Alcubierre Warp Drive: Spacetime Diagram Explained

[timmdeeg]

TL;DR Summary

Supposed Alcubierre Warp Drive is possible then which spacetime diagram shows its motion through spacetime?

https://scipost.org/SciPostPhysLectNotes.10/pdf
"Space expands behind the warp bubble and contracts in front of it, thus pushing the bubble forward at velocity v. The ship, which is at rest inside the bubble, moves along with the bubble at an arbitrarily large global velocity."

If the ship moves faster than light this would mean that in a Minkowsky diagram its world line would be straight from the origin to the destination closer to the x-axis than light world lines. This world line line would be spacelike however.
Does this make sense or would another spacetime diagram be more appropriate?
Is the distortion of spacetime produced by the warp bubble a local effect which travels through otherwise flat spacetime (as gravitational waves do) as shown in this visualization?

https://publikationen.uni-tuebingen.de/xmlui/bitstream/handle/10900/48159/pdf/09warp.html

 

[timmdeeg]

I think I found what I was searching for.

http://laplace.physics.ubc.ca/Members/mikin/thesis.pdf

Figure 4.3: A spacetime diagram of right-moving (x−) light rays for the Alcubierre metric. Here, a warp drive starts at rest and accelerates linearly. The trajectory is given by xo(t) = 1 2 a t2 with a = 0.5. The shaping function is taken to be Eq. (4.48) with σ = 1. The pink shaded region represents the inside of the warp drive and the white region is Minkowski space. When inside the warp drive, the light rays appear to be traveling faster than they do in Minkowski space.

Here the Alcubierre metric seems to be embedded in a Minkowski diagram.

Back to my question. So it seems the straight line connecting origin and destination in a Minkowski diagram isn't wrong, it just ignores the Alcubierre metric and treats the inside of the warp bubble as moving faster than light. Would you agree to that?

 

[PeterDonis]

If the ship moves faster than light this would mean that in a Minkowsky diagram its world line would be straight from the origin to the destination closer to the x-axis than light world lines. This world line line would be spacelike however.

The worldline of the ship inside the bubble is not spacelike. Fig. 3 in the lecture notes shows a spacetime diagram for the warp drive including the light cones. As that diagram makes clear, the effect of the warp drive bubble is to tip over the light cones; "faster than light" corresponds to tipping them over by more than 45 degrees. But well outside the bubble, the light cones are normal, just like they are in Minkowski spacetime. (By definition, the Alcubierre metric becomes flat, like Minkowski spacetime, far away from the bubble.)

Section 5.5 of the notes, the "Event Horizon" section, is also worth noting. As that section makes clear, the "forward" edge of the bubble is actually outside of the causal future of the ship at the center of the bubble! That means the crew of the ship cannot shut down the warp drive. Which, as the notes say, "severely limits the feasibility" of this suggested form of travel.

 

[Vanadium 50]

That means the crew of the ship cannot shut down the warp drive.

My 1976 Malibu was the same way. Kept dieseling.

I think the better answer is just a flat map is not a good representation of a curved earth, a Minkowski diagram is not a good representation of a highly curved spacetime.

 

[timmdeeg]

I think the better answer is just a flat map is not a good representation of a curved earth, a Minkowski diagram is not a good representation of a highly curved spacetime.

You say we can't say something - and thereby ignoring the warp bubble - reaches a destination faster than light and draw a straight line, right? But could we if "something" would be e.g. a tachyon (without introducing curvature)?

 

[Vanadium 50]

My point is that if you use a Mercator projection you would be misled about distance near a pole. If you are close enough to the bubble for it to matter, using a Minkowski viewpoint is likely to mislead the same way.

 

[timmdeeg]

Section 5.5 of the notes, the "Event Horizon" section, is also worth noting. As that section makes clear, the "forward" edge of the bubble is actually outside of the causal future of the ship at the center of the bubble! That means the crew of the ship cannot shut down the warp drive. Which, as the notes say, "severely limits the feasibility" of this suggested form of travel.

Thanks, I have missed this one. And I think the warp drive will remain a theoretical construct forever, quite apart from the causality issue. I'm just interested if its faster than light trip can be shown somehow in a spacetime diagram.

 

[PeterDonis]

I'm just interested if its faster than light trip can be shown somehow in a spacetime diagram.

You can certainly show something that appears to be a "faster than light trip" by ignoring the spacetime curvature of the warp bubble and pretending it's a "tachyon" and drawing what looks like a spacelike worldline for it on a Minkowski diagram. All you have to do is ignore all the actual physics involved.

 

[Ibix]

You say we can't say something - and thereby ignoring the warp bubble - reaches a destination faster than light and draw a straight line, right? But could we if "something" would be e.g. a tachyon (without introducing curvature)?

I think that if you zoom out to the scale that you can only really see the warp bubble as a worldline, not a worldtube, it would appear to be a spacelike line, yes. (Formally, I'm proposing to excise a small region of spacetime around the bubble and assert that the rest is nearly flat and a flat patch over the excised region would be long in one spacelike direction and short in the other three directions.) However, that's hiding all of the detail of how the warp bubble works. Spacetime in the bubble is very much not flat, and the spaceship itself continues on a timelike worldline - it's just that the bubble tips lightcones over so that the timelike worldlines inside the bubble don't (globally) behave the same way as those outside the bubble.

I don't know how to draw the bubble properly on a diagram. I've seen pictures (like the one at Wikipedia), but I suspect that is something like Flamm's paraboloid as a representation of the Schwarzschild metric - true, but misleading.

 

[PeterDonis]

I don't know how to draw the bubble properly on a diagram.

Figure 3 in the lecture the OP linked to, which I referred to in post #3, is probably the best that can be done in trying to represent the bubble on a feasible spacetime diagram.

I've seen pictures (like the one at Wikipedia), but I suspect that is something like Flamm's paraboloid as a representation of the Schwarzschild metric - true, but misleading.

Yes, it's misleading in much the same way the Flamm paraboloid is--it only shows space at one instant of time, and does not show time at all.

 

[timmdeeg]

Thanks to all of you for helpful answers.

Another question in this context regards causality.

Miguel Alcubierre states himself

https://arxiv.org/abs/2103.05610
In particular, we will discuss the violation of the energy conditions associated with these spacetimes, as well as some other interesting properties such as the appearance of horizons for the superluminal case, and the possibility of using a warp drive to create closed timelike curves.

He says "the possibility" which could mean (not sure about that) with some caveats. Is it proofed that warp drive would create closed timelike curves? If yes, how can this be shown, by math only, heuristically or by a spacetime diagram?

 

[PeterDonis]

Is it proofed that warp drive would create closed timelike curves?

As noted in the paper (section VI), the original Alcubierre spacetime does not contain CTCs, but a simple modification of it does.

If yes, how can this be shown, by math only, heuristically or by a spacetime diagram?

If you are willing to ignore all the complications in how the curve is arrived at, in a Minkowski-type spacetime diagram a CTC, in a modified Alcubierre spacetime that allowed one, would just look like a circle, or perhaps a circular "tube" since the warp bubble has finite size. In terms of Figure 3 in the lecture linked to in the OP, the light cones would be tipped over sideways inside the circular tube, and the light cones would have to transition between that tipped over state and vertical in the boundary region between the tube and the rest of the spacetime.

 

[timmdeeg]

Thanks for your enlightening comment.