Causality violation without time-travel: closed lightlike paths in G\"{o}del's universe

We revisit the issue of causality violations in G\"{o}del's universe, restricting to geodesic motions. It is well-known that while there are closed timelike curves in this spacetime, there are no closed causal geodesics. We show further that no observer can communicate directly (i.e.\ using a single causal geodesic) with their own past. However, we show that this type of causality violation can be achieved by a system of relays: we prove that from any event $P$ in G\"{o}del's universe, there is a future-directed lightlike path - a sequence of future-directed null geodesic segments, laid end to end - which has $P$ as its past and future endpoints. By analysing the envelope of the family of future directed null geodesics emanating from a point of the spacetime, we show that this lightlike path must contain a minimum of eight geodesic segments, and show further that this bound is attained. We prove a related general result, that events of a time orientable spacetime are connected by a (closed) timelike curve if and only if they are connected by a (closed) lightlike path. This suggests a means of violating causality in G\"{o}del's universe without the need for unfeasibly large accelerations, using instead a sequence of light signals reflected by a suitably located system of mirrors.