Is the Alcubierre Warp Drive possible?

[Heikki Tuuri]

https://en.wikipedia.org/wiki/Faster-than-light
The Alcubierre drive allows superluminal communication, and according to Wikipedia, that is equivalent to time travel. Then we will have all the paradoxes of time travel. And that is in an asymptotic Minkowski space - we would not need any wormhole to make a causal loop. The Alcubierre drive is enough.

 

[PeterDonis]

according to Wikipedia

Wikipedia is not a valid source. You need a textbook or peer-reviewed paper.

 

[PAllen]

This is not the Alcubierre warp drive spacetime.

True, but if more than one drive in the universe is possible, it suggests CTCs would be possible.

Here is the paper:

http://exvacuo.free.fr/div/Sciences...tt - Warp drive and causality - prd950914.pdf

 

[DrStupid]

For a causality violation, you would need two separate inertial frames, in relative motion with respect to each other and sending FTL messages back and forth.

A single message could be sufficient. With two events A and B, where one of them is the cause of the other, causality is violated if the order of the events is frame dependent. In Minkowski space this would be the case if the interaction is faster than a light signal traveling in vacuum directly between A and B. However, in the current discussion this would be only helpful if the Alcubierre warp drive can be used in a way that keeps the spacetime between A and B flat. PeterDonis sounds like this is not possible.

 

[PeterDonis]

this would be only helpful if the Alcubierre warp drive can be used in a way that keeps the spacetime between A and B flat. PeterDonis sounds like this is not possible.

Of course it's not possible. Alcubierre spacetime is not flat. If an Alcubierre drive travels between A and B, then spacetime between A and B is not flat.

 

[Heikki Tuuri]

https://arxiv.org/abs/gr-qc/9810026
The paper by Matt Visser et al. suggests that the Alcubierre drive breaks the Null Energy Condition.

The paper is not peer-reviewed. The fact that if you can do unrestricted superluminal travel then you can make causal loops, is folklore that everybody seems to know, but it is hard to find a peer-reviewed reference. The math is easy and can be found from my own blog, year 2013.

If you restrict the superluminal travel, then you can prevent causal loops. One may, for example allow a superluminal trip on Earth as long as it takes you forward in the UTC time coordinate. Causal loops cannot happen because the UTC time always advances on every trip.

 

[DrStupid]

If an Alcubierre drive travels between A and B, then spacetime between A and B is not flat.

My question was not limited to this special case. The ship can also go a long way around, running the warp drive only far away from A and B and flying the rest with conventional engines.

 

[George Jones]

https://arxiv.org/abs/gr-qc/9810026
The paper by Matt Visser et al. suggests that the Alcubierre drive breaks the Null Energy Condition.

I am not disputing the fact that Alcubierre warp drive spacetime violates one or more energy conditions, but violation of an energy condition is not a (mathematically) sufficient condition for closed timelike curves.

The paper is not peer-reviewed. The fact that if you can do unrestricted superluminal travel then you can make causal loops, is folklore that everybody seems to know

What does "superluminal travel" mean? If it means "woldline outside a local lightcone", then I agree. Alcubierre warp drive spacetime, however, does not have this property. Folks (e.g., @bobob , @PeterDonis ) have noted this in this thread.

The math is easy and can be found from my own blog, year 2013.

This statement is not true.

http://meta-phys-thoughts.blogspot.com/2013/
This bolgpost most certainly does not demonstrate mathematically (i.e., using actual GR spacetimes) that, specifically, Alcubierre warp drive spacetime contains closed timelike curves.

 

[1977ub]

Of course it's not possible. Alcubierre spacetime is not flat. If an Alcubierre drive travels between A and B, then spacetime between A and B is not flat.

A small kernel of heavily warped spacetime in a large ocean of flat-ish spacetime such as between here and alpha centauri. This is why i described the vehicle to be within a black box. At the quantum level, spacetime may be heavily warped but that doesn't enable FTL communication or travel at the macroscopic scale.

Adherents of the drive accept that it will allow me to send messages to my friend 4 ly away in what he and I both agree to be 3 years. This is a causality violation, as agreed by Alcubierre and others.

 

[PeterDonis]

The ship can also go a long way around, running the warp drive only far away from A and B and flying the rest with conventional engines.

