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.

arXiv comment14 pages, 3 figures

References in corpus (11)

• Gravito-electromagnetism
• Type D Spacetimes and the Weyl Double Copy
• The self-dual classical double copy, and the Eguchi-Hanson instanton
• Visualizing Spacetime Curvature via Frame-Drag Vortexes and Tidal Tendexes I. General Theory and Weak-Gravity Applications
• Janis-Newman algorithm: simplifications and gauge field transformation
• Classification of Electromagnetic and Gravitational Hopfions by Algebraic Type
• New knotted solutions of Maxwell's equations
• Linked and Knotted Gravitational Radiation
• Knotted optical vortices in exact solutions to Maxwell's equations
• Classification of complex and real, vacuum spaces of the type $[\text{N}]\otimes [\text{N}]$
• On twisting type $[\text{N}]\otimes [\text{N}]$ Ricci flat complex spacetimes with two homothetic symmetries

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