On the smoothness of the critical sets of the cylinder at spatial infinity in vacuum spacetimes

TL;DR

This paper investigates the conditions under which the critical sets of Friedrich's cylinder at spatial infinity in vacuum spacetimes are smooth, focusing on the role of radiation fields and mass aspects in the appearance of logarithmic terms.

Contribution

It establishes necessary and sufficient conditions for the smoothness of the critical sets based on radiation field behavior and mass aspects in vacuum spacetimes.

Findings

Logarithmic terms vanish if the radiation field is zero at all orders.

Vanishing of the dual ADM mass aspect is necessary for smoothness.

Constant ADM mass aspect ensures smooth representation at critical sets.

Abstract

We analyze the appearance of logarithmic terms at the critical sets of Friedrich's cylinder representation of spatial infinity. It is shown that if the radiation field vanishes at all orders at the critical sets no logarithmic terms are produced in the formal expansions. Conversely, it is proved that, under the additional hypothesis that the spacetime has constant (ADM) mass aspect and vanishing dual (ADM) mass aspect, this condition is also necessary for a spacetime to admit a smooth representation at the critical sets.