Does the Alcubierre drive shorten distances?

[Jaime Rudas]

TL;DR Summary

Would the distance between Earth and Alpha Centauri be much less than 4.3 light years if measured along a path through the Alcubierre warp bubble?

In this post, Peter Donis says:

If an Alcubierre warp drive could actually be built (which, since it requires exotic matter, it probably can't), it would allow you to, for example, travel to Alpha Centauri in much less than 4.3 years, even as seen by observers on Earth or Alpha Centauri. But the reason for this would be that the warp drive would drastically change the geometry of spacetime in between Earth and Alpha Centauri, such that the distance between them would be much less than 4.3 light years if you measured it along a path going through the warp bubble. So a light beam that passed through the warp bubble would get from Earth to Alpha Centauri even faster than the ship itself would.

My question is whether or not the distance between Earth and Alpha Centauri would actually be much less than 4.3 light years if measured along a path that passes through the Alcubierre warp bubble.

I understand that the Alcubierre bubble, by contracting space, shortens the distances forward, however, this shortening is permanently compensated by an expansion of the space behind, so the distance between Earth and Alpha Centauri would remain at all times invariant when measured along a path going through the warp bubble.

Wouldn't it be like that?

 

[PeterDonis]

Moderator's note: Thread moved to the relativity forum.

 

[PeterDonis]

I understand that the Alcubierre bubble, by contracting space, shortens the distances forward, however, this shortening is permanently compensated by an expansion of the space behind, so the distance between Earth and Alpha Centauri would remain at all times invariant when measured along a path going through the warp bubble.

Wouldn't it be like that?

No. The worldline of the spaceship inside the warp bubble is timelike; that means that the ship is not traveling faster than light with respect to the space inside the bubble. So if the travel time between Earth and Alpha Centauri is less than 4.3 years to the ship, that means that, with respect to the space inside the warp bubble, the distance traveled must be less than 4.3 light years.

The "shortening distances forward and expanding distances behind" interpretation, in other words, is misleading if it makes you think the overall distance is the same along paths that go through the bubble.

It might help to bear in mind that the entire path that the spaceship follows from Earth to Alpha Centauri is inside the bubble. The bubble travels with the ship and stays around it. So the "distance traveled" inside the bubble has to be very carefully interpreted, and one cannot really understand what is going on with simple intuitive pictures. The bubble itself is made of exotic matter, which has highly counterintuitive properties and also can't really be understood with simple intuitive pictures.

 

[Jaime Rudas]

No. The worldline of the spaceship inside the warp bubble is timelike; that means that the ship is not traveling faster than light with respect to the space inside the bubble.

Well, but I haven't said otherwise. In fact, I consider that the ship not only doesn't travel faster than light, but, in reality, it doesn't move, that is, while inside the bubble, the ship doesn't travel any distance, it remains static.

But my question isn't that, but whether or not the distance between Earth and Alpha Centauri would be much less than 4.3 light years if it were measured along a path going through the warp bubble.

 

[PeterDonis]

while inside the bubble, the ship doesn't travel any distance, it remains static.

With respect to the bubble, that is correct. The bubble itself is moving with respect to Earth and Alpha Centauri. Which means that, from the viewpoint of the ship inside the bubble, Earth and Alpha Centauri are moving. That doesn't change any of the logic I gave in my previous post.

my question isn't that, but whether or not the distance between Earth and Alpha Centauri would be much less than 4.3 light years if it were measured along a path going through the warp bubble.

My previous post should make it clear that that is the question I am answering. I even gave you the logic behind the answer.

Here's another way of getting the answer (which I described in the previous thread you linked to): suppose the ship emits a light beam towards Alpha Centauri. Since the ship's worldline is timelike, the light beam will arrive at Alpha Centauri before the ship itself does. Since the ship takes less than 4.3 years to make the trip, the light beam must take an even shorter time to make it by the ship's clock. That means the distance, according to the ship (which means as measured through the warp bubble) must be less than 4.3 light years.

