Loedel Diagram illustrating the Relativistic Doppler effect
- Red set of axes: source reference frame. Velocity
- Blue set of axes: observer reference frame. Velocity
- Light gray set of axes: auxiliary reference frame, at rest.
The velocity of the source relative to the observer is .
The source sends a signal to the observer with some frequency (segment is the world line of one of such signals).
The observer receives it at a lower frequency ().
Simple geometric considerations:
- triangle :
- triangle :
- , where
Therefore .
This means ,
i.e. the expected result for the Doppler effect: the observed frequency is less than the emitted one, because the observer and the source moving away from each other.
The last equality comes from observing that, since , .