Does the Alcubierre drive shorten distances?

[Dale]

Yes, it's true, I made a mistake when I said I was referring to the length of the path traveled, when what I had in mind was the integral of peculiar velocity with respect to time.

So then, since the peculiar velocity is always $c$ the length of the path is just $c$ times the time.

So, if you use the time coordinate inside the bubble you will get one distance and if you use the time coordinate outside the bubble you will get a different distance. And you can change to any distance that you want simply by changing coordinates. And this is true for flat spacetime too, not just curved spacetimes.

 

[Jaime Rudas]

That's basically the argument Alcubierre appeared to be making in what you quoted from him earlier in this thread. The issue I see with it, as I pointed out before, is that he is relying on space being "Euclidean"--and it's not, not when the bubble and its boundaries are taken into account. In the diagram I referred to, this is obvious: a horizontal line, which is what corresponds to the "Euclidean space" Alcubierre is talking about, isn't even spacelike inside the bubble. So any intuitions about such a line representing a "space", or "distance" along it, simply aren't valid. One would actually have to do the math to see what happens when you try to integrate along an actual spacelike line that goes from the left edge of that diagram to the right edge and passes through the bubble on the way.

Taking this into account, could you indicate the calculation to support your claim that the distance between Earth and Alpha Centauri would be much less than 4.3 light-years if you measured it along a path going through the warp bubble?

 

[PeterDonis]

In the diagram I referred to, this is obvious: a horizontal line, which is what corresponds to the "Euclidean space" Alcubierre is talking about, isn't even spacelike inside the bubble.

Hm. The diagram does show that, but looking at the math in the same paper, I'm not sure the diagram shows the light cones correctly.

The slope of the forward edge of the light cone, i.e., the one that points "to the right" of the timelike path of the observer at the center of the bubble, is $1 + v f$. This slope is $dz / dt$, i.e., the change in $z$ as $t$ increases. This will always be positive. But for the light cone edge in that direction to "slope down" so that a horizontal line is not spacelike, $dz / dt$ would have to be negative. So the diagram can't be quite right; a horizontal line in the coordinates given does always have to be spacelike, even inside the bubble.

Given that, I have to withdraw the previous concern I expressed with interpreting such a horizontal line. Its physical meaning in the presence of the bubble is a different matter, which I still want to think more about, but integrating along such a line is indeed simple; for $dt = 0$ (and $dx = dy = 0$, since we're only looking at the $z$ spatial direction), the metric is simply $dz$.

 

[PeterDonis]

could you indicate the calculation to support your claim that the distance between Earth and Alpha Centauri would be much less than 4.3 light-years if you measured it along a path going through the warp bubble?

See my post #63 just now, which crossed yours in the mail, so to speak. The diagram I was relying on in that paper isn't actually correct--at least that's how it looks to me.

 

[Jaime Rudas]

So then, since the peculiar velocity is always $c$ the length of the path is just $c$ times the time.

I agree.

So, if you use the time coordinate inside the bubble you will get one distance and if you use the time coordinate outside the bubble you will get a different distance. And you can change to any distance that you want simply by changing coordinates. And this is true for flat spacetime too, not just curved spacetimes.

I'm not entirely clear on the relationship of this to what I mentioned about peculiar velocity, but it seems to me that the time coordinate inside the bubble is equal to the time coordinate outside the bubble.

 

[PeterDonis]

could you indicate the calculation to support your claim that the distance between Earth and Alpha Centauri would be much less than 4.3 light-years if you measured it along a path going through the warp bubble?

As I said, the "calculation" was simple: inside the bubble, spacetime is flat, so "distance equals speed times travel time" works fine. So since the travel time is much less than 4.3 years, the distance must be much less than 4.3 light years.

However, given what I said in post #63, my reason for relying on logic like I just described, instead of looking at "horizontal distance in the diagram" as a "distance", is no longer valid. There are still multiple notions of "distance" that can be defined in this scenario, but I can't say now that the simple one of "horizontal distance in the diagram" isn't one of them--and that's true even for horizontal lines in the diagram that pass through the bubble (which I thought before could not be the case, as I thought such lines were not everywhere spacelike).

 

[PeterDonis]

it seems to me that the time coordinate inside the bubble is equal to the time coordinate outside the bubble.

More precisely, in the coordinates used in the paper I've been referencing, with the diagram in Figure 3 that I posted about in post #63, the "time coordinate" is just "up" in that diagram--the time "grid lines" are always vertical, even inside the bubble.

What is weird is that, for an observer who always remains in the center of the bubble, the time coordinate $t$ also represents their proper time--the time elapsed on their clocks. That's weird because the worldline of such an observer, while the bubble is moving at "warp speed", is very, very far from being vertical. But because of the way spacetime is curved in the bubble, that observer's clock still ticks at the same rate, relative to the $t$ coordinate, as clocks way outside the bubble where spacetime is flat. That's the sense in which time inside the bubble is "the same" as time outside.

