Transferring information faster than the speed of light

I didn't understand the comparison. With a push, you didn't move a particle from one side to the other, you just made the particles move (the particle on the other side only moved a few meters, no faster than the speed of light). How exactly did you make something go faster than the speed of light? And how exactly do you transfer information from one place to another like that?

“Pushing it in a certain direction will apparently immediately move its other end”.

The movement of the other end will not be immediate.

Yishai
I don't have a good answer to the question raised, but on the surface your suggestion isn't a solution either. The effect on the position of the far end of the stick seems like a way to transmit a signal. You can imagine, for example, two friends standing at either end of the stick who agreed in advance that a slight movement of the stick on their side to the right means "yes" and a movement to the left means "no". And here information has been transmitted at a speed greater than that of light. At least the paradox seems to be valid.

Doron,
When you manage to move a rod 300 thousand km long in less than a second, we will have something to talk about. In practice, the rod will not move and neither will you.

Rabbi Michi: Thank you, I didn't know. Where can I find a place that discusses this?
Yishai: See Doron's answer
To whom: Why not?
Doron: Thank you, that's what I was referring to
Y.D.: This is not an acceptable answer to questions of this type. The question is at the level of theoretical perception, and technical difficulties are not an answer.

Search for example long rod light speed

Well, I searched the net and found a solution to the paradox, apparently…. The claim is that material bodies (rods) are never completely rigid and therefore even short objects do not move simultaneously. When I push a body, there is inevitably (due to the laws of physics) a disturbance in the medium that travels from one end of the rod to the other. This is actually the movement of a signal or information. When it comes to a rod that long, the disturbance will move no more than the speed of light anyway (and possibly much slower).
But…. Here the question arises: does this barrier to the speed of light have a logical or physical status? That is, it is possible that in another possible world where slightly different laws govern, there are ideally rigid rods and therefore there it is actually possible to cross the speed of light. And more: How does the special theory of relativity see the last question? To the best of my understanding, relativity blurs the line between a priori (logical) considerations and empirical physical considerations, and as such, it maneuvers itself into a situation that is uncomfortable for itself. Because according to its internal logic, there is not even a logical status for such a possible world. If that is indeed what it claims (albeit only implicitly), perhaps there is a problem with it as well.

Here, the thought experiment has a perhaps more successful version: a very powerful flashlight casts a shadow on a body at an astronomical distance. The question is whether moving the flashlight will immediately change the position of the shadow on that body?

Thanks to all the answers, I also found the answer in a detailed explanation:
https://physics.stackexchange.com/questions/2175/is-it-possible-for-information-to-be-transmitted-faster-than-light-by-using-a-ri

Moving a lantern will never change a shadow faster than the speed of light.
Besides, what's the problem with assuming that you can transmit messages faster than the speed of light with a pole?

Regarding the rod, I came up with an explanation that seems very plausible to me as to why there is a problem in overtaking the speed of light. At least a physical problem. Regarding the flashlight and the shadow, it seems that this limitation is explained in the same way.

There may be a physical limit. The point is, if there were no limit, there would be no question about anything.
In fact, did the limit also exist in pushing forward? According to this, if we push a rod forward at a threshold speed of the speed of light, it will shrink to zero size. According to the link, this is a much lower speed. Fast planes and missiles should shrink significantly in the slow wave of pushing.

A shadow is the absence of light, and moving a flashlight should not move the light that has already come out of it, so there is no room for questioning.

In my opinion, you are mixing up two different issues (which are both related in special relativity). The first is a change in the measured dimensions of a moving object as predicted by relativity, and the second is the phenomenon of elasticity of bodies, whose rigidity is never absolutely ideal.
As I understand it, even before Einstein, we knew how to say that there is an apparent paradox. But it was not between the speed of light as an upper limit and the simultaneous movement of the two ends of the rod, but between the intuitive (erroneous) assumption that the two ends must move simultaneously and their law of elasticity that prohibits them from doing so. The solution to the old paradox is also good for the new one and there is no need to introduce relativistic considerations here. That's how I understand it.

My perception of things is a little different than what you understood

And I didn't quite understand why you call incorrect intuition a paradox

Do you agree with Mishk about the flashlight?

Regarding the flashlight, you might be right. I brought the example from memory from a YouTube clip I saw a long time ago. Maybe I remembered incorrectly.

Regarding the fundamental issue:
I said that even in the world before Einstein, they could perhaps think that there was a paradox here. After all, there are two seemingly conflicting opinions here: on the one hand, our intuition says that both ends of a rod, even a hypothetical rod no matter how long it is, move at the same time (if you pushed one end, the far end will inevitably move immediately); on the other hand, they knew even then that the material is not completely rigid and that there is a “delay” in the motion passing through it from one side to the other (the far end will start moving some time after the near end has already begun its movement).
In this case, I suppose that even in the pre-Einstein period, they understood that this was only an apparent paradox. The intuition that the ends of the rod move simultaneously is created by mistake because in reality the rods are so short that it is impossible to actually notice that supposed delay. In contrast, in our thought experiment it is not difficult to see that the elasticity of the material ceases to be negligible and then a real “delay” is created.

Take a long rod, tie it to one side so it can't move, and put a machine on the other side that will pull it all the time. As soon as you release it from the tie, it will start moving.

I've already written before that I agree with Yishai that there is no problem with this, but anyone who sees a problem with this is welcome to look for an answer.
Yeshiva Guy

For the original example of a shadow moving faster than the speed of light (in English)
http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/FTL.html#3