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13. ABOUT "v" VALUE IN (c+v) (c-v) MATHEMATICS

v value in (c+v) (c-v) mathematics carries two meanings:
    • It is the approach or move-away speed of two reference systems relative to each other.
    • It is the amount of deviation from the speed of light. 

Motions in the examples I have given so far always took place on X axis. Therefore, without finding it necessary to calculate “v” speed value between reference systems, we were able to use it in (c+v) (c-v) mathematics. But this is often not the case. 

In the figure below, how to calculate the v value for a reference system that is in motion in any direction is described.

We see three separate visuals in the figure. Planes are moving in different directions (in the direction of the arrow) in each visual. We are examining the event from the reference system of the observer on the ground. 

Let me talk about the visual on the left. When the plane is at point B, a signal sent from point A reaches the plane at point C. We are drawing an arc whose center is A and whose radius is AB line. Then we are drawing another line connecting point A and C and intersecting the arc. OC distance is the length giving us the value v in (c+v) (c-v) mathematics. Why?

We are applying the principles. Let’s assume that the plane stands still at point B. A signal is sent to it from point A. As the speed of this signal coming to the plane itself is c, we find the signal’s travel time to the plane. This time is Δt = AB/c. However, if we look at it carefully, the signal follows AC line and reaches C relative to the observer on the ground. Here we find the speed of the signal going towards the plane relative to the observer on the ground. This speed will be c' = AC/Δt.
Because of AC=AO+OC, we can define a v value for OC distance that determines the change in the speed of the signal and we can write OC = v.Δt. Because AO=AB, AO=c. Δt is obtained. 
Therefore; 
We can write AC = c . Δt + v . Δt = Δt . (c+v). And (c+v) becomes the speed of the signal relative to the observer on the ground. 

OC distance also shows how much two reference systems move away from (or approach) each other during Δt time. 

In this way, by using “time”, we have obtained (c+v) (c-v) mathematics and verified it. v value is, in conclusion, the speed (v) of two reference systems relative to each other and at the same time the amount of change in the speed of light (c±v).

To make it easier for you to follow the topic, I added the same figure on this page too.

Let’s talk about a bit about point C, as well. The place of point C is determined by Δt time elapsed between the emission of the signal and its arrival to the plane. If we call the speed of the plane “u”, the plane will cover BC = u . Δt distance in the direction of the arrow during this time.

13.1. DOPPLER TRIANGLE

If we are careful enough, we can see that ABC triangle in the figure is a very special one. I named this special triangle “Doppler Triangle”. The travel time of the signal "Δt" is determinative for the side lengths of a “Doppler Triangle”. 
AB = c . Δ
BC = u . Δ
AC= (c+v). Δt (for the left visual on the figure)
The place of point O is, naturally, very special. OC = v . Δ
Doppler Shift calculations should be done based upon this special triangle. 

Let’s do the calculations in a similar way for the visual on the right.
AB = c . Δ
BC = u . Δ
OC = v. Δt
AC= c . Δt - v. Δt = Δt . (c-v)
Therefore, the speed of the signal going towards the plane on the right visual will be = c-v.

If we are careful;
AB length is the distance between the signal source and arrival target of the signal at the moment of signal emission. 
AC length is the distance between the signal source and arrival target of the signal at the moment of the arrival of the signal.
Let’s represent both the figure on the right and on the left by writing AC= (c±v). Δ
Because of AB = c . Δt , if we divide the first equation by the second equation:
We get:

This helps us reach a very important result. We can express this equation as follows:

However, I have eventually noticed that there is something missing in the wording of the expression of the equation above. I think the way the same equation is presented below is more appropriate. I’d like to state that these equations are valid between inert reference systems.

Let’s keep examining the same figure. When thought with the principles, the equation becomes easier to understand. I’m making use of the examples in the figure while saying:

  • Distance OUTGOING signal covers relative to the transmitter: Relative to the reference system of the transmitter, the signal follows the AC line and reaches the plane at point C. Therefore, the distance that the signal covers relative to the reference system of the transmitter is AC.

