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

TL;DR

This paper analyzes the long-term behavior of solutions to Einstein's vacuum equations in wave coordinates, revealing wave-like asymptotics at null infinity and homogeneity at timelike infinity, with implications for gravitational radiation and black hole models.

Contribution

It provides new asymptotic descriptions of Einstein vacuum solutions in wave coordinates with small data, linking null infinity behavior to Schwarzschild geometry and energy conservation.

Findings

Solutions exhibit wave-like behavior at null infinity.

Outgoing null hypersurfaces approach Schwarzschild solutions.

Radiated energy equals initial mass.

Abstract

We give asymptotics for Einstein vacuum equations in wave coordinates with small asymptotically flat data. We show that the behavior is wave like at null infinity and homogeneous towards time like infinity. We use the asymptotics to show that the outgoing null hypersurfaces approach the Schwarzschild ones for the same mass and that the radiated energy is equal to the initial mass.