The derivation of the relativistic Doppler effect can be done by applying the conservation laws to the physical pair-annihilation process.

(This is an abridged version of the derivation of the relativistic Doppler effect. For the detailed version, click here).

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

The decay of the 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 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, and f is the electromagnetic frequency associated with the photons.

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 seen by the resting observer, leads to the relationships of the relativistic optical Doppler effect for an approaching light source:

\[f_1=f\sqrt{\frac{c+v}{c-v}}\]
Derivation of the relativistic Doppler effect - Light source is approaching - Classical and Relativistic Frequency Shift

and for a light source that is moving away:

\[f_2=f\sqrt{\frac{c-v}{c+v}}\]
Derivation of the relativistic Doppler effect - Light source goes away - Classical and Relativistic Frequency Shift

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

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

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