Scenario allowing a signal to travel at superluminous speeds?

[TheGoodDoctor]

Hi, new here. I'm a physics major undergraduate and just got through a class dealing with special relativity. I also learned about it a bit in high school.

My question is regarding Einstein's second postulate that, paraphrased, grants that no particle can be accelerated to a speed greater than c. But I've been thinking and came up with a scenario where you have a string of gears all arranged together. That is, you have one gear, like a sprocket, engaged to another gear to it's right and another to its right and so on until you have millions of gears that span some great distance, like the circumference of the Earth. What happens when you torque the first gear in the series?

Assuming these gears are "ideal", meaning there is no loss of momentum between the teeth of contact of two adjacent gears, and assuming Newton's 3rd law to be accurate in this case (relativity would agree, considering each gear would rotate at a relatively small angular velocity), wouldn't each gear move completely in sync, in effect rotating the final gear at the same instant the first gear is torqued? Doesn't this allow a signal to be transmitted instantly between two points regardless of the distance separating, as long as such a system is in place? Doesn't this imply the signal moves at a speed v=infinite?

I'm assuming many people have thought of(and shot down) a scenario like this, but I haven't been able to find an anything for or against it. Even the internet has failed me. Help please.

 

[mgb_phys]

It's basically the same as the pushing a light-year long rod to prod the recipient.
The push only travels at the speed of sound in the material, it's the same with a gear chain (although a little harder to picture).

It would go faster than light of you had an infinitely stiff material - so relativity puts an upper limit on the speed of sound in a material and so on it's Young's modulus.