In deriving Special Relativity equation, we modify the length parallel to motion in a way that outside observer would agree that light moving in perpendicular and in parallel with motion would meet in the same time and spot (the same event). But the same effect could be achieved if instead we make the length perpendicular to motion becoming longer and set the length parallel to motion fixed.
I’m referring to this derivation from Feyman Lectures on Physics’ The Special Theory of Relativity (https://www.feynmanlectures.caltech.edu/I_15.html):
The first equation in question is:
$t_1 + t_2 = \frac{2L/c} {1 - u^2 / c^2}$ (Eq. 1)
which is the time needed for light to go from $B$ to $E^\prime$ and coming back to $B^\prime$.
The second equation is:
$2 t_3 = \frac{2L/c} {\sqrt{1 - u^2 / c^2}}$ (Eq. 2)
which is the time needed for light to go from $B$ to $C^\prime$ and coming back to $B^\prime$.
In order to make it equal so the light would arrive at $B^\prime$ in the same time, we choose to shorten the length $L$ parallel to line of motion to $L \sqrt{1 - u^2/c^2}$, so Eq. 1 would become:
$t_1 + t_2 = \frac{2L/c} {\sqrt{1 - u^2 / c^2}}$ (Eq. 3)
where $t_1 + t_2$ would equal with $2 t_3$ from Eq. 2.
But the same effect could be achieved by lengthening line $L$ perpendicular with line of motion to $L / \sqrt{1 - u^2/c^2}$ so Eq. 2 would become:
$2 t_3 = \frac{2L/c} {1 - u^2 / c^2}$ (Eq. 4)
where again $t_3$ would equal with $t_1 + t_2$ from Eq. 1.
If we are doing that, what I know immediately from the equation is the time in the other observer would be slowed down by a factor of $\frac{1}{1 - u^2 / c^2}$ instead of $\frac{1}{\sqrt{1 - u^2 / c^2}}$.
I know there would be paradoxes when it does happen, and it would not be invariant with Maxwell’s equation, but just from Special Relativity point of view, what is wrong with this? Why don't we choose this solution?
And please do not just vote down this question, but also give reason when it's considered stupid. I have searched and no answer to find.
