Does the Alcubierre drive shorten distances?

[Jaime Rudas]

That's basically the argument Alcubierre appeared to be making in what you quoted from him earlier in this thread.

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.

 

[PAllen]

Getting back to the notion of distance, and how to make sense of it in relation to the warp drive metric, a few common conventions in GR as:

1) If you have a timelike congruence, and an orthogonal foliation for it, then defining distance in terms ot the geodesics of the induced 3-metric of the foliation makes a lot of sense. In particular, this is true FLRW spacetime and Minkowski spacetime. It is also possible for the stationary congruence outide a BH. It is rare, in the general case, and is not possible in the warp spacetime. Note that of these 3 examples, only in Minkowski spacetime are the geodesics of the slices also geodesics of the spacetime. In the other two cases, this is false.

2) A different notion, tailored to an individual observer world line, is to measure along spacetime geodesics orthogonal to the observer world line, meeting word lines of distant objects. This can be applied more generally, but only in the case of Minkowski space does it agree with the prior definition. Also, in general, it is not symmetric. That is, if you define distance using the destination object world line, starting from the same event reached by the prior geodesic, you will typically get a different distance back to the observer world line.

As far as I know, these are the most common approaches. Note, that neither of these can be made to give 13.8 billion ly for "light travel distance" in our cosmology.

Another less conventional approach I proposed in a long ago thread, also fails. This approach is to formalize the notion of an odometer. It does, in fact, idealize and generalize the way a real odometer works. The key concept to define what a road is in a general spacetime. The obvious choice is a timelike congruence. Then an integral for the amount of road passing an arbitrary timelike world line (along which an odometer operates) is given as follows. First note that the congruence defines 4-velocity vector field that I denote $\hat u({\mathbf x} )$, where the argument is an event in spacetime. Associated with an observer world line ${\mathbf x}(\tau)$, its 4-velocity tangent is $\hat v(\tau)$. Then define $\gamma(\tau) = \hat u \cdot \hat v$. Then, the odometer measures $$d=\int_{\tau_1}^{\tau_2} \sqrt{1-1/\gamma^2}\,d\tau$$
You might think this could 'solve' the issue of light travel distance as @Jaime Rudas hopes to define it, but it fails. The key to the odometer is that it is one instrument following a timelike trajectory, not an attempt to sum infinitesimal measurements each made in a different local frame by different instruments at different cosmological times. It could be applied to an observer with arbitrary peculiar velocity over time to define 'comoving road' passed over time. But it does not work for light.

Anyway, I don't see a good way to apply any of these definitions to the warp spacetime, except possibly defintion (2) above. I have not tried any calculation based on this yet.

 

[PeterDonis]

I don't understand what you mean

Consider a spacetime consisting of a static spherical shell with empty space both inside and outside. Consider two observers, one at rest at the center of the shell, and the other at rest very, very far away from the shell. Spacetime is flat locally along the worldlines of both observers.

These observers can verify that they are at rest relative to each other by exchanging round-trip light signals (we'll assume the shell is transparent to the light signals they are using), and checking that the round-trip travel time remains constant. But the round-trip travel times recorded on each of their clocks will not be the same: it will be less for the observer at the center of the shell, than for the observer far away. And if we adopt a global time coordinate for this spacetime in the natural way, then the far away observer's clock will tick at the same rate as the global time coordinate, but the clock of the observer at the center of the shell will tick slower. Even though spacetime is flat in the local regions around both of them.

 

[Dale]

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

Sure. And my point is that that is a choice of your time coordinate. You can make that choice, but you are not required to.

 

[PAllen]

An observation is that if you take the limit as peculiar velocity goes to c, of comoving odometer distance as i defined in my prior post, the result is 0, in all cases. So you can’t solve the light problem by taking the limit unless you like this answer.

 

[PeterDonis]

If you have a timelike congruence, and an orthogonal foliation for it

It is rare, in the general case, and is not possible in the warp spacetime.

