Are there better derivation for space contraction?

Question

https://web.pa.msu.edu/courses/2000fall/PHY232/lectures/relativity/dilation.html

I really love the derivation here.

It's very easy. It makes it very clear.

We know that from moving observer (relatively), the pulse have to travel longer. So the time is longer.

How much longer? That's also pretty easy. The "vertical" distance is the same, h, and that's what I am asking. Why the "vertical" distance is the same?

The horizontal distance is added by a factor of v/c. While betha is the "sin" and we want to find the "cos". Which is easy.

The derivation is so simple, I can do this on my head and would love to write a better one if someone can point me how to insert equation here.

In fact, the light do not have to "bounce". The light can go vertically from point A and B and a bit of sin cos trigonometry and it'll work out.

In fact, I can re derive that more easily if only I know how to put math equation here.

I wish I can do the same to derive space contraction.

Now, is there something as simple as that for space contraction?

Note: This question originally ask why there is no space dilation for the "vertical" or perpendicular part. However, it turns out it's already asked and answered well somewhere else. So I change the question to how to derive the horizontal space dilation easily.

You can read out the gedanken experiment relating to the perpendicular expansion . It's well written at the end of Griffiths . — , Mar 11 '18 at 18:34
Where wo persons , one standing on the platform and the other in the train , both have paint brushes with different colours and try to mark out a line at a certain height of a wall...Rest I leave for your research.... — , Mar 11 '18 at 18:36
Related: How do I derive the Lorentz contraction from the invariant interval? — John Rennie, Mar 11 '18 at 20:23
BTW, it isn't standard usage to refer to it as "space dilation." We usually say "Lorentz contraction." — , Mar 12 '18 at 01:16

Nanashi No Gombe · Answer 1 · 2018-03-11T18:51:16.047

It follows from the principle of relativity: The laws of physics are the same in all inertial frames of reference.

Imagine a train rushing towards a tunnel whose size is almost the same as the rest width of the train, so that it can stay inside the tunnel at rest without touching the walls of the tunnel. But while it is moving, if the train's cross-sectional radius had increased (assuming distances perpendicular to motion in fact increase), it would bash into the wall and there would be an accident. On the other hand, from the perspective of the passengers, the incoming tunnel would widen up (again assuming distances perpendicular to motion in fact increase), and the train would easily pass through and there would be no accident. Alternatively, if normal distances shrunk during motion, for the ground observer the train would pass through the tunnel because the width of the train would be thinner with respect to the tunnel. On the other hand, the passengers would experience a narrow tunnel, too narrow for the train to squeeze into and, therefore, witness an accident.

If you wish to make sure that all inertial observers agree on whether there is an accident or not after all, you should not have length contractions or dilations in the direction normal to relative motion.

Are there better derivation for space contraction?

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