On the Well-Posedness of the Vacuum Einstein's Equations

TL;DR

This paper proves the well-posedness of the vacuum Einstein equations in a specific Sobolev space framework under harmonic gauge, resolving previous incompatibilities between initial data and evolution equations.

Contribution

It establishes the well-posedness of Einstein's vacuum equations in a unified Sobolev space setting, addressing prior limitations in initial data compatibility.

Findings

Proves well-posedness of vacuum Einstein equations in Sobolev spaces.

Resolves the incompatibility between initial data and evolution equations.

Provides a rigorous mathematical foundation for Einstein's equations under harmonic gauge.

Abstract

The Cauchy problem of the vacuum Einstein's equations aims to find a semi-metric $g_{\alpha\beta}$ of a spacetime with vanishing Ricci curvature $R_{\alpha,\beta}$ and prescribed initial data. Under the harmonic gauge condition, the equations $R_{\alpha,\beta}=0$ are transferred into a system of quasi-linear wave equations which are called the reduced Einstein equations. The initial data for Einstein's equations are a proper Riemannian metric $h_{ab}$ and a second fundamental form $K_{ab}$. A necessary condition for the reduced Einstein equation to satisfy the vacuum equations is that the initial data satisfy Einstein constraint equations. Hence the data $(h_{ab},K_{ab})$ cannot serve as initial data for the reduced Einstein equations. Previous results in the case of asymptotically flat spacetimes provide a solution to the constraint equations in one type of Sobolev spaces, while initial data for the evolution equations belong to a different type of Sobolev spaces. The goal of our work is to resolve this incompatibility and to show that under the harmonic gauge the vacuum Einstein equations are well-posed in one type of Sobolev spaces.