Alice Law and The Relativity Theory

Chapter 7 

On (c+v)(c-v) Mathematics

Obtaining (c+v)(c-v) Mathematics 

for The Electromagnetic Theory

Han Erim

October 8, 2011

(Updated Oct 13, 2011. Some translation mistakes corrected, and a small part added)

Copyright © 2011 Han Erim All Rights Reserved.

On (c+v)(c-v) Mathematics:

A crucial error made in the electromagnetic theory has caused it to be lacking. The very same error has resulted in building the theory of relativity on an imperfect base. The theory of relativity and the electromagnetic theory are not fundamentally different. Both are theories dealing with the outcomes of electromagnetic interaction. (c+v)(c-v) mathematics lies in the foundations of both theories. The aim of this article is to show you this error.

(c+v)(c-v) mathematics of Alice Law and the mathematics used in the electromagnetic theory today are actually not different from each other. The mathematics belonging to the electromagnetic theory explains and formulates only the electromagnetic interaction between the frames that are inert according to each other for today and this corresponds to v=0 in (c+v)(c-v) mathematics. Apart from that, it also involves an exclusive situation which we will see here in the forthcoming parts. As (c+v)(c-v) mathematics covers all the electromagnetic interactions between frames either inert or in motion according to each other, it perfectly represents the mathematics of the electromagnetic theory. The actual mathematics of the electromagnetic theory is (c+v)(c-v) mathematics.

In the formulation of electromagnetic theory, the interaction between frames that are inert according to each other has been taken as basis generally, and the equations have been produced accordingly. Assuming that the speed of light is c according to all reference systems, a separate formulation for the frames that are in motion according to each other has not been conceived. This lack of the electromagnetic theory results in practical inconsistencies and deviations when it is applied to frames in motion.

Obtaining (c+v)(c-v) mathematics for the electromagnetic theory:

I will hereby use the very same example that Albert Einstein utilized while building up his theory of relativity:

Let’s think of a Frame B on X axis, in motion according to Frame A. Now we are sending a light signal from Frame A to Frame B. Let’s mark the coordinate on which Frame A is as O Point, and the coordinate on where Frame B is as P Point (Animated Figure 1).

We are leading in the topic with a question. If I ask you such a question “Has the light signal been emitted from point O according to Frame B?”, your answer will most probably be “yes”, which constitutes the source of the error. 

In order to reach the correct answer here, it is necessary to turn the scene upside down and contemplate accordingly. That is, we need to think by taking not Frame A but Frame B as basis. Let’s transform the example into a phenomenon of physics:

Frames A and B are two references systems in motion according to each other. I guess you already know that an observer standing in Frame B may regard his own Frame as inert and may assume that Frame A is in motion. It does not matter which of them is in motion.

From this point on, let’s carry our own observation point to Frame B and examine the same incident from Frame B this time. In this case, Frame B will be inert and Frame A will be in motion according to Frame B. Frame A sends the signal when it is on point O. As Frame A is in motion, it will not be on point O but on another point like O' when the signal reaches Frame B (Animated Figure 2).

We ask the same question here. "Has the light signal been emitted from point O according to Frame B?" Yes, the signal in this example has definitely been emitted from point O. However, we should notice that at the moment when the signal arrives, the place where Frame A is not where point O is as in the first figure above, but point O’. The place of point O is somewhere between points O’ and P. When we compare figures 1 and 2 by taking the arrival moment of the signal into consideration, we can clearly see this difference between the figures. Here, we need to ask this question: according to Frame B, which one is correct, the first figure above or the second figure here? The correct one is of course the second figure here. The first figure does not show how the incident occurred from the point of Frame B.

Let’s calculate the time passing until the signal reaches Frame B by taking the second figure as basis. We can obtain it if we divide OP distance into c. We already know that the signal will travel from point O towards Frame B at c speed. The speed of the signal has to be c according to Frame B. It will be t=OP/c.
However, there are two details we need to notice here:

1) The observer in Frame B will see Frame A not on point O’, but on point O when the signal reaches him, as the signal has come to him not from point O’ but from point O. As Frame A is not on point O at the moment when the signal arrives, what the observer on Frame B sees on point O is not Frame A itself, but its image (See: Ghost and Spring. Brought forward by Alice Law, the issue of "Ghost and Spring" will occupy a significant place in electromagnetic theory in the future).

