I enjoyed a lot of the discussion in the comments
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The pole would basically be a space elevator. I suspect gravity and inertia would effectively keep you from moving the stick. Even if you could move it, you'd only be able to move it at a speed that would seem like it's stationary. As such, the light would still be faster.
That would not work. Pushing an object is transmitting kinetic energy to it. The object will push back, and energy would not be distributed to the whole object at the same time.
If the object cannot be altered in any way, then the energy would not be transferred to it, and if it has enough plasticity to absorb the kinetic energy, it would be spread in a wave to the tip. A wave that would always be slower than light.
Now stop fooling around and give Ruyi Jingu Bang back to Sun Wukong.
What about the mass of that stick? Inertial doesn't care for your little silly games.
Neither do the two gravity wells the stick spans. And the earth and moon are moving relative to each other, someone would probably get their head knocked off by that stick. Before it eventually falls to the earth with quite a bit of force because earth's gravity well will win. Then it'll eventually settle into a giant teeter totter, assuming it is rigid enough to survive the impact.
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.
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.
This wouldn't work, entangled particles don't work like that. They would be disentangled the moment you do anything to either particle of the entangled pair. The only time any information can be encoded onto entangled particles is when they're created.
The only time any information can be encoded onto entangled particles is when theyβre created.
If that were the case, then we aren't really doing FTL communication, unless we manage to entangle them at a distance. No?
OIC, it's still useful if we want to make a secret key and send it somewhere. Then both sides can take a reading sometime in the future and they can then use whatever cluster of entangled particles they saw, as the symmetric key.
Any time you do something to the particles on Earth, the ones on the Moon are affected also
The no-communication theorem already proves that manipulating one particle in an entangled pair has no impact at al on another. The proof uses the reduced density matrices of the particles which capture both their probabilities of showing up in a particular state as well as their coherence terms which capture their ability to exhibit interference effects. No change you can make to one particle in an entangled pair can possibly lead to an alteration of the reduced density matrix of the other particle.