A and B are points in spacetime, not space. You are stipulating that a light ray not going through the warp bubble could not reach B from A. That means the warp bubble has to affect the spacetime geometry between those events.

 

[PeterDonis]

A small kernel of heavily warped spacetime in a large ocean of flat-ish spacetime such as between here and alpha centauri.

See my response to @DrStupid just now.

 

[PeterDonis]

The math is easy and can be found from my own blog

Your blog is not a valid reference. Please do not reference it again. If you do you will receive a warning.

 

[PeterDonis]

Adherents of the drive accept that it will allow me to send messages to my friend 4 ly away in what he and I both agree to be 3 years.

A lot of complexities that you are blithely ignoring are lurking underneath your simple-looking statement "in what he and I both agree to be 3 years". As has already been pointed out multiple times, nothing moves outside the light cones in Alcubierre spacetime. You cannot just wave your hands and ignore the spacetime curvature involved.

 

[DrStupid]

You are stipulating that a light ray not going through the warp bubble could not reach B from A.

No, I'm stipulating, that a light ray could reach B from A, not going throuth the warp bubble. There may also be light rays, reaching B from A, going through the warp bubble. But I'm not interested in them because I'm currently checking causality in the flat parts of the space-time.

 

[Ibix]

Alcubierre's 1994 paper notes that his spacetime is globally hyperbolic and therefore does not contain any causal loops.

Everett's paper (linked upthread) points out that Alcubierre's solution can be written as $\eta_{\mu\nu}+h_{\mu\nu}$ and that $h$ is very small except near the bubble. So you can have two warp bubbles traveling in opposite directions essentially as a linear superposition of two Alcubierre spacetimes - i.e. $\eta_{\mu\nu}+h_{\mu\nu}+h'_{\mu\nu}$ where the two $h$s are warp bubbles moving in $\pm x$ directions and offset by some distance in the $y$ direction so that they do not interfere with each other. Furthermore, he argues that there's no reason that the two warp bubbles have to share a notion of "at rest" and both can arrive at their destinations in arbitrarily short coordinate times for their rest frames. Thus he says you can build a tachyonic anti-telephone with two warp drives, although you can't do it with one.

I don't know whether or not Everett's analysis is correct. It seems plausible on the face of it, but if I understand correctly Alcubierre's paper assumes a spacetime in which CTC's are impossible before showing that the warp drive is a solution. So it's not completely clear to me that simply adding two warp bubbles can work quite as Everett claims - he seems to have ended up with a spacetime that isn't consistent with the one Alcubierre used.

I may be wrong - this is just my impression from reading the papers this morning.

 

[PeterDonis]

I'm stipulating, that a light ray could reach B from A, not going throuth the warp bubble.

Then there is no "violation of causality" even in the (mistaken) sense you are using the term, since events A and B can be connected by light rays in the absence of any warp bubble, so even "causality in the flat parts of the spacetime" is not violated (that would require events A and B to not be connected by any timelike or null paths that do not go through the warp bubble).

 

[PeterDonis]

it's not completely clear to me that simply adding two warp bubbles can work quite as Everett claims

Since the EFE is nonlinear, we should not expect simply adding together two solutions to give a solution. So it's quite possible that the scenario Everett describes is not in fact a solution of the EFE.

 

[martinbn]

Since the EFE is nonlinear, we should not expect simply adding together two solutions to give a solution. So it's quite possible that the scenario Everett describes is not in fact a solution of the EFE.

It will be a solution with a different stress energy tensor. As long as the sum is still a Lorentzian metric, which it need not be in general. The paper is very sketchy, I am not sure that he achieves what he claims.

 

[PeterDonis]

It will be a solution with a different stress energy tensor. As long as the sum is still a Lorentzian metric

Yes, it's true that you can take any Lorentzian metric and call it a "solution" by simply computing its Einstein tensor and dividing it by $8 \pi$ and calling that the "stress-energy tensor".

 

[Ibix]

Since the EFE is nonlinear, we should not expect simply adding together two solutions to give a solution. So it's quite possible that the scenario Everett describes is not in fact a solution of the EFE.

