Question

you saw a green light on a stoplight while driving at a constant high speed. At the same moment, a policeman at rest saw the light as red. Using the concept of relativity, how are you going to defend yourself from a policeman who reprimanded you and gave you a ticket for speeding?

Answer 1

Accoring to the relativistic longitudinal Doppler effect the change in colour of the stoplight is given as:

$$\dfrac{f_g}{f_r} = \sqrt{\dfrac{1+\frac{v}{c}}{1-\frac{v}{c}}}$$

where $f_r = 476\times10^{12}Hz$ is the frequency of red light, $f_g = 550\times 10^{12}Hz$ is the frequency of green light, and $v$ is the speed of the car. Solving for $v/c$, obtain:

$$\dfrac{v}{c}=\dfrac{\left( \dfrac{f_g}{f_r} \right)^2-1}{\left( \dfrac{f_g}{f_r} \right)^2+1}$$

$$\dfrac{v}{c} = \dfrac{\left( \dfrac{550\times 10^{12}}{476\times 10^{12}} \right)^2-1}{\left( \dfrac{550\times 10^{12}}{476\times 10^{12}} \right)^2+1} \approx 0.14$$

Thus, if the speed of car is 0.14 of speed of light, the colours of stoplight can indeed be confused. But since this is very hight speed, the ticket for speeding was fair.