Proper Distance: What is Relativistic Distance?

[HALON]

Meter sticks laid out around the rim of a rotating disk will foreshorten by the Lorentz factor. This means more of them will fit around the disk in the reference frame of an observer on the moving rim.

Q: In physics, what name is given to the longer measured length by a foreshortened meter stick?

I've already thought of "proper distance" and "relativistic distance" but these may not be acceptable. Of course there is always "wrong distance"...but that would imply there is a "wrong time" too.

 

[PAllen]

I've never heard of a term for it, but note that a good argument can be made that it is not a distance in any reasonable sense. The alleged length measured this way does not correspond to proper length along a closed curve in a spacelike hypersurface. One normally wants a length to represent a measurement in some reasonable simultaneity surface, with the further expectation that for a 'rim' you have a closed curve. This oft described measurement corresponds to no such thing. Thus "wrong length" is not such a bad idea, but it won't sell as accepted terminology.

Note, that there is no unique or preferred way to specify a meaningful length for rotating frame in SR because there is no simultaneity surface that meets all requirements one would like; thus each choice gives up something, so none is preferred. Further, any of the popular choices produce either the same or shorter length than that measured in the inertial frame in which the problem is posed (rather than longer length). The longer length comes from the fact that it does not correspond to a closed spacelike curve.

 

[jedishrfu]

Maybe we could call it the "wright distance" since we are placing the boards on a wheel and wheels were typically made by wheel wrights. (Some miscellaneous humor : - )

 

[WannabeNewton]

Meter sticks laid out around the rim of a rotating disk will foreshorten by the Lorentz factor. This means more of them will fit around the disk in the reference frame of an observer on the moving rim.

Yes but bear in mind this is entirely due to the choice of time coordinate. In your case the time coordinate is non-integrable. If you don't keep in mind this fact then you will be led to some rather interesting paradoxes regarding the shortening of infinitesimal tangential meter sticks laid out along the rim of the disk. For example consider a rotating ring placed in a non-rotating frictionless circular groove with the same circumference as measured in the inertial frame. Then in the frame corotating with the ring the length of the groove will be measured to be less than that of the ring which is contradictory as the ring has to fit inside the groove.

The resolution of course is that the time-coordinate being used to measure circumferential lengths in the rotating frame is non-integrable. Another way to say this is, imagine we build a curve of events whose spatial coordinates describe a closed curve around the ring in the rotating frame but which are locally Einstein simultaneous events, meaning each event on the curve is simultaneous with the event infinitesimally close to it in e.g. the counter-clockwise direction according to the respective momentarily comoving inertial frames. While the spatial curve around the ring is closed the simultaneity curve will not be closed because the time-coordinate is non-integrable. The result will be a nonsensical measure of circumferential lengths such as that occurring in the case of the rotating ring in the non-rotating circular groove.

longer[/I] measured length by a foreshortened meter stick?

I know of no standard terminology. In the case you are referring to it is due to the hyperbolic geometry of the disk associated with the non-integrable time-coordinate in the rotating frame.

 

[Ich]

Q: In physics, what name is given to the longer measured length by a foreshortened meter stick?

The example of the circumference of a rotating disk is (more than) a bit tricky, as PAllen and WannabeNewton explained.
But generally, the length measured by a meter stick, foreshortened or not, is called "length". If said length is meant to be its own, or of another thing momentarily at rest with it, it's called "proper length".
If we were talking about linear motion instead of circular, this is obvious. The extension to circular motion is straightforward: the combined length of the meter sticks is the proper length of a spacelike curve orthogonal to the worldlines of the respective sticks. So no problem there, the length of this curve is expected to be more than the circumference at rest.
The conceptual problem arises if one assumes incorrectly that this spacelike curve is somehow to be identified with the (rest-) circumference. That would mean that the longer meter sticks measure the shorter circumference as of equal length, i.e. stretched. But this is not the case: using (correctly) the simultaneity definition of the meter sticks, one finds that they measure more that 360° of the circumference. So they see measured distance shrinked, as it should be.

The incorrect assumption that the combined length of the meter sticks is fostered by the fact that, in the "space" of a rotation coordinate system, they form a closed curve. You'd think there can be only one closed curve of a given radius, which is why you identify it with the circumference.
The problem is: the "space" of a rotating coordinate system is not really a space. Going round "closed" curve you don't arrive at the spacetime event where you started. You arrive at a sooner or later time, which means the curve is not closed at all. So there's no reason to identify with exactly 360° of the circumference.

 

[HALON]

Thanks for the viewpoints about simultaneity, closed curves and the non-integrability of time. And the wright length joke. (Too bad the pun doesn't work in German, it might have caught on).

I didn’t want to convey that I thought lengths were longer for a moving body. But I thought lengths measured in this way deserved a special name. Einstein evidently thought the longer measured circumference was a good enough concept to suggest space is non-Euclidian, which is kind of special.

Also, the error is not confined to rotations. A passenger in a straight moving train could theoretically lay out measuring rods from point A to point B along the track. As each rod is “shorter” (by a factor of γ) there are more to lay out (by a factor of 1/γ) than in a rest frame. But it is wrong to conclude AB is longer merely because it is apparent to the passenger.

 

[WannabeNewton]

But it is wrong to conclude AB is longer merely because it is apparent to the passenger.

There is nothing apparent about it given that we have no absolute notion of length. See my comment here, https://www.physicsforums.com/showpost.php?p=4766303&postcount=3, and replace "illusion" with "apparent".

 

[HALON]

There is nothing apparent about it given that we have no absolute notion of length. See my comment here, https://www.physicsforums.com/showpost.php?p=4766303&postcount=3, and replace "illusion" with "apparent".

I understand the concept of no absolute notion of length. We need atomic clocks to establish S.I. units of metre and of course time is relative. No argument from me.

When I use the term “apparent” it means “first evidence”. I could have added that to accept the passenger's conclusion we would need to formulate a theory in which it is equally apparent to a stationary observer too. If that can be done then it supports the hypothesis of “absolute Lorentzian space-time”, and that space and time is not just an illusion.

 

[WannabeNewton]

If that can be done then it supports the hypothesis of “absolute Lorentzian space-time”, and that space and time is not just an illusion.

No that is not necessary. Special relativity already incorporates these things.