That's kind of my point. Everett points out that the disturbances are small except near the warp bubble, and hence that the superposition ought to be near a solution (even if not actually a solution) as long as the warp bubbles don't get too close. However, "the components of a tensor are small" is not a coordinate independent statement and I don't think Everett shows that the components of one disturbance are small in the coordinate system in which the components of the other are small. So I don't think it's completely clear that adding the solutions is necessarily "nearly right" for a spacetime containing two warp drives. At least, not from that paper.

 

[DrStupid]

Then there is no "violation of causality" even in the (mistaken) sense you are using the term

Do you have a reference for the sense you are using the term?

that would require events A and B to not be connected by any timelike or null paths that do not go through the warp bubble

I am talking about events A (e.g. departure of the spaceship) and B (e.g. arrival of the spaceship at the destination) that cannot be connected by any timelike or null paths that do not go through the warp bubble.

 

[PeterDonis]

Do you have a reference for the sense you are using the term?

Do you?

I am talking about events A (e.g. departure of the spaceship) and B (e.g. arrival of the spaceship at the destination) that cannot be connected by any timelike or null paths that do not go through the warp bubble.

Then you are contradicting yourself, since you said:

I'm stipulating, that a light ray could reach B from A, not going throuth the warp bubble.

If a light ray could reach B from A, not going through the warp bubble, then that light ray's worldline is a null path connecting A and B.

 

[Ibix]

I am talking about events A (e.g. departure of the spaceship) and B (e.g. arrival of the spaceship at the destination) that cannot be connected by any timelike or null paths that do not go through the warp bubble.

Alcubierre's solution has a globally applicable notion of "forward in time", picked out by his initial choice of foliation. So it cannot include causal paradoxes.

Everett, as far as I understand it, points out that the choice of foliation is not unique. Thus causal paradoxes become possible in a more general spacetime that includes multiple warp bubbles. However I am not convinced by his argument because it does not appear clear to me that you can combine two Alcubierre solutions in the way he does (or, at least, that it must necessarily have the properties he ascribes to such a combination). I could, of course, be missing something.

 

[PeterDonis]

Alcubierre's solution has a globally applicable notion of "forward in time", picked out by his initial choice of foliation.

The particular notion of "forward in time" that he describes might be foliation dependent, but as you pointed out in a previous post, the key property for the absence of causal loops is that the spacetime is globally hyperbolic, and that property is not foliation dependent; it's an invariant geometric property.

 

[PAllen]

Alcubierre's solution has a globally applicable notion of "forward in time", picked out by his initial choice of foliation. So it cannot include causal paradoxes.

Everett, as far as I understand it, points out that the choice of foliation is not unique. Thus causal paradoxes become possible in a more general spacetime that includes multiple warp bubbles. However I am not convinced by his argument because it does not appear clear to me that you can combine two Alcubierre solutions in the way he does (or, at least, that it must necessarily have the properties he ascribes to such a combination). I could, of course, be missing something.

It seems to me the only thing missing from Everett's paper is a more explicit demonstration that the combined metric tensor he constructs is a valid metric tensor. It would not be expected to preserve the global hyperbolicity of the one bubble metric. I would find it totally convincing if there were an argument that the candidate metric he writes in equation (10) [ confusingly using upper case G for something that is clearly meant as a metric] has the same signature everywhere. Everything else would hang together, IMO, if this were demonstrated. Continuity and such are already demonstrated (and G being symmetric is also obvious), and there are very minimal requirements to simply declaring some tensor on a manifold be treated as the fundamental tensor. Unchanging signature is the only nontrivial requirement that would not obviously hold in this case.

 

[DrStupid]

Do you?

https://en.wikipedia.org/wiki/Causality

Then you are contradicting yourself, since you said:
"I'm stipulating, that a light ray could reach B from A, not going throuth the warp bubble."

That referred to points A and B in space, not in spacetime (you already confused that in #45). I have to admit it was not a good idea to reuse the same symbols for events in a later post. In order to fix that, let me explain it again with a better notation:

Let's say we have four points A, B, C, D in space (not spacetime!). The points form a rectangle. The long edges A-D and B-C have a length of 5 LYs. The short edges A-B and C-D have a length of 1 LY. All points remain at rest in a frame of reference K.

At the time t = 0 a light signal is submitted from A to D. At the same time a spaceship starts from A to B at a speed of c/2. This is event X.
After 2 years the ship arrives at B, starts its warp engine and arrives C halve a yeat later. Than it goes with c/2 to D and arrives at t = 4.5 a. That is event Y.
The light signal (which is assumed to remain in flat space) reaches D at t = 5 a. That is event Z.

The events X and Y cannot be connected by any timelike or null paths that do not go through the warp bubble.

 

[PeterDonis]

https://en.wikipedia.org/wiki/Causality

Wikipedia is not a valid reference.

In any case, the word "causality" is not the important point; the physics is.

The events X and Y cannot be connected by any timelike or null paths that do not go through the warp bubble.

Ok, that makes it clearer what scenario you intended.

 

[PeterDonis]

With two events A and B, where one of them is the cause of the other, causality is violated if the order of the events is frame dependent.

Now that you've clarified your scenario, I can respond to this. Your statement is only true in flat spacetime. The spacetime in your scenario is not flat. The fact that you have a "basically flat" region in it that the light signal between events X and Y traverses does not make the spacetime as a whole flat.

What you have is a spacetime where there is a pair of events, X and Y, which (a) are connected by a timelike path (through the bubble), and (b) have a frame-dependent ordering in some "inertial" frame ("inertial" is in quotes because there are no global inertial frames in a curved spacetime, but we can construct frames that, outside the warp bubble, are "inertial enough" given the asymptotically flat nature of the spacetime--we have to stipulate some restriction on the frame because it is always possible to construct non-inertial frames with different orderings for some chosen pair of events). In flat Minkowski spacetime, this would not be possible: any pair of events whose ordering is frame-dependent in different inertial frames cannot be connected by any timelike or null path.

Your interpretation of this is that "causality" is violated in Alcubierre spacetime. But the proper interpretation of this is that your definition of "causality" is too limited, since it only works for spacetimes that satisfy a condition (the one I just described above) which Alcubierre spacetime violates. (Which just illustrates why you should not get your definitions from Wikipedia.) Since you asked for a reference earlier, if you want the definitive treatment of "causality" for general curved spacetimes, check out Hawking & Ellis, which treats the subject in exhaustive detail. They point out that there are multiple possible causality conditions on spacetimes; of those, globally hyperbolic is the strongest, and it is the one satisfied by Alcubierre spacetime, as was pointed out earlier in the thread. The brief summary in the Wikipedia article is not bad as a quick overview, but of course leaves a lot out:

https://en.wikipedia.org/wiki/Causality_conditions

 

[PAllen]

It seems to me the only thing missing from Everett's paper is a more explicit demonstration that the combined metric tensor he constructs is a valid metric tensor. It would not be expected to preserve the global hyperbolicity of the one bubble metric. I would find it totally convincing if there were an argument that the candidate metric he writes in equation (10) [ confusingly using upper case G for something that is clearly meant as a metric] has the same signature everywhere. Everything else would hang together, IMO, if this were demonstrated. Continuity and such are already demonstrated (and G being symmetric is also obvious), and there are very minimal requirements to simply declaring some tensor on a manifold be treated as the fundamental tensor. Unchanging signature is the only nontrivial requirement that would not obviously hold in this case.

Thinking a little more about this, I think the above issue is dealt with in the paper, if a bit obliquely. The two bubbles are arranged to be displaced from each other, with essentially flat spacetime between them. The closed timelike curve takes a timelike path out of one bubble and into another. The non overlap of the each bubble’s contribution to the metric makes it obvious that there is no signature change issue. Thus I claim there are no substantive issues with the paper. Obviously, it passed peer review to appear in Phys. Rev. D.

 

[PeterDonis]

The closed timelike curve takes a timelike path out of one bubble and into another. The non overlap of the each bubble’s contribution to the metric makes it obvious that there is no signature change issue.

The problem I see with this reasoning is that, if the bubbles really don't overlap, then the spacetime containing two bubbles should be globally hyperbolic since the spacetime containing one bubble is. But you can't have CTCs in a globally hyperbolic spacetime.

The part I think it might be worth focusing in on is the "timelike path out of one bubble and into another". I'm not sure this is actually possible in a way that allows a CTC to form given the other restrictions involved (that the bubble's don't overlap and that spacetime is basically flat outside the bubbles).

