On the asymptotic behavior of solutions to the Einstein vacuum equations in wave coordinates
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.
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