(a) An observer at rest in some inertial frame receives a photon that was emitted in direction n by an atom moving with ordinary velocity v (Fig. 2.6). The photon frequency and energy as measured by the emitting atom are ν_em and E_em; those measured by the receiving observer are ν_rec and E_rec. By a calculation carried out solely in the receiver’s inertial frame (the frame of Fig. 2.6), and without the aid of any Lorentz transformation, derive the standard formula for the photon’s Doppler shift:

(b) Suppose that instead of emitting a photon, the emitter produces a particle with finite rest mass m. Using the same method as in part (a), derive an expression for the ratio of received energy to emitted energy, E_rec/E_em, expressed in terms of the emitter’s ordinary velocity v and the particle’s ordinary velocity V (both as measured in the receiver’s frame).

Fig 2.6

Vrec Vem 1-v 1-v.n (2.33)