Question

Show that if an emitter of radio waves is receding from the receiver with a speed V, the frequency v₀ of the radio wave in the reference frame of the emitter and the frequency v in the reference frame of the receiver are related by...

Show that if an emitter of radio waves is receding from the receiver with a speed V, then the frequency vo of the radio wave in the reference frame of the emitter and the frequency v in the reference frame of the receiver are related by 1- V [Hint: In Newtonian physics, the frequencies would be related by the Doppler for- mula for a moving emitter and a stationary receiver, - vo/(1 + V). Modify this formula to take into account the relativistic time-dilation effect.]

Answer 1

I'll use the 4-vector and Lorentz transformation concepts.

The 4-momentum of light emitted in the rest frame of emitter

$p^{\mu}=(h\nu_0,h\nu_0,0,0)$

The 0th component gives the energy of the photon. Transform this into the frame of receiver as follows

$\Lambda^{\, \mu'}_{\mu}=\begin{bmatrix} \gamma & -\gamma v & 0 & 0\\ -\gamma v & \gamma & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}$

Thus the 4-momentum of photon in the frame of receiver is

$p^{\mu'}=\Lambda_{\mu}^{\mu'}p^{\mu}= (\gamma h\nu_0(1-v),\gamma h\nu_0(1-v),0,0)$

Again the 0th component is the energy, also set this to apparent energy of frequency $\nu$

$h\nu=\gamma h\nu_0(1-v)$

$\Rightarrow \nu=\gamma\nu_0(1-v)=\nu_0\sqrt{\frac{1-v}{1+v}}$