First of all, I want to tell how I get the time dilation formula by this method. Imagine if we at S' reference framework that moves at speed of v in x-axis, then we shoot a light to y-axis. According to the S reference framework the light will move diagonally while to S' reference framework the light will move vertically as follows:
Since the speed of light is the same at any reference framework, we can get the relationship between t and t' like the following: $$(ct)^2=(vt)^2+(ct')^2$$ $$t^2=\frac{(ct')^2}{(c^2-v^2)}$$ $$t = \frac{t'}{\sqrt{1-\frac{v^2}{c^2}}}$$ Now let's define some premises for the next scheme:
- S' reference framework moves at speed of $0.866c$ and indirectly cause $t = 2t'$.
- Using Lorentz contraction the length that S reference sees will be a half of the length S' reference sees.
- The "final" state is when S' reference framework experiences $1s$ and S reference experiences $2s$.
The problem goes here if at the same speed instead of shooting the light to y-axis, we shoot the light to x-axis so the light will move like the following:
Since S reference experiences $2s$ the light also will move with distance of $2C$ as just like by dropping the red diagonal arrow to the horizontal x-axis. And for the S' which experiences $1s$ the light will move with $1C$. Now I thought this is when Lorentz contraction playing a role. If we analyze the "final" state there is a difference between "distance of object to light's endpoint"($2C-2V$) that S reference looks and "distance of object to light's endpoint"(C) that S' reference looks. This difference of length is explained by Lorentz contraction. But The problem is if we use that logic the length's result won't be a half of the "origin length" according S' reference which violates the Lorentz contraction formula. Can someone help me finding where are mistakes in my story? and how $x' = \gamma(x-vt)$ is defined like that with an intuitive explanation?
