Date of Award
2014
Document Type
Dissertation
Publication Status
Version of Record
Submission Date
October 2026
Thesis/Dissertation Advisor [Chair]
Beetle, Christopher
Abstract
In general relativity, angular momentum of the gravitational field in some volume bounded by an axially symmetric sphere is well-defined 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 second-order 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.
Extent
87 p.
Recommended Citation
Wilder, Shawn M., "General relativistic quasi-local angular momentum continuity and the stability of strongly elliptic eigenvalue problems" (2014). Electronic Theses and Dissertations 2010 - 2019. 1017.
https://digitalcommons.fau.edu/etd_general_6/1017
Identifier
FA00004235
Comments
Includes bibliography.|Dissertation (Ph.D.)--Florida Atlantic University, 2014.|FAU Electronic Theses and Dissertations Collection|Includes bibliography.