I have been studying relativity from a decent amount of time and all the books have a common thing.That is, They explain time dilation in a simple independent way but for length contraction they explain it using muons reaching earth or they use the lorentz transformation to directly derive it,Or use time dialation for mathematical derivation ,What I want is not a derivation.I want length contraction to pop up intuitively..this is what I mean(I will explain time Dialation)
Relativity is there because we accept that velocity of light($c$) is constant and won't change for different frames, This simply directly pops out time dilation
If we study the two mirrors thought experiment in which we measure $c$ in a frame at rest (wrt the observer) by reflecting light on the mirrors and then measure it from a observer moving wrt the mirrors and find out that the distance that light has to travel In the later is greater than the distance it has to travel in the former,so to keep $c$ a constant, we know $c=d/t$ ,so If d is increased then t will also to keep $c$ constant and hence the time in the moving frame is greater than time in the rest frame..
Now I am unable to find an intuitive way like this to pop up length contraction directly just by saying velocity of light is constant, This is what I have tried.. but not able to pop out length contraction
Let's say two mirrors separated by a distance $L$(now the mirrors are like this >> | | and will move toward left or light along their length , now for a stationary observer, we measure the length through light going from mirror 1 to mirror 2 And $c=L/t$
Now doing the same for a observer moving with velocity $v$ wrt the mirror Now $c=(L' + vt')/t' - L'$ for contracted length and $t'$ for improper time, we know $t'$ is greater than $t$ (I have included time dilation ) . can we conclude from here that $L'$ Is less than $L$?