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

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.