A new class of obstructions to the smoothness of null infinity

TL;DR

This paper investigates the smoothness of null infinity in asymptotically Euclidean, conformally flat spacetimes, revealing that non-vanishing Newman-Penrose constants and higher order quantities obstruct smoothness, implying initial data must be Schwarzschildean for smooth null infinity.

Contribution

It introduces a new analysis of gravitational field expansions near infinity using conformal geodesics, identifying obstructions to null infinity smoothness and linking them to initial data conditions.

Findings

Null infinity is non-smooth unless Newman-Penrose constants vanish.

Higher order quantities also obstruct smoothness.

Smooth null infinity implies initial data is Schwarzschildean.

Abstract

Expansions of the gravitational field arising from the development of asymptotically Euclidean, time symmetric, conformally flat initial data are calculated in a neighbourhood of spatial and null infinities up to order 6. To this ends a certain representation of spatial infinity as a cylinder is used. This set up is based on the properties of conformal geodesics. It is found that these expansions suggest that null infinity has to be non-smooth unless the Newman-Penrose constants of the spacetime, and some other higher order quantities of the spacetime vanish. As a consequence of these results it is conjectured that similar conditions occur if one were to take the expansions to even higher orders. Furthermore, the smoothness conditions obtained suggest that if a time symmetric initial data which is conformally flat in a neighbourhood of spatial infinity yields a smooth null infinity, then the initial data must in fact be Schwarzschildean around spatial infinity.