Is length contraction real or just a matter of perspective?

[Dale]

I don't see the purpose of the word "coordinates" in the response, because the Rindler observer determines light speed without the use of coordinates. Say there is a row of Rindler observers, and one at one end lays a meter stick along the row and measures the distance to the other end to be one meter. Then he sends a signal to the other end, which reflects and returns. The observer uses time determined by his own wristwatch (proper time) for the only measure of time (the round trip time for the signal), and distance determined by a meter stick for the only measure of distance.

That is still using coordinates defined by the length of the meter stick. As PAllen points out, that is not the only reasonable definition and indeed it has its own problems.

The use of the word "coordinates" instead of "observer" avoids anthropomorphizing too much or attributing the relativistic effects to the presence or absence of a conscious observer. It is also the more mathematically accurate term to use.

Personally, I find the overuse of the term "observer" to be a problem for students learning relativity. It encourages several mistaken notions:
1) That observers must use coordinate systems where they are at rest
2) That there is a unique preferred coordinate system for every observer
3) That relativistic effects require a conscious observer
4) That relativistic effects are mental distortions rather than physical phenomena
5) Etc.

 

[PAllen]

Thanks. Is there a special meaning to using the word "coordinates" in the response? As opposed to, for example, saying "Light signals traveling faster than c as measured by accelerating observers is completely routine."

I don't see the purpose of the word "coordinates" in the response, because the Rindler observer determines light speed without the use of coordinates. Say there is a row of Rindler observers, and one at one end lays a meter stick along the row and measures the distance to the other end to be one meter. Then he sends a signal to the other end, which reflects and returns. The observer uses time determined by his own wristwatch (proper time) for the only measure of time (the round trip time for the signal), and distance determined by a meter stick for the only measure of distance. So why use the word "coordinates?"

I think my prior answer didn't get at what you were asking in the best way.

Coordinates are just a system of labels attached to events in spacetime meeting certain conditions. Coordinates can be chosen that have little direct relation to any measurements. Coordinate speed in arbitrary coordinates will have no direct relation to measurements. To use general coordinates to extract observables, you have a metric tensor defined on them such that invariants computed with this tensor produce the same results as if you transformed to standard inertial coordinates and used standard SR formulas.

However, we were discussing setting up coordinates based on measurements - each coordinate value corresponds to measurements made by one or a family of specified observers. For these, a statement about a coordinate quantity translates to a statement about computing that quantity using the measurement scheme underlying the coordinates.

Thus, for the type of coordinates we were discussing, I could just as well have used your words:

"Light signals traveling faster than c [over substantial distances] as measured by accelerating observers is completely routine." with the addendum that you get different answers for different measurement methods.

 

[JVNY]

Thanks DaleSpam and PAllen. The question was driven in part by the prior posts and in part by some language in the Dolby and Gull radar article referred to in an earlier post. That article states that it shows a method "to define hypersurfaces of simultaneity . . . independent of choice of coordinates . . ." (quotation from the abstract), and it makes many statements about radar time and the like being independent of coordinates. The most important takeaway for me is that you get different answers for different measurement methods, none of which is preferred. It is irrelevant whether you define coordinates first, or measure and then assign coordinates to the measurement, or even define a method that is independent of coordinates. The answers will be different, and none will be preferred.

 

[Dale]

That article states that it shows a method "to define hypersurfaces of simultaneity . . . independent of choice of coordinates . . ." (quotation from the abstract), and it makes many statements about radar time and the like being independent of coordinates.

I like that article, and have cited it many times, but I also feel that the "independent of choice of coordinates" comment goes a little too far.

The most important takeaway for me is that you get different answers for different measurement methods, none of which is preferred. It is irrelevant whether you define coordinates first, or measure and then assign coordinates to the measurement, or even define a method that is independent of coordinates. The answers will be different, and none will be preferred.

I think that is the bottom-line.

 

[WannabeNewton]

The answers will be different, and none will be preferred.

