The Stability of the Spacetime Penrose Inequality in Spherical Symmetry

This dissertation explores the stability of the Spacetime Penrose Inequality within the framework of general relativity, focusing on spherically symmetric, asymptotically flat initial data that adheres to the dominant energy condition. The research investigates the scenario where the ADM mass is closely aligned with the half area radius of the outermost apparent horizon. Employing the generalized Jang equation approach, this work demonstrates that under these conditions, the initial data must closely resemble an isometric embedding into a static spacetime. Specifically, the time-slice is shown to be near a Schwarzschild time-slice in terms of volume-preserving intrinsic flat distance, static potentials are close in L², and the initial data's extrinsic curvature approximates the second fundamental form of the embedding in L². The Penrose inequality itself posits that the mass of a universe model is bounded below by a function of the area of its boundary black holes. This inequality is crucial in general relativity for understanding the cosmic censorship conjecture, which suggests that singularities (points where spacetime is ill-defined) are hidden behind black holes. If the inequality were false, it could imply unguarded singularities, violating this conjecture. The rigidity aspect of the inequality further suggests that equality implies the universe is a time-slice of Schwarzschild spacetime, the simplest black hole model.

Curriculum

The dissertation is structured into three main chapters, preceded by acknowledgements and an introduction. The chapters cover the formalization of the problem, the setup and background theories, and the presentation of the stability result.

1. Chapter 1: Introduction
   • Formalization of the Penrose Inequality
   • History and Motivation of the Penrose Inequality and Cosmic Censorship
2. Chapter 2: Setup and Background
   • The Jang Equation for Inequality and Equality
   • Intrinsic Flat Distance for Stability Analysis
3. Chapter 3: Stability Result
   • Problem Setup and Preliminaries
   • Convergence of Warping Factor and Consequences
   • Application of VADB with Boundary (Volumes, Areas, Diameters, Metric Estimate, Volume of Bad Region, Distance Estimate)
   • Proof of Theorem

Careers

While this dissertation focuses on theoretical research in mathematics and general relativity, the skills developed are applicable to various advanced research and academic positions.

• Researcher in theoretical physics or mathematics
• Academic positions (Professor, Postdoctoral Researcher)
• Data Scientist
• Cryptographer