On the well-posedness of the vacuum Einstein’s equations

Abstract

The Cauchy problem of the vacuum Einstein’s equations aims to find a semi-metric g αβ of a spacetime with vanishing Ricci curvature R α,β and prescribed initial data. Under the harmonic gauge condition, the equations R α,β = 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 the present article 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.