The measurement of the flux of muons at the Earth's surface shows that many more muons are detected than would be expected, based on their mean half-lifetime of 2,2 microseconds. This is a good proof for the time dilatation as predicted by the special relativity theory.
QUESTION: does there exist a similar proof for the Lorentz contraction?
As far as I know no-one has ever directly measured the length of a relativistic object. However indirect measurements of Lorentz contraction have been made at the RHIC. This collides nuclei together at relativistic speeds and analyses the resulting shrapnel. The nuclei are far too small to see, but from the results of the collisions we can infer that the nuclei must have collided as disks not spheres:
So while the Lorentz contraction of the nuclei isn't seen directly it is inferred from the results.
Remember that any inertial frame is equally valid.
You've been looking at the muons and mountain experiment from the POV of a person standing on the ground (or) on the mountain. In that frame the muon cover a distance equal to the mountain's height and take several times the muons (uncontracted) lifetime to go that distance, but they make it because the person see's their internal clock as running slow by a factor of $\gamma$.
Now imagine that you are riding on one of the muons. In this frame of reference the muons are at rest and the Earth rushed toward you at large speed. Because the muons are at rest relative you they will decay on their un-contracted schedule, which means that there isn't enough time for the Earth to move by a distance equal to the height of the mountain. None-the-less you crash into the ground before many of the muons have decayed because the mountain was length contracted from your point of view.
The point is that both length contraction and time dilation are expressions of the fact that the interval $$ s^2 = (c \Delta t)^2 - (\Delta x)^2 $$ between events is measured to be the same in all inertial frames of reference (this is a re-statement of the constancy of the speed of light).
That is we can say that $$ (c \Delta t)^2 - (\Delta x)^2 = (c \Delta t')^2 - (\Delta x')^2 \,$$ where the un-primed frame represents the time and distance measurements of a person standing on the ground and the primed frame those of a person riding a muon.
The un-primed frame see the full height of the mountain, but a dilated time period. The primed frame see the un-dilated (therefore shorter) time period but see the mountain's height as contracted allowing the difference to be the same.
Imagine for each observer there is a grid in space that maps how far light travels in an amount of time. You can measure an object at rest in this grid by how long it takes light to travel its length and back again (this is important to measure it both ways). Since light speed is a constant, the object needs to change shape to keep its 'light length/duration' the same. It is hard to measure length and duration contraction directly, but you can sort of get evidence through secondary effects, but these could possibly be from other causes.
