Two layered discs rotating at relativistic angular velocities

[Alexrose12345]

Hypothetically, if you had an object on top of a disc on Earth that was rotating clockwise incredibly quickly such that the object had a tangential velocity of almost c, and this disc sat on another disc rotating anticlockwise with the same angular velocity, would the object feel the effects of special relativity with respect to someone standing outside of the two discs?

Also, would it feel the effects of the coriolis and centrifugal "forces"?

 

[pervect]

Is the second disk rotating anticlockwise in an absolute sense, or relative to the other disk, in which case it would be standing still?

Is the object on the first disk or the second?

Finally, given that the clock is on one of the two disks, why is the other disk even in the problem - is it supposed to be massive or something? YOu didn't specify it, and if it's not massive it seems like its not part of the problem, really.

 

[Alexrose12345]

The top disc is sitting on top of the bottom disc, rotating in the opposite direction from it at the same angular velocity, so in classical mechanics it would be static to someone standing outside of both discs. The object is on the top disc.

The other disc is there because I want to determine whether special relativity would act in this scenario. In other words, if you put an object on this hypothetical top disc and spun it at almost c clockwise, and then spun the bottom disc at almost c anticlockwise, such that the object would be static relative to the rest of Earth, would the object age much more slowly than someone on Earth outside of the discs?

 

[pervect]

No - if the object is static in classical mechanics, it'll be static in special relativity too. And being static means it's velocity is zero.

For a formal demonstration, if you use the SR velcoity addition formula (for velocities in the same direction) which is:

v_add = (v1 + v2) / (1 - v1 * v2)

if v1 = +v and v2 = -v, the added velocity, v_add, is still zero. And given that the disks are rotating at equal speeds in opposite directions, v1 and v2 will be equal and opposite and in the same direction.

Formally again, v1 = r x w, v2 = - r x w, where x is the vector cross product.

 

[jacoblost]

^Although I could be wrong, I would disagree.

Real world example of relativistic effects: GPS satellites must be corrected for time dilation do to their relative velocity to someone using a GPS device on the earth. IE the clocks of two rotating objects someone on the Earth and GPS satellite would be different. If we compared them to someone who would see the two objects as stationary, say someone on the sun, all three clocks, earth, satellite and sun would be different. Minutely different, GPS satellites are corrected for a dilation of 40ns per day, but still different.

 

[pervect]

I'm really not sure how you go from the observation that GPS, doing it's calculations in the usual Earth-Centered-Inertial frame, has to take into account the velocities of the orbiting satellite clocks (which is true), to the statement that a non-rotating disk would experience any sort of relativistic effects in its own inertial non-rotating frame, which is false.

It's possible I misunderstood the original question, that seems to be the essence of it as I understood it.

 

[jacoblost]

Hmm... Somehow, I really don't know how this happens, I think I completely misunderstood the OPs question.

Sorry. You're correct. Somehow I understood that both discs were spinning opposite each other and still spinning relative to someone observing. I didnt see the OPs clarification that the discs were connected making the top disc stationary to the observer. Oops :)