# How does a world line of an Alcubierre drive look like?

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?

For the interested, the interactive app I created to make these diagrams is here. All velocities are relative to the Earth. Note that positive velocity for the return trip going to the left is an artifact of negative time such that v = -x/-t = +v.

If the Alcubierre drive could achieve FTL in the true sense of the phrase, then its worldline would look like the green line in the time space diagram below.

(Earth is at the origin of the chart.) In the reference frame of another observer travelling at 0.7c to the right, the warp drive appears to arrive at destination before it left Earth. (See diagram below):

Now staying in the rest frame of this moving observer, the drive turns around and returns to Earth at a modest 1.8c and arrives back at Earth before it left. This is obviously going to cause causality issues.

For completeness the round trip as seen by the observer at rest in the Earth reference frame is shown below:

If the Alcubierre drive can go backwards in time in one reference frame, then it can go backwards in time in any reference frame, because the laws of physics are the same in any reference frame.

If an object can even go only 1% greater than the speed of light then the consequences are:

1. There is always a reference frame where the velocity is infinite and the drive can reach anywhere in the universe in zero time.

2. There is always a reference frame where the drive can go backwards in time and return to its launch site before it the time it launched.

3. There is a reference frame where the drive remains stationary and only travels backwards in time.

4. It is not necessary to build the drive or rocket because it possible to return to a point in time before the drive was built.

5. If the drive is launched sometime after it is built and returns to a point in time after it was built but before it was launched, there are now two rockets at the base for the price of one (and two sets of astronauts).

6. By repeating step 5, it is possible to produce infinitely many rockets for the price of building one.

There are other time paradoxes involved. FTL = Causality violation.

• The fact that something is a paradox does not mean that it cannot exist, but that we do not understand it. For now. An eminent Hungarian physicist, Béla Lukács, made this point, and I believe it to be absolutely true. Commented Jan 23 at 19:07
• @AttilaJanosKovacs I agree a paradox means either we do not understand it or it does not exist.
– KDP
Commented Jan 23 at 19:41
• With inclusive or exclusive disjunction? Commented Jan 23 at 20:10
• I have written a separate answer to your points. @KDP Commented Jan 27 at 14:58
• @AttilaJanosKovacs I have posted a response to your response.
– KDP
Commented Jan 27 at 19:10

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.

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.

Good question. It is worth splitting the problem into two parts.

1. The space-time bubble itself is moving. It moves along a spacelike worldline if v > c. This does not violate anything in relativity, since the bubble is not a massive object obeying the geodesic equation. It is just a specific curvature of spacetime itself.
2. The spaceship inside the bubble. It travels along a timelike wordline inside the bubble, as expected from a massive object. The light cones have a “tilt” which is proportional to v. Even though the spaceship travels faster than light globally, it is locally still within its own light cone along the path, so no relativistic rules are broken. Here is an illustrative diagram about this, the source is very useful and clears up a lot of things.

This is a friendly response to the thoughtful response directed to me by Attila and I hope no one minds our side discussion. He has taken the time to post a detailed reasoned response and I thought it only polite to reply.

Point 4) I agree the ship that returns is not exactly the same as the ship that was launched. It will obviously requiring refueling (see point 5) and some minor repairs, but the main structure of the ship will not have to built and we get that for free.

Point 5) I agree that when I said we get two ships for the cost of building one that we need to also include the cost of FTL travel itself which can be thought of as the cost of fuel. However, one outcome from point 6 is that we can get potentially infinite ships for cost of one plus fuel. Creating a potentially infinite amount of mass requires a lot of energy. This is a strong indication and informal justification that the required amount of fuel or energy is also potentially infinite which has already been alluded to in much more formal calculations and discussions of the Alcubierre drive. This is based on the informal "no free lunch" theorem.

Point 6) It is possible if the loop repeats, than not only does an additional ship return, but a potentially infinite number of ships all return simultaneously with each other. With all that mass in one place, there are strong gravitational implications.

Point A) I disagree that Minkowski flat spacetime cannot be used to analyse this problem. We can make the travel distance arbitrarily large, such that the the warp bubble is an arbitrarily small portion of the diagram. What goes on inside the bubble probably does require general relativity to analyse, but we can isolate the bubble and its interface from the flat spacetime and analyse what happens outside. By the way I think the diagram, you posted of the light cones inside the bubble is useful and I am giving you +1 for that. What goes on outside the bubble is important and if we are not travelling FTl relative to flat spacetime the drive becomes pointless.

Point B) When I said "There is always a reference frame where the drive can go backwards in time and return to its launch site before it the time it launched." there are in fact infinitely many reference frames where the journey is going back in time and this cannot be ignored. If there is any reference frame where the drive is going backwards in time relative to flat spacetime (and the laws of physics are the same in all reference frames), then the drive can go backwards in time in any reference frame. It follows that the backwards in time time travel is possible on both the outward and return trip.

There is a way around this last problem, but it requires throwing out the principle of relativity. We can restrict the FTL drive to only going forward in time in any inertial reference frame. This requires have a preferred reference frame or effectively an aether. Noe Einstein did not rule out a preferred reference frame. He basically said it is undetectable and there is "no need of it". If someone does manage to build a Alcubierre drive and it does not violate causality and does not go backwards in time (in any reference frame) that would in fact constitute a proof of the aether.

Let me respond to KDP's previous answer, as the original question in the post has in fact been answered; but he has raised some interesting issues on the subject. Of the 6 points he wrote down as a consequence, I think 4, 5, 6 are incomplete. Why?

