Question

The most common use of relativity in every day life involves what is called
the ‘Doppler effect.’ This is an effect in which the frequency (= rate at
which crests pass by you) of a wave depends on the reference frame. There
is also a Doppler effect in Newtonian physics, but Newtonian physics would
predict a different amount of the effect. The Doppler effect is used in
various devices such as ‘Doppler radar’ and police radar guns to measure
the speed of storm systems or cars. For this problem, derive the formula
for the relativistic Doppler effect
τR =
r
c + v
c − v
τS . (3.6)
where v is the velocity of the source (S) away from the receiver (R). Here,
I have expressed the formula using the period τ (the time between wave
crests) as measured by each observer.

Answer 1

Non-relativistic Doppler effect for a moving source S and resting receiver R can be written mathematically as

$$\frac{\tau_R}{\tau_S}=\frac{c}{c-v}.$$

But if the source moves away with velocity close to the speed of light, the time will change:

$$\tau_{S(rel)}=\tau_S\sqrt{1-v^2/c^2}.$$

Substitution gives:

$$\frac{\tau_R}{\tau_{S(rel)}}=\frac{c\sqrt{1-v^2/c^2}}{c-v}=\sqrt{\frac{c+v}{c-v}}.$$