Best name for the length of a moving rod measured by a stationary observer

[bernhard.rothenstein]

I find in the literature of the subject different names for the length of a moving rod measured by an observer relative to whom it moves. Please let me know if ther is a standard name. Thanks

 

[Fredrik]

I don't know if I've ever heard a name for it. Does it need one? Why not just give the frame a name, e.g. "F", and call it the "length of the rod in frame F"?

 

[bernhard.rothenstein]

I don't know if I've ever heard a name for it. Does it need one? Why not just give the frame a name, e.g. "F", and call it the "length of the rod in frame F"?

Thanks. I think that would be "proper length".

 

[JesseM]

Thanks. I think that would be "proper length".

Proper length is defined in terms of a pair of events with spacelike separation (it's the distance between them in the frame where they are simultaneous), not a physical object. Of course you can pick two events on either end of the rod which are simultaneous in the frame F where it's moving if you want, and then the proper length between those events will be the same as the length of the rod in frame F, but just saying "proper length" isn't specific enough.

 

[bernhard.rothenstein]

Proper length is defined in terms of a pair of events with spacelike separation (it's the distance between them in the frame where they are simultaneous), not a physical object. Of course you can pick two events on either end of the rod which are simultaneous in the frame F where it's moving if you want, and then the proper length between those events will be the same as the length of the rod in frame F, but just saying "proper length" isn't specific enough.

I think I was not enough explicit. Consider
L=L(0)/g
g Lorentz factor. If L(0) represents a proper length then what is the name of L?

 

[Fredrik]

I think I was not enough explicit. Consider
L=L(0)/g
g Lorentz factor. If L(0) represents a proper length then what is the name of L?

L and L(0) are both proper lengths. They are the proper lengths of two different spacelike curves. I would call L the "length" and L(0) the "rest length". What I call "length" is just the difference between the spatial coordinates of the rod's endpoints in some frame. What I call "rest length" is just the "length" in the frame where the rod is at rest.

Note that although "proper length" is a coordinate independent concept, different spacelike curves have different proper lengths, and in this case the frame is what determines what spacelike curve you should be using, so the "proper length of the rod" actually depends on the frame (and is equal to what I just called "length").

 

[bernhard.rothenstein]

L and L(0) are both proper lengths. They are the proper lengths of two different spacelike curves. I would call L the "length" and L(0) the "rest length". What I call "length" is just the difference between the spatial coordinates of the rod's endpoints in some frame. What I call "rest length" is just the "length" in the frame where the rod is at rest.

Note that although "proper length" is a coordinate independent concept, different spacelike curves have different proper lengths, and in this case the frame is what determines what spacelike curve you should be using, so the "proper length of the rod" actually depends on the frame (and is equal to what I just called "length").

Then "REST LENGTH" and "LENGTH". I find for the last one "Coordinate length", "Measured length", Apparent length".
Thanks

 

[Phrak]

From the terms "time dilation" and "spatial contraction", indicative names are "dilated time interval" and "contracted length", respectivly.

 

[Fredrik]

Then "REST LENGTH" and "LENGTH". I find for the last one "Coordinate length", "Measured length", Apparent length".
Thanks

Yes, "length", "coordinate length" and "measured length" are clearly the same. Which one you use only depends on what aspect you want to emphasize. (It's like eigenvector/eigenket/eigenstate/eigenfunction in QM). I don't really like the term "apparent length" though. It suggests that Lorentz contraction is an illusion.

 

[Meir Achuz]

I don't really like the term "apparent length" though. It suggests that Lorentz contraction is an illusion.

Isn't it? Taking a picture shows this.

 

[JesseM]

Isn't it? Taking a picture shows this.

Visually it doesn't appear contracted, because of the Penrose-Terrell effect which actually compensates for length contraction (this is just a consequence of the fact that light from parts of the object at different distances from you will take different times to reach you, distorting the visual appearance). But if you measure the length of a rod using local measurements on rulers and clocks which are at rest and synchronized in your frame, you find the rod is shrunk--for example, if the rod is 10 light-seconds long in its own rest frame, and it's moving at 0.6c in your frame, then if at the moment the back end passes the 2-light-second mark on your ruler the clock at that mark reads t=5 seconds, then you'll find that at the moment the front end passes the 10-light-second mark on your ruler the clock at that mark will also read t=5 seconds, so you conclude the two ends are "really" 8-light seconds apart at any given moment in your frame.

