Assumptions:
☛ Two reference systems: S (β=cost; inertial) , and S' (β=cost; β≅c)
☛ S' emits a light pulse every τ meters of time
☛ S' is moving wrt S with speed β≅c
☛ S listens to the signal emitted by S' (which, as said, has a fixed period τ), and measures different periods, depending on β (the speed of S').
☛ We need only τ, whose value as a coordinate refers to the S' system of reference, and β :
First, we set S time and space coordinates to {0,0} and consider that a first flash of light is sent by S' at that space-time position; so this "first flash" will happen at coordinates {0,0} of both S and S'. Now, let :
be the 4-position vector, in the S' system, of the event "Light pulse sent out", which we will call "A";
and also let :
be the 4-position of event A in the S system.
To get the coordinates of A {τ,0} in the S system, we just need to apply a "reverse" Lorentz transformation from S' to S:
BUT the first component of X is still NOT the time that A measures!
To get the total time, one has to add the time the light takes to cover the (increasing due to β) distance from S.
So one can write:
This is because, as light speed is always =1, it takes exactly x time to get to position x (refer to the picture!); so light traces a 45° inclined worldline.
Now, rearranging the previous equation, we may write
that just resembles the acoustic Doppler effect:
this is the relativistic Doppler effect!
We can get two very important relations from this picture:
-the shift of the period, from τ to t, seen by S; it is also a shift in frequency. It is not fixed!
-the velocity β of S' as a function only of t and τ:
Final, synthetic way of writing the whole thing:
