# I Is the Alcubierre Warp Drive possible?

#### PeterDonis

Mentor
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

Mentor
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

Mentor
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.

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#### 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

Mentor
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

Mentor
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.

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#### DrStupid

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 space ship 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

Mentor
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

Mentor
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:

#### 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

Mentor
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.

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#### PeterDonis

Mentor
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

Mentor
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?

#### PeterDonis

Mentor
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.
I'm not making this claim as a general claim; as a general claim it would be way too strong. I'm only making it in the specific case of Everett's construction.

Can you try to clarify why you think there should be a problem with Everett's construction?
I'm not saying there's a problem with the construction in itself. I'm questioning whether the construction Everett describes will actually contain CTCs. I need to go through the paper again and see if I can fill in the details he leaves out.

#### PAllen

I'm not making this claim as a general claim; as a general claim it would be way too strong. I'm only making it in the specific case of Everett's construction.

I'm not saying there's a problem with the construction in itself. I'm questioning whether the construction Everett describes will actually contain CTCs. I need to go through the paper again and see if I can fill in the details he leaves out.
One thing I found is that several papers published in major journals decades after Everett's paper cite it as an established, non-controversial result.

#### DrStupid

Your interpretation of this is that "causality" is violated in Alcubierre spacetime.
I know it is limited to Minkowski spacetime. That's why I added above that it is only works within the spacetime that is not affected by the Alcubierre warp drive. It just means that an observer who is not aware of the bubble would conclude that causality is violated and that the space ship travels faster than light. For some observers it would even look like the ship travels back in time. In that case it should always be possible for the ship to go all the way back and connect event X at position D with another event X' at position A with X' preceding X.

#### PeterDonis

Mentor
In that case it should always be possible for the ship to go all the way back and connect event X at position D with another event X' at position A with X' preceding X.
This is a spacetime with two warp bubbles, not one. One warp bubble can only go in one direction.

#### DrStupid

This is a spacetime with two warp bubbles, not one.
Yes, I assumed that to be obvious.

"Is the Alcubierre Warp Drive possible?"

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