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.
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.
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).
investigations into the interactions involving topological defects, such as
magnetic monopoles and strings, that may have been produced in the early
universe. I performed numerical studies on the interactions of twisted
monopole-antimonopole pairs in the 't Hooft-Polyakov model for a range of
values of the scalar to vector mass ratio. Sphaleron solution predicted by
Taubes was recovered, and I mapped out its energy and size as functions of
parameters. I also looked into the production, and decay modes of $U(1)$ gauge
and global strings. I demonstrated that strings can be produced upon evolution
of gauge wavepackets defined within a certain region of parameter space. The
numerical exploration of the decay modes of cosmic string loops led to the
conclusions that string loops emit particle radiation primarily due to kink
collisions, and that their decay time due to these losses is proportional to
$L^p$, where $L$ is the loop length and $p \approx 2$. In contrast, the decay
time due to gravitational radiation scales in proportion to $L$, and I
concluded that particle emission is the primary energy loss mechanism for loops
smaller than a critical length scale, while gravitational losses dominate for
larger loops. In addition, I analyzed the decay of cosmic global string loops
due to radiation of Goldstone bosons and massive scalar ($\chi$) particles.
The length of loops I studied ranges from 200-1000 times the width of the
string core. I found that the lifetime of a loop is approximately $1.4L$. The
energy spectrum of Goldstone boson radiation has a $k^{-1}$ fall off, where $k$
is the wavenumber, and a sharp peak at $k\approx m_\chi/2$, where $m_\chi$ is
the mass of $\chi$. The latter is a new feature and implies a peak at high
energies (MeV-GeV) in the cosmological distribution of QCD axions.
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).