class of algebraic links includes not only all torus knots and links thereof, but also more intricate cable knots. While the unknot has been considered before, the solutions presented here show that more general knotted structures can also arise as optical vortices in exact solutions to Maxwell's equations.

math-ph↗

arXiv2014

Gravitational Hopfions

The electromagnetic hopfion (EM hopfion) is a topologically nontrivial solution to the vacuum Maxwell equations with the property that any two field lines belonging to either the electric, magnetic, or Poynting vector fields (EBS fields) are closed and linked exactly once. The most striking and characteristic feature of this peculiar field configuration is the existence of an exceptional constant-time hyperplane wherein the EBS fields are tangent to three orthogonal Hopf fibrations. Using twistor methods we find the spin-$N$ generalization of the EM hopfion. Furthermore, we analyze the spin-2 solution within the framework of gravito-electromagnetism and show that the topology is manifest in the tendex and vortex lines. By decomposing the spin-2 field into spatial gravito-electric and gravito-magnetic tensors, we characterize its topological structure and evolution in terms of the EM hopfion.

gr-qc↗