I think you mean "more correct" as opposed to "preferred". For a given physical system, the symmetries of the system will always determine a "preferred" coordinate system even if there is no "more correct" coordinate system.

The observer uses time determined by his own wristwatch (proper time) for the only measure of time (the round trip time for the signal), and distance determined by a meter stick for the only measure of distance. So why use the word "coordinates?"

This is true you don't need any coordinates to determine the two-way speed of light. But do keep in mind that you do need a coordinate system in order to talk about the one-way speed of light.

 

[PAllen]

I think you mean "more correct" as opposed to "preferred". For a given physical system, the symmetries of the system will always determine a "preferred" coordinate system even if there is no "more correct" coordinate system.

I wouldn't go that far. You can say that coordinate systems that manifest such symmetries are preferred, but that still leaves a huge choice. For example, for Rindler observers, radar coordinates built from a particular Rindler observer, Rindler coordinates with horizon at the origin, and Fermi-Normal coordinates build from a particular Rindler observer, all share the same foliation and the same congruence for 'constant position' world lines. Isn't that enough to say they manifest the same symmetries? They differ on trivial matters (where the origin is) and more substantive ones (distance scale for Radar). But how could they not manifest the same symmetries if they share the same foliation and congruence?

 

[WannabeNewton]

That article states that it shows a method "to define hypersurfaces of simultaneity . . . independent of choice of coordinates . . ." (quotation from the abstract)...

If you use the Einstein simultaneity convention when employing radar in order to assign 'radar time' and 'radar distance' to a distant simultaneous event then for any given observer in arbitrary motion the local simultaneity surfaces can be characterized by the orthogonality of the simultaneity surface to the observer's 4-velocity. This is a purely geometric characterization that makes no reference to coordinates whatsoever.

If we have a family of observers at rest with respect to one another and they form an irrotational tangent field in space-time then we can "patch together" their local simultaneity surfaces into global simultaneity surfaces that are characterized by orthogonality with the tangent field (at least locally depending on the topology of space-time). Again this is purely geometric.

 

[WannabeNewton]

But how could they not manifest the same symmetries if they share the same foliation and congruence?

Point taken. When I said symmetries I was indeed thinking of the symmetries determined by the tangent field to the congruence (e.g. the time-like killing field tangent to the Rindler congruence or the linear combination of time-like and axial killing fields tangent to the congruence of observers atop a rotating disk) and the congruence alone determines a foliation by standard simultaneity surfaces (assuming the tangent field is irrotational of course). But nothing here makes reference to coordinates so yes I agree with you that the presence of various coordinate systems all equally well adapted to the congruence rules out the existence of a single "preferred" coordinate system.

 

[JVNY]

A follow up. Given that the answers are different, do you agree that a lot of other things that we rely in inertial motion in SR will be different for the Rindler observers? For example, both the radar method and the ruler method agree on which observer is in the center of a row of observers where they are in inertial movement. But if the observers are in Born rigid motion, the methods will not agree. The ruler method will identify the same observer as still being in the center. But the radar method will identify a Rindler observer closer to the rear as being at the center (because light flashes sent simultaneously from the front toward the rear, and from the rear toward the front, will cross farther back in the row).

Also, the method for each Rindler observer to use light flashes to determine simultaneity will differ, as the same example makes clear. Each Rindler observer agrees on the simultaneity of events, but simultaneous light flashes from the ends strike one observer at the same time when the observers are in inertial movement, but a different one at the same time when in Born rigid movement.

Again, this does not say that any of these is more correct or preferred, but just that they are different, and the observers will have to take them into account when drawing conclusions about events.

 

[PAllen]

A follow up. Given that the answers are different, do you agree that a lot of other things that we rely in inertial motion in SR will be different for the Rindler observers? For example, both the radar method and the ruler method agree on which observer is in the center of a row of observers where they are in inertial movement. But if the observers are in Born rigid motion, the methods will not agree. The ruler method will identify the same observer as still being in the center. But the radar method will identify a Rindler observer closer to the rear as being at the center (because light flashes sent simultaneously from the front toward the rear, and from the rear toward the front, will cross farther back in the row).

