Alice Law and The Relativity Theory
Chapter 7
On (c+v)(cv) Mathematics
Obtaining (c+v)(cv) 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)(cv) Mathematics:
Obtaining (c+v)(cv) mathematics for the electromagnetic theory:


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.


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.
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)(cv) 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’=cv for the speed of the signal according to Frame A. Therefore, we have achieved (c+v)(cv) mathematics for the electromagnetic theory.
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)(cv) 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.
Importance of frame use in (c+v)(cv) mathematics: 

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.
OP'=OP+PP' c'.t=c.t+v.t c'=c+v
(c+v)(cv) mathematics between two frames moving to anyway and in any speed:
Animated Figure 4 below is designed for the purpose of analyzing (c+v)(cv) 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:
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.
OO'= V1.t PP' = V2.t
Speed of the signal for each frame considering travel time (t) is calculated by:
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)(cv) 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'/tSP'/t c'=c.t/tv.t/t c'=cv
By using above equalities, we can write OP/O'P'=c/(c±v) equality for (c+v)(cv) mathematics. Due to the fact that it is a very important equality, I named it Alice Equality.
(c+v)(cv) 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)(cv) mathematics, the calculation is needed to be redone for each new position of frames.


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