Regularity conditions at spatial infinity revisited

TL;DR

This paper revisits regularity conditions at spatial infinity for vacuum Einstein equations, ensuring smooth extensions of solutions through critical sets, generalizing Friedrich's analysis to data with non-zero second fundamental form.

Contribution

It extends Friedrich's analysis by establishing regularity conditions for initial data with non-zero second fundamental form, allowing solutions to extend smoothly at spatial infinity.

Findings

Regularity conditions ensure smooth extension at spatial infinity.

Generalization of Friedrich's analysis to non-zero second fundamental form.

Solutions extend analytically through critical sets.

Abstract

The regular finite initial value problem at infinity is used to obtain regularity conditions on the freely specifiable parts of initial data for the vacuum Einstein equations with non-vanishing second fundamental form. These conditions ensure that the solutions of the propagation equations implied by the conformal Einstein equations at the cylinder at spatial infinity extend smoothly (and in fact analytically) through the critical sets where spatial infinity touches null infinity. In order to ease the analysis the conformal metric is assumed to be analytic, although the results presented here could be generalised to a setting where the conformal metric is only smooth. The analysis given here is a generalisation of the analysis on the regular finite initial value problem first carried out by Friedrich, for initial data sets with non-vanishing second fundamental form.