Question 4

The space-time coordinates of a particle according to inertial observers $O$ and $O'$ are related by the equations

$$x'_1 = x_1,$$

$$x'_2 = x_2,$$

$$x'_3 = {x_3 - Vt\over \sqrt{1-V^2/c^2} },$$

$$t' = {t - Vx_3/c^2 \over \sqrt{1-V^2/c^2} } .$$

If the particle has momentum ${\bf p}= (p_1,p_2,p_3)$ and energy $E$ according to observer $O$, what are its momentum and energy according to observer $O'$ ?

If the particle is a photon, find the relation between the frequencies of the photon as perceived by observers $O$ and $O'$ when it is moving (a) in the ${\bf e}_3$ direction and (b) in the ${\bf e}_1$ direction.

[Hints can be found here, but do not follow this link until you have attempted the question on your own.]