Under what conditions would it be possible to create closed timelike curves, assuming an Alcubierre drive could exist? Would it be possible to have the latter without the former?
See here for information on the chronology protection conjecture.
[Physics] The Alcubierre drive and closed timelike curves
faster-than-lightgeneral-relativityspacetimetime-travelwarp-drives
Best Answer
If somehow, it is only possible to create one alcubierre drive, and it can never turn around, then you won't get closed timelike curves. Otherwise, any construction is going to have them.
The reason is that you can "zoom out" far enough that the distortions to spacetime caused by the drive are no longer present, and the person flying the drive then just looks like a spacelike curve in the spacetime. At that point, you're just travelling along spacelike curves, and all the special-relativistic causality objections apply.