Asymptotic description of the formation of black holes from short-pulse data

TL;DR

This thesis advances understanding of black hole formation from short-pulse data by proposing a desingularized framework and analyzing curvature blowup in Einstein vacuum solutions.

Contribution

It introduces a new geometric approach using blowup techniques and provides formal solutions and curvature analysis for short-pulse initial data in Einstein's equations.

Findings

Existence of formal solutions on a desingularized manifold

Identification of curvature blowup at a hypersurface

Development of a short-pulse tangent bundle framework

Abstract

In this thesis we present partial progress towards the dynamic formation of black holes in the four-dimensional Einstein vacuum equations from Christodoulou's short-pulse ansatz. We identify natural scaling in a putative solution metric and use the technique of real blowup to propose a desingularized manifold and an associated rescaled tangent bundle (which we call the "short-pulse tangent bundle") on which the putative solution remains regular. We prove the existence of a solution solving the vacuum Einstein equations formally at each boundary face of the blown-up manifold and show that for an open set of restricted short-pulse data, the formal solution exhibits curvature blowup at a hypersurface in one of the boundary hypersurfaces of the desingularized manifold. This thesis is intended to be partially expository. In particular, this thesis presents an exposition of double-null gauges and the solution of the characteristic initial value problem for the Einstein equations, as well as an exposition of a new perspective of Christodoulou's monumental result on the dynamic formation of trapped surfaces.