Anti-self-dual spacetimes, gravitational instantons and 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. † s.sabharwal@umail.leidenuniv.nl ‡ Dalhuisen@physics.leidenuniv.nl 1 arXiv:1904.06030v3 [hep-th]

The Newman-Penrose map and the classical double copy

“…which when written in light-cone coordinates (u, v, ζ,ζ) (defined below in (3.1)) reproduces the Plebański equation [20,30,46,47]…”

Weyl doubling

“…The Schwarzschild solution was shown to single copy to an electric charge [44], the Taub-NUT JHEP09(2020)127 solution to a magnetic monopole [45] and the Eguchi-Hanson solution mapped to a selfdual gauge field [46], for example. More general topologically non-trivial solutions have been double copied in the work of [47]. Other work examining symmetries of the linearised double copy is [48][49][50][51][52][53][54][55][56].…”

The Weyl double copy from twistor space

“…We have yet to show whether such a double copy is meaningful, although we will see later that this is indeed the case, provided one restricts to linearised level in some cases. Since its inception, the Weyl double copy has been used to study certain topologically non-trivial electromagnetic solutions ("Hopfions") [133], and a tensorial formulation of the relationship between the electromagnetic field strength and Weyl tensor has been presented in refs. [71,73], allowing a generalisation to higher dimensions.…”