 

[Jaime Rudas]

Here's another way of getting the answer (which I described in the previous thread you linked to): suppose the ship emits a light beam towards Alpha Centauri. Since the ship's worldline is timelike, the light beam will arrive at Alpha Centauri before the ship itself does. Since the ship takes less than 4.3 years to make the trip, the light beam must take an even shorter time to make it by the ship's clock. That means the distance, according to the ship (which means as measured through the warp bubble) must be less than 4.3 light years.

Yes, that is correct, but that would be the distance from the ship (that is, from inside the bubble) to Alpha Centauri and not the distance from Earth to Alpha Centauri measured along a path through the warp bubble.

 

[PeterDonis]

that would be the distance from the ship (that is, from inside the bubble) to Alpha Centauri and not the distance from Earth to Alpha Centauri measured along a path through the warp bubble.

If the ship is at Earth (or just left it) when it emits the light, they're the same thing.

 

[Nugatory]

Yes, that is correct, but that would be the distance from the ship (that is, from inside the bubble) to Alpha Centauri and not the distance from Earth to Alpha Centauri measured along a path through the warp bubble.

And how are they different if the light signal is emitted when the ship and the bubble are at the earth?

 

[Jaime Rudas]

If the ship is at Earth (or just left it) when it emits the light, they're the same thing.

The ship cannot be on or near Earth because the bubble requires enough empty space behind it to expand. It is precisely that empty space that expands that makes the difference.

 

[PeterDonis]

The ship cannot be on or near Earth because the bubble requires enough empty space behind it to expand.

First, if this is true, your whole scenario is impossible.

Second, why do you think it is true? Where in the math (as opposed to your handwaving description) does this restriction occur?

It is precisely that empty space that expands that makes the difference.

Please back this up with math.

 

[PeterDonis]

This previous thread is relevant:

https://www.physicsforums.com/threads/alcubierre-warp-drive-spacetime-diagram-explained.1001700/

 

[Jaime Rudas]

First, if this is true, your whole scenario is impossible.

Second, why do you think it is true? Where in the math (as opposed to your handwaving description) does this restriction occur?

Please back this up with math.

In this paper, Alcubierre explains how the perturbation of space-time occurs in the vicinity of a bubble of radius R where the tidal forces are very large. Taking this into account, he proposes the following procedure to avoid the impact of these large tidal forces:

To see how one can use this metric to make a round trip to a distant star in an arbitrary small time, let us consider the following situation: Two stars A and B are separated by a distance D in flat spacetime. At time To, a spaceship starts to move away from A at a speed V < 1 using its rocket engines. The spaceship then stops at a distance d away from A. I will assume that d is such that R ≪ d ≪ D. It is at this point that a disturbance of spacetime of the type described, centered at the spaceship’s position, first appears. This disturbance is such that the spaceship is pushed away from A with a coordinate acceleration that changes rapidly from 0 to a constant value a. Since the spaceship is initially at rest (Vs = 0), the disturbance will develop smoothly from flat spacetime (see equation (8)).
When the spaceship is halfway between A and B, the disturbance is modified in such a way that the coordinate acceleration changes rapidly from a to −a . If the coordinate acceleration in the second part of the trip is arranged in such a way as to be the opposite to the one we had in the first part, then the spaceship will eventually find itself at rest at a distance d away from B, at which time the disturbance of spacetime will disappear (since again Vs = 0 ). The journey is now completed by moving again through flat spacetime at a speed V."

 

[Vanadium 50]

cannot be on or near Earth because the bubble requires enough empty space behind it

How much is "enough" is a quantitative statement. That needs a calculation to back it up. Please show it,.

The alternative is saying "I can't do the calculation myself, but I am sure you are wrong". This seldom goes well.

 

[PeterDonis]

In this paper, Alcubierre explains how the perturbation of space-time occurs in the vicinity of a bubble of radius R where the tidal forces are very large. Taking this into account, he proposes the following procedure to avoid the impact of these large tidal forces:

First, none of this contradicts anything I've said, or supports the claims you have made.

