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

TL;DR

This paper explores the connection between knotted electromagnetic fields, anti-self-dual spacetimes, and the Weyl tensor, revealing new insights into gravitational instantons and the structure of spacetime with knotted zeroes.

Contribution

It introduces a superpotential for knotted electromagnetic fields, links these to anti-self-dual Kerr-Schild spacetimes, and formulates conditions for solutions with Hopf fibration structures.

Findings

Weyl double copy relates electromagnetic Hopfions to specific spacetimes.

Conditions for linearized Einstein solutions with Hopf fibrations are established.

Numerical analysis suggests stability of knotted zeros of the Weyl tensor.

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.