Everyone knows about the doppler effect with sounds. When a train approaches, the pitch of its sound increases. After it passes, the pitch seems to drop off. This is because the sound waves are compressed (wavelength shortened/frequency increased) ahead of a moving sound source. The sound waves expand (wavelength increased/frequency decreased) as the source moves away. The faster the sound source moves, the greater the change in pitch.

The doppler effect happens with all types of waves, not just sound. Light waves can be affected by the speed of the observer in the same matter. If you drive fast enough, you can change a red light to appear green to the driver. How fast would you have to be driving to make a red light look green?

The speeds necessary to achieve a noticeable change in light are on the order of the speed of light. These velocities need to take account of relativistic transformations of the moving systems. The relativistic doppler effect of wavelength for systems approaching each other can be expressed by the formula

where

λ_{R} is the wavelength seen by the receiver

λ_{S} is the wavelength of the source

β = v/c = velocity / speed of light

We can solve this for the velocity in a few steps. First, divide both sides by λ_{S}

Square both sides

Cross multiply each side

λ_{R}^{2}( 1 + β) = λ_{S}^{2}( 1 – β)

Multiply out both sides

λ_{R}^{2} + λ_{R}^{2}β = λ_{S}^{2} – λ_{S}^{2}β

Add λ_{S}^{2}β to both sides

λ_{R}^{2} + λ_{R}^{2}β + λ_{S}^{2}β = λ_{S}^{2}

Subtract λ_{R}^{2} from both sides

λ_{S}^{2}β + λ_{R}^{2}β = λ_{S}^{2} – λ_{R}^{2}

Factor out β from the left side of the equation

β (λ_{S}^{2} + λ_{R}^{2}) = λ_{S}^{2} – λ_{R}^{2}

Finally, divide both sides by (λ_{S}^{2} + λ_{R}^{2})

Now we can find the velocity using the relationship: β = v/c.

Now we can plug in some numbers for red lights and green lights. Let’s take the wavelength of a red light to be 650 nm and a green light to be 540 nm. The source light is red and the received light is green. λ_{S} = 650 nm and λ_{R} is 540 nm. Plug these values into the above equation.

β = 0.183

β = v/c

v = βc

v = 0.183c

If we take the speed of light to be 3 x 10^{5} km/s, then you would have to be driving 54,900 km/s to shift a red light to look green. Another way to view it is that you’d need to be travelling 18.3% of the speed of light.

Multiply this value by 3600 s/hr to convert to km/hr, you get 197,640,000 km/hr. While you won’t get a citation for running a red light, you will get one for speeding.

If you do get pulled over, respect the police officer that managed to catch up to you.