Second, Alcubierre says "the disturbance is modified" to allow the spaceship to travel in the way he describes, without specifying how the modification happens. If you read the previous thread I linked to, and the papers referenced there, you will see that this "modification" cannot be done by the crew of the spaceship: it is literally impossible for them to causally affect the spacetime geometry in the required region.

When you work through things even more, you find that the only way to bring such a spacetime geometry into existence at all is to do it at some event far enough in the past that the entire bubble region, all the way from Earth to Alpha Centauri in your scenario, is in the future light cone of the event where the spacetime geometry is generated. In other words, saying "the disturbance is modified" in the middle of the trip is misleading: the entire "disturbance" has to be already in place before the trip ever takes place.

 

[PeterDonis]

Taking this into account, he proposes the following procedure to avoid the impact of these large tidal forces

To the extent the description you quote is relevant to this thread at all, it means you need to modify the scenario you posed in the OP to take it into account. It's no problem for those of us who are responding to you; you're the one who proposed the scenario, so it's on you to make sure the scenario as you pose it is consistent with whatever theory you're basing it on.

As for how such a modification would affect the discussion here, it wouldn't to any significant extent. What you quoted explicitly specifies that $d$, the distance from the starting point that the ship would have to travel before entering the bubble, is much smaller than $D$, the flat spacetime (i.e., without going through the bubble) distance from the starting point to the end point. So the trips at the start and end of distance $d$ at $V < 1$ don't significantly affect the discussion; we can just as easily talk about the segment from bubble entry to bubble exit, of flat spacetime length $D - 2d$, which is not significantly different from $D$ since $d$ is much smaller than $D$.

 

[Jaime Rudas]

First, none of this contradicts anything I've said,

And I didn't claim that it contradicts anything you had said.

or supports the claims you have made.

I consider it supports the fact that empty space behind the bubble is required.

Second, Alcubierre says "the disturbance is modified" to allow the spaceship to travel in the way he describes, without specifying how the modification happens. If you read the previous thread I linked to, and the papers referenced there, you will see that this "modification" cannot be done by the crew of the spaceship: it is literally impossible for them to causally affect the spacetime geometry in the required region.

When you work through things even more, you find that the only way to bring such a spacetime geometry into existence at all is to do it at some event far enough in the past that the entire bubble region, all the way from Earth to Alpha Centauri in your scenario, is in the future light cone of the event where the spacetime geometry is generated. In other words, saying "the disturbance is modified" in the middle of the trip is misleading: the entire "disturbance" has to be already in place before the trip ever takes place.

None of this contradicts anything I've said.

 

[Jaime Rudas]

How much is "enough" is a quantitative statement. That needs a calculation to back it up. Please show it,.

The alternative is saying "I can't do the calculation myself, but I am sure you are wrong". This seldom goes well.

There is a third alternative: I can't do the calculation myself and I haven't claimed that you or anyone else is wrong. Furthermore, the fact that I can't do the calculation myself doesn't mean that I am wrong that an empty space behind the bubble is required.

 

[Jaime Rudas]

To the extent the description you quote is relevant to this thread at all, it means you need to modify the scenario you posed in the OP to take it into account. It's no problem for those of us who are responding to you; you're the one who proposed the scenario, so it's on you to make sure the scenario as you pose it is consistent with whatever theory you're basing it on.

As for how such a modification would affect the discussion here, it wouldn't to any significant extent. What you quoted explicitly specifies that $d$, the distance from the starting point that the ship would have to travel before entering the bubble, is much smaller than $D$, the flat spacetime (i.e., without going through the bubble) distance from the starting point to the end point. So the trips at the start and end of distance $d$ at $V < 1$ don't significantly affect the discussion; we can just as easily talk about the segment from bubble entry to bubble exit, of flat spacetime length $D - 2d$, which is not significantly different from $D$ since $d$ is much smaller than $D$.

