Does the Alcubierre drive shorten distances?

[PAllen]

Yes, I get that, but it seems to me that there is a fundamental tension between using that as the definition of "the road" and trying to interpret what's going on as motion, or lack thereof, in "Euclidean space", because in the "Euclidean space" of the usual warp coordinates, at least some Eulerian worldlines are moving--they do not have constant spatial coordinates in that chart. That's also true in the modified chart using $\xi$ as the spatial coordinate instead of $x$.

If you want a measure of travel distance inside the bubble, the only simple choice is relative to Eulerian world lines. Other simple choices have you measuring almost all outside the bubble.

I suppose one could try to just construct a chart using the Eulerian worldlines themselves as the timelike "grid lines" and see what "space" (surfaces of constant time) looks like in that chart. It might be that even in such a chart (which would be neither of the ones we've discussed so far, but sort of a mismash of the two), "space" would still be Euclidean, since those horizontal surfaces are indeed orthogonal to all the Eulerian worldlines. If that's true, this whole business of transforming between these different coordinate charts would be something like a Galilean transformation in Newtonian physics--but with the "relative speed" of the transformation varying in space.

To me, this was a key insight some time ago - the Euclidean slices are orthogonal to the Eulerian world lines, everywhere. Thus this is the only orthogonal foliation. The metric would thus be diagonal in this chart. However, a benefit of my odometer definition is there is no need to transform to these coordinates. It would change nothing about the computation. The result would still be that the odometer relative to the Eulerian road only measures distance during the short trip into the bubble and out of the bubble.

 

[PeterDonis]

If you want a measure of travel distance inside the bubble, the only simple choice is relative to Eulerian world lines. Other simple choices have you measuring almost all outside the bubble.

I'm not sure what you mean by this. There are Eulerian worldlines outside the bubble as well, and using them as the "road" doesn't restrict the "road" to just the bubble and its boundaries.

 

[PeterDonis]

the odometer relative to the Eulerian road only measures distance during the short trip into the bubble and out of the bubble.

I'm not sure what you mean by this either. If we want to find a $\hat{v}$ with nonzero odometer distance by your definition, we just need to construct it so that it has $\gamma > 1$ by your definition. That's straightforward. I'll work the construction for $\gamma = 2$, which gives for your odometer distance

$$
d = \frac{\sqrt{3}}{2} t
$$

where $t$ is the elapsed coordinate time in the warp coordinates given in the first paper (the one with the diagram we've referred to).

We have $\hat{u} = \left( 1, v_s f \right)$, where I'll give vector components in $(t, x)$ form. We want to find $\hat{v}$ such that $\hat{u} \cdot \hat{v} = 2$ and $| \hat{v} |^2 = 1$ (i.e., $\hat{v}$ is a timelike unit vector--I'm using the timelike signature convention in this post, which is a sign switch from the spacelike convention used in the paper, but makes this computation easier).

Let $\hat{v} = (v^t, v^x)$. The line element is

$$
ds^2 = dt^2 - \left( dx - v_s f dt \right)^2
$$

Then we have

$$
\hat{u} \cdot \hat{v} = v^t
$$

(because the last term on the RHS of the line element conveniently vanishes when we take any dot product with $\hat{u}$), and

$$
| \hat{v} |^2 = \left( v^t \right)^2 - \left( v^x - v_s f v^t \right)^2
$$

Since we want $\gamma = 2$, that gives $v^t = 2$, and plugging that into the equation for the squared norm of $\hat{v}$, and choosing $v^x$ so that it's moving to the left outside the bubble, we get $\hat{v} = \left( 2, 2 v_s f - \sqrt{3} \right)$. This describes a worldline that, far outside the bubble, moves to the left (the minus $x$ direction) at 87% of the speed of light, but inside the bubble, gets "dragged" to the right, but is still moving to the right (in the Euclidean space of the warp coordinates in the paper) slower than the Eulerian observers inside the bubble, so it's still accumulating leftward odometer distance relative to them.

 

[PAllen]

I'm not sure what you mean by this. There are Eulerian worldlines outside the bubble as well, and using them as the "road" doesn't restrict the "road" to just the bubble and its boundaries.

I mean that with the other distance definitions we used, the existence of the bubble was mostly irrelevant. The experience inside the bubble made minimal contribution. This is fine from the point of view of non bubble observers, but it doesn’t capture the experience of the observer mostly inside the bubble.

 

[PAllen]

I'm not sure what you mean by this either. If we want to find a $\hat{v}$ with nonzero odometer distance by your definition, we just need to construct it so that it has $\gamma > 1$ by your definition. That's straightforward. I'll work the construction for $\gamma = 2$, which gives for your odometer distance

$$
d = \frac{\sqrt{3}}{2} t
$$

where $t$ is the elapsed coordinate time in the warp coordinates given in the first paper (the one with the diagram we've referred to).

