Isn't time travel in the Godel universe supposed to be a non-starter?
After having discussions with Einstein in the late 1940s, Godel postulated a ROTATING universe in order to achieve time travel .. this was before we knew that the universe doesn't rotate .. so my (limited) understanding is that Godel's solution for time travel doesn't correspond to reality. That's why others since then have come up with vast rotating masses (cylinder shaped) INSIDE the non-rotating universe - it's apparently equivalent to Godel's set-up.
That's all I know though so I'll download this article and keep it for a time when I might actually understand it ..