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arXiv · 2109.03270

Global stability of Minkowski space for the Einstein-Maxwell-Klein-Gordon system in generalized wave coordinates

Christopher Kauffman·Hans Lindblad
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Abstract

We prove global existence for Einstein's equations with a charged scalar field for initial conditions sufficiently close to the Minkowski spacetime without matter. The proof relies on generalized wave coordinates adapted to the outgoing Schwarzschild light cones and the estimates for the massless Maxwell-Klein-Gordon system, on the background of metrics asymptotically approaching Schwarzschild at null infinity in such coordinates, by Kauffman. The generalized wave coordinates are obtained from a change of variables, introduced by Lindblad, to asymptotically Schwarzschild coordinates at null infinity. The main technical advances are that the change of coordinates makes critical components of the metric decay faster, making the quasilinear wave operator closer to the flat wave operator, and that commuting with modified Lie derivatives preserves the geometric null structure, improving the error terms. This improved decay of the metric is essential for proving our stability result, and will likely be useful in other contexts as well.

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Christopher Kauffman, Hans Lindblad. 2021-09-07. Global stability of Minkowski space for the Einstein-Maxwell-Klein-Gordon system in generalized wave coordinates. https://arxiv.org/abs/2109.03270

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