Does the Alcubierre drive shorten distances?

[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.

 

[Jaime Rudas]

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.

By definition, a light-year is the length of the path travelled by light in vacuum during a time interval of one (Julian) year. From this, I deduce that if a photon has traveled for 13.8 Gy, then it has traveled 13.8 Gly.

 

[PeterDonis]

By definition, a light-year is the length of the path travelled by light in vacuum during a time interval of one (Julian) year.

Under appropriate circumstances, yes. "Appropriate circumstances", when you dig into the details, turns out to mean flat spacetime.

From this, I deduce that if a photon has traveled for 13.8 Gy, then it has traveled 13.8 Gly.

And, as I said, this deduction is wrong because the spacetime in question is curved.

 

[Jaime Rudas]

Under appropriate circumstances, yes. "Appropriate circumstances", when you dig into the details, turns out to mean flat spacetime.


And, as I said, this deduction is wrong because the spacetime in question is curved.

So in curved spacetime the speed of light is not c?

 

[PeterDonis]

So in curved spacetime the speed of light is not c?

if you mean the coordinate speed of light, that doesn't even have to be $c$ in flat spacetime, if you use non-inertial coordinates.

In curved spacetime, in general, no, there are no coordinates in which the speed of light is $c$ everywhere, though you can find coordinates that make it $c$, at least to a good enough approximation, on a local patch of spacetime (by choosing local inertial coordinates on that patch).

All this is why in GR we don't talk about "the speed of light" being $c$; instead we talk about the light cone structure of spacetime, which is the actual invariant that "the speed of light is $c$" is referring to in the cases where it's true.

 

[Jaime Rudas]

if you mean the coordinate speed of light

No, what I'm referring to is the speed of light understood as the length of the path travelled by light in vacuum per unit of time.

 

[PeterDonis]

No, what I'm referring to is the speed of light understood as the length of the path travelled by light in vacuum per unit of time.

And what, specifically, do you mean by "the path travelled by light"? Think carefully. In a curved spacetime, particularly one that is not stationary, like the spacetime of our expanding universe, there actually is no such thing as what I suspect you're thinking of--what anyone would intuitively think of in flat spacetime.

 

[Jaime Rudas]

And what, specifically, do you mean by "the path travelled by light"? Think carefully. In a curved spacetime, particularly one that is not stationary, like the spacetime of our expanding universe, there actually is no such thing as what I suspect you're thinking of--what anyone would intuitively think of in flat spacetime.

Locally (that is, at each point), in an expanding universe, the speed of light is c. If I integrate it with respect to time, I get the length of the path travelled by light.

 

[PAllen]

Locally (that is, at each point), in an expanding universe, the speed of light is c. If I integrate it with respect to time, I get the length of the path travelled by light.

So you are summing the separate local observations of many (infinite) local inertial observers, each at a different cosmological time and place (and these observers are not maintaining a static distance from each other as their world lines are tracked). You may accept this as a definition of distanced traveled, but it is very different from an everyday definition, and many another would not find it meaningful.

 

[PAllen]

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".

This is a fundamental point. It is clear by inspection the t coordinate is spacelike, not timelike inside the bubble when it is operating as a warp drive. This means that the interval between two Euclidean slices, while timelike outside the bubble, is spacelike inside the bubble. Thus this is not a valid foliation, as normally defined.

 

[Ibix]

Locally (that is, at each point), in an expanding universe, the speed of light is c. If I integrate it with respect to time, I get the length of the path travelled by light.

I think you are thinking of a process something like this. Draw a spacetime diagram in some coordinate system and add the (orange) worldline of the light pulse:

Pick a series of events in some sense evenly spaced along the line (you might choose fixed intervals of an affine parameter along the null path, or fixed coordinate time intervals) and draw timelike dividing lines between small regions (black):

Measure the spacelike distance between each dividing line and the next along a line orthogonal to the divider (green):

Sum the spacelike distances. Take the limit as the even spacing between dividing lines reduces to zero.

Am I right? If so, the issue appears to be the freedom to choose the direction of the timelike dividing lines. I don't think there's a unique choice (even in flat spacetime) and the final result will depend on your choice of scheme for picking them.

 

[Jaime Rudas]

So you are summing the separate local observations of many (infinite) local inertial observers, each at a different cosmological time and place (and these observers are not maintaining a static distance from each other as their world lines are tracked). You may accept this as a definition of distanced traveled, but it is very different from an everyday definition, and many another would not find it meaningful.

