No-faster-than-light in the special relativity

[gxu]

I saw the following question in the Physics-SE, and thought it is interesting.

"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. 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?"

Anyone, any ideas?

Thanks,
gxu

 

[Topolfractal]

Adapt the linear velocity addition formula to an angular formula

 

[Topolfractal]

Your key is to find such a formula

 

[PeterDonis]

The fact that nothing can travel faster than light in SR is not a consequence of the relativistic velocity addition formula, either translational or rotational. It's the other way around. The structure of spacetime in SR is what keeps objects from traveling faster than light. The relativistic velocity addition formula is a consequence of the structure of spacetime.

 

[Topolfractal]

He is asking pretty much is there a formula that works the same for angular velocities as linear velocities and whether it can be made relativistic and support the no faster than light principle.

 

[PeterDonis]

He is asking pretty much is there a formula that works the same for angular velocities as linear velocities and whether it can be made relativistic and support the no faster than light principle.

And the answer is yes, of course, because the no faster than light principle is more fundamental than any formula for addition of velocities.

 

[gxu]

The tricky, and non-trivial, part of this question is, dealing with the rotational velocity seems to inevitably involve the non-inertial rotating frame. Or, could there be any clever way to avoid that, and keep the technical details in the theory of SR? By the way, the GR is beyond my knowledge domain.

 

[gxu]

And the answer is yes, of course, because the no faster than light principle is more fundamental than any formula for addition of velocities.

I am not totally convinced by the statement that, "the no faster than light principle is more fundamental than any formula for addition of velocities". The two fundamental postulates in the SR didn't explicitly (or precisely) contain the no-faster-than-light principle. The spacetime structure, and the implied the no faster than light "principle", seemed to be a more or less derived consequence, if I haven't missed some points.

 

[PeterDonis]

The two fundamental postulates in the SR didn't explicitly (or precisely) contain the no-faster-than-light principle.

Not explicitly, but one of the two postulates is that the speed of light is invariant, the same for all observers. What I'm calling "spacetime structure"--the fact that there are three fundamentally different kinds of vectors in spacetime, timelike, null, and spacelike--is a direct consequence of that postulate. The no-faster-than-light principle is just the fact that objects can't change categories: once a timelike (slower than light) object, always a timelike object. This is just part of what "spacetime structure" means. The formula for relativistic velocity addition comes further along in the logical sequence.

 

[Dale]

I think that "fundamentalness" is pretty subjective. After all, you can generally arrive at the same set of experimental predictions from different sets of postulates.

However, I would tend to agree with Peter Donis on this one. In the usual modern understanding, the metric is the most fundamental object. The "no faster than light" bit can be expressed purely in terms of the metric. The velocity addition formula requires also the specification of local inertial coordinates and the transforms between them.