For time dilation, both observers see the other as experiencing a time dilation symmetrically based on relative velocity to each other. In the twin paradox, however, one must be careful of which twin is the "proper" or rest time, and this is based off of acceleration to know who is truly moving. I know this, yet my question is: does the Relativistic Doppler Effect follow these same rules?
Does the doppler effect take into account who is the proper time or is the rest frequency based solely on the velocity of the two frames, irregardless of "proper time"? Because for example I would think that in the twin paradox, both twins would see each other's signals as red shifted since they are moving away from each other. But how could this be the case if their time dilation aren't symmetrical due to only one being in an inertial reference frame, considering that the relativistic doppler shift is originally based off of Lorentz Transformations / Time Dilation in the form of
Is this T0 a different proper time than in the normal time dilation?