October 20: The Relativistic Doppler Effect
Meaning of Equation
The wavelength of light that is received by an observed moving away from the source is more than the wavelength that is actually emitted. In other words, the wavelength gets red-shifted when it reaches the observer who is receding away from the source.
We all know the classical Doppler effect. We experience it in day to day life. Suppose a fire truck is coming towards you with its siren turned on. What do you notice? The frequency of the siren starts increasing as the truck comes closer. It reaches its peak when it is closest to you and again the frequency decreases when it recedes away from you. This is the Doppler effect with the sound waves. A similar effect exists for the light waves. But if the source of the observer moves with a velocity comparable to that of light, we need to consider the effect of time dilation too. This is known as the Relativistic Doppler Effect.
To understand the relativistic Doppler effect, let us first understand the concept of wavelength and frequency in brief. Wavelength is the distance that a wave travels between two successive wave emissions. Suppose first wave started from source at time t = 0 and the speed of light c and second wave started at time t = 5 seconds. The distance, d= 5c that the first wave travelled is known as the wavelength of the wave. The image below shows this.
Alternatively, wavelength can also be defined as the distance between two peak points (crests) or two valley points (troughs) of a wave as shown. The next important term is frequency f which is nothing but the number of waves emitted per second. If more number of waves are emitted in lesser time (high frequency), then obviously the wavelength will be lesser as the first wave would have covered a small distance before the next wave started. The relation between the speed of light, its wavelength and frequency is: c = λf.
There is one piece of information that I feel is important to add here: The electromagnetic spectrum. The EM spectrum is the band of wavelengths and frequencies of the electromagnetic waves as shown below.
That's it. We now have all the important information that we need to understand the relativistic Doppler effect. Now suppose there is a stationary source of light and a receiver who is receding away from the source. In the frame of reference of the source, wavelength λ = c/f where f is the frequency of EM wave emitted. Now if the receiver was stationary, the wave-front would have reached it easily without such difficulty. But now it is moving away from the source with velocity v. So the wave needs to travel extra distance to reach it. Thus the equation becomes λ + vt = ct. The vt term is the distance travelled by the receiver and since it is moving away, it is added to λ. If the receiver moves towards the source, the amount of distance travelled by the wave would have decreased and the equation would have been λ - vt = ct.
I have solved the derivation in the image below with steps:
The only difference that arises is from the time dilation term. Time measured in the reference frame of the source will be different from the observer's frame. This adds to the equation and modifies it.