doppler effect

Relativistic Doppler effect

Relativistic Doppler effect. 

There is an apparent shift in the observed frequency of any electromagnetic wave (light) when there is any relative motion between the source of light and the observer. This can be easily determined by using the 4-vector formulation of theory of relativity.

Lets discuss the details of this phenomena under two situations.

A. Source is at rest and observer is in motion. 
Lets us consider two inertial frames S and S’. S’ is moving wrt S, along the x-axis with speed v = (beta) c where the observer is at rest in S’ frame but the source is at rest in the  S frame.

Moving clocks run faster in the Doppler hemisphere of approach

Moving clocks run faster in the Doppler hemisphere of approach, run slower in the Doppler hemisphere of recession and run slower in the Doppler plane of intersection of hemisphere of approach and recession. The GPS satelites eg are in constant free fall towards earth and located in the Doppler plane of intersection, they never approach us as they never recede away, hence they induce a constant time dilation or a frequency redshift. That is the more time is spent the more the incorrectness grows.
As I had claimed a week ago, from binomial analysis I have proved that all inertial special relativistic effects (Doppler shift) are time dilation = red shift for transverse motion and time dilation = red shift for longitudinal recession and time contraction = violet/blue shift for longitudinal approach. In the transverse case which is exactly a time-dilation formula as taught (moving clocks run slower) is only a special case of time dilation/contraction. Moving clocks run faster if they are approaching us along our line of sight a thing not said enough times to our students. In other words for an observer inside a sphere of reference the general motion of a clock is one hemishphere (hemisphere of approach) Lorentz contracted and one hemi-sphere Lorentz dilated. (hemisphere of recession) The total sphere you can call a Doppler sphere. The satelites in free fall towards earth are neither in approach nor in recession hence they represent the transverse Doppler and always in red-shift or time dilation. This is a property of the binomial numbers.

I have the related diagrams and the calculations. I will update the calculations first and after some more additions the diagram.