4: apparently what he means in this point is that if the warp drive rocket returns to its starting point, but at an earlier time before it was built, then there is "no need" to build it. Because this returned rocket can be launched from its original location. But that would not be a repeat of what happened! The causal chain of events does not come full circle. The returned rocket is not the same object that was launched. This is obvious for many reasons. For example, it has worn out en route, its parts have aged, changed; not to mention the astronauts who were travelling in it. This is a different rocket, not the one that was newly built. In the case of people, it is immediately obvious: if you launch a 30-year-old and return a 32-year-old, you are not doing the same thing if you launch the 32-year-old instead of the original 30-year-old. Ergo, if it is necessary to repeat the original event of building and launching the warp drive, then it is certainly necessary to build the warp drive rocket as it was originally built. If the returned rocket is used for this, then a different event is implemented. And it may have different immediate consequences than what was already accomplished once.

5:"If the drive is launched sometime after it is built and returns to a point in time after it was built but before it was launched, there are now two rockets at the base for the price of one (and two sets of astronauts)." - Not for the price of one. Well, the cost of FTL travel with a warp drive rocket? Why not include that? After all, that is a huge cost - because of the large and special energy required - and at that cost the rocket could return and there could be 2 of them at that time. That cost was incurred, just not at the time the cost of building the original rocket was incurred. The flaw in point 6 is obvious from this - the cost of the FTL warp drive trip has to be included for each return.

Regardless of the foregoing, I have one more comment, perhaps more important. I think that Special Relativity is not suitable for analysing time travel from FTL. For two reasons:

A) The spacetime diagrams used in the previous answer are clever and useful. They illuminate some important features. But they are still only Minkowski spacetimes, flat spacetimes, and they differ only in the velocities of the inertial observers travelling in them. And as far as we are concerned, warp drive FTL travel can only be achieved with highly curved spacetimes, not flat ones. Strong space expansion and contraction takes place at the walls of the warp drive, or other types of strong curvatures. We cannot therefore say that we give a valid description of warp drive travel in this way. This description is valid for tachyons travelling in Minkowski spacetime, but we should not confuse them with warp drive travel. To model the latter, we need General Relativity and a description of warped spacetimes, expanding and contracting 3d spaces.

B) In addition, consider the following: what types of conclusions or claims can we make based on the Minkowski spacetime description? Such as: "There is always a reference frame where the drive can go backwards in time and return to its launch site before it the time it launched." That is, that in principle there exists another inertial frame from which such and such is happening. But there is also one in which it doesn't happen that way, and you can look at it from many different inertial reference frames. Viewed and experienced from the original reference frame,it does not arrive before departure, but only an FTL journey takes place.

So what is reality? Can we arbitrarily select one inertial frame among many, claiming that the "reality" is that we calculate from it; this is valid? Why would it be stronger in terms of reality than frames in which there is no effect backwards in time? I don't think any such argument is correct or valid. Based on the Minkowski diagrams of the Special Theory of Relativity, we can only obtain equivalent theoretical descriptions. One of the basic postulates of the theory is that all inertial systems are equivalent. We have no right or reason to designate those that describe "reality" and to tell another frame that it does not describe reality. This again shows that SR is inadequate and incomplete for handling FTL and time travel. For that, it is minimally necessary to switch to the tool system of the General Theory of Relativity. Maybe even that is not enough.

Let's continue discussing the previous interesting problems; unfortunately, there is not enough to say in small comments. @KDP Minkowski spacetime is good for something, but it does not move us forward on the most pressing issues, it only creates deadlock.

"When I said "There is always a reference frame where the drive can go backwards in time and return to its launch site before it the time it launched." there are in fact infinitely many reference frames where the journey is going back in time and this cannot be ignored. If there is any reference frame where the drive is going backwards in time relative to flat spacetime (and the laws of physics are the same in all reference frames), then the drive can go backwards in time in any reference frame. It follows that the backwards in time time travel is possible on both the outward and return trip"-wrote KDP.

It is interesting food for thought and I may not understand it clearly. The "possible" is significantly different than what is happening in my view. My take is this: it is certainly true that by switching between inertial frames of reference there are frames from which, written down, the warp drive rocket returns to the launch location, but earlier than it was launched. An infinite number of these are possible, of course. But that doesn't remove the validity of the fact that there are reference frames in which this is not the case, described as no time travel, just FTL travel. Such is the frame in which it is tied to the launch station. So what is the reality? Does the rocket return earlier or does it not happen? We can't answer this with certainty based on Minkowski spacetime analysis. We can only reach the ambiguous impasse described above.

Let us compare this with the question of length. Because of the relativistic length contraction, let's take the length of the rocket, say 40 m, as it is built, so measured in its own system. However, by switching between inertial frames, the length as seen and measured from a different frame is no longer 40 m, but less. Let's say there is one frame with only 20 m, another with 10 m, etc. What is the reality? What is the real length of the rocket? On the ground of SR we solve this by stating that length is not absolute but relative. There is the proper length, which is what is measured in the inertial frame in which the rocket is at rest. And there are the lengths of motion, measured from the other systems moving relative to it; the contraction depends on the relative velocity of the two systems according to the well-known formula. The question of which is "the real", the true length, is not a precise and clear-headed one. Each length is real according to the rules of SR. Of course, most physicists and experts believe that the proper length is the "most real" of all, if only because there is only one of these lengths, while there are infinitely many other lengths of motion, which differ from each other.

Can a similar analysis be applied to our problem, i.e. what is actually happening: did the rocket return earlier than its own launch or not? The similar solution does not work. After all, whether the rocket that has already returned is standing at the launch pad when the builders go to start construction cannot be relative in the same way that a number that measures the length of an object is relative. It's a tough story, it's either there or it's not. Something that is only half there, or 0.875% there, is not seriously considered. Or / or. So what is the reality?