Actually, it is possible, even in the warp spacetime. For the Alcubierre metric, if we define the timelike congruence of "Eulerian" observers, as defined in equation (201) of the paper I referenced, then these worldlines are everywhere orthogonal to surfaces of constant coordinate time $t$ in that chart--i.e., to the "horizontal" surfaces in the spacetime diagram in Figure 3 in the paper! And those surfaces are Cauchy surfaces that foliate the spacetime.

Note that that means the diagram in Figure 3 is misleading in another way besides the one I described in an earlier post, that it incorrectly shows the light cones. The diagram makes it seem that inside the bubble, the spacelike surfaces orthogonal to the worldlines of the in-bubble observers are perpendicular to the bubble's path, so to speak. But they're not. They're the horizontal lines in the diagram. So the bubble doesn't really "tilt" things, as the Figure would lead one to believe, so much as "squash" things in the direction it's moving--in that direction, the tangents to the bubble observers' worldlines can look really, really close to horizontal, even though they're still orthogonal, in the sense of the spacetime metric, to the horizontal lines.

 

[PAllen]

Actually, it is possible, even in the warp spacetime. For the Alcubierre metric, if we define the timelike congruence of "Eulerian" observers, as defined in equation (201) of the paper I referenced, then these worldlines are everywhere orthogonal to surfaces of constant coordinate time $t$ in that chart--i.e., to the "horizontal" surfaces in the spacetime diagram in Figure 3 in the paper! And those surfaces are Cauchy surfaces that foliate the spacetime.

Note that that means the diagram in Figure 3 is misleading in another way besides the one I described in an earlier post, that it incorrectly shows the light cones. The diagram makes it seem that inside the bubble, the spacelike surfaces orthogonal to the worldlines of the in-bubble observers are perpendicular to the bubble's path, so to speak. But they're not. They're the horizontal lines in the diagram. So the bubble doesn't really "tilt" things, as the Figure would lead one to believe, so much as "squash" things in the direction it's moving--in that direction, the tangents to the bubble observers' worldlines can look really, really close to horizontal, even though they're still orthogonal, in the sense of the spacetime metric, to the horizontal lines.

Those light cones are, indeed off, but only that the right bound of the cones can approach horizontal, but never reach it. That is a minor adjustment to the diagram (which visually seems to have them going below horizontal). With corrected light cones, it becomes clear that horizonal lines are spacelike, and that vertical lines inside the bubble may also be spacelike. It is just that the Eulerian observers (except to the very left and right of fig. 3) bend along to the right, threading the light cones. These don't look orthogonal to the horizontal slices, but mathematically they are.

This is interesting, because now all 3 of my proposed distance conventions could be applied.

 

[PAllen]

So, then, it is obvious that for a traveler in the center of the bubble, using Eulerian observers to define a road, the odometer distance traveled is 0.

Meanwhile, using the Eulerian congruence with the orthogonal slices, and computing distance within a slice, you get 4 light years for our example.

Next I'll try looking at orthogonal spacetime goedesics.

 

[PeterDonis]

That is a minor adjustment to the diagram

I'm not sure I'd say it's minor, since it can mislead the viewer about whether the horizontal surfaces are spacelike, which is an important point.

 

[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.

Yes, that's the case I was referring to.

 

[Jaime Rudas]

Sure. And my point is that that is a choice of your time coordinate. You can make that choice, but you are not required to.

I agree

 

[Jaime Rudas]

An observation is that if you take the limit as peculiar velocity goes to c, of comoving odometer distance as i defined in my prior post, the result is 0, in all cases. So you can’t solve the light problem by taking the limit unless you like this answer.

The fact that the peculiar speed of light is c is something I took from Davis & Lineweaver, and my knowledge of the subject isn't sufficient to discern whether you or they are correct.

 

[PAllen]

The fact that the peculiar speed of light is c is something I took from Davis & Lineweaver, and my knowledge of the subject isn't sufficient to discern whether you or they are correct.

No we are in agreement that the peculiar velocity is c. The difference is the notion of distance traveled. You have not provided any part of their discussion that uses your notion of distance traveled. The section you quoted earlier make no mention distance. A feature of the comoving odometer definition is that even though peculiar velocity is approaching c, proper time along the journey is approaching 0, so amount of comoving road going by approaches 0.