 

[bernhard.rothenstein]

Isn't it? Taking a picture shows this.

Taking a picture (snapshot) you can obtain contraction, dilation and no distorsion of the length. The same situation could take place in the case of the radar detection,

 

[Meir Achuz]

Right on. What you measure depends on how you measure it.

 

[Dale]

I don't really like the term "apparent length" though. It suggests that Lorentz contraction is an illusion.

Isn't it? Taking a picture shows this.

That depends on your definition of "illusion". It is an experimentally measurable, but coordinate dependent, effect. Personally, I wouldn't call that "illusion" since it is measurable, but I also wouldn't object to someone who doesn't want to call it "real" since it is coordinate dependent. Same with time dilation.

 

[yuiop]

Isn't it? Taking a picture shows this.

It is a myth that you cannot in principle photograph a moving rod as length contracted. The Penrose-Tyrell rotation only works for spheres which have the same cross section whichever angle you look at them from. When you rotate a rod it looks shorter. If you had a large sheet of photgraphic film and array of flash lights timed to go off simulataneously you would capture a length contracted shadow of the passing rod on the film.

As for the length definitions it is my understanding that proper length is synomous with rest length. In other words proper length is the length measured by an observer at rest with the rod or co-moving with the rod. The length of the rod as measured by an observer moving relative to the rod is probably best described as its contracted length or possibly its relative length.

 

[Meir Achuz]

"If you had a large sheet of photgraphic film and array of flash lights timed to go off simulataneously you would capture a length contracted shadow of the passing rod on the film."
You are describing the same illusion as when a rotated stick looks shorter.
The moving stick is just rotated into time and not into space.

 

[yuiop]

"If you had a large sheet of photgraphic film and array of flash lights timed to go off simulataneously you would capture a length contracted shadow of the passing rod on the film."
You are describing the same illusion as when a rotated stick looks shorter.
The moving stick is just rotated into time and not into space.

You are right that the rod is only rotated in time and not into space, but you are wrong if you are implying that that a photograph of the moving rod would not capture a length contracted shadow.

Here is another experiment that might make it clearer. Imagine we have a long narrow hollow cylinder with an array of ink jets embedded on the inside, in a line parallel to the long axis of the cylinder. The construction is designed to leave an imprint on the opposite internal wall, of any object passing through the cylinder . The ink jets are timed to fire simultaneously in the rest frame of the cylinder. The diameter of the hollow cylinder is designed so that a rod can pass through it, but is narrow enough that the rod can not rotate and and has to remain parallel to the long axis of the cylinder. When the rod passes through the cylinder at relativistic speeds the ink jets will capture a length contracted imprint of the rod.

The rotation in time results in the spatial length contration without any actual rotation of the rod in space.

There is a possibility that in the future, instruments of sufficient accuracy and speed might be developed that a similar experiment could actually be carried out.

 

[RandallB]

Proper length is defined in terms of a pair of events with spacelike separation (it's the distance between them in the frame where they are simultaneous), not a physical object. Of course you can pick two events on either end of the rod which are simultaneous in the frame F where it's moving if you want, and then the proper length between those events will be the same as the length of the rod in frame F, but just saying "proper length" isn't specific enough.

No, you cannot pick a distance in the moving F as “proper”.
By definition that frame cannot correctly define “simultaneous” only the reference frame “A” can define “simultaneous” where “proper” lengths and distances can be defined.

That is why http://en.wikipedia.org/wiki/Proper_velocity" based on a distance traveled measured in “A” divided by the “proper time” on the clock doing the traveling will always be faster than the speed measured in the stationary frame A (where simultaneous is defined as correct). Even giving the appearance of FTL to the traveling clock in some cases.

To put "Proper Distance" in the F frame, the F frame Clocks and Rods must be considered stationary (with simultaneous synchronization in the “stationary” POV) and other frames like Frame A would be in motion with “Proper Times” running slower than the F clocks.

 

[JesseM]

No, you cannot pick a distance in the moving F as “proper”.
By definition that frame cannot correctly define “simultaneous” only the reference frame “A” can define “simultaneous” where “proper” lengths and distances can be defined.

Both "proper length" and "proper time" are defined in such a way that it doesn't matter what frame you choose to use. If you pick two events along the worldline of an object, then the proper time is simply the time measured between those events by an clock moving along that worldline, you can calculate it in any frame and you'll get the same proper time. Likewise, the "proper length" between two events with a spacelike separation is always defined as the distance between them in the frame where they are simultaneous, so it has nothing to do with what frame you are using.

