On smoothness-asymmetric null infinities

We discuss the existence of asymptotically Euclidean initial data sets to the vacuum Einstein field equations which would give rise (modulo an existence result for the evolution equations near spatial infinity) to developments with a past and a future null infinity of different smoothness. For simplicity, the analysis is restricted to the class of conformally flat, axially symmetric initial data sets. It is shown how the free parameters in the second fundamental form of the data can be used to satisfy certain obstructions to the smoothness of null infinity. The resulting initial data sets could be interpreted as those of some sort of (non-linearly) distorted Schwarzschild black hole. Its developments would be so that they admit a peeling future null infinity, but at the same time have a polyhomogeneous (non-peeling) past null infinity.