In the relativistic Dopper shift, which frame is and which is , and which is proper time? Why is that one proper?
This is worth some careful attention. There are two frames here: the frame of the source of light, , and that of the observer moving at speed towards the source, the observer frame . The source emits light with some frequency : the light wavefronts are emitted in the same place (at the source) in the source’s frame , so the frequency in is the proper frequency, . Now, still in the source’s frame, consider the time interval between the observer hitting one wavefront, and hitting the next wavefront. That time interval is , seen in the source’s frame.