Time-symmetric integration of partial differential equations with applications to black hole physics

The detection of gravitational waves from merging binary black holes has heralded a new era of astronomy: gravitational radiation has joined electromagnetic radiation and elementary particles as a channel through which we may observe the universe around us. This has created the need for accurate, robust models and calculations describing the predictions that general relativity and other theories make for a whole new class of exotic, complicated phenomena. One such phenomenon is the ``extreme mass ratio inspiral,'' wherein a star or stellar-mass black hole plunges into a supermassive black hole found in a galactic nucleus and emits gravitational waves. These events present the possibility to learn about the objects found under such conditions, which would reveal information about galaxy formation and stellar evolution and probe the validity of general relativity in these regions' strong gravitational fields. Such phenomena are incredibly complicated, with only the most trivial problems possessing closed-form solutions. The aim of this work is to present a series of tools to facilitate the numerical study of these phenomena. First, we will emphasize the need for numerical methods which accurately track the physics of the problem: i.e. respect the qualitative features of the evolution equations--symmetry under time-reversal and symplecticity--and accurately track conserved quantities of interest in gravitational wave astronomy: energy and angular momentum. Next, we will discuss the problem of representing a distributional source term, i.e. a point mass, on a discrete grid, and consider the effect this has on discrete operations like numerical differentiation and integration. After providing a brief overview of the relevant aspects of black hole theory, we will discuss how to extract gravitational waves at infinite distances from the black hole using numerical methods. Finally, we will demonstrate the methods' utility in working with problems of black hole theory, showing that conservative, geometric evolution is possible. Second, we will discuss a problem related to the astrophysics of neutron stars. A current goal of gravitational wave astronomy is the constraint of the neutron star thermodynamic equation of state, which will inform the fundamental theory of nuclear interactions. It is necessary to parametrize the equation of state to compare to observation. We present an extension of the popular piecewise polytrope formalism that allows for a continuous sound speed in the nuclear matter. We found that in addition to capturing the thermodynamics of many microscopic models, it more accurately predicts the astrophysically observable properties of neutron stars -- mass, radius, and tidal deformability -- when compared to the original model.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Physics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• O'Boyle, Michael F.

Contributors dc:contributor

• Gammie, Charles F
• Markakis, Charalampos M
• Ricker, Paul M
• Witek, Helvi

• Geometric integrators
• numerical analysis
• black hole perturbation theory
• hyperbolic partial differential equations
• hyperboloidal compactification
• gravitational waves
• neutron star equation of state

Statement dc:rights

• Copyright 2022 Michael F O'Boyle

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/116185

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F116185