Is the train in the circular track paradox similar to the barn/pole paradox?

[pervect]

Oh, as far as deformation goes, I should mention that that's what the shear tensor I mentioned compute - how things deform. But if you're not familiar with it, I'll probably have to draw some diagrams to (hopefully) explain it. It might fit best in a different thread.

 

[pervect]

Are you referring to the case of a train going at a constant angular velocity around the circle, or of the train accelerating while in a circle?

In the former case (constant angular velocity), the train car would not rotate as you describe. In the latter case, I'm not sure whether it would or not.

It's not an exotic relativity effect like Thomas precession. It's just what normally happens when a train goes around a loop. See the diagram below.

[add]
If you imagine a gyroscope mounted on the train, the train body rotates relative to the gyroscope, that's why it's rotating. If you imagine a round train sliding around the loop without rotating, then there would be no rotation.

The physics point I'm trying to make is just this. It's possible to have a Born-rigid non-rotating object, and it's possible to have Born-rigid rotating object, but you can't Born-rigidly make a non-rotating object into a rotating one, or vice-versa.

 

[PeterDonis]

It's just what normally happens when a train goes around a loop.

In that case, the rotation is entirely in the plane of the loop. It looked to me like what you were describing was rotation in a plane perpendicular to the plane of the loop.

It's possible to have a Born-rigid non-rotating object, and it's possible to have Born-rigid rotating object, but you can't Born-rigidly make a non-rotating object into a rotating one, or vice-versa.

Agreed.

 

[pervect]

As far as the curvature of the track goes - if you consider the proper frame of the train, i.e a comoving inertial frame, the track is elliptical due to Lorentz contraction, as in the sketch below. So at the location where the train is, the track is more curved - since the track isn't circular, the curvature isn't constant.

 

[JVNY]

Oh, as far as deformation goes, I should mention that that's what the shear tensor I mentioned compute - how things deform. But if you're not familiar with it, I'll probably have to draw some diagrams to (hopefully) explain it. It might fit best in a different thread.

I am very unlikely to understand the math, but I suspect the other posters would. Diagrams would be great.

 

[A.T.]

As far as the curvature of the track goes - if you consider the proper frame of the train, i.e a comoving inertial frame, the track is elliptical due to Lorentz contraction, as in the sketch below. So at the location where the train is, the track is more curved - since the track isn't circular, the curvature isn't constant.

Is this the instantaneous inertial frame of a train car? Doesn't the proper centripetal acceleration of the car affect the geometry in the actual non-inertial rest frame of the car? Is there even clear way to determine that, or does it again depend on the simultaneity conventions in non-inertial frames?

 

[pervect]

Is this the instantaneous inertial frame of a train car? Doesn't the proper centripetal acceleration of the car affect the geometry in the actual non-inertial rest frame of the car? Is there even clear way to determine that, or does it again depend on the simultaneity conventions in non-inertial frames?

Yes, this is the instantaneous inertial frame of the train. And I assume the simultaneity convention of that co-moving observer. The issue I was concerned with was how to transport the basis vectors to create what is called in my textbook (MTW) "a proper frame" for the observer, when I realized that it was irrelevant to what the shape of the track was "now" at a point co-located with the train. Note that I didn't think in detail about other possible simultaneity conventions, though my current thinking is that the Einstein convention is expected.

As far as the shape of the track goes, basically, since we know what the basis vectors are for the train observer, we know the spatial geometry from the point of view of that co-moving observer - and it's just the Lorentz transform.