2) We see that if Frame A is in motion according to Frame B, whether it is inert, or it is actually in motion or inert is not important. For Frame B, the incident ends up as if it took place in an inert frame. Therefore, the equalities belonging to the electromagnetic theory and used today are not erroneous for Frame B. This is the exceptional condition I have mentioned above (Here, Frame B represents the frame located on the target destination of a signal).

Making measurements by standing on the target at which light will arrive does not reveal the presence of (c+v)(c-v) mathematics.

Now, by keeping our position on Frame B, let’s calculate the speed of the signal according to Frame A. According to Frame B, Frame A emits the signal on point O and gets away. During the time period passing until the signal arrives, Frame A reaches point O’ from point O. OP’ distance in the 1st figure above and O’P distance in the 2nd figure are equal. Depending on this information, we can calculate the speed of the signal for Frame A. The arrival time of the signal does not change for any of the two frames. According to Frame A, as the signal travels O’P distance in t time and as c=OP/t, it is supposed to travel O’P distance at c’=O’P/t speed. Here it is c’>c. Let’s calculate c’ value:

c' = O'O/t + OP/t    c' = v.t/t + c.t/t    c' = c+v

As both frames get away from each other, we have obtained c+v here for the speed of the signal according to Frame A. If they moved closer to each other, we would obtain c’=c-v for the speed of the signal according to Frame A. Therefore, we have achieved (c+v)(c-v) mathematics for the electromagnetic theory. 

• The speed of the signal is c according to Frame B.

• The speed of the signal is c+v according to Frame A.

Here, c= speed of light constant and v=the speed of each frame according to each other. (You will find the exact explanation of v value at the end of this chapter.)

Since the speed of a light signal is accepted as c in the equalities belonging to the electromagnetic theory used today, the equalities of the electromagnetic theory are only valid for Frame B. For the frames in motion according to each other, calculations based on Frame A have erroneous results. If we want to observe the existence of (c+v)(c-v) mathematics, the speed of the signal must be measured from Frame A, the side which sends the signal.

Due to this deficiency and error, Albert Einstein had to accept that the signal would travel at c speed according to both Frame A and Frame B. Naturally, this deficiency in the foundation of the electromagnetic theory has not only caused it to be lacking significantly, but also dragged the theory of relativity somewhere it should have never gone. We can clearly see in the example here that Frame B is the only determinant for the signal. The signal travels at c speed according to Frame B, independently of Frame A. This condition does not change, no matter what the speed of Frame A or Frame B is. The signal travels at c speed according to Frame B, and at (c+v) speed according to Frame A.

Thus, Alice Law has corrected a crucial mistake of the electromagnetic theory and shown that (c+v)(c-v) mathematics is the one that belongs to the electromagnetic theory. It has already been told since the beginning of the article series that the mathematics of the theory of relativity is (c+v)(c-v) mathematics. Consequently, the theory of relativity and the electromagnetic theory, which utilize the same mathematics, have been combined by Alice Law.

Importance of frame use in (c+v)(c-v) mathematics: 

As long as the figure 1 above is used as is, logical reasoning based on Frame A cannot clarify in what way the (c+v)(c-v) mathematics is occurred. 

If we have to analyze the incident from the side which the signal is transmitted, that is to say Frame A, we have to make use of THE FIELD CONCEPT. Adding a FIELD to Frame B which is on the arrival target of the signal, and thinking that this signal will travel with the c speed in this field is enough to get the correct result. Using field, allows us to think Frame B as a stationary frame despite it is on motion and get the result in figure 2 simply and accurately. It is clearly expressed in below presentation that the signal will travel in (c+v) speed. (Animated figure 3)

Note that, used names for points in a figure are peculiar to the figure, pay attention while comparing figures.

