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

Understand

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.