Relativistic Doppler Effect for LightConsider two objects: the light source and the "listener" (or observer). Since light waves traveling in empty space have no medium, we analyze the Doppler effect for light in terms of the motion of the source relative to the listener.
We set up our coordinate system so that the positive direction is from the listener toward the source. So if the source is moving away from the listener, its velocity v is positive, but if it is moving toward the listener, then the v is negative. The listener, in this case, is always considered to be at rest (so v is really the total relative velocity between them). The speed of light c is always considered positive.
The listener receives a frequency fL which would be different from the frequency transmitted by the source fS. This is calculated with relativistic mechanics, by applying necessary the length contraction, and obtains the relationship:
fL = sqrt [(c - v)/(c + v)] * fS
Red Shift & Blue ShiftA light source moving away from the listener (v is positive) would provide an fL that is less than fS. In the visible light spectrum, this causes a shift toward the red end of the light spectrum, so it is called a red shift. When the light source is moving toward the listener (v is negative), then fL is greater than fS. In the visible light spectrum, this causes a shift toward the high-frequency end of the light spectrum. For some reason, violet got the short end of the stick and such frequency shift is actually called a blue shift. Obviously, in the area of the electromagnetic spectrum outside of the visible light spectrum, these shifts might not actually be toward red and blue. If you're in the infrared, for example, you're ironically shifting away from red when you experience a "red shift."
ApplicationsPolice use this property in the radar boxes they use to track speed. Radio waves are transmitted out, collide with a vehicle, and bounce back. The speed of the vehicle (which acts as the source of the reflected wave) determines the change in frequency, which can be detected with the box. (Similar applications can be used to measure wind velocities in the atmosphere, which is the "Doppler radar" of which meteorologists are so fond.)
This Doppler shift is also used to track satellites. By observing how the frequency changes, you can determine the velocity relative to your location, which allows ground-based tracking to analyze the movement of objects in space.
In astronomy, these shifts prove helpful. When observing a system with two stars, you can tell which is moving toward you and which away by analyzing how the frequencies change.
Even more significantly, evidence from the analysis of light from distant galaxies shows that the light experiences a red shift. These galaxies are moving away from the Earth. In fact, the results of this are a bit beyond the mere Doppler effect. This is actually a result of spacetime itself expanding, as predicted by general relativity. Extrapolations of this evidence, along with other findings, support the "big bang" picture of the origin of the universe.