Alice Law and Relativity Theory

Chapter 7

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

Han Erim

October 8, 2011

On the (c+v)(c-v) mathematics:

A very important error made in electromagnetic theory has left electromagnetic theory incomplete. The same error has also caused relativity theory to be built on an incorrect foundation. Relativity theory and electromagnetic theory are not fundamentally different theories. Both are theories about the results of electromagnetic interaction. At the foundation of both lies the (c+v)(c-v) mathematics. The purpose of this article is to show you this error.

Added on October 13, 2011.

Definition of (c+v)(c-v) Mathematics:

“(c+v)(c-v) Mathematics” is a naming. The (c+v)(c-v) in the name is not a multiplication. This mathematics may also be called “c±v Mathematics”.

In frames moving relative to each other, the value obtained for the speed of a light signal differs depending on from which frame the measurement is made. The (c+v)(c-v) mathematics explains why and how this difference arises. The v value in the expression is the amount by which the speed of the light signal deviates from c. (c is the speed-of-light constant.)

“(c+v)(c-v) Mathematics” is the mathematics that reveals that the speed of light is relative.

Alice Law’s (c+v)(c-v) mathematics and the mathematics used today in electromagnetic theory are not actually different mathematics. The mathematics of electromagnetic theory, for today, explains and formulates only the electromagnetic interaction between frames that are at rest relative to each other. This corresponds to the v = 0 case in (c+v)(c-v) mathematics. Besides that, it also covers one additional special case, which we will see in the following part.

Since (c+v)(c-v) mathematics covers all electromagnetic interactions between frames that are at rest relative to each other or moving relative to each other (v ≠ 0), it represents the mathematics of electromagnetic theory completely. The true mathematics of electromagnetic theory is the (c+v)(c-v) mathematics.

In the formulation of electromagnetic theory, interaction between frames that are at rest relative to each other has generally been taken as the basis, and mathematical equalities have been produced for this situation. Based on the assumption that the speed of light is c with respect to all reference frames, no separate formulation has been considered for frames moving relative to each other. This deficiency of electromagnetic theory, when applied to moving frames in practice, causes inconsistencies and deviations.

Deriving the (c+v)(c-v) mathematics for electromagnetic theory

I use the same example that Albert Einstein used while building his relativity theory:

On the X axis, let us consider a B Frame moving relative to an A Frame, and we send a light signal from A Frame to B Frame. Let us mark the coordinate of A Frame as point O, and the coordinate of B Frame as point P. (Animated Figure 1)

We begin with a question: If I asked you, “According to the B Frame, was the light signal emitted from point O?” your answer would most likely be yes. The source of the mistake is right here.

To find the correct answer here, one must reverse the situation and think accordingly. That is, one must think not by taking the A Frame as the basis, but by taking the B Frame as the basis. Let us turn the example into a physical event:

A and B Frames are two reference systems moving relative to each other. You already know that an observer in B Frame can accept their own frame as at rest and assume A Frame as moving relative to them. Which one is in motion is not important.

From this point, let us move our observation point onto the B Frame and examine the same event this time from the B Frame. In this case the B Frame will be at rest and the A Frame will be moving relative to the B Frame. The A Frame sends the signal when it is at point O. Since A Frame is moving, at the moment the signal arrives at B Frame, A Frame will no longer be at point O but at another point such as O′. (Animated Figure 2)

We ask the same question again here: “According to the B Frame, was the light signal emitted from point O?” Yes, in this example the signal was definitely emitted from point O according to the B Frame.

But note that, at the moment the signal arrives, the position of A Frame is not at the place where point O is located as in the first figure above, but at point O′. The location of point O remains somewhere between points O′ and P. When we compare Figures 1 and 2 by taking the arrival moment into account, we clearly see this difference between the figures.

Here one must ask: According to the B Frame, which of the above is correct—the first figure, or the second figure here? The correct one is of course the second figure here. Figure 1 cannot show how the event develops from the perspective of the B Frame.

Based on the second figure, let us calculate the arrival time of the signal at the B Frame. If we divide the OP distance by c, we find the travel time of the signal. We already know that the signal will reach the B Frame from point O with speed c. The speed of the signal according to the B Frame must be c. We will have t = OP / c.

However, there are two important details we must pay attention to here:

1) When the signal reaches the observer in B Frame, the observer will see the image of A Frame not at O′, but at O, because the signal came to them not from O′ but from O. Since A Frame is not at point O at the arrival moment, what the observer in B Frame sees at point O is not A Frame itself, but its image (Ghost). (See GHOST and SPRING. The “Ghost and Spring” topic introduced by Alice Law will hold an important place in electromagnetic theory later.)

