*f*₀, and let a detector move away from it at a speed

*v*, as messured in the reference frame of the source. If that speed

*v*is close to

*c*(the speed of light in a vacumm), then Special Relativity claims the meassured frequency in that detector is relativistic, so:

(1) |

But, under classical mechanics assumptions, that Doppler shift is computed as follows:

(2) |

*v*=

*u*+

*u*, and place an antenna in the intermediate reference frame that moves at

*u*with respect to the source. The detector is placed in a final third collinear reference frame, which is moving away at a speed

*u*with respect to the second one. Then, this detector meassures a double Doppler effect, which in classical mechanics is:

(3) |

*f*₁ is not equal to

*f*in [2], and that means the Doppler effect under classical mechanics assumption is not self-similar. so it is incomplete. But, let’s see now how the prediction under Special Relativity assumptions is for that double Doppler:

(4) |

*f*₂ is not equal to

*f*in [1] either, so Special Relativity predicts a non self-similar Doppler, an incorrect one, since that theory pretends to be complete for every inertial system. A complete Doppler must be strictly self-similar. So, how can we design a model that can predict a complete Doppler? Let’s do the following:

*v*into two halves,

*u*. Let’s now divide

*v*into

*n*parts,

*v*=

*u*+

*u*+ … +

*u*. Then, we compound those $n$ parts, under classical mechanics, assuming there is an antenna in each intermediate reference frame that relays the signal to the next collinear one. Then, we have:

(5) |

If we now compute the limit of when *n* tends to infinity, we get:

(6) |

*v*of the detector with respect to the source can be split into so many parts as you like, and each part will correspond to an intermediate antenna that relays the signal to the next collinear antenna, so the compound Doppler will match the simple one, that is directly observed between source and final detector.

**canonical sum of velocities**,

*v*=

*v*₁ +

*v*₂ , but if we apply the convention of Einstein addition of velocities we can achieve that equation [1] becomes self-similar. Why that amazing feat?. Actually, it is not any amazing feat at all. Any theory of relativity that owns a formula for Doppler effect of electromagnetic waves can be proclaimed as self-similar if we define a method of how the composition of velocities must be. Any theory of relativity exhibits the following generic Doppler formula:

(7) |

where obviously, $latex\beta=\frac{v}{c}$ and is a function of . Since Special Relativity has the following equation for Doppler effect:

(8) |

that means the function must be:

(9) | |

(10) | |

(11) | |

(12) |

*β*,

*β*₁ and

*β*₂, then we have:

(13) | |

(14) | |

(15) |

so, we definitively get:

(16) |

as the well-known Einstein’s addition of velocities.

Above, we have seen that Doppler equation under classical mechanics is

(17) |

(18) | |

(19) |

So the sum of betas would be now:

(20) | |

(21) | |

(22) | |

(23) | |

(24) |

(25) | |

(26) |

Regards