Davis & Lineweaver present their equations 19 and 20 as follows:

$$\dot D= \dot R\chi+R \dot \chi$$
$$v_{tot}=v_{rec}+v_{pec}$$
This explains the changing slope of our past light cone in the upper panel of Fig. 1. The peculiar velocity of light is always $c$ (Eq. 21) so the total velocity of light whose peculiar velocity is towards us is $v_{tot} = v_{rec} − c$ which is always positive (away from us) when $v_{rec} > c$.

As I see it, the velocity I defined is nothing other than what D&L call peculiar velocity.

 

[PeterDonis]

the velocity I defined is nothing other than what D&L call peculiar velocity.

Pretty much, yes, although they are expressing it in global comoving coordinates instead of local inertial coordinates, and they haven't emphasized the caveat that this only works within the confines of a local inertial frame centered on a particular event.

However:

If I integrate it with respect to time, I get the length of the path travelled by light.

Please write down this integral explicitly and evaluate it for the case of a CMB photon emitted at the surface of last scattering and just reaching Earth now. What do you get?

 

[PAllen]

Davis & Lineweaver present their equations 19 and 20 as follows:

As I see it, the velocity I defined is nothing other than what D&L call peculiar velocity.

But they are defining a local relative velocity. Do they also turn around and define a distance traveled by a body with peculiar motion? That’s the step I find dubious, because you are summing local measurements of observers at different cosmological time and place, with non static distance between them. That is what makes it hard for me to accept as a distance traveled.

 

[PeterDonis]

you are summing local measurements of observers at different cosmological time and place, with non static distance between them. That is what makes it hard for me to accept as a distance traveled.

My question is whether the sum (integral) that is being implicitly done here actually evaluates to the distance that was claimed, namely, the travel time times $c$. That's why I asked for the integral to be explicitly written down and evaluated. Once we have an explicit integral to look at, then we can discuss what, if anything, it actually means physically.

 

[PAllen]

So, @Jaime Rudas , if you apply your notion of distance traveled for light in an FLRW solution, to a traveler in the warp bubble, and use as the timelike congruence relative to which you are summing distance as that of comovers within the bubble, you get that an in bubble traveler has traveled zero distance.

 

[PAllen]

I haven't had a chance to study this in relation to the warp metric, but are there any timeilike paths that enter and leave the Bubble? Alcubierre's description in the paper of the bubble 'appearing' around traveler, is obviously not realizable within GR.

 

[PeterDonis]

are there any timeilike paths that enter and leave the Bubble?

The diagram in Figure 3 of the lecture notes linked to in the OP of this previous thread would seem to me to show that there are:

T

Undergrad Thread 'Alcubierre Warp Drive: Spacetime Diagram Explained'

Apr 5, 2021

https://scipost.org/SciPostPhysLectNotes.10/pdf
"Space expands behind the warp bubble and contracts in front of it, thus pushing the bubble forward at velocity v. The ship, which is at rest inside the bubble, moves along with the bubble at an arbitrarily large global velocity."

If the ship moves faster than light this would mean that in a Minkowsky diagram its world line would be straight from the origin to the destination closer to the x-axis than light world lines. This world line line would be spacelike however.
Does this make sense or would another spacetime diagram be...

• timmdeeg
• Alcubierre warp drive Diagram Drive Spacetime Spacetime diagram Warp Warp drive
• Replies: 12
• Forum: Special and General Relativity

While inside the bubble, such paths would have to be moving "in the opposite direction" from the bubble, from the standpoint of observers at rest at the center of the bubble. From the standpoint of observers far away from the bubble, such paths, from what I can tell, would appear to be "dragged along" with the bubble while they were inside it. These are very heuristic descriptions, of course, and I haven't done any actual calculations to verify what I think the diagram is telling me.

 

[PAllen]

One simple calculation is that if $v_s$ is 2, then the slope of light cones well within the bubble is appx. $2 \pm \sqrt 3$ (considering the x t surface). In other words, in the future direction, backward light rays slope to the right in the coordinates.

[edit: I made a miscalculation, which is doesn't change the general observation. The light cones are given by $2\pm 1$.]

 

[PeterDonis]

in the future direction, backward light rays slope to the right in the coordinates.

Yes.