 

[PeterDonis]

I agree.

Note, though, that by this definition, using the notion of "length" of "spacelike distance along a horizontal line in the diagram", the "peculiar velocity" of light is not always $c$ (or $1$ in the units being used in the paper). It's $vf - 1$ for "left-moving" light (which doesn't always move to the left) and $vf + 1$ for "right-moving" light. The former can actually be $+1$ if $vf = 2$ (which it will be at the center of the bubble for the values given in Figure 3 of the paper--but note that this means "left-moving" light is actually moving to the right at $c$, not to the left). But the latter will always be greater than $1$ (it's $3$ at the center of the bubble for the values given in the paper).

And note that that notion of "length" is the one in which the distance from Earth to Alpha Centauri is 4.3 light years, even along a horizontal line that goes through the bubble. If there is any notion of length in this spacetime that does make the "peculiar velocity" of light always $c$ (or $1$ in the natural units of the paper), it will be a different one from that. As far as I can tell, there will be no notion of "distance" for which the peculiar velocity of light is always $c$ and the distance from Earth to Alpha Centauri is 4.3 light years.

 

[Jaime Rudas]

As I said, the "calculation" was simple: inside the bubble, spacetime is flat, so "distance equals speed times travel time" works fine. So since the travel time is much less than 4.3 years, the distance must be much less than 4.3 light years.

It seems to me that this applies to distances inside the bubble, but not to distances along a path going through the bubble.

 

[PeterDonis]

It seems to me that this applies to distances inside the bubble, but not to distances along a path going through the bubble.

Please, read all of my latest posts before responding.

 

[Dale]

it seems to me that the time coordinate inside the bubble is equal to the time coordinate outside the bubble

There isn’t any one “the time coordinate”. You can choose time coordinates pretty arbitrarily, both inside and outside. Whatever you want the answer to be (within limits) you can choose a time coordinates pretty arbitrarily to get that answer.

There is nothing inherently wrong with that. It is just a consequence of how you chose to measure distance. The distance you chose is just $c$ times the time, and the time depends on your coordinates. Pick a distance you like and you can choose a time coordinate to get that distance. Any choice you make is valid.

 

[Jaime Rudas]

What is weird is that, for an observer who always remains in the center of the bubble, the time coordinate $t$ also represents their proper time--the time elapsed on their clocks. That's weird because the worldline of such an observer, while the bubble is moving at "warp speed", is very, very far from being vertical. But because of the way spacetime is curved in the bubble, that observer's clock still ticks at the same rate, relative to the $t$ coordinate, as clocks way outside the bubble where spacetime is flat. That's the sense in which time inside the bubble is "the same" as time outside.

It doesn't seem weird to me because spacetime at the center of the bubble is flat. The extremely curved spacetime is that of the "walls" of the bubble, not its center.

 

[Jaime Rudas]

Please, read all of my latest posts before responding.

Yes, it seems we're having problems with the issue of relative simultaneity.

 

[PAllen]

It doesn't seem weird to me because spacetime at the center of the bubble is flat. The extremely curved spacetime is that of the "walls" of the bubble, not its center.

Note the spacetime inside a spherical shell of matter is flat, but proper time for the at rest observer inside is NOT the same as for an at rest, far away observer outside (where the spacetime is also flat). Thus, your “intuition” is based on something that is, most case, false, but happens to be true in this one case.

 

[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

Note that the paper I referenced previously references another paper by Natario which constructs a warp drive spacetime in which the expansion is zero everywhere:

https://arxiv.org/pdf/gr-qc/0110086

This is, of course, a different metric from the Alcubierre one.

 

[Jaime Rudas]

Note the spacetime inside a spherical shell of matter is flat, but proper time for the at rest observer inside is NOT the same as for an at rest, far away observer outside (where the spacetime is also flat). Thus, your “intuition” is based on something that is, most case, false, but happens to be true in this one case.

I don't understand what you mean, nor do I see how it relates to what I posted.

 

[Jaime Rudas]

There isn’t any one “the time coordinate”. You can choose time coordinates pretty arbitrarily, both inside and outside. Whatever you want the answer to be (within limits) you can choose a time coordinates pretty arbitrarily to get that answer.

What I meant is that time inside the bubble flows at the same rate as outside (and far away from) the bubble.

 

[PAllen]

I don't understand what you mean, nor do I see how it relates to what I posted.

It relates exactly to what you said. You said it was intuitive that proper time inside the bubble was the same as outside because both were flat spacetime. My example shows this to be false expectation, and that it is, fact false in most cases. In fact, its ability to be true in the warp bubble case is related to the shell being exotic matter.

 

[PAllen]

What I meant is that time inside the bubble flows at the same rate as outside (and far away from) the bubble.

And this is an expectation which is false for almost any type of bubble or shell you can construct, where inside and far away are both flat. The given warp metric is the rare case where it is true.