5. (15 pts) Relativistic transformation of 4-wave-vector. Show that for plane waves in vacuum, (w/c, kj, kl1, k12) is a four-vector where w/c = [k| and kj (11, 12) are the parallel (perpendicular) components of k to the direction of motion between reference frames. (i) Write the transformation equations explicitly for a frame moving at speed v. Use primes for variables in the moving frame. (ii) From this transformations, derive directly the effects of light aberration (see figure below, what is O'?, Doppler shift, and reflection from moving mirrors. k k' V V