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Annals of PDE

, 4:12 | Cite as

Stability of Minkowski Space-Time with a Translation Space-Like Killing Field

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Abstract

In this paper we prove the nonlinear stability of Minkowski space-time with a translation Killing field. In the presence of such a symmetry, the \(3+1\) vacuum Einstein equations reduce to the \(2+1\) Einstein equations with a scalar field. We work in generalised wave coordinates. In this gauge Einstein’s equations can be written as a system of quasilinear quadratic wave equations. The main difficulty in this paper is due to the decay in \(\frac{1}{\sqrt{t}}\) of free solutions to the wave equation in 2 dimensions, which is weaker than in 3 dimensions. This weak decay seems to be an obstruction for proving a stability result in the usual wave coordinates. In this paper we construct a suitable generalized wave gauge in which our system has a “cubic weak null structure”, which allows for the proof of global existence.

Keywords

Einstein equations Nonlinear stability Wave coordinates 2 dimensional wave equations 

Notes

Acknowledgements

This paper has benefit from the insight of many people. The author would like to thank in particular Jérémie Szeftel, Qian Wang, Spyros Alexakis, Mihalis Dafermos and Igor Rodnianski for the interesting conversations.

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CMLSEcole PolytechniquePalaiseauFrance

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