Asymptotic simplicity and static data

TL;DR

This paper investigates the conditions under which time symmetric initial data for vacuum Einstein equations extend smoothly through critical sets at spatial infinity, emphasizing the unique role of static data in achieving a smooth conformal compactification.

Contribution

It proves that only static initial data near infinity allow solutions to extend smoothly through the critical sets at null infinity.

Findings

Solutions extend smoothly iff initial data are static near infinity

Static data uniquely enable smooth conformal compactification

Highlights the special role of static data in Einstein's equations

Abstract

The present article considers time symmetric initial data sets for the vacuum Einstein field equations which in a neighbourhood of infinity have the same massless part as that of some static initial data set. It is shown that the solutions to the regular finite initial value problem at spatial infinity for this class of initial data sets extend smoothly through the critical sets where null infinity touches spatial infinity if and only if the initial data sets coincide with static data in a neighbourhood of infinity. This result highlights the special role played by static data among the class of initial data sets for the Einstein field equations whose development gives rise to a spacetime with a smooth conformal compactification at null infinity.