Relativistic Rolling Ball

I’ve been thinking about this problem all morning.

Imagine a homogeneous spherical mass that rolls along a flat plane without slipping, with uniform velocity v in the lab frame. Consider the limit as v/c goes to 1. How does the motion of the ball change, if at all? And in particular, what are the angular kinematics?

If the ball were simply sliding, then it would obviously Lorentz contract in the lab frame, into an oblique “grape-like” shape. But my question is:

Why doesn’t the relativistic rolling ball wobble like a rolling grape?

One easy answer is that the principle of relativity forbids it. For if it did wobble, then you could detect absolute motion by setting up a detector that beeps just in case the maximum height above the surface is changing — which is not allowed.

However, this argument doesn’t really get at the mechanism of what is going on. It is an example of what I call a “principled argument,” and it isn’t very satisfying. What I’d like is an explanation that teaches me how the angular kinematics of a spinning object change at relativistic speeds.

Notably, the usual tools (moment/torque) don’t seem to be doing the job, since the usual classical analysis implies that the ball will wobble like a grape. So something must be different here.

This leads one to think more generally about what happens when a sphere spins with angular velocity close to the speed of light. First of all, is there any contraction from this motion alone? And if so, what determines it? Both the light postulate and the principle of relativity, which generate special relativity, are defined so as to account for observers in transverse (inertial) motion. How are we to make use of them in the case of angular motion?

There may be a compelling theoretical argument out there which indicates how this relativistic rolling balls behave on the basis of classical relativity. But in the absence of one, we may have to resort to empirical observation to determine what’s going on. For the moment, I remain perplexed.

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