arXiv subjects

Alan R. Parry

Publications and source records attributed to Alan R. Parry.

6 recordsLinked to original sources

arXiv2020

Parametrizations of the Poisson-Schrödinger Equations in Spherical Symmetry

We consider the asymptotically flat standing wave solutions to the Poisson-Schrödinger system of equations known as static states. These solutions can be parameterized using a variety of choices of two continuous parameters and one discrete parameter, each having a useful physical-geometrical interpretation. The values of the discrete variable determines the number of nodes (zeros) in the solution. We use numerical inversion techniques to analyze transformations between various informative choices of parametrization by relating each of them to a standard set of three parameters. Based on our computations, we propose explicit formulas for these relationships. Our computations also show that for the standard choice of continuous variables, the zero-node ground state yields a minimum value of a geometrically natural discrete variable. We give an explicit formula for this minimum value. We use these results to confirm two related observations from previous work by the author and others, and suggest additional applications and approaches to understand these phenomena analytically.

math-ph↗

arXiv2015

Spherically Symmetric Static States of Wave Dark Matter

In this paper, we present two observations about static spherically symmetric solutions of the Einstein-Klein-Gordon equations. The first is a comment extending the well-known result of the existence of static states (i.e. standing wave solutions) of the Einstein-Klein-Gordon equations. The second more important observation shows that, in the low field limit, the mass profiles of these static states lie along hyperbolas of constant $Υ$, the fundamental constant of the Einstein-Klein-Gordon equations.

gr-qc↗

arXiv2014

A Survey of Spherically Symmetric Spacetimes

We survey many of the important properties of spherically symmetric spacetimes as follows. We present several different ways of describing a spherically symmetric spacetime and the resulting metrics. We then focus our discussion on an especially useful form of the metric of a spherically symmetric spacetime in polar-areal coordinates and its properties. In particular, we show how the metric component functions chosen are extremely compatible with notions in Newtonian mechanics. We also show the monotonicity of the Hawking mass in these coordinates. As an example, we discuss how these coordinates and the metric can be used to solve the spherically symmetric Einstein-Klein-Gordon equations. We conclude with a brief mention of some applications of these properties.

gr-qc↗

arXiv2013

A Classification of Real Indecomposable Solvable Lie Algebras of Small Dimension with Codimension One Nilradicals

This thesis was concerned with classifying the real indecomposable solvable Lie algebras with codimension one nilradicals of dimensions two through seven. This thesis was organized into three chapters. In the first, we described the necessary concepts and definitions about Lie algebras as well as a few helpful theorems that are necessary to understand the project. We also reviewed many concepts from linear algebra that are essential to the research. The second chapter was occupied with a description of how we went about classifying the Lie algebras. In particular, it outlined the basic premise of the classification: that we can use the automorphisms of the nilradical of the Lie algebra to find a basis with the simplest structure equations possible. In addition, it outlined a few other methods that also helped find this basis. Finally, this chapter included a discussion of the canonical forms of certain types of matrices that arose in the project. The third chapter presented a sample of the classification of the seven dimensional Lie algebras. In it, we proceeded step-by-step through the classification of the Lie algebras whose nilradical was one of four specifically chosen because they were representative of the different types that arose during the project. In the appendices, we presented our results in a list of the multiplication tables of the isomorphism classes found.

math.DG↗

arXiv2013

Wave Dark Matter and Dwarf Spheroidal Galaxies

We explore a model of dark matter called wave dark matter (also known as scalar field dark matter and boson stars) which has recently been motivated by a new geometric perspective by Bray. Wave dark matter describes dark matter as a scalar field which satisfies the Einstein-Klein-Gordon equations. These equations rely on a fundamental constant $Υ$ (also known as the "mass term" of the Klein-Gordon equation). Specifically, in this dissertation, we study spherically symmetric wave dark matter and compare these results with observations of dwarf spheroidal galaxies as a first attempt to compare the implications of the theory of wave dark matter with actual observations of dark matter. This includes finding a first estimate of the fundamental constant $Υ$. The majority of this thesis has also been presented by the author in three separate shorter papers with arXiv reference codes [arXiv:1210.5269 [gr-qc]], [arXiv:1212.6426 [gr-qc]], [arXiv:1301.0255 [astro-ph.GA]]; note that [arXiv:1301.0255 [astro-ph.GA]], and hence part of Chapter 4, represents joint work with Hubert Bray.

gr-qc↗

arXiv2013

Modeling Wave Dark Matter in Dwarf Spheroidal Galaxies

This paper studies a model of dark matter called wave dark matter (also known as scalar field dark matter and boson stars) which has recently also been motivated by a new geometric perspective by Bray [arXiv:1212.5745]. Wave dark matter describes dark matter as a scalar field which satisfies the Einstein-Klein-Gordon equations. These equations rely on a fundamental constant Upsilon (also known as the "mass term" of the Klein-Gordon equation). In this work, we compare the wave dark matter model to observations to obtain a working value of Upsilon. Specifically, we compare the mass profiles of spherically symmetric static states of wave dark matter to the Burkert mass profiles that have been shown by Salucci et al. [arXiv:1111.1165] to predict well the velocity dispersion profiles of the eight classical dwarf spheroidal galaxies. We show that a reasonable working value for the fundamental constant in the wave dark matter model is Upsilon = 50 yr^{-1}. We also show that under precise assumptions the value of Upsilon can be bounded above by 1000 yr^{-1}.

astro-ph.GA↗