The Physics Philes, lesson 116: Relativistic Doppler Shift
I know, I know…I didn’t write anything about physics last week. It hurt me more than it hurt you, I promise. Let’s see. Where did we leave off? Oh yes, Doppler shift.
As we found last week, sound waves can be shifted if either the source of the sound or the listener is moving. After our derivation, we found that the frequency, as heard by the listener, can be found by this equation:
However, this only works when we can measure the velocities measured with respect to a medium. Not all waves need a medium to travel through. Electromagnetic waves can travel through the vacuum of space with no problem. Does the Doppler shift apply to these waves?
The short answer is yes, but we can’t use the above equation to describe it. The problem is that, as the wave travels faster and faster – approaching the speed of light – classical mechanics no longer holds. At these super-fast speeds, we enter the realm of relativity.
Einstein based his theory of special relativity on two assumptions: 1. That the law of physics have the same form in all inertial reference frames and 2. The speed of light in a vacuum has the same value in all inertial frames, regardless of the velocity of the observer or source. One of the consequences of this is called time dilation. Think of ticks on a clock. If you are at rest with respect to the clock, then the time we measure between ticks is called the proper time. But what if someone flies by at some percentage of the speed of light? You might think that they would measure same time as you do as you are sitting with the clock at rest. But she won’t. The time the person moving with respect to the clock will measure a time between ticks that is longer than the proper time you measure sitting right next to the clock. This is time dilation.
It’s pretty weird and counterintuitive on its own, but the consequences are far-reaching. One of those consequences is the Doppler shift of electromagnetic waves. Unlike the Doppler shift of sound waves, the Doppler shift of electromagnetic waves depends on the relative motion of the source of the wave and the receiver of the wave. In order to measure the time between the arrival of wave crests to the eyes of an observer, we need to integrate time dilation into the equations. The final result relies only on the frequency of the wave at the source, the relative velocity of the source, and the speed of light c:
When the source is moving away from an observer, v is positive and the frequency the receiver sees is less than the frequency at the source. When the source is moving toward the observer, v is negative and the observed frequency is greater than the frequency at the source.
This effect is super important for astronomy. When we look at far-away galaxies, we notice that the light emitted from them have longer wavelengths. This is called redshift. Since the wavelengths look longer to us on Earth, we can determine that most galaxies are moving away from us. Edwin Hubble used relativistic Doppler shift to confirm that the universe is expanding.
All of this is pretty cool, and I’ve love to talk about special and general relativity forever. But it’s time to move on. Next time we’ll look at shock waves.
Featured image credit: Wikipedia