Misunderstanding the lorentz transformation

A laser goes wherever it is pointed, and pointing it vertically in one frame is pointing it at a slight angle in the other. You might find that easier to understand if you take an example of a person climbing up and down a vertical cliff.

Suppose the person climbs at a meter per second, and you are at the x=0 origin of your frame of reference. If you stand at the foot of the cliff while the person climbs vertically up it, then the person's y coordinate increases by 1 for every second they are climbing up, and decreases by one for every second they are climbing down, but their x coordinate doesn't change, so they are moving orthogonally to your x axis. Now suppose the person repeats the climb while you (the coordinate origin) are walking at a meter per second along the bottom of the cliff. Now the person's y coordinate changes by 1 every second, but so does their x coordinate, so relative to your walking frame they trace out a path at 45 degrees to your x-axis, and their up and down climb is a triangular path.

In order for the person to move vertically in your walking frame, they would have to follow a 45 degree path in the frame of the cliff.

The direction of all motion is therefore frame dependent. Shining a light so that its motion is vertical in one frame, means that it's not vertical in another, and vice versa.

In the thought experiments, the light is shone so it moves vertically in one frame and follows an angled path in the other. It doesn't matter which way round you do it.

You might find it interesting to consider a version of the thought experiment in which a person moving in the x direction shines two lights, one to follow a vertical up-and-down path in their frame (which is a triangular path in yours) and the other to follow a triangular path in their frame which is straight up-and-down in yours. You can now see how time dilation is reciprocal. The time the person measures for the return of the up-and-down light in their frame has to be less than the time you measure for it (so their measurement is time dilated relative to you), whereas the time they measure for the return of the angled light in their frame has to be more than the time you measure for it, so your measurement of that is time dilated in their frame.

Pretty much all the textbooks and videos describe this in a same way

That is very unfortunate, but it mostly just shows that you have not considered enough books / videos.

This is not a glib statement. I think all introductions of the Special Theory of Relativity (SR) that follow Einstein's original derivation should just stop. Instead, if you consult a textbook or video that starts from Minkowski diagrams first, then you will never have this confusion in the first place.

why would[n't] the light beam follow the $y$-axis?

Really, what is happening is that the man on the train has aligned his laser and mirror and detector in such a way that, for the man in the train, the light beam follows his $y$-axis. Everything should be fine here.

We already know, from the train frame of reference, that the light successfully bounces at the mirror and ends up at the detector. Everything is aligned for the train's frame of reference, everything works just fine.

But "things working just fine" is a physical statement. This means that the fact that the light reaches the detector is a physical statement that everybody has to agree upon.

So, the alignment of the $y$-axis on the train is slightly off from the $y$-axis on the ground. The laser is slanted a little bit along the $x$-axis for the ground reference frame.

That is all there is to it.

Also, when Einstein first introduced SR, there was not yet lasers, even though his annus mirabilis papers laid the foundations for them. He was talking about lamps, which are not as directionally tight as lasers are. The lamp light just goes everywhere.

the light beam too "lands" at the very same point it was shot from. My question is, why?

It doesn’t have to land there. The man could aim it at another location. Or the mirror could be angled at a different angle. If the man didn’t aim correctly and if the mirror wasn’t angled correctly then the light would not land at the same point.

So in the end, it lands there because we chose to have it land there. We chose to have it land there because it makes the rest of the analysis easier. We could have made a different choice, but that just would have unnecessarily complicated a difficult concept.

Why does it have to follow the movement of the platform?

If it didn’t then it would miss the mirror. And we already established that we have chosen to have the light beam hit the mirror. The two facts go together.

Now, this is possible because the light beam sort of moves with the platform.

I think you have this a bit backwards. Instead of saying that it is possible for the light to hit the mirror because it sort of moves with the platform, I would say it is necessary for the light to move with the platform because we chose to have the light hit the mirror.

It may help you to consider that the "pure time" direction in one frame is a mixture of time and space in a different frame. The Lorentz transform can be written:

$$cT = \gamma ~ct - \gamma \beta ~x $$

$$X = \gamma ~x - \gamma \beta ~ct $$

or if c=1:

$$T = \gamma ~t - \gamma \beta ~x $$

$$X = \gamma ~x - \gamma \beta ~t $$

You can see that each "time" direction in one frame contains components of "time" and "space" in a different frame. Just like if you have two coordinate axes $(x,y)$ and $(x',y')$, one rotated with respect to another, the pure $x$ direction will be a mixture of the $x'$ and $y'$ directions, which can be shown via a right triangle and the Pythagorean theorem.

All this to say, I think any (linear) coordinate transformation will involve right triangles and each coordinate being a linear combination of components in the other frame.

Note: $\beta = v/c$, $\gamma = 1/\sqrt{(1-\beta^2)}$

Why would the light beam go upwards and also "follow" the platform?

The man has aimed the laser so that the beam returns to him, and all observers must agree that the beam returns to him. Changing reference frames just changes coordinates. It doesn't change what actually occurs.

Let's say that the returning beam hits a particular spot on his hand. All observers must agree that the light hits that spot, regardless of what time and space coordinates they assign to the events of the light pulse leaving the laser, reflecting off the mirror, and hitting his hand.

Now, this is possible because the light beam sort of moves with the platform.
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If it were a ball being thrown upwards, it would make sense since that ball already has the momentum of the boat, so it will cover a track that looks like a triangle from our perspective. Why would the light beam also do that?

The light beam does that because the laser inherits the horizontal momentum of the platform, and so do the photons it emits. A photon has no mass, but it does have momentum, and it behaves in accordance with the law of conservation of momentum, just like everything else.