This is all correct, but even taking into account that $d$ is much smaller than $D$, a light beam emitted at Earth will not be able to pass through the warp bubble. That is, if the warp bubble is activated, a light beam emitted on Earth will not be able to pass through it and will reach Alpha Centauri 4.3 years later.

 

[PeterDonis]

And I didn't claim that it contradicts anything you had said.

Ok.

I consider it supports the fact that empty space behind the bubble is required.

Only a small amount, much smaller than the distance from Earth to Alpha Centauri. Which means, as I pointed out, that it's not significant for this discussion.

None of this contradicts anything I've said.

It means that if you are going to claim that "empty space behind the bubble is required", then you need to update your specification of the scenario to account for that.

There is a third alternative: I can't do the calculation myself and I haven't claimed that you or anyone else is wrong.

You're making a claim which the rest of us think is wrong: that the distance through the warp bubble is still 4.3 light years even though the ship takes much less than 4.3 years to travel that distance. The calculation you are being asked to do is the calculation to support that claim. If you can't support the claim with a calculation, you should withdraw it.

Furthermore, the fact that I can't do the calculation myself doesn't mean that I am wrong that an empty space behind the bubble is required.

The "empty space behind the bubble" issue is irrelevant to the claim I described above that the rest of us think is wrong.

even taking into account that $d$ is much smaller than $D$, a light beam emitted at Earth will not be able to pass through the warp bubble.

You need to work on your reading comprehension. Nobody has talked about a light beam emitted from Earth. We have talked about a light beam emitted by the ship inside the bubble just as the ship is starting up. Whether the start-up is right at Earth or a distance $d$ away from Earth is insignificant, since $d$ is much smaller than 4.3 light years.

@Jaime Rudas this discussion does not appear to be going anywhere. Either you can respond to the concerns raised by myself and others in this thread about the claim I described above, or you can't. If you can, you need to do so, not clutter the thread with irrelevancies. If you can't, just say so and we can close the thread. Or, if you are now withdrawing the claim, you can say that. But you need to do one of those things if this discussion is going to go anywhere.

 

[Jaime Rudas]

@Jaime Rudas this discussion does not appear to be going anywhere. Either you can respond to the concerns raised by myself and others in this thread about the claim I described above, or you can't. If you can, you need to do so, not clutter the thread with irrelevancies. If you can't, just say so and we can close the thread. Or, if you are now withdrawing the claim, you can say that. But you need to do one of those things if this discussion is going to go anywhere.

I want to respond to some concerns raised in this thread, but I need time to do so, therefore, I ask that you do not close it.

 

[PeterDonis]

I want to respond to some concerns raised in this thread, but I need time to do so, therefore, I ask that you do not close it.

That's fine, take whatever time you need.

 

[Jaime Rudas]

It means that if you are going to claim that "empty space behind the bubble is required", then you need to update your specification of the scenario to account for that.

By dealing with the Alcubierre bubble, we are implicitly assuming its characteristics, one of which is having this space behind. Furthermore, The original post talks about the distance from Earth to Alpha Centuari following a path through the Alcubierre bubble. Saying "through" implies that there is a space between the Earth and the bubble. Thus, I consider that those who modify the initially proposed scenario are those who propose that the bubble starts from the Earth.

You're making a claim which the rest of us think is wrong: that the distance through the warp bubble is still 4.3 light years even though the ship takes much less than 4.3 years to travel that distance. The calculation you are being asked to do is the calculation to support that claim. If you can't support the claim with a calculation, you should withdraw it.

In the original post I did not categorically claim that the distance from Earth to Alpha Centuari following a path through the Alcubierre bubble was 4.3 light years, but rather I raised it as a doubt. On the other hand, the only one who has stated that this is wrong has been you, but you have not explained where the error is. In post #3 you argued that drawing conclusions from simple intuitive images can be misleading, however, this does not mean that these conclusions are necessarily wrong.

You need to work on your reading comprehension. Nobody has talked about a light beam emitted from Earth. We have talked about a light beam emitted by the ship inside the bubble just as the ship is starting up. Whether the start-up is right at Earth or a distance $d$ away from Earth is insignificant, since $d$ is much smaller than 4.3 light years.

