Hyun Lim (PhD Dissertation, August 2019,
Advisor: Eric Hirschmann
)

Abstract

[Abstract]

Ryan Hatch (Senior Thesis, August 2018,
Advisor: Eric Hirschmann
)

Abstract

The full eigenvector decomposition for the flux Jacobian matrices of the equations of general relativistic magnetohydrodynamics (GRMHD) are found. The matter equations for GRMHD in the ideal limit can be written in a system of balance law equations. These equations can then be framed as a single 8$\times$8 matrix equation written in terms of the fluid and magnetic field variables. Obtaining the full decomposition allows for the implementation of sophisticated numerical techniques. In this paper we provide the right and left eigenvectors as well as the wave speeds for use in such numerical modeling.

Abstract

We show our work to refine the process of evolutions in general relativistic magnetohydrodynamics. We investigate several areas in order to improve the overall accuracy of our results. We test several versions of conversion methodologies between different sets of variables. We compare both single equation and two equations solvers to do the conversion. We find no significant improvement for multiple equation conversion solvers when compared to single equation solvers. We also investigate the construction of initial data and the conversion of coordinate systems between initial data code and evolution code. In addition to the conversion work, we have improved some methodologies to ensure data integrity when moving data from the initial data code to the evolution code. Additionally we add into the system of MHD equations a new field to help control the no monopole constraint. We perform a characteristic decomposition of the system of equations in order to derive the associated boundary condition for this new field. Finally, we implement a WENO (weighted non-oscillatory) system. This is done so we can evolve and track shocks that are generated during an evolution of our GRMHD equations.

Abstract

Cosmological data provide new constraints on the number of neutrino species and neutrino masses. However these constraints depend on assumptions related to the underlying cosmology. Since a correlation is expected between the number of effective neutrinos N_eff, the total neutrino mass \sum m_\nu, and the curvature of the universe \Omega_k, it is useful to investigate the current constraints in the framework of a nonflat universe. We provide an introduction to modern cosmology, placing an emphasis on topics relevant to constraining neutrino parameters with the latest cosmic microwave background (CMB) anisotropy data. The theoretical framework is designed to condense a wide variety of resources and make them available for students with similar research interests. We consider how cosmological flatness affects neutrino properties by providing theoretical arguments for correlation and performing statistical analyses on cosmological models involving neutrinos. We place new constraints on N_eff and \Omega_k, with N_eff = 4.03 \pm 0.45 and 10^3 \Omega_k = -4.46 \pm 5.24. Thus, even with a nonflat universe, N_{eff} = 3 is still disfavored with 95% confidence. We then investigate the correlation between neutrino mass \sum m_\nu and curvature \Omega_k that shifts the 95% upper limit of \sum m_\nu < 0.45 eV to \sum m_\nu < 0.95 eV. Thus, the impact of assuming flatness should be an essential consideration for future neutrino analysis in experimental cosmology.

Pedro Acosta (Honors Thesis, May 2010,
Advisor: Eric Hirschmann
)

Abstract

For any non-degenerate, quasihomogeneous superpotential W and an admissible group of diagonal symmetries G_W, Fan, Jarvis and Ruan have constructed a quantum cohomological field theory (FJRW-theory) that gives, among other things, a Frobenius algebra HFJRW(W;G_W;C) ((a,c) ring) and n-point correlation functions associated with the superpotential. This construction is analogous to a theory of the Gromov-Witten type. The FJRW-theory is a candidate for the mathematical structure behind N = 2 superconformal Landau-Ginzburg orbifolds. In this work I present an overview of this theory and give a proof of the Berglund-Hubsch-Krawitz mirror symmetry conjecture: For a given invertible superpotential W there exists an invertible superpotential W^T such that the Frobenius algebra H_{FJRW}(W;Gmax_W ;C) is isomorphic to the (c,c) ring of WT , and the Frobenius algebra H_{FJRW}(WT ;Gmax_W^T ;C) is isomorphic to the (c,c) ring of W.

Abstract

We present a method for solving the Einstein-Maxwell equations in a five dimensional, asymptotically flat, black hole spacetime with three commuting Killing vector fields. In particular, we show that by reducing the dimension of the Einstein-Maxwell equations in a Kaluza-Klein like manner we can determine the components of the metric and vector potential which lie in the direction of the Killing vector fields. These components are determined by nine scalar fields each of which satisfy a partial differential equation in two variables. These equations take the form of an elliptic operator set equal to a nonlinear source. We find evidence that particular combinations of these fields satisfy Dirichlet boundary conditions, and are well suited to numerical solution using Green functions. Using this method we generate numerical solutions to the 4+1 Einstein-Maxwell equations corresponding to charged generalizations of the Myers-Perry solution. We also discover symmetry relations among the scalar equations which constrain their functional forms and posit the existence of two rigidity-theorem-like relations for electrovac spacetimes and sketch how their use generalizes our method to N + 1 dimensions.

