Geometric Analysis Problems in Mathematical Relativity

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

The purpose of this project is to study several problems in general relativity, using the tools and techniques of differential geometry and mathematical analysis. General relativity is a geometric theory of gravity proposed by A. Einstein over a century ago. It is fundamental to our understanding of the large scale structure of the universe, and has many practical applications such as to the fine tuning of global positioning system (GPS) technology. Within this theory, singularities are known to generically form in the evolution of spacetime. A central conjecture due to R. Penrose, which is tied to the predictive power of general relativity, asserts that these singularities must be hidden behind the horizons of black holes. One of the main objectives of this project is to investigate this conjecture, known as weak cosmic censorship, by establishing a range of geometric inequalities relating total mass of a spacetime to properties of the black holes it contains. This project will also involve a significant impact in the educational arena at all levels, through the training of at least five Ph.D. students under the PI's supervision to the development of STEM talent in high schools by mentoring local students' research projects and science competition entries. The PI will develop ideas surrounding a new proof of the positive mass theorem based on level sets of spacetime harmonic functions. This new quantitative approach suggests methods to complete the stability program, and establish more general mass lower bounds. With collaborators, the PI will continue investigations of the harmonic maps with prescribed singularities which naturally arise from the stationary Einstein equations under symmetry conditions, and use this theory to find new black hole solutions with exotic topologies. This work also suggests a novel method to construct Riemannian Einstein metrics, as well as a possible classification for stationary axisymmetric black holes. Recently, together with Alaee and Yau the PI has introduced a new quasi-local mass with advantageous attributes, and will fully explore its properties with the goal of applying it to the trapped surface/hoop conjecture associated with gravitational collapse, as well as the proposed Bekenstein bounds concerning the entropy/information within a relativistic body which has implications from thermodynamics to computer science. Additionally, this project will advance new interactions, and encourage interdisciplinary efforts between pure mathematics and physics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.