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General relativistic quasilocal angular momentum continuity and the stability of strongly elliptic eigenvalue problems
 Date Issued:
 2014
 Summary:
 In general relativity, angular momentum of the gravitational field in some volume bounded by an axially symmetric sphere is welldefined as a boundary integral. The definition relies on the symmetry generating vector field, a Killing field, of the boundary. When no such symmetry exists, one defines angular momentum using an approximate Killing field. Contained in the literature are various approximations that capture certain properties of metric preserving vector fields. We explore the continuity of an angular momentum definition that employs an approximate Killing field that is an eigenvector of a particular secondorder differential operator. We find that the eigenvector varies continuously in Hilbert space under smooth perturbations of a smooth boundary geometry. Furthermore, we find that not only is the approximate Killing field continuous but that the eigenvalue problem which defines it is stable in the sense that all of its eigenvalues and eigenvectors are continuous in Hilbert space. We conclude that the stability follows because the eigenvalue problem is strongly elliptic. Additionally, we provide a practical introduction to the mathematical theory of strongly elliptic operators and generalize the above stability results for a large class of such operators.
Title:  General relativistic quasilocal angular momentum continuity and the stability of strongly elliptic eigenvalue problems. 
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Name(s): 
Wilder, Shawn M., author Beetle, Christopher, Thesis advisor Florida Atlantic University, Degree grantor Charles E. Schmidt College of Science Department of Physics 

Type of Resource:  text  
Genre:  Electronic Thesis Or Dissertation  
Date Created:  2014  
Date Issued:  2014  
Publisher:  Florida Atlantic University  
Place of Publication:  Boca Raton, Fla.  
Physical Form:  application/pdf  
Extent:  87 p.  
Language(s):  English  
Summary:  In general relativity, angular momentum of the gravitational field in some volume bounded by an axially symmetric sphere is welldefined as a boundary integral. The definition relies on the symmetry generating vector field, a Killing field, of the boundary. When no such symmetry exists, one defines angular momentum using an approximate Killing field. Contained in the literature are various approximations that capture certain properties of metric preserving vector fields. We explore the continuity of an angular momentum definition that employs an approximate Killing field that is an eigenvector of a particular secondorder differential operator. We find that the eigenvector varies continuously in Hilbert space under smooth perturbations of a smooth boundary geometry. Furthermore, we find that not only is the approximate Killing field continuous but that the eigenvalue problem which defines it is stable in the sense that all of its eigenvalues and eigenvectors are continuous in Hilbert space. We conclude that the stability follows because the eigenvalue problem is strongly elliptic. Additionally, we provide a practical introduction to the mathematical theory of strongly elliptic operators and generalize the above stability results for a large class of such operators.  
Identifier:  FA00004235 (IID)  
Degree granted:  Dissertation (Ph.D.)Florida Atlantic University, 2014.  
Collection:  FAU Electronic Theses and Dissertations Collection  
Note(s):  Includes bibliography.  
Subject(s): 
Boundary element methods Boundary value problems Differential equations, Elliptic  Numerical solutions Differential equations, Partial  Numerical solutions Eigenvalues Spectral theory (Mathematics) 

Held by:  Florida Atlantic University Libraries  
Sublocation:  Digital Library  
Persistent Link to This Record:  http://purl.flvc.org/fau/fd/FA00004235  
Use and Reproduction:  Copyright © is held by the author, with permission granted to Florida Atlantic University to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.  
Use and Reproduction:  http://rightsstatements.org/vocab/InC/1.0/  
Host Institution:  FAU  
Is Part of Series:  Florida Atlantic University Digital Library Collections. 