An iterative approach to twisting and diverging, type-N, vacuum Einstein equations: A (third-order) resolution of Stephani's 'paradox'

In 1993, a proof was published, within this journal, that there are no regular solutions to the linearized version of the twisting, type-N, vacuum solutions of the Einstein field equations. While this proof is certainly correct, we Show that the conclusions drawn from that fact were unwarranted, namely that this irregularity might cause exact, twisting solutions of Petrov type N to be unable to truly describe pure gravitational waves. In this paper, we resolve the paradox-since such first-order solutions must always have singular lines in space for all sufficiently large values of r-by showing that if we iterate the solution up to the third order, there do exist acceptable, regular solutions. That these solutions become Bat before they become non-twisting tells us something interesting concerning the general behaviour of solutions describing gravitational radiation from a bounded source.