In the answer to this question: ergosphere treadmills Lubos Motl suggested a straightforward argument, based on the special theory of relativity, to argue that light passing through a strong gravitational region which reaches infinity cannot reach infinity faster than a parallel light ray which does not pass through the strong gravitational region.
The argument is just the standard special relativistic one, that if you can go faster than light, you can build a time-machine. To do this, you just have to boost the configuration that allows you to go faster than light, and traverse it in one direction, then boost it in the other direction, and go back, and you have a closed timelike curve.
In the condition of the question, where the strong gravitational region is localized, you can do this using pre-boosted versions of the solution. Preboost two far away copies of the solution in opposite direction, one boosted to go very fast from point B to point A (at infinity) and the other boosted to go from point A to point B.
Then if you traverse a path from A to B, crossing through the first boosted solution, then cross the second on the way back, you can make a CTC.
I was skeptical of this argument, because the argument I gave required the Weak energy condition, not a no CTC condition. The argument from null energy just says that a lightfront focuses only, it doesn't defocus, the area doesn't ever spread out. Any violation of weak energy can be used to spread out a light front a little bit, and this corresponds to the geodesics passing through the violating point going a little bit faster than light, relative to the parallel geodesics nearby.
So if both arguments are correct (and although I was skeptical of Lubos's argument at first, I now believe it is correct), this means that a generic violation of the weak energy condition which is not hidden behind a horizon could be turned into a time machine. Is this true?
Question: Can you generically turn a naked violation of weak energy condition into a closed timelike curve?
A perhaps more easily answerable version: If you have an asymptotically flat solution of GR where there is a special light ray going from point A to point B which altogether outruns the asymptotic neighbors, can you use this to build a time machine, using boosted versions of the solution?