Your push would travel at the speed of sound in the stick. You could think of hitting a pipe with a hammer, the sound of the hit would travel at the speed of sound, same is true for you pushing the stick.
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I don't see this mentioned in any of the other comments: the repulsion between atoms that causes the movement to propagate through the stick is actually communicated via photons. So your push really generates the same kind of particles that your light torch is generating, and they travel at the same speed. Except in the stick it is slowed down by repeated absorption and excitation by the electrons of the atoms.
why wouldn't this work
because bullets are faster than whatever the fuck speed stickman is achieving
and even bullets are slower than light
That's what he meant by we'll use sticks on the other side
Nah, I prefer using quantum spookiness for that. Send a steady stream of entangled particles to the other person on the moon first. Any time you do something to the particles on Earth, the ones on the Moon are affected also. The catch is that this disentangles them, so you have only a few limited uses. This is why you want a constant stream of them being entangled.
You also cannot choose the spins of entangled particles, they collapse randomly in either direction when interacted with, meaning you cannot send messages. If you can figure out how to directly influence the spin of generated subatomic particles then BAM you have FTL communication.
But you would be amazed how many obstacles the universe throws in front of you when you try to break the speed of causality. Faster than light communication isn't possible because it makes no sense when you understand it. It's like "getting answers faster than questions." It's nonsense.