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It can look dumb, but I always had this question as a kid, what physical principles would prevent this?
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.
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.
Wouldn't that still be normal light speed communication from earth to two places on the moon, not FTL communication between two places on the moon?
Because you put the apostrophe in the wrong place?
Move a sheet up and down rapidly
You can see the wave travel across it