We have $\hat{u} = \left( 1, v_s f \right)$, where I'll give vector components in $(t, x)$ form. We want to find $\hat{v}$ such that $\hat{u} \cdot \hat{v} = 2$ and $| \hat{v} |^2 = 1$ (i.e., $\hat{v}$ is a timelike unit vector--I'm using the timelike signature convention in this post, which is a sign switch from the spacelike convention used in the paper, but makes this computation easier).

Let $\hat{v} = (v^t, v^x)$. The line element is

$$
ds^2 = dt^2 - \left( dx - v_s f dt \right)^2
$$

Then we have

$$
\hat{u} \cdot \hat{v} = v^t
$$

(because the last term on the RHS of the line element conveniently vanishes when we take any dot product with $\hat{u}$), and

$$
| \hat{v} |^2 = \left( v^t \right)^2 - \left( v^x - v_s f v^t \right)^2
$$

Since we want $\gamma = 2$, that gives $v^t = 2$, and plugging that into the equation for the squared norm of $\hat{v}$, and choosing $v^x$ so that it's moving to the left outside the bubble, we get $\hat{v} = \left( 2, 2 v_s f - \sqrt{3} \right)$. This describes a worldline that, far outside the bubble, moves to the left (the minus $x$ direction) at 87% of the speed of light, but inside the bubble, gets "dragged" to the right, but is still moving to the right (in the Euclidean space of the warp coordinates in the paper) slower than the Eulerian observers inside the bubble, so it's still accumulating leftward odometer distance relative to them.

Not remotely what I meant. Considering the famous fig. 3, I assume the world line of earth is vertical and just to the left of the bubble region, Alpha Centauri vertical and just to the right of it. I want travel from one to the other. Starting from point on earth world line, i move at modest speed (slanted to the right in the diagram) arriving at the bubble center just as it begins to accelerate. Then stay put until the bubble center has decelerated near appliance centauri. Then move to the right a small amount at modest speed. My odometer distance traveled will be completely determined by the short journies into and out of the bubble.

 

[PeterDonis]

I mean that with the other distance definitions we used, the existence of the bubble was mostly irrelevant. The experience inside the bubble made minimal contribution.

Ah, I see. Well, to the extent that your odometer distance inside the bubble is a lot less than, say, 4.3 light years, that's still true for your odometer definition. But I would agree that with your definition, it's a lot easier to understand the relative contributions, since they're driven basically by how much time a given worldline spends inside vs. outside the bubble, which makes sense.

 

[PAllen]

Ah, I see. Well, to the extent that your odometer distance inside the bubble is a lot less than, say, 4.3 light years, that's still true for your odometer definition. But I would agree that with your definition, it's a lot easier to understand the relative contributions, since they're driven basically by how much time a given worldline spends inside vs. outside the bubble, which makes sense.

Right, and my definition justifies the intuition that travel distance using the bubble to get from earth to Alpha Centauri is small.

 

[PeterDonis]

Not remotely what I meant.

Then I'm really confused about what you meant, since I just took your odometer distance definition and constructed a worldline with a particular $\gamma$ by your definition. How is that not what you meant?

Considering the famous fig. 3, I assume the world line of earth is vertical and just to the left of the bubble region, Alpha Centauri vertical and just to the right of it. I want travel from one to the other.

Sure, just use the worldline I gave, starting from the right edge of the diagram and going to the left edge. In that diagram there will be a segment of the worldline that gets dragged to the right by the bubble, but it will still make it to the left edge eventually. Just compute how much coordinate time that takes and plug into the formula I gave.

If you want to go from Earth to Alpha Centauri instead of Alpha Centauri to Earth, then we can just construct a worldline with the same $\gamma$ as I gave, but moving to the right instead of to the left. Since the bubble itself moves to the right, this worldline will take less time to go across the diagram, so its odometer distance will be less--which means that the odometer distance from Earth to Alpha Centauri by your definition is not isotropic, but that's to be expected since you're using the Eulerian worldlines as the "road" and they're not isotropic.

 

[PeterDonis]

my definition justifies the intuition that travel distance using the bubble to get from earth to Alpha Centauri is small

It would if, for example, a worldline with the $\gamma$ I gave, or at least some reasonable $\gamma$, but moving to the right instead of to the left, spends most of its time inside the bubble, not outside. So the time it takes such a worldline to accumulate enough odometer distance (to the right this time) to cross the bubble would have to be about the same as the time it takes the bubble to make the trip. I'll have to run numbers to see what that looks like when I get a chance.

 

[PeterDonis]

Then stay put

But if you stay put, obviously you're accumulating zero odometer distance--you're just letting the bubble carry you. That seems to me like trying to get out on a technicality, so to speak.

