The classical double copy maps exact solutions of general relativity to exact solutions of U(1) Yang-Mills theory and suggests a hitherto unknown connection between gravity and gauge theory. In this thesis I study three problems using the Kerr-Schild and Weyl formulations of the classical double copy. Using the Kerr-Schild double copy, I analyze the single copy of a rotating nonsingular black hole and analyze its horizon structure to probe the relationship between the presence of horizons on the gravity side and the single copy field on the gauge theory side. In the second problem I describe the mapping between the surface gravity of static spherically symmetric black holes and the force on a test particle due to the single copy field of the black hole. I also describe potential routes to extending this map to rotating black holes. Finally, inspired by the extended Weyl double copy for spacetimes possessing sources, I reinterpret the single copy of the Taub- NUT metric as being comprised of two terms each being sourced by a separate parameter (the mass and the NUT charge).

This document is a guide that can be used by undergraduate physics students alongside Richard J. Jacob and Professor Emeritus’s Tutorials in the Mathematical Methods of Physics to aid in their understanding of the key mathematical concepts from PHY201 and PHY302. This guide can stand on its own and be used in other upper division physics courses as a handbook for common special functions. Additionally, we have created several Mathematica notebooks that showcase and visualize some of the topics discussed (available from the GitHub link in the introduction of the guide).

In a hypothetical Grand Unified Theory, magnetic monopoles are a particle which would act as a charge carrier for the magnetic force. Evidence of magnetic monopoles has yet to be found and based off of their relatively high mass (4-10 TeV) will be difficult to find with current technology. The goal of my thesis is to mathematically model the magnetic monopole by finding numerical solutions to the equations of motion. In my analysis, I consider four cases: kinks, cosmic strings, global monopoles, and magnetic monopoles. I will also study electromagnetic gauge fields to prepare to include gauge fields in the magnetic monopole case. Numerical solutions are found for the cosmic string and global monopole cases. As expected, the energy is high at small distance r and drops off as r goes to infinity. Currently numerical solutions are being worked towards for electromagnetic gauge fields and the magnetic monopole case.

This paper outlines the design and testing of a z-scan spectrometer capable of measuring the third order refraction index of liquids. The spectrometer underwent multiple redesigns, with each explored in this paper with their benefits and drawbacks discussed. The first design was capable of measuring the third order refraction index for glass, and found a value of 8.43 +- 0.392 x 10^(-16) cm^2/W for the glass sample, with the literature stating glass has a refraction index between 1-100 x 10^(-16) cm^2/W. The second design was capable of measuring the third order refraction index of liquids, and found values of 1.23 $\pm$ 0.121 $\e{-16}$ and 9.43 +- 1.00 x 10^(-17) cm^2/W for water and ethanol respectively, with literature values of 2.7 x 10^(-16) and 5.0 x 10^(-17) cm^2/W respectively. The third design gave inconclusive results due to extreme variability in testing, and and the fourth design outlined has not been tested yet due to time constraints.