Suppose I have a stationary point charge $q$ in frame $S'$ where $S'$ is the frame moving toward the $x$ direction of frame $S$ with velocity $v$. I can write its four potential $A^{\mu}{'}$ but since it's a stationary point charge the only non-zero term is its time component. It should be$$\frac{q}{4\pi\epsilon r'}$$ Suppose I want to calculate the potential in $S$ frame, the procedure is just performing tensor transformation. And in $S$ frame it becomes:
$$\gamma \frac{q}{4\pi\epsilon r'}$$
Next I should express $r'$ using coordinate of $S$. Like what they did for stationary electric field (the following link). http://www.physicsinsights.org/moving_charge_1.html
But the instead my teacher use the relation:$$r'=ct'$$ and $$t'=\gamma(t-\vec{v} \cdot \vec{r}/c^2)$$ which is Lorentz transformation for time. Using these two formula, he gets expression for potential in $S$ frame. $$ \frac{q}{4\pi\epsilon (r-\vec{v} \cdot \vec{r}/c)}$$
My question is why we use such way to transform the potential? Why we need to consider the time for the potential to travel, while for the Electric field, we don't consider retarded time?
