On the choice of a conformal Gauss gauge near the cylinder representing spatial infinity

A convenient approach to analyze spatial infinity is to use a cylinder representation $I$ and impose a gauge based on a congruence of conformal geodesics. This so-called conformal Gauss gauge comes along with the freedom to specify initial data for the conformal geodesics. Such a gauge has been constructed from an ordinary Cauchy surface and from past null infinity ${\mathcal{J}}^{-}$, respectively. The purpose of this note is to compare these gauges near the critical set $I^{-}$, where $I$ “touches” ${\mathcal{J}}^{-}$, as it turns out that they are related in a somewhat unexpected intricate way.