Abstract

We examine the question of chaotic particle orbits in general relativity. In particular we consider the dynamics of a spinning test particle in a black hole spacetime. We describe the model of a spinning test particle in general relativity and determine whether or not an orbit is chaotic. For chaotic orbits we calculate the Lyapunov exponent and give an improved method of comparing the exponents of dierent orbits. We nd a class of orbits that are chaotic for physical spin values. We also nd new lower bounds on the spin required for more general orbits to be chaotic.

Abstract

A short overview of string phenomenology is presented which motivates the search of the free-fermionic heterotic string models. The process of generating these models is discussed along with the use of group theory in describing the Standard Model. All possible order 2 and 4 boundary vectors (BV’s) are generated with the constraint that they can be added to the NAHE set and still form a consistent MSSM string model. Combining the BV’s with the NAHE set results in new string models which will be analyzed in group theoretic terms to classify them according to their individual Standard Models.

Abstract

The problem of modeling rotating astronomical objects is one central to astrophysics. Because most, if not all, astronomical bodies are rotating, an understanding of this problem has universal application. We will be considering two rotating astronomical systems. The first system we will consider will be a rotating galaxy. We will use the van Stockum metric [1] in order to find an equation for the tangential velocity of a galaxy. There are problems associated with the van Stockum metric including the occurrence of closed timelike curves in the presence of matter. These problems prevent the van Stockum metric from being used to describe a physical system. The second system that we will consider will be a rotating neutron star, which is the type of star that creates a pulsar. Following the work of Cook, Shapiro and Teukolsky (CST) [2] we will discuss the methods used and derive the Green’s function that can be used to find the coefficients of the metric. We will also derive the source terms using the Einstein Equations. Then we will consider an extension to the problem done by CST by relaxing the assumption of circularity. We will create the corresponding Einstein tensors and other associated tensors.

Abstract

We investigate spherically-symmetric gravitational collapse in the presence of a single “large” extra dimension through the use of analytical and numerical techniques. This has bearing on higher-dimensional ideas concerning hypothetical objects called “black strings,” or black holes extending into an extra circular dimension, which dimension we herein label ζ. These putative objects were first seriously considered as elements of string theory but have relevance in simpler, higher-dimensional generalization of Einstein’s general relativity. We assume a universe of topology M2 × S 2 × S 1 (where M2 is a two-dimensional Lorentzian manifold; S 2 is the sphere; and S 1 is the circle). We model the formation of a uniform black string via two modes—the collapse of a massless scalar field, and of pure gravitational waves consisting of (gaussian) distortions in the extra dimension. We report on and discuss two aspects of the nonlinear dynamics, viz., that in five dimensions larger-amplitude fields appear to collapse more slowly than their loweramplitude cousins; and that ζ-wave collapse exhibits signs of self-similarity at the threshold between black string formation and dispersal of the collapsing fields.

Abstract

A numerical approach to including electromagnetism with general relativity is developed using GRAXI [1] as a starting point. We develop a mathematical model describing electromagnetism coupled to a scalar field in an evolving axisymmetric spacetime. As there are numerous formulations of electromagnetism, we evalute different formulations in a limited flat space case. The full curved space system is then developed, using the flat case as a guide to implementing electromagnetism. This model is then implemented using GRAXI as a code base.

Robert Steven Millward (PhD Dissertation, April 2004,
Advisor: Eric Hirschmann
)

Abstract

This dissertation is a description of a variety of methods of solving the Einstein equations describing the gravitational interaction in different, mathematical and astrophysical settings. We begin by discussing a numerical study of the Einstein-Yang-Mills-Higgs system in spherical symmetry. The equations are resented along with boundary and intimal conditions. An explanation of the numerical scheme is then given. This is followed by a discussion of the solutions obtained together with an interpretation in the context of gravitational collapse and critical phenomena at the threshold of black hole formation. Following this, we generalize the same system to axisymmetric. The full, gravitational equations are presented along with a short discussion of the problems we encountered are presented along with a short discussion of the problems we encountered in trying to solve these. As a first step we considered evolving the matter fields in flat space. The simplified equations are given and the numerical scheme implemented to solve them discussed. We then consider some analytical techniques to understanding the Einstein equations and the gravitating systems they should describe. One such is to change the space-time dimension. This we do in considering magnetic solutions to the (2 + 1) Einstein-Maxwell-Dilaton system with nonzero cosmological constant. The solutions are investigated to determine whether these correspond to “solution” – like solutions or black holes. As another example of this general approach, we introduce an extra time-like coordinate into the spherically symmetric vacuum system, and attempt to find a solution comparing the result to the more well-known Schwarzschild solution. Finally, we give a short description of some preliminary work which will combine some of these numerical and analytical techniques. This approach simply takes the matter fields as weak and propagates them on a fixed space-time background. In our particular case, we intend to study the evolution of Maxwell fields in the Schwarzschild geometry. We provide motivation for this as well as present the equations describing the system.

Jay M. Call (Honors Thesis, June 2003,
Advisor: Eric Hirschmann
)

Abstract

Eric Baird (Senior Thesis, July 2002,
Advisor: Eric Hirschmann
)

Abstract