Global existence for the Einstein vacuum equations in wave coordinates
In this paper the authors revisit the problem of global stability of Minkowski space for the Einstein vacuum equations in harmonic coordinates. More precisely, for the set of restricted data coinciding with the Schwarzschild solution in the neighborhood of space-like infinity, they prove that the harmonic coordinates are stable in the large: the reduced Einstein equations have a global solution \(g\) defining a future causally geodesically complete space-time. The metric \(g\) in wave coordinates \(x\) approaches the Minkowski metric \(m\): sup\(_{x\in \mathbb{R}^3}| g(t,x)-m|\to 0\) as \(t\to\infty\). The paper provides an alternative approach to the problem of global stability of Minkowski space for the Einstein vacuum equations originally solved by \textit{D. Christodoulou} and \textit{S. Klainerman} [The global nonlinear stability of the Minkowski space, Princeton Univ. Press (Princeton, NJ) (1993; Zbl 0827.53055)]. Though being less precise about the asymptotic behavior of the solution, this work is focused more on developing a technically relatively simple approach to the proof of stability of Minkowski space in a physically interesting wave coordinate gauge. In order to establish their result, the authors use the crucial fact that the Einstein vacuum equations in harmonic coordinates fulfil the weak null condition, a notion introduced by the authors for general quasilinear systems in a previous paper. This observation is combined with the vector field method based on the symmetries of the standard Minkowski space.
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