Hint 4

The 4-momentum $(p_1,p_2,p_3,iE/c)$ is a 4-vector and so it transforms just like
$(x_1,x_2,x_3,ict)$ under a Lorentz transformation.

For a photon $E=\vert{\bf p}\vert c=h\nu$, where $\nu$ is the frequency of the photon and $h$ is Planck's constant. If it is moving in the ${\bf e}_3$direction, then it has $E=p_3c=h\nu$ (and $p_1=p_2=0$) as perceived by $O$. Boosting this to the frame of $O'$, you should find that the energy there corresponds to

$$\nu'=\sqrt{{1-V/c\over 1+V/c}}\;\nu,$$

in agreement with the relativistic expression for the (longitudinal) Doppler shift.

A photon moving in the ${\bf e}_1$ direction (in the frame of $O$) has $E=p_1c=h\nu$ (and $p_2=p_3=0$) according to $O$. Boosting to the frame of $O'$, you should find

$$\nu'={1\over\sqrt{1-V^2/c^2}}\;\nu.$$

[What is the direction is the photon moving in according to $O'$?]