Relativistic Doppler Shift

The standard formula for relativistic Doppler shift is: 

√((1 + v/c)/(1 - v/c)).

This multiplication factor describes the stretching out of wavelengths (reduction in energy) for light emitted from a body moving directly away from you, or the increase in the frequency (increase in energy) of light coming directly towards you. In this model, a positive velocity is moving away from you, and negative one towards you. To get the factor for frequency instead of wavelength, negate the signs for the velocity or take the inverse of the output.

For a body moving away you at 0.5c, a photon’s wavelength will increase by a factor of 1.732051. For a body moving towards you at the same speed, a photon’s frequency will increase by the same factor.

How does this work in the geometry of the particle model of gravity and motion? The factor for the wavelength of a particle moving away you (thus, decreasing in energy) is determined by taking the gamma factor (the stretching out of time, showing energy loss for a photon) and adding the inertial energy of the particle: 

1/cos(asin(v)) + tan(asin(v)).

The get the factor for the wavelength of a particle moving towards you (increasing energy), subtract the energy from the gamma factor. Remember, a wavelength factor smaller than one shows an increase in energy.

To get the factor for frequency instead of wavelength, reverse the sign or take the inverse of the output. This really makes sense, because a photon’s frequency is directly related to its energy. It makes sense that a high gamma factor, showing a reduction in the perceived time of the source, would reduce the energy of any photon it emitted (regardless of direction), just as its inertial energy in your direction would add to that photon’s energy, and reduce it when traveling away from you.

QED

And as always, Copenhagen interpretation delenda est!