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In my recent question ”Speed of light and warp drives in general relativity” I asked exactly how an Alcubierre drive worked and exactly what "FTL travel" meant.

One of the comments I got stated that:

[…] if it [the Alcubierre drive] moved on a straight path at FTL speed, it would arrive for all 3 observers [on Earth, in the bubble and at the destination] before the light beam.

Now, this is a very interesting answer. If the Alcubierre drive would arrive at the destination before a light beam sent alongside the bubble, how does its worldline look like?

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I think the comment was talking about a race type situation.

For instance if you had a 50m dash with a bunch of runner lined up horizontally and running towards a horizontal finish line with some tape across the finish line they could race and see who hits the tape first.

It is exactly (and only that sense) that the warp drive arrives first.

Let's say after that first race you felt sorry for the light pulse so you put them in the same track and just let the light have a tiny head start. In that case the light gets there first.

On that second race the curvature in front of the bubble helps the light too. In the first race the bubble helps only the warp drive and not the light running next to (along side) the warp drive.

You are the one that said along side (before the comment), not in front of. If the light was in front, then the light gets there first. In fact then the light is one those things that gets swept up en route, one of those things responsible for the super deadly radiation that results if you slow down a warp drive.

Every thing in front gets swept up, and like a snow plow the stuff gets dragged around and shoots forward if the brakes are slammed.

Enough about the comment.

Since you asked about the world line just draw Minkowski space and pick a global inertial frame, and draw all the x,y,z, And t axis as orthogonal lines in the regular Euclidean geometry of 4d space (not because it is physical, but because later I want to say words like straight line and parallel and I want to refer to this 4d euclidean geometry because I'm trying to describe the geometry of the 4d picture we are drawing).

Draw all the light cones at every point opening up symmetrically about the family of world lines parallel to that inertial frame's $t$ axis.

Now pick a FTL world line (any spacelike line between two events) label the line segment L and extend it not just to a line segment but extend it everywhere in a straight line S. Now thicken that line and label it blue then thicken it again like a coaxial cable (or a thick coat of paint on a pencil) and label the newly added region red.

Leave all the uncolored light cones alone.

Replace all the light cones in the blue region with new ones that open symmetrically about lines parallel to the line S. Now in the red region smoothly tilt each light cone in a smooth interpolation between the tilted ones on the inside and the untilted ones on the outside.

That is entirely the sense where it is FTL we are literally travelling the curve in 4d that used to be a FTL path. But it isn't a spacelike cirve anymore, it is timelike.

But look at what happens to a null line that was heading in the same direction as the warp bubble and was aligned, i.e. the ones that in the Minkwoski space version travelled along the same space as the line segment L and just traveled it at earlier times. These null curves in the warp bubble space get bent towards the direction of the warp bubble and actually never cross the line L because they keep getting to places before the warp bubble does.

The warp bubble hoovers up space debris like bugs on a windshield and the bugs get there before the people behind the windshield get there. The light beam with the head start is the bug on the windshield.

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