 

[PAllen]

The problem I see with this reasoning is that, if the bubbles really don't overlap, then the spacetime containing two bubbles should be globally hyperbolic since the spacetime containing one bubble is. But you can't have CTCs in a globally hyperbolic spacetime.

I don’t think this claim is true. Consider two flat topologically trivial Minkowski spaces. Cut a section of each and out them together right and you have Minkowski space with CTCs due to nontrivial topology. Everetts’s construction joins a cut of two globally hyperbolic solutions together, the cut being in the essentially flat region of each. There is no expectation this will necessarily preserve global hyperbolocity. The physical claim is that the procedure simply amounts to building two warp bubbles using exotic ingredients at different relative speed in near flat spacetime. If one is possible, why not two, arranged as described?

They discuss the timelike path from one bubble to the other in some detail.

 

[PeterDonis]

Cut a section of each and out them together right and you have Minkowski space with CTCs due to nontrivial topology.

It is known that this can be done, but I don't think it's relevant here since Everett's solution does not appear to involve nontrivial topology.

Everetts’s construction joins a cut of two globally hyperbolic solutions together, the cut being in the essentially flat region of each. There is no expectation this will necessarily preserve global hyperbolocity.

In general there is no theorem that says what kind of cutting and joining of globally hyperbolic solutions will preserv global hyperbolicity, true. But in this specific case I think it should.

Global hyperbolicity is equivalent to the existence of a Cauchy surface for the spacetime. In flat Minkowski spacetime, any surface of constant coordinate time in an inertial frame is a Cauchy surface. In a spacetime with a single warp bubble, otherwise flat, the same should be true: any surface of constant coordinate time in a frame which is inertial far away from the bubble should be a Cauchy surface. But if this is true for one bubble, it should also be true for two bubbles that do not overlap. So the two-bubble spacetime Everett describes should also have a Cauchy surface.

The physical claim is that the procedure simply amounts to building two warp bubbles using exotic ingredients at different relative speed in near flat spacetime. If one is possible, why not two, arranged as described?

I'm not disputing that a two warp bubble spacetime could be created if a one warp bubble spacetime could. The question is only about whether a two warp bubble spacetime can contain CTCs.

 

[PAllen]

In general there is no theorem that says what kind of cutting and joining of globally hyperbolic solutions will preserv global hyperbolicity, true. But in this specific case I think it should.

Global hyperbolicity is equivalent to the existence of a Cauchy surface for the spacetime. In flat Minkowski spacetime, any surface of constant coordinate time in an inertial frame is a Cauchy surface. In a spacetime with a single warp bubble, otherwise flat, the same should be true: any surface of constant coordinate time in a frame which is inertial far away from the bubble should be a Cauchy surface. But if this is true for one bubble, it should also be true for two bubbles that do not overlap. So the two-bubble spacetime Everett describes should also have a Cauchy surface.

I disagree with this. It is precisely the existence of two shortcuts with the right relation to each other that breaks global hyperbolicity. That is, it is the two bubbles boosted relative to each other that allow what was a Cauchy surface for one bubble to have two intersections by a causal curve, thus making it no longer a Cauchy surface.

 

[PeterDonis]

it is the two bubbles boosted relative to each other that allow what was a Cauchy surface for one bubble to have two intersections by a causal curve

I'll have to work through the paper again, because I'm not really seeing how this can work. I understand the author is claiming basically this, but the paper is too sketchy for me to accept that claim at face value.

 

[PAllen]

I'll have to work through the paper again, because I'm not really seeing how this can work. I understand the author is claiming basically this, but the paper is too sketchy for me to accept that claim at face value.

Note that the 2 bubble CTCs only cause only 'some' partial cauchy surfaces to have two intersections with a causal curve. 'Most' partial Cauchy surfaces fail be be Cauchy surfaces because some points in the future or past of the surface have causal curves through them that don't intersect the partial Cauchy surface at all. This makes the surface fail the condition that D+ U D- U S be the whole manifold, so it is not a Cauchy surface. In this case, the CTCs created by the two bubbles will not be in this set, for most candidate Cauchy surface.

I really don't see any basis to claim that you can't join two sections of spacetimes each globally hyperbolic in such a way that the result is not. The more I think about it, the less reason I see for any objection.

Can you try to clarify why you think there should be a problem with Everett's construction?