Derivation of the relativistic Doppler effect

The derivation of the relativistic Doppler effect plays a key role in the traditional derivation of the Theory of Relativity.

In fact, Einstein – after deriving it from the laws of electrodynamics – used this principle in his fourth paper in the year 1905 to substantiate the hypothesis of the dependence of the inertia of bodies on their energy content.

With the following article we will see how a simple demonstration of the relativistic Doppler effect can be made by applying the principles of conservation of energy and momentum to the physical process of pair annihilation.

Derivation of the Relativistic Doppler Effect in reduced form

(For the detailed version of the derivation we refer to the fifteenth chapter of the book “Newton and Relativity“).

After the annihilation of the electron-positron pair, an unstable particle is formed.

Subsequently, the decay of this particle can be followed by the emission of two photons in opposite directions:

Click repeatedly on the rectangular surface to be shown the process

With this process, the entire mass of the particle is transformed into the 2hf energy of the two photons emitted. Therefore:

For an observer at rest

For an observer who is at rest with the particle, the following relationship of energy conservation applies before and after the particle decay:

\[ m_0c^2= 2hf \]

Where:

m₀ is the mass of the unstable particle formed after annihilation

c is the speed of light in a vacuum

h is Planck’s Constant

f is the electromagnetic frequency associated with the photons

For an observer in motion

For a second observer who instead moves at the speed of v, the following relations of energy:

\[ \frac{m_0c^2}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=hf_1+hf_2\]

and momentum conservation are valid:

\[ \frac{m_0v}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\frac{hf_1}{c}-\frac{hf_2}{c}\]

… before and after the particle decay.

Solving the three equations with respect to the frequencies f1 and f2 measured by the moving observer, as a function of the frequency f detected by the resting observer, leads to the relationships of the relativistic optical Doppler effect for an approaching light source:

\[f_1=f\sqrt{\frac{c+v}{c-v}}\]

and for a light source that is moving away:

\[f_2=f\sqrt{\frac{c-v}{c+v}}\]

Also for this, as for all the other proofs on this site, the Lorentz transformation is not used:

In fact, only the principles of conservation of energy and momentum are used.

The detailed version of the derivation of the relativistic Doppler Effect is reported in the fifteenth chapter of the book “Newton and Relativity“.

______________

With the Relativistic Calculator you can calculate the Frequency Ratio as a function of speed.

Continue on the alternative path of relativistic proofs: Alternative derivation of relativistic acceleration.

Go to homepage