The Mathematical Analysis of Extremal Black Holes and Gravitational Radiation

Introduction

The Einstein equations constitute a system of geometric, nonlinear partial differential equations that describe gravitational dynamics in the framework of Einstein's theory of general relativity. The last decade has seen tremendous progress towards understanding dynamical aspects of the Einstein equations.

At the mathematical level, great insight has been gained due to recent advances in the study of partial differential equations, differential geometry, and microlocal analysis. The present proposal builds upon these advances in the context of the following two mathematical problems.

Stability and Instability of Extremal Black Holes

Extremal Kerr black holes describe rapidly rotating solutions to the Einstein equations. They sit at the transition between black holes and "naked singularities" and exhibit critical geometric features.

This proposal addresses the stability and instability properties of extremal Kerr black holes and is motivated by recent advances by the PI, which cover linear and nonlinear aspects. A successful resolution would give fundamental, new insights into the fate of perturbed extremal black holes and the transition between black holes and naked singularities.

Late-Time Analysis of Gravitational Radiation

Gravitational radiation provides an observational window into deep mathematical aspects of general relativity. In this proposal, we investigate a key feature that is amenable to mathematical analysis: the existence of late-time tails in gravitational radiation.

Recent work by the PI and collaborators has led to the first proof of the existence of late-time tails in a toy model setting, also known as Price's Law. This proposal considers the full setting of the nonlinear Einstein equations via the analysis of late-time tails in the dynamics of perturbations of both flat spacetime and black hole spacetimes. A successful resolution would have important implications for the Strong Cosmic Censorship conjecture.