I tried to derive the Lorentz contraction by using relativity in simultaneity.
A '2L' long cart is moving with a velocity of 'V'.
Events A and B are simultaneous w.r.t observer O1
Since cart is moving in frame of O2, therefore, event A and B are not simultaneous w.r.t O2
In frame of O2
Let's assume a light source at the center of cart,
So, time required for photon to hit at back of cart be $t_1$
$ct_1=L'-Vt_1$,
$t_1=\frac{L'}{c+V}$
And similarly, time required to hit the front part cart be $t_2$
$ct_1=L'+Vt_2$,
$t_2=\frac{L'}{c-V}$
This implies event A occurs before event B w.r.t observer O2, and the time lag between these two events is given by $t_2-t_1$
$$t_1-t_2=\frac{L'}{c-V}-\frac{L'}{c+V}$$
$$t_1-t_2=\frac{2VL'}{c^2-V^2}$$
Now,observer O2 tempts to measures the length of the cart,Since events at the at the both ends of cart are not simultaneous,So at the moment when the observer O2 observe end A indeed for him end B will seem quite shifted towards rights side due to lack in simultaneity which causes length contraction.
Length by which the cart contracts w.r.t O2
=velocity of cart* time lag
$$2L-2L'=\frac{2V^2L'}{c^2-V^2}$$
$$L=\frac{c^2L'}{c^2-V^2}$$
The above result deviates a lot from the standard result of length contraction.Does this implies relativity in simultaneity is not the core cause behind Lorentz contraction?
