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Space-time diagram: The "explosion problem": relativistic Doppler effect

Suppose a situation as the one depicted here. Red dashed line is the path of **light**, the blue axes is the **high speed frame**, and the black axes is an **Inertial frame** of reference; green dots represent the signal:
__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:
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