How fast a person must be going in a car to make a red light appear green?

Asked by: Katherine

Answer

Well, I would simply say 'relativistic speeds which a car cannot achieve with the current technology', and leave it at that. But, for the curious PhysLink reader, let me actually find out _how_ fast:

This is simply the relativistic Doppler shift for light. The following formula can be found in any textbook:

f' = f sqrt(1 + beta)/sqrt(1-beta)

where f is the frequency of the light for an observer at rest with respect to the light source, f' is the the frequency of the light for an observer approaching the source with velocity v, and beta is just v/c, where c is the speed of light.

Note that this equation does not come out of the blue. It can be derived from the Lorentz transformation for the four-momentum of a photon, (as well as otherwise) but I will not go into that here.

First, we need some data, or 'trivia'. The frequency of red light is about 4.5x10¹⁴ Hz, and the frequency of green light is about 5.5x10¹⁴ Hz.

Second, we need to express beta in the above equation in terms of f and f'. This takes a few steps of algebra, but finally we get the following equation:

beta = {1 - (f/f')²} / {1 + (f/f')²}

Plugging in 4.5x10¹⁴ for f, and 5.5x10¹⁴ for f', we arrive at the desired answer:

beta = 0.198

which means

v = 0.198 c

... or about one fifth of the speed of light. Considering the speed of light is about 300,000 kilometers per _second_ or 186,000 miles per _second_, that's pretty darn fast. Calculating the exact figures is left as an exercise for the PhysLink reader.
Answered by: Yasar Safkan, Ph.D. M.I.T., Software Engineer, Istanbul, Turkey