(a) Consider a wave with dispersion relation (k). Show that the group velocity u = k transforms under a Lorentz transformation exactly like

(a) Consider a wave with dispersion relation ω(k). Show that the group velocity u = ∇_kω transforms under a Lorentz transformation exactly like a particle velocity.
(b) Show that the phase velocity u_p = (ω/k)k̂ transforms under a Lorentz transformation exactly like a particle velocity when ω = c|k|.