  • Distance INCOMING signal covers relative to the receiver: We should remember the principles. Let’s think that the plane is motionless at point B. In this case, the one in motion is the transmitter. The transmitter sends the signal when the transmitter is at point A. In this case, the signal comes towards itself following AB line relative to the reference system of the plane. Therefore, the distance that the signal covers relative to the reference system of the plane is AB.

  • The speed of the OUTGOING signal relative to the transmitter: I guess we have been informed about this sufficiently topic so far. The speed of a signal going to a target in motion is always different than “c” relative to the reference system sending the signal. There is only one exception. And we will see it in a minute.

  • The speed of the INCOMING signal relative to the receiver: We have also seen that, relative to a reference system that receives signal, the speed of the signal coming to it is always constant and equals “c”.

We continue studying the same figure.

What are Blue Shift, Dead Point, and Red Shift in Doppler Shift?

We have seen that, if an electromagnetic wave sets out to a target in motion, the wavelength in its factory setting during the emission changes. Wavelength change also means a change in the energy of the electromagnetic wave. Shift to Blue is the increase and Shift to Red is the decrease in the energy. 

Red Shift
It is the visual on the left in the figure. It appears as the increase in the length of signal wavelength. It is represented with (c+v). The conditions below are formed in this case: 
• AC > AB
• Speed of signal emission > The speed of light constant
• Speed of the OUTGOING signal relative to the transmitter > Speed of the INCOMING signal relative to the receiver

Dead Point
The visual in the middle shows “Dead Point”.
Even if both the frames are in motion relative to each other, there is a moment when Doppler Shift doesn’t occur. In this case, there is no change in signal wavelength and the following conditions are formed:
• AC = AB
• Speed of signal emission = The speed of light constant
• Speed of the OUTGOING signal relative to the transmitter = Speed of the INCOMING signal relative to the receiver = The speed of light constant

Blue Shift
It is the visual on the right in the figure. It emerges as the shortening of signal wavelength. It is represented by (c-v). Below conditions are formed in this case:
• AB > AC
• The speed of light constant > Speed of signal emission 
• Speed of the INCOMING signal relative to the receiver > Speed of the OUTGOING signal relative to the transmitter 

13.2. DOPPLER QUADRANGLE

In the above figure, relative to the reference system of the man on the ground, we see planes moving in any direction and at any speed. Two situations are represented on the left and right side of the figure. On the right side of the figure, the planes are moving in the opposite directions.

Let’s find out how Doppler Shift takes place between these the two planes. In the figure;
• Points A and C are the coordinates of the planes at the moment they send signals to each other. 
• Points B and D are the coordinates at the moment the signals reach the planes. 
• AB and CD are the distances covered by planes during the travel time of the signals.
• AC is the distance that the incoming signal covers relative to the receivers.
• BD is the distance that the outgoing signal covers relative to the transmitters. 
• The travel time of the signal for both the planes is Δt = AC/c.

Let’s now deal with the visual on the left side of the figure. We see ABCD quadrangle formed here. The travel time of the signal "Δt" is the time determining the side lengths of ABCD quadrangle formed. I named this special quadrangle “Doppler Quadrangle”. 

Side lengths of ABCD Doppler Quadrangle 
AB = Δt . u1
CD = Δt . u2
AC = Δt . c
BD = Δt . (c±v)
u1 and u2 are the speeds of the frames (the planes).
v is the amount of change in the speed of the signal. If AC>BD, v gets minus; if AC<BD, v gets plus values. 

We see ABCD Doppler Quadrangle in the visual on the right side of the figure as well. However, as the planes are moving in the opposite directions, it is folded on its two sides. The same equations above apply for the side lengths of this quadrangle as well. If we see an event from the outside and if both the reference systems are in motion relative to us, it is necessary to think with Doppler Quadrangle in mind.

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