# On the no-faster-than-light in special relativity

In the special relativity it is well established that, in the vacuum no one can ever travel faster than light, due to the relativistic velocity addition formula.

Recently I saw some silly statement claiming, that this conclusion holds for the translational speed addition doesn't imply it holds for the rotational speed. I think it is silly since the statement didn't give any justification reasoning.

However, it is legitimated to ask whether the no-faster-than-light derived from the translational speed addition could be extended, or be equivalent to, the no-faster-than-light for the rotational speed as well. I had the hunch that, there should be a proof to show the equivalency between the two. How to prove, or disprove (unlikely though), this equivalency?

• It's not a trivial question, by any means. The tangential velocity is simply a velocity, so it not exceed the speed of light for massive bodies. The problem with finding an "rpm addition formula" is that rotating frames are non-inertial, so the coordinate transformation effects become dependent on the radius at which they are being measured. – CuriousOne Jun 13 '15 at 23:47
• – user12262 Jun 14 '15 at 6:00

There is no such thing as "rotational speed". There is angular momentum, which has units of $\frac{\mathrm{kg} \cdot \mathrm{m}}{\mathrm{s}}$ (and so is not a speed), and angular velocity which has units of $\frac{1}{\mathrm s}$ (and so is not a speed either). There is no maximum angular velocity in special relativity. So long as $\omega r\lt c$ (the velocities of the involved particles stay slower than light), you're fine.
Finally, unfortunately, a rigid body cannot rigidly spin up or spin down in special relativity! The key word here is "Born rigidity and angular acceleration". The quick explanation is that a disk whose edge spins near the speed of light will be length contracted. So a tape measure spinning with the edge would read more than $2\pi r$. Therefore, if you want to apply an angular acceleration to the disk and not cause any compressive/stretching forces, you have to add material to the disk!