## Rotational velocity and the speed of light

We all know that, as an object rotates about an axis in constant circular motion ($$\omega$$ is constant), the linear velocity increases the further the object is from the axis (v increases as r increases, v =$$\omega * r$$)

Let's say you build a scyscraper. The taller the skyscraper, the faster the tip of the skyscraper moves (linear velocity) as it rotates about the earth's axis.

Increase the height of the skyscraper, you increase the linear speed of the top of the skyscraper.

So, let's say you increase the height of the skyscraper to, say, 4,125,296,124,942 meters. Then, the very top of the skyscraper would be traveling at the speed of light. And increase the height a little more and you would be traveling faster than the speed of light.
You could be carrying information at the top of the skyscraper faster than the speed of light.

How is all of this refuted?

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 Mentor Blog Entries: 9 Once again non physical assumptions lead to non physical conclusions. You cannot build the structure you are hypothesizing so of what interest is your thought experiment. A key to good, meaningful, thought experiments is completely physical assumptions.
 Recognitions: Science Advisor Ignoring the practical limitations, the main reason that it won't work is that as the tip of the skyscraper goes faster, it increases in mass. This would force the earth to slow down. The net result - the earth could never rotate fast enough to get the tip to rotate as fast as the speed of light.

## Rotational velocity and the speed of light

 Originally posted by mathman Ignoring the practical limitations, the main reason that it won't work is that as the tip of the skyscraper goes faster, it increases in mass. This would force the earth to slow down. The net result - the earth could never rotate fast enough to get the tip to rotate as fast as the speed of light.
No, it will not exceed the speed of light because of the inter-atomic communication time/speeds between adjacent atoms. This will affect the "rigidity" of the building in question (it will "bend" it).

As you will find in numerous other threads on this forum, mass does not increase with increasing velocity -- a notion which I have seen pop up so many times in the past three days that it is tantamount to spam at this point!

 Mentor Blog Entries: 9 Would it require a mass increase to slow the earth down, wouldn't such a huge structure slow it down just due to conservation of angular momentum. Just like a spinning ice skater spreading her arms.

 Originally posted by Integral Would it require a mass increase to slow the earth down, wouldn't such a huge structure slow it down just due to conservation of angular momentum. Just like a spinning ice skater spreading her arms.
If we really want to argue this "physically", you can't build a building which is 4,125,296,124,942 meters tall in the first place. So the point is pretty academic.

Recognitions:
 Originally posted by Integral Would it require a mass increase to slow the earth down, wouldn't such a huge structure slow it down just due to conservation of angular momentum. Just like a spinning ice skater spreading her arms.
It would depend exactly how you constructed the building. In any case, GRQC gave the right explanation. Perfectly rigid bodies can't exist because they imply infinitely fast information transfer.

 Mentor Blog Entries: 9 So we have come full circle to my original post. Non physical assumption leads to no information.

Mentor
 Originally posted by Integral So we have come full circle to my original post. Non physical assumption leads to no information.
To examine the non-physical assumption a little more, carrying that last brick up to the top of the tower would require an infinite amount of energy.

 to answer your question, it is impossible, since the particules at the top would get an infinite mass, so the intermolecular forces couldn't hold the building together anymore, no mather how strong the material you used is. Even if this is only a thought experiment and you use a completely rigid imaginary object, you simply wouldn't be able to apply enough torque to give it enough angular velocity for the tip to reach the speed of light. Wether the acceleration you give an object is linear or angular, the mass always goes to infinity as it reaches the speed of light, which means it's impossible to accelerate it faster than c. Finally, I think that complete rigidity is incompatible with GR.
 I like the angular momentum argument best, because it works even in the classical limit. The higher you make the tower, the slower the world turns. THEN take into account the relativistic effects on mass and kinetic energy. Finally, realize that the earth-tower system would rotate about its center of mass--as you build the tower taller and taller, the center of mass is creeping up behind you like your shadow. So there are more than enough phenomena conspiring to ensure that the top of the tower doesn't move faster than the speed of light. HOWEVER: if the tower is massless, then all of these arguments break down. For this reason, you can make a 'tower of light' (laser pointing straight up) and run its point along the surface of the moon at faster than c. P

 Originally posted by rocketcity HOWEVER: if the tower is massless, then all of these arguments break down. For this reason, you can make a 'tower of light' (laser pointing straight up) and run its point along the surface of the moon at faster than c. P
woah! elaborate!

but in this case, information truly would not be moving faster than c, right?

Mentor
 Originally posted by brum but in this case, information truly would not be moving faster than c, right?
No, it wouldn't. Information is still transferred in the beam, not along the point of contact.

 Relative speeds under GR are not restricted to luminal or subluminal velocities. For instance, compare the velocities of oppositely expanding CBR horizons. The "tower" situation may be framed as the rotational analog of the horizon problem. I hypothesize that two oppositely rotating light beams, as an extention of the equivalence principle, eventually separate by an relative maximum velocity near twice light speed.