Yes, it is much smaller than 4.3 light years, but it is not insignificant compared to the distance between the bubble and Alpha Centauri which you claim is much less than 4.3 light years. Additionally, you calculate the path taken by the bubble to Alpha Centauri, but this is not the same as distance between departure and arrival, because we are dealing with an expanding and contracting space. For example, the CMB photons that we received today have traveled about 13.8 Gly, however, the site from which they were emitted is at a distance of about 46 Gly. For the case at hand, you disregard the first section $d$, arguing that its size is much smaller than the total distance between the two stars, however, you do not take into account that it is precisely that section that expands to occupy the greatest proportion of that distance.

 

[PeterDonis]

I consider that those who modify the initially proposed scenario are those who propose that the bubble starts from the Earth.

It's your scenario; it's up to you to specify it the way you want it specified. If you want to specify that the ship starts at some distance $d$ from Earth and ends at some distance $d$ from Alpha Centauri, then just say so. Claims like the one quoted above add nothing to the discussion and are just wasting everyone's time.

In the original post I did not categorically claim that the distance from Earth to Alpha Centuari following a path through the Alcubierre bubble was 4.3 light years, but rather I raised it as a doubt.

You ended your OP with the question: "Wouldn't it be like that?" That means you think the answer is yes, it would.

it is much smaller than 4.3 light years, but it is not insignificant compared to the distance between the bubble and Alpha Centauri which you claim is much less than 4.3 light years.

First, this is a quibble. Suppose the distance $d$ is 0.1 light year. Then we have a start point 0.1 light year from Earth, and an end point 0.1 light year from Alpha Centauri, and the distance between them, measured outside the bubble, is 4.1 light years. Fine. Then just substitute "start point" for "Earth", "end point" for "Alpha Centauri", and "4.1" for "4.3", and we have the same situation as before. From the viewpoint of the ship inside the bubble, the distance from start point to end point must be much less than 4.1 light years, because the ship takes much less than 4.1 years to cover it and the ship's worldline is timelike.

Second, no, that's not what I claimed. Which you recognize in your very next sentence:

Additionally, you calculate the path taken by the bubble to Alpha Centauri, but this is not the same as distance between departure and arrival, because we are dealing with an expanding and contracting space.

What you are really saying here is that there is no such thing as "the" distance between Earth and Alpha Centauri (or between the start point and end point, if we make the change I described above to account for the distance $d$), because we can specify different paths through spacetime that could be called such a "distance", and they have different lengths.

Which is fine as a starting point for investigation, but if you want to take this approach, then you need to specify which path through spacetime between Earth and Alpha Centauri you are interested in. At what event on Earth's worldline does the path start? At what event on Alpha Centauri's worldline does the path end? At what event does it enter the bubble? At what event does it exit the bubble? Why is the term "distance from Earth to Alpha Centauri, as measured through the bubble" appropriate to describe this path? (Drawing a spacetime diagram can be helpful for addressing these questions--note that there is one, at least a heuristic one, in one of the papers I referenced before.)

The last becomes particularly important when you compare whatever path this is with the one I described, that you call "the path taken by the bubble to Alpha Centauri", and which, from the point of view of the ship inside the bubble, is the natural thing to call "the distance from Earth to Alpha Centauri as measured through the bubble".

 

[PeterDonis]

the CMB photons that we received today have traveled about 13.8 Gly

No, the CMB photons that we receive today were emitted about 13.8 Gy ago. Saying that "the speed of light" means they must have traveled 13.8 Gly is not justified, because the spacetime they traveled in is curved.

the site from which they were emitted is at a distance of about 46 Gly.

The site from which they were emitted 13.8 Gy ago is at a distance of about 46 Gly now. But when the CMB was emitted, that site was much less than 13.8 Gly away from us. (In fact, using a CMB redshift of approximately 1100, that site was about 42 million light years from us when the CMB was emitted.)

