A sharp counterexample to local existence of low regularity solutions to Einstein's equations in wave coordinates

TL;DR

This paper presents a counterexample demonstrating that initial data with Sobolev regularity s=2 do not guarantee local existence of solutions to Einstein's equations in wave coordinates, challenging previous assumptions.

Contribution

The authors construct a specific example of low regularity initial data in Sobolev space H^2 that fails to produce a local solution, highlighting limitations in existing well-posedness results.

Findings

Counterexample in Sobolev space H^2 showing no local solution exists

Challenges the assumption that s>2 is necessary for local existence

Highlights the sharpness of regularity thresholds for Einstein's equations

Abstract

We are concerned with how regular initial data have to be to ensure local existence for Einstein's equations in wave coordinates. Klainerman-Rodnianski and Smith-Tataru showed that there in general is local existence for data in Sobolev spaces H^s with regularity s>2. We give an example of data in Sobolev spaces with regularity s=2 for which there is no local solution in this space.