Is length contraction real or just a matter of perspective?

[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.