In other words, you can't possibly do a correct analysis if you leave out time.

For the case at hand, you disregard the first section , arguing that its size is much smaller than the total distance between the two stars, however, you do not take into account that it is precisely that section that expands to occupy the greatest proportion of that distance.

Again you have left out time in your analysis.

This "expansion", if we assume for the sake of argument that that viewpoint is justified, happens after the bubble passes a particular point. So if $d$ is 0.1 light year, the value I gave in my last post, when the ship starts out, it doesn't matter that this "section" expands later on to a larger distance (if it in fact does). It's 0.1 light year when the ship starts out, and that's what matters from the ship's point of view. Similarly, the $d$ at the other end is 0.1 light year when the ship arrives; the fact that that "section" might have contracted previously from a much larger distance (again if we assume for the sake of argument that that viewpoint is justified) is irrelevant from the ship's point of view. From the ship's point of view, the "outside the bubble" distance it would have had to cover is 4.1 light years (4.3 minus 0.1 at the start minus 0.1 at the end), but it goes from start point to end point in much less than 4.1 years, so the "inside the bubble" distance it has to cover must be much less than 4.1 light years, because the ship's worldline is timelike.

 

[Ibix]

Is the underlying point here that the spacetime isn't static, so there isn't really a unique answer to how far is it from A to B? There's no definition of space that's agreeable to everyone, and you can probably get very different answers by making different definitions. Is there even a spacelike line from A to B that is "in" the warp bubble for the whole journey, or even a high percentage of it?

 

[PeterDonis]

Is the underlying point here that the spacetime isn't static, so there isn't really a unique answer to how far is it from A to B?

I think that is at least part of the picture, yes.

Is there even a spacelike line from A to B that is "in" the warp bubble for the whole journey, or even a high percentage of it?

I think this depends on which point of view you adopt, the ship's or the Earth/Alpha Centauri one. (It might also depend on how large the superluminal bubble "velocity" $v$ is.)

Another point that I think is important is to look at lines that are spacelike inside the bubble and think about how they look from outside the bubble. Do they still look spacelike? Or do they look timelike?

 

[Ibix]

I think this depends on which point of view you adopt, the ship's or the Earth/Alpha Centauri one.

As in, again, there's some flexibility about which regions of spacetime are "inside" the warp bubble? It seems pretty clear for an eternal bubble, (at least in the high-$\sigma$ case), but it isn't clear to me what a forming warp bubble looks like.

 

[PeterDonis]

there's some flexibility about which regions of spacetime are "inside" the warp bubble?

There is some unavoidable imprecision in where the "boundary" of the bubble is as it's traveling, but that imprecision is negligible for this discussion.

it isn't clear to me what a forming warp bubble looks like.

See the previous thread I linked to in post #11, and in particular the spacetime diagram in Fig. 3 of the lecture notes linked to in the OP of that thread.

Basically, if you think of the warp bubble as a region where the light cones can "tip over" by more than 45 degrees, then forming the warp bubble just means starting from 0 tipping and tipping the light cones inside the bubble over to whatever the final angle is, and stopping the warp bubble just means "untipping" the light cones inside the bubble back to 0 degrees. That's what the diagram shows. The "exotic matter" part comes in at the "walls" of the warp bubble, where the highly tipped light cones have to continuously "untip" through the boundary back to where they are in the flat spacetime outside the bubble--doing that requires stress-energy in the walls that violates the energy conditions.

 

[Jaime Rudas]

You're making a claim which the rest of us think is wrong: that the distance through the warp bubble is still 4.3 light years even though the ship takes much less than 4.3 years to travel that distance. The calculation you are being asked to do is the calculation to support that claim. If you can't support the claim with a calculation, you should withdraw it.

Regarding this, I asked Miguel Alcubierre the following:

In a physics discussion forum, Peter Donis claims that an Alcubierre warp drive would allow travel to Alpha Centauri in much less than 4.3 years because it would drastically change the geometry of spacetime. Thus, the distance between Earth and Alpha Centauri would be much less than 4.3 light-years if measured along a path going through the warp bubble.

