Asklemmy

Big stick energy

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.

So have to ask what a solid is to answer this question.

Sticks are quite complex, so lets consider a simpler solid: an elementally pure iron rod.

You can imagine said rod as if it were a fixed array of crystalline atomic cores surrounded by a jelly-like substance. In this 'jellium' model the atomic cores have a positive charge, they are the protons and neutrons, and the jelly has a negative charge. The jelly is the wavefunction that represents the electron structure in bulk. If that makes no sense, congrats on knowing your limits.

You've probably seen the more modern model of an atom where there's a nucleus and around it is an electron fuzz with discrete energy levels. Or if you've studied at uni strange geometry representing a threshold in probability of finding the electron/s there on a given measurement (if not familiar under certain conditions reality kinda unfocuses it's eyes and things that we often think of as points become volumes of possible effect). This is a good model of a single atom, but when we bring atoms together they change each other's properties and the result is that these density functions (the weird electron cloud/shape things) start to blur together.

In our iron rod the electrons delocalize sufficiently we can kinda think of it as a weird jelly. A real stick is more complex, but can kinda be thought of as a stack of smaller jelly treats packed against each other.

When you push on the rod you're mashing the jelly of your hand into the jelly of the rod, this causes a shockwave that begins to spread, it propagates like a ripple in a skipping rope or a bounce on a trampoline. But since it's moving 'amount of electron like properties here'. That makes some areas more negatively charged which drags the positively charged atom cores slowly after it. It moves much slower than the speed of light as we aren't considering individual electrons which can move energy between them via photons, but the propagation of a disturbance in the collective arrangement of many that are tightly linked (we say coupled).

We can't imagine a stick that is perfectly rigid because we would be proposing a kind of matter that does not exist, one which isn't made of a lot of fuzzy electron jelly stuff but something else entirely. We can imagine matter where the jelly is very stiff, and consequently less energy goes into wobbling it all about and the squish moves forward very fast but that speed is still much slower than light because of this collective behaviour.

Move a sheet up and down rapidly

You can see the wave travel across it

Because you put the apostrophe in the wrong place?

Objects like an unbreakable stick are still composed of atoms suspended in space and held together by the fundamental forces of nature. When you push on one end, the other end doesn't immediately move with it but rather the object experiences a wave of compression traveling through it. This wave of compression travels faster than we can perceive but still cannot travel faster than light.

Look up why arrows bend after they've been released by a bow, it's essentially the same mechanic.

I could've sworn I saw a video about this and the gist is that it's called "speed of push" and is essentially the speed of sound. When you push something, you're compressing the molecules of it and that will travel like a wave through it. Light travels faster than that wave.

I'm probably explaining wrong because it's something I'm half remembering from a video I could've seen over a decade ago, but that's the quick explanation.

Well no. As others have said the force in the pole will travel at the speed of sound.

Though if you were to wiggle the flashlight back and forth really fast the spotlight on the moon would travel "faster" than the speed of light.

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.

Something about objects don't move instantaneously but at the speed of sound that material has, so the stick would move way later. If you think about it, speed of sound inside a medium is basically how fast the particles inside that medium can send energy from one another.

I enjoyed a lot of the discussion in the comments

...so the thing is that, after accounting for time dilation, light is instantaneous and perhaps better-described as the speed of causality...even a 'perfect stick' comprising quantum-crystal wonder-material can't move before it's pushed, so you'd find that it, too, transmits information at the speed of light...

The issue is, that kind of stick wouldn't even exist. You'd have better luck with between some dwarf planet and its satellite, since the stick would break under its mere weight.

The stick would only move at the speed of sound. Or the speed the molecules can push against each other, which is the speed of sound in that material.

it wouldn't work, because there is no unbreakable, unfoldable stick. the stick will have flex, and the force transmitted will occur much more slowly through the molecular chain of the stick than light's travel time.

reality is much more woobly and spongy than you know.