Stationary solutions and asymptotic flatness I

TL;DR

This paper proves that vacuum strictly stationary space-times representing isolated bodies are necessarily asymptotically flat with Schwarzschild fall-off, even without assuming asymptotic flatness beforehand.

Contribution

It demonstrates that such space-times are inherently asymptotically flat, removing the need for prior assumptions about their asymptotic structure.

Findings

Vacuum strictly stationary space-times are necessarily asymptotically flat.

The space-times exhibit Schwarzschild fall-off behavior.

The result holds without initial assumptions on asymptotic conditions.

Abstract

In this article and its sequel we discuss the asymptotic structure of space-times representing isolated bodies in General Relativity. Such space-times are usually required to be asymptotically flat (AF), and thus to have a prescribed type of asymptotic. Despite all the "reasonable" that the requirement is, it seems to be against the spirit of General Relativity where the global structure of the space-time should be also considered as a variable. It is shown here that, even eliminating from the definition any a priori reference or assumption about the asymptotic, the space-times of isolated bodies are unavoidably and a posteriori AF. In precise terms, between the two articles it is proved that any vacuum strictly stationary space-time end whose (quotient) manifold is diffeomorphic to R^3 minus a ball and whose Killing field has its norm bounded away from zero is necessarily AF with Schwarzschidian fall off. The "excised" ball would contain (if any) the actual material body, but this information or any other is not necessary to reach the conclusion. Physical and mathematical implications are also discussed.