I argued that the Alcubierre bubble, by contracting space, shortens the distances forward, however, this shortening is permanently compensated by an expansion of the space behind, so the distance between Earth and Alpha Centauri would remain at all times invariant when measured along a path going through the warp bubble.

After a lengthy discussion, Peter Donis informs me that forum participants consider wrong my claim that the distance through the warp bubble is still 4.3 light years even though the ship takes much less than 4.3 years to travel that distance. He also suggests that I must support my claim with a calculation or withdraw it.

Could you please tell me who you think is correct?

And if it's not too much to ask, could you perform a calculation of the distance between Earth and Alpha Centauri, measured along a path going through the warp bubble?

Alcubierre answered:

Normally I don't answer these types of messages, but on this occasion I will make an exception. You are right, at all times the distance between Alpha Centauri and Earth remains constant in my model, since three-dimensional space is always Euclidean. What happens with the warp bubble is that in front of the bubble the volume elements (or distance if you prefer) contract, while behind it they expand, so that the total distance remains unchanged.

No “calculation” needs to be done to show this. The original metric in my article clearly shows (by construction) that the geometry of three-dimensional space is always perfectly Euclidean.

 

[PeterDonis]

Alcubierre answered

It's neat that he responded!

I would be interested to see his take on the spacetime diagram I mentioned in post #28. I would particularly be interested to see his description of how the Euclidean 3-dimensional spacelike slices he refers to are drawn on that diagram (they would be 1-dimensional spacelike lines on the diagram, which only considers motion in one spatial direction, but that would be sufficient to illustrate his meaning).

I would also be interested in his answer to the issue I have raised several times now: the ship's worldline is timelike, and it takes much less than 4.3 (or 4.1 if we allow for the distance $d$ at the start and end) years for the ship to go from the start point to the end point, so the distance the ship travels must be much less than 4.3 (or 4.1) light years.

 

[Jaime Rudas]

It's neat that he responded!

And the superluminal velocity with which he did it.

I would be interested to see his take on the spacetime diagram I mentioned in post #28. I would particularly be interested to see his description of how the Euclidean 3-dimensional spacelike slices he refers to are drawn on that diagram (they would be 1-dimensional spacelike lines on the diagram, which only considers motion in one spatial direction, but that would be sufficient to illustrate his meaning).

I would also be interested in his answer to the issue I have raised several times now: the ship's worldline is timelike, and it takes much less than 4.3 (or 4.1 if we allow for the distance $d$ at the start and end) years for the ship to go from the start point to the end point, so the distance the ship travels must be much less than 4.3 (or 4.1) light years.

Well, with his "normally I don't answer", I wouldn't want to abuse his kindness.

 

[PeterDonis]

The original metric in my article clearly shows (by construction) that the geometry of three-dimensional space is always perfectly Euclidean.

There is a comment to be made about this as well. What he means here by "three-dimensional space" is "a surface of constant $t$ in the coordinates in which the metric is standardly written". But while it is true that one can "read off" from his metric, without requiring any calculation, that the metric of such a surface is $dx^2 + dy^2 + dz^2$ (which makes it look like Euclidean 3-space), the intuition that makes us call that surface "three-dimensional space" is that it is a "surface of constant time", i.e., a surface with $dt = 0$. But for the "warp" case $v > 1$, the $t$ coordinate is not timelike! That is, $t$ is not a valid "time" coordinate, and surfaces of constant $t$ are not valid "surfaces of constant time".

 

[PeterDonis]

with his "normally I don't answer", I wouldn't want to abuse his kindness.

Yes, I understand that. What I was actually hoping is that someone would be able to find papers in the literature where the issues I have raised are already addressed. Unfortunately I have so far not been able to find any. That is somewhat surprising to me, but it might be that the questions we are discussing have simply not come up in a way that would generate a paper addressing them. "Warp drive" physics is something of a niche field and there might not be many physicists actually taking the time to look at the details.