  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.
