On the Local Existence for the Characteristic Initial Value Problem in General Relativity

TL;DR

This paper extends Rendall's theorem by proving local existence of solutions to the vacuum Einstein equations in a neighborhood of intersecting null cones, emphasizing the importance of initial constraint satisfaction and null structure.

Contribution

It demonstrates that solutions exist in a neighborhood of the cones if initial constraints are satisfied, utilizing energy estimates and null structure techniques.

Findings

Solutions exist in a neighborhood of null cones when initial constraints are satisfied.

Energy estimates and null structure are crucial for proving local existence.

The approach extends previous results to a broader setting.

Abstract

Given a truncated incoming null cone and a truncated outgoing null cone intersecting at a two sphere $S$ with smooth characteristic initial data, a theorem of Rendall shows that the vacuum Einstein equations can be solved in a small neighborhood of $S$ in the future of $S$. We show that in fact the vacuum Einstein equations can be solved in a neighborhood in the future of the cones, as long as the constraint equations are initially satisfied on the null cones. The proof is based on energy type estimates and relies heavily on the null structure of the Einstein equations in the double null foliation.