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 , .