Weakly decaying asymptotically flat static and stationary solutions to the Einstein equations

TL;DR

This paper demonstrates that weak initial conditions on static or stationary solutions to Einstein's equations inevitably lead to the standard Schwarzschildian decay and analyticity near infinity, reinforcing the rigidity of such solutions.

Contribution

It proves that minimal differentiability and decay assumptions suffice to ensure solutions are analytic and exhibit Schwarzschild-like falloff at infinity.

Findings

Solutions are analytic near infinity.

Solutions exhibit standard Schwarzschildian falloff.

Weak initial conditions are sufficient for strong asymptotic behavior.

Abstract

The assumption that a solution to the Einstein equations is static (or stationary) very strongly constrains the asymptotic behaviour of the metric. It is shown that one need only impose very weak differentiability and decay conditions {\it a priori} on the metric for the field equations to force the metric to be analytic near infinity and to have the standard Schwarzschildian falloff.