Classification of Electromagnetic and Gravitational Hopfions by Algebraic Type

TL;DR

This paper generalizes hopfions to higher spin fields, constructs electromagnetic and gravitational hopfions via Penrose transform, and analyzes their topological and algebraic properties, including propagation and energy radiation characteristics.

Contribution

It introduces a unified framework for classifying electromagnetic and gravitational hopfions by algebraic type using the Penrose transform, revealing their topological structures and dynamics.

Findings

Null and type N fields propagate at light speed.

Type D fields radiate energy outward.

Type III gravitational hopfion exhibits unique features.

Abstract

We extend the definition of hopfions to include a class of spin-$h$ fields and use this to introduce the electromagnetic and gravitational hopfions of different algebraic types. The fields are constructed through the Penrose contour integral transform, thus the singularities of the generating functions are directly related to the geometry of the resulting physical fields. We discuss this relationship and how the topological structure of the fields is related to the Robinson congruence. Since the topology appears in the lines of force for both electromagnetism and gravity, the gravito-electromagnetic formalism is used to analyze the gravitational hopfions and describe the time evolution of their tendex and vortex lines. The correspondence between fields of different spin results in analogous configurations based on the same topological structure. The null and type N fields propagate at the speed of light, while the non-null and type D fields radiate energy outward from the center. Finally we discuss the type III gravitational hopfion, which has no direct electromagnetic analog, but find that it still exhibits some of the characteristic features common to the other hopfion fields.