5 Relativistic Doppler Effect

Using the identity $\chi=\sqrt{c^2-v^2}$ for the time-speed variable in the wavelength equation for the relativistic Doppler effect

$$\lambda'=\lambda_0 \sqrt{\frac{1+v/c}{1-v/c}}$$ (20)

simplifies this expression to

$$\lambda'=\lambda_0(c+v)/\chi$$ (21)

It is possible to identify the individual contributions of the factors $v$ and $\chi$ to the total Doppler effect by considering $\chi=c$ (which isolates the effect of the spatial speed) and $v=0$ (which isolates the effect of the time-speed).

Setting $\chi=c$ results in:

$$\lambda'_v=\lambda_0 (1+v/c)$$ (22)

which is the regular equation for the acoustic Doppler effect with moving source and stationary receiver. Setting $v=0$ results in:

$$\lambda'_\chi=\lambda_0 c/\chi$$ (23)

which simply makes the photon's frequency proportional to the time-speed of the emitting particle.

The relativistic Doppler effect can thus be interpreted as a combination of the normal 'acoustic' Doppler effect in space and a frequency shift that results from the lower time-speed.