O' point in Figure 3 corresponds to O point where the signal is emitted according to Frame B in Figure 2. The point (the red triangle on the field) where the signal enters the field is the coordinate of signal’s transmission according to Frame B (note that O' point is defined with respect to Frame B) . The distance of O' point to Frame B is OP (OP=O'P'). According to Frame B, Frame A (Spring) is placed in O point. At the moment of signal arrival, observer in the Frame B will see the image of Frame A (Ghost) on O' point. 

Location of O' point according to Frame A is calculated by utilizing of OP=O'P' equality or from arrival time of the signal. Duration of arrival time of the signal is OP/c=t. Hence OO' distance is acquired with the help of v.t=OO' equality. 

Speed of signal according to Frame A is acquired by OP/c=OP'/c' equality.

OP'=OP+PP'    c'.t=c.t+v.t    c'=c+v

(c+v)(c-v) mathematics between two frames moving to anyway and in any speed:

Animated Figure 4 below is designed for the purpose of analyzing (c+v)(c-v) mathematics. Positions, direction of movements and speeds of frames in reference to each other can be set for any kind of situation through red colored control points. The animation shows how the event occurs from the sides of both frames during the time when the signal begins emitting until it delivers. It is possible to analyze the occurrence by using the slider control.

Explanation of Animated Figure 4: 

As is seen, during the signal is going to Frame B, both frames are proceeding toward their direction of movement. 

We can see movement direction of signal for both frames.
According to Frame A, signal goes in FrameAQ direction. The direction of FrameAQ is parallel to the direction of O'P'. And according to Frame B signal goes in GFrameB direction. Both Q and G points are relative points. Q point is defined with respect to Frame A. And G point is defined with respect to Frame B.

Observer in the Frame B will see the image of Frame A (Ghost) in G point when the signal is received to him. G point is the point where the signal enters the field. GFrameB distance is equal to OP distance. Upon the arrival moment of the signal, Spring (Frame A) is on the O' point according to Frame B. 

As we see in Figure 2, travel time of signal is t=OP/c and this duration is invariable for both frames. t time also determines the place of O' and P' points on frames directions of movement. If we say that the speed of Frame A is V1 and the speed of Frame B is V2, then OO' and PP' distances become:

OO'= V1.t 

PP' = V2.t    

Speed of the signal for each frame considering travel time (t) is calculated by: 
Speed of the signal according to Frame A c'=O'P'/t 
Speed of the signal according to Frame B c= OP/t 

We see a circle in the figure, of which center is O' point, radius is equal to OP distance, and which is passing through S and R point. So, it is OP=O'R=O'S. Whether the P' point stays in the circle or not shows us how the (c+v)(c-v) mathematics occurs. If;
P' point stays in the circle, then (c-v) is occurred. O'S>O'P' (OP>O'P' state) 
P' point stays out of the circle, then (c+v) is occurred. O'S<O'P' (OP<O'P' state)

Yellow arrow which connects S and P' points is the criterion that shows how much the signal diverged from c speed according to Frame A. In other words, it represents the size of the v value in (c+v)(c-v) statement. Extent of the v value is calculated by v=SP'/t equality with the help of travel time. If the direction of arrow is towards O' point (OP>O'P') v value gets negative sign, if it is towards outside (OP<O'P') v value gets the positive sign.   

Speed of the signal according to Frame A: 

O'S<O'P'  state      c'=O'P'/t c'=O'S'/t+SP'/t c'=c.t/t+v.t/t c'=c+v   

O'S>O'P'  state      c'=O'P'/t c'=O'S'/t-SP'/t  c'=c.t/t-v.t/t  c'=c-v   

By using above equalities, we can write OP/O'P'=c/(c±v) equality for (c+v)(c-v) mathematics. Due to the fact that it is a very important equality, I named it Alice Equality.

(c+v)(c-v) mathematics is a dynamic mathematics. Only one signal’s state is dealt in this example here. While dealing with continuous movements, movement of the frames changes the place of O and P points, therefore the distance of OP, OO', PP' and O'P' changes perpetually. Hence in (c+v)(c-v) mathematics, the calculation is needed to be redone for each new position of frames.

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