You are confusing special relativity and general relativity. In special relativity you are ( sort of) right. In general relativity you can find shortcuts in the way we measure distances. The trick is, that speed can either denote a local property, as measured by your speedometer, or a global property, that is distance traveled divided by time. And the light speed limit only applies to the local property. So if you can find a spacetime that contains a shortcut, then you can increase the distance traveled faster than the speed of light. ( And there is nothing in general relativity to stop you, since it only requires you to move locally slower than the speed of light.) An example for this would be a expanding universe ( and for simplicity let me assume that the expansion only starts a some time t1.) So you travel from t0 to t1 with constant velocity v < c. And you are then at some point x=v(t1-t0). Then you stop and expansion sets in. You stay in place at x, as measured relative to some point nearby, but the distance to your original starting point increases, since space itself is expanding. And there is nothing to stop this increase to be faster than the speed of light.
The other thing is the "sort of" parenthesis, the Lorentz transformation is directly linked to the causal structure of special relativity. A statement like "if* you could go faster than the speed of light, then you could violate causality" is somewhat nonesensical, since if you could go faster than the speed of light, then you would break local Lorentz invariance and therefore you break your very definition of causality.
Thank you for explaining the GR vs SR aspect of this. I was somewhat aware that the no-FTL rule was different in GR but I didn't really understand how. I'll have to look into this more and now I know where to start. I'm not sure I follow the second part, about Lorentz invariance, but my physics is decades old, so that's not surprising. Time to do more reading.
Sorry for the botched formatting (HN is for some reason not showing me a edit link :/ )
A quite interesting paper on the topic is McMonigal & al. [1], it is a bit technical but they discuss the Alcubierre warp drive quite nicely.
To elaborate on the second paragraph about special relativity, one way to define Lorentz transformations is, that they are the transformations which preserve the causal structure of Minkowsky space. On the other hand, the speed limit is a direct consequence of the Lorentz transformations. So the causal structure and the speed limit are directly related. To look again at the argument that a velocity larger than the speed of light implies causality violation: The moment you argue that a speed faster than the speed of light is possible, you implicitly change the meaning of causality. Then you use the Lorentz transformations to conclude that in on frame of reference event A precedes B and that event B precedes A in another. So you first deny the special status of the Lorentz transformation, just to go on and use Lorentz transformations. And my take on this is a strong maybe, the moment you start talking about local Lorentz invariance violation, anything goes. Perhaps there is causality violation, perhaps not, but this depends on your specific meaning of 'causality' and 'faster than light' and not very much on special relativity.
The other thing is the "sort of" parenthesis, the Lorentz transformation is directly linked to the causal structure of special relativity. A statement like "if* you could go faster than the speed of light, then you could violate causality" is somewhat nonesensical, since if you could go faster than the speed of light, then you would break local Lorentz invariance and therefore you break your very definition of causality.