Type
Text
Type
Dissertation
Advisor
Anderson, Michael | Khuri, Marcus A. | Chen, Xiu-Xiong | Wang, Mu-Tao.
Date
2015-08-01
Keywords
Mathematics
Department
Department of Mathematics.
Language
en_US
Source
This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree.
Identifier
http://hdl.handle.net/11401/76389
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
Abstract
In this dissertation, we study geometric inequalities for black holes, mainly the angular momentum-mass inequality and the angular momentum-mass-charge inequality. Firstly, we show how to reduce the general formulation of the angular momentum-mass inequality, for (non-maximal) axially symmetric initial data of the Einstein equations, to the known maximal case. This procedure is based on a certain deformation of the initial data which preserves the relevant geometry, while achieving the maximal condition. More importantly, we compute the scalar curvature formula for the deformation of initial data, which shows that the dominant energy condition holds in a weak sense. Through this procedure, we develop a geometrically motivated system of quasi-linear elliptic equations which is conjectured to admit a solution. The primary equation bears a strong resemblance to the Jang-type equations studied in the context of the positive mass theorem and the Penrose inequality. Secondly, in a similar sense, we show how to reduce the general formulation of the angular momentum-mass-charge inequality, for (non-maximal) axially symmetric initial data of the Einstein-Maxwell equations with zero magnetic field, to the known maximal case, whenever there exists a solution for the system of quasi-linear elliptic equations. Lastly, we combine these two results and the area-angular momentum inequality to show the lower bound of the area in terms of ADM mass, angular momentum, and charge for black holes under the same assumptions. | 112 pages
Recommended Citation
Cha, Ye Sle, "Deformations of Axially Symmetric Initial Data and the Angular Momentum-Mass Inequality" (2015). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 2312.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/2312