The question, to me, would be whether a worldline that did not stay put--that still tried to move to the right, even inside the bubble--would still accumulate a very small amount of odometer distance (roughly the spatial extent of the bubble, plus a small amount for moving in and out of it). My previous post just now described how I would approach that question.

 

[PAllen]

Then I'm really confused about what you meant, since I just took your odometer distance definition and constructed a worldline with a particular $\gamma$ by your definition. How is that not what you meant?


Sure, just use the worldline I gave, starting from the right edge of the diagram and going to the left edge. In that diagram there will be a segment of the worldline that gets dragged to the right by the bubble, but it will still make it to the left edge eventually. Just compute how much coordinate time that takes and plug into the formula I gave.

If you want to go from Earth to Alpha Centauri instead of Alpha Centauri to Earth, then we can just construct a worldline with the same $\gamma$ as I gave, but moving to the right instead of to the left. Since the bubble itself moves to the right, this worldline will take less time to go across the diagram, so its odometer distance will be less--which means that the odometer distance from Earth to Alpha Centauri by your definition is not isotropic, but that's to be expected since you're using the Eulerian worldlines as the "road" and they're not isotropic.

There is no reason for gamma to be constant. My definition was, in fact, partly motivated by a bug scampering over the surface of a spinning record, both with arbitrary speeds. Problem: describe what an ideal odometer would measure.
So the world line i describe is what I would take to be the natural definition of travel from earth to Alpha Centauri using the bubble. You could use yours, but mine captures what I would think the average person would want for a travel experience.

 

[PAllen]

But if you stay put, obviously you're accumulating zero odometer distance--you're just letting the bubble carry you. That seems to me like trying to get out on a technicality, so to speak.

The question, to me, would be whether a worldline that did not stay put--that still tried to move to the right, even inside the bubble--would still accumulate a very small amount of odometer distance (roughly the spatial extent of the bubble, plus a small amount for moving in and out of it). My previous post just now described how I would approach that question.

Not a technicality. It is the natural result of using the bubble to efficiently get to your intended destination. Note, this would be visually clear in the chart where all Eulerian world lines are vertical, and the metric is diagonal. Note that I think light cones inside the bubble in this chart would be extremely skinny, so you would have no choice but to “move” near vertical inside the bubble.

 

[PeterDonis]

There is no reason for gamma to be constant.

Of course worldlines exist where it isn't constant. But if you're trying to capture the idea of "moving at a certain speed along the road" with your definition, $\gamma$ captures what that speed is. And that's the simplest case of trying to understand what your odometer distance is telling you.

If you let $\gamma$ vary, you're letting your speed relative to the road vary--but that should also make the time it takes you to travel vary. A lower speed takes a longer time to cover the same odometer distance. Again, you can certainly analyze such a case, but it's more complicated. I wanted to keep it simple.

 

[PeterDonis]

It is the natural result of using the bubble to efficiently get to your intended destination.

The "push" to the right that the bubble gives any worldline, regardless of $\gamma$, of course helps you to get to the destination faster. That's not the technicality.

The technicality is having your observer who's trying to get to the destination just stop, relative to the road, and then claiming that, well, that makes the odometer distance smaller. Of course it does. But that's the technicality. To really see the effect of the bubble, you want to have an observer who doesn't stop--who keeps moving along the road--but also gets the extra push because the bubble is moving (or, more pertinently, the road itself is moving) in the same direction the observer is trying to go, and therefore still covers a lot less odometer distance.

The bubble, in other words, is kind of like a moving walkway--yes, you can just stop on it and still get to your destination, and cover zero "odometer distance" in doing so. But again, that's what seems to me to be a technicality. It seems to me that a better question would be whether you can keep walking along the moving walkway, but still only cover an odometer distance about the same as the spatial extent of the bubble.

 

[PAllen]

Of course worldlines exist where it isn't constant. But if you're trying to capture the idea of "moving at a certain speed along the road" with your definition, $\gamma$ captures what that speed is. And that's the simplest case of trying to understand what your odometer distance is telling you.

If you let $\gamma$ vary, you're letting your speed relative to the road vary--but that should also make the time it takes you to travel vary. A lower speed takes a longer time to cover the same odometer distance. Again, you can certainly analyze such a case, but it's more complicated. I wanted to keep it simple.

That's fine, but then, as you noted, the relevant travel direction is to the right in terms of that fig.3. It is now a case of "I like my world line, you like yours, we don't have to agree". Generally, for any "road definition" distance traveled will be essentially independent of relative speed until the relative gamma becomes large, matching simple SR intuition. As relative gamma goes to infinity, odometer distance goes to zero because the integrand is bounded above, while the bounds of integration go to zero.