Correct me if I'm wrong, but it is meaningless to talk about a "Lorentz Frame" on a spacetime manifold unless its perfectly flat, which it isn't in our case. You are allowed to look at physical Lorentz Transformations as the transformation of the frame fields at the intersection of two wordlines, which in of themselves must have velocities within the local lightcone to be observers which we could boost in the first place. There is no global notion of Lorentz Invariance in curved spacetimes, even if it is asymptotically flat. You only Lorentz transform locally...do you agree?

Future directed observers within these new "Lorentz-transformed" coordinates, will see the bubble moving forward in time either way. Please note that the motion is purely non-local.

@Joeseph123 Yes, there is no global Lorentz invariance, but there is "something resembling Lorentz invariance". You could think of it this way: if an Alcubierre warp tube is physically realizable, it must be possible to construct a machine that makes one. Can you construct two of those machines? Can you give them positions and velocities of your choice using conventional rocket engines before turning them on? What is the resulting spacetime geometry, supposing the machines are far enough apart that they don't interfere with each other's operation?

Could you help me sum this up constructively and definitively? If you believe in the author's argument, could you specify, without the confusing mention of Lorentz transformations of primed and unprimed coordinates, a Closed Timelike Curve?

@Joeseph123 I don't know if this is what you're asking, but the CTCs are $\bowtie$ shaped. The vertical edges are outside the warp tubes (traveling from one mouth to another) and the diagonal edges (which don't intersect each other) are inside. The mouths could be placed anywhere in the spacetime as long as they're reachable from each other by timelike worldlines.

No I mean a parametrised wordline in the regular unprimed coordinates, which is both timelike and returns to the same point in spacetime. $X^{\mu}(\lambda)=()$?

@Joeseph123 See my edit to the answer.

The tachyonic anti-telephone would need local violation of SR, which is not our case.

Ah...but then you would be constructing a warp bubble with a past-directed timelike trajectory for the crew on board...this would of course easily create a CTC, but it relies on the trajectory being past-directed within the spacetime...just as if I consider a past-directed trajectory here on earth in the vicinity of a perfectly physical Schwarzschild spacetime.

@Joeseph123 Are you talking about my construction? There is no point on my CTC at which it reverses time direction. You seem to want the CTC to never go backwards relative to some globally defined time coordinate, but no spacetime could ever have a CTC by that criterion. Note that the entrance and exit of Alcubierre warp tubes are geometrically distinguishable. The boosted tubes can only be entered at $t=0$ and exited at $t=-1$.

Sorry at this point I am completely lost: all I am sure of is this: the Alcubierre spacetime: $$ds^2=-dt^2+(dx-vfdt)^2+dy^2+dz^2$$ so the proper time elapsed by an observer still in the ship is given by $$\tau = \int_0^t \sqrt(1-(\frac{dx}{dt}-vf)^2)$$, since $dx/dt = v, \tau = t$ regardless of how the warp drive accelerates. This I am completely sure of, and I think we could both agree on this. I do not see how the "Lorentz-boosted" drive is actually a warp drive, but only a muddled coordinate transformation which we, with undue assumption, assume is also a warp drive.

By match I mean place them both in the same spacetime.

@Joeseph123 "Remember, that now asymptotically (far away from the drive) unprimed spacetime no longer looks like 𝜂𝜇𝜈, but a boosted metric Λ𝜇𝜈≠𝜂𝜇𝜈 determined by your Lorentz Boost." --> I haven't followed everything in this thread but this is not correct and I think a key point. Asymptotically far away the metric is Minkowski, and if you do a coordinate transformation that is asymptotically a Lorentz boost, then under this change the asymptotic metric will still be Minkowski, since $\Lambda^T \eta \Lambda = \eta$ is the definition of $\eta$.

I haven't read the paper in detail so YMMV, but in GR there is no "global time." Here, there is an asymptotic time (or to be more precise an asymptotic light cone). However it may be tricky to compare the time coordinate near the Alcubierre drive with the asymptotic time coordinate (think Schwarzschild black hole). I think understanding how the CTC can arise in special relativity is useful. My impression is that this paper proves an analogous argument works here. Showing that there is a CTC is an invariant, coordinate independent statement, and is likely easier than comparing time coordinates.

I am aware of this, but you could by convention foliate spacetime hypersurfaces according to an arbitrary time coordinate, and in our case we just so happen to choose our "global time" to coincide with the departure site...I am assuming the ADM Formulation to be clear. CTC's must be invariant which I perfectly agree with, but the paper's argument, or at least my understanding thereof, contains coordinate artifacts of some sort, from which the author deduces a nonexistent CTC.

If you send a warp drive on a two way trip, the time elapsed in the ship is exactly equal to however much time passes at the launch site, by definition of the Alcubierre spacetime. There is no way that consideration of the round trip in "boosted" coordinates could change the causal structure of the drive spacetime such that the ship arrives at an earlier time. Contradiction.

There are no CTC's in special relativity, you mean the lightcones as defined by the metric structure of spacetime within GR are open in such a way as to allow for a timelike trajectory to return to the same point in spacetime.

@Joeseph123 Here's yet another way of looking at it: my construction has two tubes, one in the $y<\frac14$ half-space and one in $y>\frac14$. Each of these half spaces can be causally foliated: they're standard Alcubierre tubes in nonstandard coordinates, as you observed at the beginning. But there's no causal foliation of the whole space. There isn't enough room between the tubes to join up the hyperplanes from either side while keeping them spacelike everywhere. If there were, the mouths would be too far apart to reach the entrances from the exits, and there would be no CTCs.

You could foliate the whole space either way if you consider the round trip in standard Alcubierre coordinates, you will find the total elapsed time is equal to the launch site local time. Our conclusions about CTC's cannot be coordinate dependent. Could you fully describe the spacetime of the full round trip, without mention of primed coordinates? Could you please write down an Alcubierre Metric of a full round trip, and show me how a CTC could be formed, with no mention of change of coordinates? The coordinates cannot change the physics, nor the causal structure.

Unless we adulter the spacetime with coordinate artefact (the Lorentz Boost).