On the asymptotic behavior of solutions to the Einstein vacuum equations in wave coordinates
type: Item
instance of: scholarly article
title: On the asymptotic behavior of solutions to the Einstein vacuum equations in wave coordinates
zbMATH Open document ID: 1373.35308
author: Hans Lindblad
published in: Communications in Mathematical Physics
publication date: 2017-06-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1606.01591
review text: The Einstein vacuum equations in wave coordinates become a system of nonlinear wave equations for the metric, called the reduced Einstein equations. The initial data for the reduced Einstein equations are assumed to be asymptotically flat. \textit{Y. Fourès-Bruhat} [Acta Math. 88, 141--225 (1952; Zbl 0049.19201)] proved local existence for Einstein's equations in wave coordinates. \textit{D. Christodoulou} and \textit{S. Klainerman} [The global nonlinear stability of the Minkowski space. Princeton, NJ: Princeton University Press (1993; Zbl 0827.53055)] proved global existence of small solutions in a coordinate invariant way. It was assumed that the wave coordinates behaved badly for large times. The author and \textit{I. Rodnianski} [Ann. Math. (2) 171, No. 3, 1401--1477 (2010; Zbl 1192.53066)] proved global existence in wave coordinates. In the present paper, the authors studies the precise asymptotic behaviour of solutions to the reduced Einstein equations that are small perturbations of the Minkowski metric. The author shows that the behaviour is wave like at null infinity and homogeneous towards time like infinity. He uses the asymptotics to show that the outgoing null hypersurfaces approach the Schwarzschild hypersurfaces for the same mass and that the radiated energy is equal to the initial mass.
Mathematics Subject Classification ID: 35Q76
Mathematics Subject Classification ID: 83C05
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 6728519
zbMATH Keywords: Einstein vacuum equations
zbMATH Keywords: wave coordinates
zbMATH Keywords: small data
zbMATH Keywords: Schwarzschild hypersurfaces
reviewed by: Dmitry Pelinovsky
MaRDI profile type: Publication
OpenAlex ID: W3101983577
arXiv ID: 1606.01591
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DOI: 10.1007/S00220-017-2876-Z
description: scientific article