Topics in Numerical Relativity: The Periodic Standing-Wave Approximation, the Stability of Constraints in Free Evolution, and the Spin of Dynamical Black Holes

Author: Owen, Robert Philip

Year: 2007

Degree: Dissertation (Ph.D.)

Advisors: Lindblom, Lee A.; Price, Richard H.; Thorne, Kip S.

Committee Members: Lindblom, Lee A.; Cutler, Curt J.; Libbrecht, Kenneth George; Thorne, Kip S.

Option: Physics

DOI: 10.7907/464A-4Y76

Abstract

This thesis concerns numerical relativity, the attempt to study Einstein's theory of gravitation using numerical discretization. The goal of the field, the study of gravitational dynamics in cases where symmetry reduction or perturbation theory are not possible, finally seems to be coming to fruition, at least for the archetypal problem of the inspiral and coalescence of binary black hole systems. This thesis presents three episodes that each bear some relationship to this story.

Chapters 2 and 3 present previously published work in collaboration with Richard Price and others on the so-called periodic standing-wave (PSW) approximation for binary inspiral. The approximation is to balance outgoing radiation with incoming radiation, stabilizing the orbit and making the problem stationary in a rotating frame. Chapters 2 and 3 apply the method to the problem of co-orbiting charges coupled to a nonlinear scalar field in three dimensions.

Chapters 4, 5, and 6 concern the stability of constraint fields in conventional numerical relativity simulations. Chapter 4 (also previously published work, in collaboration with the Caltech numerical relativity group, along with Michael Holst and Lawrence Kidder) presents a method for immediately correcting violations of constraints after they have arisen. Chapters 5 and 6 present methods to "damp" away constraint violations dynamically in two specific contexts. Chapter 5 (previously published work in collaboration with the Caltech numerical relativity group and Lawrence Kidder) presents a first-order linearly degenerate symmetric hyperbolic representation of Einstein's equations in generalized harmonic gauge. A representation is presented that stabilizes all constraints, including those that appear when the system is written in first-order form. Chapter 6 presents a generalization of the Kidder-Scheel-Teukolsky evolution systems that provides much-improved stability. This is investigated with numerical simulations of a single black hole spacetime.

Finally, chapter 7 presents work in progress to implement code to calculate the spin of black holes in numerical simulations. This requires a well-defined generalization of the concept of "rotation generators" on topological two-spheres that may not have any true Killing vectors. I present a new method for defining these fields, and results of a numerical code that computes them.

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