The causal structure of microlocalized Einstein metrics

TL;DR

This paper develops geometric analysis of the Eikonal equation for microlocalized rough Einstein metrics, advancing the understanding of very rough solutions to Einstein's vacuum equations in wave coordinates.

Contribution

It introduces a novel geometric analysis framework for microlocalized rough Einstein metrics, crucial for decay estimates in rough solution theory.

Findings

Established geometric analysis techniques for microlocalized Einstein metrics.

Derived decay estimates essential for rough solution existence.

Enhanced understanding of the structure of very rough Einstein solutions.

Abstract

This is the second in a series of three papers in which we initiate the study of very rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and Sobolev inequalities. In this paper we develop the geometric analysis of the Eikinal equation for microlocalized rough Einstein metrics. This is a crucial step in the derivation of the decay estimates needed in our first paper.