I also don't know what you mean by "by definition that frame cannot correctly define 'simultaneous'". In relativity each frame has its own definition of "simultaneous", and none is more correct than any other. Another frame's definition of simultaneity will be different than that of the frame A that you're using, just as another frame's definition of an object's velocity will be different than the same object's velocity in frame A, but relativity says there's no basis for saying one frame's opinions about frame-dependent quantities are more correct in any objective sense than any other frame.

That is why http://en.wikipedia.org/wiki/Proper_velocity" based on a distance traveled measured in “A” divided by the “proper time” on the clock doing the traveling will always be faster than the speed measured in the stationary frame A (where simultaneous is defined as correct). Even giving the appearance of FTL to the traveling clock in some cases.

As I've mentioned before, "proper velocity" is a pretty rarely-used term, and the term was invented long after "proper time" and "proper length" so it has no bearing on what the adjective "proper" was supposed to suggest in those earlier terms. "Proper velocity" is unlike proper time and proper length in that it can have different values in different frames.

To put "Proper Distance" in the F frame, the F frame Clocks and Rods must be considered stationary (with simultaneous synchronization in the “stationary” POV) and other frames like Frame A would be in motion with “Proper Times” running slower than the F clocks.

It's meaningless to talk about "proper time" or "proper distance" independent of a specific choice of two events that you want to measure the proper time between (if they are timelike separated events that lie on some object's worldline) or the proper distance between (if they are spacelike separated events). This is just how "proper time" and "proper distance" are defined, in terms of pairs of events. And once you have specified the two events, then the proper time or proper distance between them is independent of what reference frame you're using.

 

[RandallB]

Likewise, the "proper length" between two events with a spacelike separation is always defined as the distance between them in the frame where they are simultaneous, so it has nothing to do with what frame you are using.

I wasn't using frame F for the rod your defining you are. And you defined the "proper length" of the rod in F as it is measured in the F frame. Now you’re saying that "proper length" has nothing to do with the F frame you are using. Which one is right in your view?

As I've mentioned before, "proper velocity" is a pretty rarely-used term, and the term was invented long after "proper time" and "proper length" so it has no bearing on what the adjective "proper" was supposed to suggest in those earlier terms. "Proper velocity" is unlike proper time and proper length in that it can have different values in different frames.

I don’t find anything inconsistent between the “proper” terms as defined in Wiki, with “velocity” coming directly from “proper length” and “proper time” there. I take your comments to mean they have something wrong.
If you think they have something wrong, correct it, but I they seem specific and consistent too me. While your definition of proper length or distance seems to vary randomly by comparison to the Wiki definitions.
So for now I'll stick with Wiki.

 

[JesseM]

I wasn't using frame F for the rod your defining you are. And you defined the "proper length" of the rod in F as it is measured in the F frame. Now you’re saying that "proper length" has nothing to do with the F frame you are using. Which one is right in your view?

The phrase "proper length of the rod" isn't really meaningful, as I understand the way "proper length" is defined. You can only talk about the "proper length" or "proper distance" between two events, and if you just say "the rod" it's not clear what events on either end of the rod you're referring to. If you specify that you're talking about two events on either end of the rod which are simultaneous in the rod's rest frame, then we can talk about the "proper length" between them (which is the same as the rod's rest length), but you can equally well specify two events on either end of the rod that are simultaneous in some other frame.

Consider the fact that the wikipedia page on proper length specifies that it's defined in terms of the distance between two events in the frame where they are simultaneous:

In relativistic physics, proper length is an invariant quantity which is the rod distance between spacelike events in a frame of reference in which the events are simultaneous.

Do you disagree with this definition?

I don’t find anything inconsistent between the “proper” terms as defined in Wiki, with “velocity” coming directly from “proper length” and “proper time” there. I take your comments to mean they have something wrong.

I have no problems with the proper velocity wiki page, but nor do I see them even mentioning the term "proper length" or "proper distance" on that page, so in what way do you think their definition of proper velocity is "coming directly from proper length and proper time"? As far as I can tell they are defining proper velocity in terms of proper time divided by coordinate distance in whatever arbitrary frame you choose to use.

While your definition of proper length or distance seems to vary randomly by comparison to the Wiki definitions.

In what way do you think my definition different from the definition I quoted above? Since I'm pretty sure I'm using an identical definition, I think you're either misunderstanding my definition, or misunderstanding the definition on the wikipedia page.