Take a look at this paper on "Warp Drives and Causality:" https://doi.org/10.1103/PhysRevD.53.7365
The author attempts to argue that the Alcubierre Drive spacetime could exhibit Closed Timelike Curves, if one is to consider a "Lorentz Transformation of the coordinates." In such coordinates, one finds that a light ray emitted on board the ship could travel backwards in time in such coordinates.
The author seems to confuse unphysical coordinate artifact with actual phenomenology. As far as I understand, in GR, one has the freedom to label points on the spacetime manifold by whatever $t, x, y, z$ they like, but the metric tensor remains a physical invariant, regardless of what you call your coordinates. However, its representation in a coordinate system will of course depend on your choice of coordinates.
To be mathematically precise, one cannot perform a Lorentz Transformation on the coordinates within the context of a curved spacetime as in GR. If we want to of course, we could transform Alcubierre coordinates, to coordinates which look like a Lorentz Boost, but then we must transform the representation of the metric as well, and in our case we don't have the Minkwoskian structure as we do in SR.
A light beam propagating inside the vessel will remain within its local light cone, regardless of what you call your coordinates, as specified in Alcubierre's original paper. The "Lorentz transformation" attempted by the author does not hold.
Generally speaking, to find a CTC, we are looking for a Timelike trajectory for a particle in the spacetime that returns to the same point on the spacetime manifold. In summary, I do not see that a CTC arises, but that the paper concludes the existence of CTC from a violation of causality in special relativity which is exactly what a CTC does not entail AND an attempt to apply special relativity beyond its domain of applicability. Lorentz Boosts could only be defined Locally at a point in GR, and not the entire spacetime.
I am only a student, and the author is surely more experienced in GR than me, but I can't see where I am wrong: could someone kindly explain to me where my attempted refutation fails? Sorry for the excessive text...