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Linked and Knotted Fields in Plasma and Gravity

  • Author(s): Thompson, Amy V.
  • Advisor(s): Bouwmeester, Dirk
  • et al.

Hopfions are a class of fields whose topology is derived from the Hopf fibration, with field lines that are linked circles which lie on a set of space-filling nested toroidal surfaces. In this thesis, we use analytic and computational methods to study hopfions and their generalization to field configurations based on torus knots in both plasma and gravity.

The Kamchatnov-Hopf solution to the magnetohydrodynamic (MHD) equations is a topological soliton, a field configuration that is stable due to a conserved topological quantity, with linking number one. By realizing that the angular momentum can also serve as a secondary stabilizing mechanism for certain field configurations, we generalize this solution to a class of topological solitons with linking number greater than one and show they are stable in ideal MHD. When studied in full resistive MHD, the results of simulations indicate that the non-zero linking number serves to inhibit the decay of the magnetic field energy.

The vacuum Maxwell and linearized Einstein equations take a similar form when expressed as spin-N field equations, suggesting that electromagnetic and gravitational radiation possess analogous topologically non-trivial field configurations. The solutions to the massless spin-N equations can be found by complex contour integral transformations with generating functions in terms of twistor variables. Using these methods, we construct the null electromagnetic hopfion and the analogous Petrov Type N gravitational solution. The fields are decomposed into tendex and vortex lines, the analog of the electromagnetic field lines, to investigate their physical properties and characterize their topology. For both electromagnetism and gravity the topology manifests in the lines of force.

We then show that the hopfion solutions are the simplest case in a class of Type N linked and knotted gravitational solutions based on torus knots. Reparameterizing the twistor generating functions in terms of the winding numbers of the field lines allows one to choose the degree of linking or knotting of the associated field configuration.

Finally, we discuss how the singularity structure of the twistor generating functions determines the spinor classification of the fields in Minkowski space. Since the solutions are constructed by contour integral transformations, the poles of the generating functions are directly related to the geometry of the physical fields. By modifying the generating functions, we extend the construction of the Type N gravitational hopfion to find the analogous spin-2 solutions of different Petrov classifications and characterize their tendex and vortex line configurations.

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