Also, the method for each Rindler observer to use light flashes to determine simultaneity will differ, as the same example makes clear. Each Rindler observer agrees on the simultaneity of events, but simultaneous light flashes from the ends strike one observer at the same time when the observers are in inertial movement, but a different one at the same time when in Born rigid movement.

Again, this does not say that any of these is more correct or preferred, but just that they are different, and the observers will have to take them into account when drawing conclusions about events.

Correct, overall. Trying to construct coordinates based on a non-inertial frame in SR raises most of the issues and complexities of local frames versus global coordinates as GR; and also brings in the GR issue that there is no unique way to talk about non-local distances, precisely because different methods you 'expect' to agree differ. What is still simpler than GR is that you can still talk, objectively, about relative velocity because the spacetime still flat. In GR, you cannot attach any unique meaning to the relative velocity of distant objects (even if you specify a chosen global coordinate chart). This is because parallel transport (the way to bring vectors together to compare) is path dependent in curved spacetime.

 

[JVNY]

And one more, if I can. In the case of change of direction acceleration (specifically, a rotating cylinder), everyone seems to agree that there is only one valid measure of the length of the circumference in the cylinder's frame (although they disagree on what that length is, whether less than, equal to, or greater than 2 pi r). Is there a simple explanation of why there aren't different measures of length in this case, each equally valid?

 

[PAllen]

And one more, if I can. In the case of change of direction acceleration (specifically, a rotating cylinder), everyone seems to agree that there is only one valid measure of the length of the circumference in the cylinder's frame (although they disagree on what that length is, whether less than, equal to, or greater than 2 pi r). Is there a simple explanation of why there aren't different measures of length in this case, each equally valid?

The proper accelerations are all the same, and the geometry and speeds are unchanging in any inertial frame; finally, there is high symmetry. In contrast, the Born rigid uniformly acceleration rod differs in each of these: proper acceleration varies front to back, geometry and speed are changing in any inertial frame; there is much less symmetry.

 

[JVNY]

Cool, thanks.

 

[PAllen]

And one more, if I can. In the case of change of direction acceleration (specifically, a rotating cylinder), everyone seems to agree that there is only one valid measure of the length of the circumference in the cylinder's frame (although they disagree on what that length is, whether less than, equal to, or greater than 2 pi r). Is there a simple explanation of why there aren't different measures of length in this case, each equally valid?

I thought it would be interesting to ask about this measurement in a different way, based on ideas from my relativistic odometer thread. I dispense with any notion of cylinder frame.

Assume we have a ring spinning around its center such that the speed of the rim in an inertial frame such that the center is not moving is v1. Now to measure the length from a rim dweller's perspective, I use a direct, mechanical method: a assume the rim dweller walks along the rim at some constant, slow speed relative to the rim. They know their speed, and simply time how long it takes them to get around the rim. I'm not sure which is simpler, but I chose to give this second speed in the inertial frame of the center (this is important to specifiy), call it v2. Let's call the circumference as measured in the inertial frame C. We want to know how v2 walker will measure the circumference, in the limit as v2->v1. Note, v2 > v1, by construction.

In the inertial frame, the time it takes O2 (the rim walker) to get around is simply C/(v2-v1). For the rim walker, this time is C/((v2-v1)γ2). Now the speed for rim relative to the walker is NOT v2-v1. The velocity addition formula must be used, and because we are dealing in local, asymptotically straight, measurements, we can use it in its linear form, getting: (v2-v1)/(1 - v1v2/c^2). Thus the rim walker's length measurement comes out:

[C/((v2-v1)γ2)] * (v2-v1)/(1 - v1v2/c^2)

Algebra and limiting then leads to Cγ as the measurement made by a slow walker. Thus, they find the circumference longer than inertial observer measures.