0
$\begingroup$

Let's consider two inertial reference frames S and S', S' moving with velocity V relative to S (velocity V to the right with respect to S considered stationary, in X-axis direction). We have a rotating rod at rest in S (strictly speaking, only one of the endponts of the rod is at rest, the others are rotating with constant angular velocity $\omega$).

If we assume that this rod is capable of undergoing relativistic velocities without breaking, what would be seen from S is simply a rod which rotates with a really high velocity. No length contraction, no bending. However, what would be seen from S' is a rod such that:

(1) it translates to the left -since S' moves with velocity V to right with respect to S-

(2) it changes its length as it rotates, the length being contracted when the rod is rotating near the X-axis - which can be understood in terms of the length contraction caused by relative velocity V being close to c-

(3) it bends!!!

I cannot understand why (3) happens. When $\omega$ isn't relativistic, there would be no bending but only contraction (2). When $\omega$ is relativistic, we perceive all of (1), (2) and (3) as seen in S'.

I would like to find an explanation for (3) which rests upon solid arguments and definitions. Can (3) be explained only in terms of Lorentz Transformations, which is the approach I find most clarifiying?

Additional comments:

I know that this scenario is similar to the one in Ehrenfest paradox about a rotating disk, but I don't see how the paradox regarding the perimeters of the rotating 'rigid' disk relates to the bending of the rod I'm studying -whatever 'rigid' means-.

I also know that this problem is related to the concept of 'rigidity'. So far I haven't even manged to find a good definition of this concept in the context of special relativity. What I've read about this concept seems to me quite dense and obscure.

$\endgroup$

1 Answer 1

1
$\begingroup$

I cannot understand why (3) happens.

(3) happens because of the relativity of simultaneity. If the axis of rotation is parallel to the axis of the boost then events on the rod that are simultaneous in S are also simultaneous in S' and the rod is unbent in both frames. If the axis of rotation is perpendicular to the axis of the boost then events on the rod that are simultaneous in S are not simultaneous in S' and the rod is bent in S'.

If you look in spacetime the rod forms a double helix. When you slice a double helix perpendicular to the axis then you get a straight line. But when you slice a double helix diagonally you get a curve. Geometrically this is what is happening. In other frames you are slicing the helix along a diagonal.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.