UC Santa Barbara

Building off senior and master thesis work, we present a class of topological plasma configurations characterized by their toroidal and poloidal winding numbers, $n_t$ and $n_p$, respectively. The case of $n_t=1$ and $n_p=1$ corresponds to the Kamchatnov-Hopf soliton, a magnetic field configuration everywhere tangent to the fibers of a Hopf fibration so that the field lines are circular, linked precisely once, and form the surfaces of nested tori. We show that for $n_t \in \mathbb{Z}^+$ and $n_p=1$, these configurations represent stable, localized solutions to the magnetohydrodynamic equations for an ideal incompressible fluid with infinite conductivity. Furthermore, we extend our stability analysis by considering a plasma with finite conductivity and estimate the soliton lifetime in such a medium as a function of the toroidal winding number.

Torus knot topology is also inherent in electromagnetic and gravitational radiation. We show this by constructing spin-$N$ fields based on the elementary states of twistor theory. The twistor functions corresponding to the elementary states admit a parameterization in terms of the torus knots' poloidal and toroidal winding numbers, allowing one to choose the degree of linking or knotting of the associated field configuration. We describe the topology of the gravitational fields and their physical interpretation in terms of their tidal and frame drag phenomenology. Using the gravito-electromagnetic formalism, we show that the torus knot structure is exhibited in the tendex and vortex lines for the analogous linearized gravitational solutions.

We extend the definition of hopfions to include a larger class of spin-$h$ fields and use this to classify 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. We thus obtain similar configurations based on the same topological structure but varying spin-$h$ algebraic types. 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 characteristic features common to the other hopfion fields.

In the complete non-linear theory we present an exact solution of Einstein's equation that describes the gravitational shockwave of a massless particle on the horizon of a Kerr-Newman black hole—generalizing the Dray-'t~Hooft solution to the case of a rotating background. The back-reacted metric is of the generalized Kerr-Schild form and is Type II in the Petrov classification. We show that if the background tetrad is aligned with shear-free null geodesics, and the background Ricci tensor satisfies a simple condition, all nonlinearities in the perturbation will drop out of the curvature scalars. We reformulate Einstein-Hilbert gravity in the first-order formalism of Elie Cartan and find that the Riemann curvature tensor is the Yang-Mills gauge curvature of a local Lorentz group gauge connection. We derive and make heavy use of the method of spin coefficients (the Newman-Penrose formalism) in its compacted form (the Geroch-Held-Penrose formalism).