2) We see that, according to the B Frame, it does not matter whether A Frame is moving or at rest, nor whether B Frame itself is moving or not. For the B Frame, the event concludes as if it is happening in a stationary frame. Therefore the equalities of electromagnetic theory used today do not give an incorrect result for the B Frame. This is the special case I mentioned above. (Here, B Frame represents the destination frame of a signal.)

Measuring while being located at the target where the light will arrive does not reveal the existence of (c+v)(c-v) mathematics.

Now, keeping our position on frame B, let us calculate the speed of the signal according to A Frame. According to the B Frame, A Frame is moving away after sending the signal at point O. During the time until the signal arrives, A Frame moves from point O to point O′. The OP′ distance in the first figure above and the O′P distance in the second figure are equal. Using this information, we can calculate the signal speed for A Frame.

The travel time of the signal will not change for either frame. Since, according to A Frame, the signal will traverse the O′P distance in the same time t, and since c = OP / t, the signal must traverse the longer O′P distance with speed c′ = O′P / t. Here c′ > c. Let us calculate c′:

c′ = O′O / t + OP / t    c′ = v·t / t + c·t / t    c′ = c + v

Since the two frames are moving away from each other, we obtained c+v here for the signal speed according to A Frame; if they were approaching each other, we would obtain c′ = c - v for the signal speed according to A Frame. Thus, we have derived the (c+v)(c-v) mathematics for electromagnetic theory.

• According to the B Frame, the speed of the signal is c.
• According to the A Frame, the speed of the signal is c+v.

Here c = the speed-of-light constant, and v is the relative speed between the two frames (you will find the full explanation of v at the end of this chapter).

Since, in the equalities of electromagnetic theory used today, the speed of a light signal is assumed to be c, the equalities of electromagnetic theory are valid only for the B Frame. In frames moving relative to each other, calculations made by taking the A Frame as the basis give incorrect results.

 If we want to observe the existence of (c+v)(c-v) mathematics, the measurement of the signal speed must be made from the sending side, that is, from the A Frame.

 Due to this error or deficiency within electromagnetic theory, Albert Einstein had to accept that the signal would travel at speed c both with respect to the A Frame and with respect to the B Frame. This deficiency at the foundation of electromagnetic theory has both left a major incompleteness in electromagnetic theory and dragged relativity theory to places it should not have gone.

In this example, we clearly see that the only determinant for the signal is the B Frame. The signal travels at speed c according to the B Frame, independent of the A Frame. No matter what the speed of A Frame or B Frame is, this does not change. The signal travels at speed c according to the B Frame, and at speed (c+v) according to the A Frame.

The importance of using the field concept in (c+v)(c-v) mathematics

If we use the above Figure 1 as it is in logical reasoning based on the A Frame, it cannot show how (c+v)(c-v) mathematics takes place.

If we must examine the event from the side where the signal is emitted, that is from the A Frame, we need to make use of the field concept. Adding a field to the destination B Frame and thinking that the signal will travel within that field at speed c is sufficient to reach the correct result.

Using the field concept enables us to consider the B Frame as a stationary frame even though it is moving, and to obtain the results in Figure 2 easily and correctly. In the representation below, it is explicitly stated that according to the A Frame the signal will travel at speed (c+v). (Animated Figure 3)

Note: The names given to points in the figures are specific to each figure. Please pay attention to this when comparing the figures.

Explanation of Animated Figure 4

As can be seen, while the signal is traveling toward the B Frame, both frames are also moving along their directions of motion. We can see the direction of travel of the signal for both frames.

According to A Frame, the signal travels in the direction FrameA Q. The line FrameA Q is parallel to the line O′P′. According to B Frame, the signal travels in the direction G FrameB.

Both points Q and G are relative points. Point Q is defined with respect to A Frame. Point G is defined with respect to B Frame.

When the signal reaches the observer in B Frame, the observer will see the image of A Frame (Ghost) at point G. Point G is the point where the signal enters the field. The distance G FrameB is equal to the distance OP. At the arrival moment of the signal, the Spring (A Frame) is at point O′ with respect to B Frame.

As we saw in Figure 2, the travel time of the signal is t = OP / c, and this time does not change for either frame. The time t also determines the positions of points O′ and P′ on the direction lines of motion of the frames.

If we denote the speed of A Frame by V₁ and the speed of B Frame by V₂:

OO′ = V₁ · t

PP′ = V₂ · t

Taking the travel time t into account, the signal speed for each frame is calculated as follows:

According to A Frame, the signal speed c′ = O′P′ / t

According to B Frame, the signal speed c = OP / t.

 In the figure, we see a circle centered at point O′, with radius equal to the distance OP, passing through points S and R. That is, OP = O′R = O′S. Whether point P′ lies inside or outside the circle shows how the (c+v)(c-v) mathematics is realized. If:

If point P′ remains inside the circle, (c - v) is realized. O′S > O′P′ (the case OP > O′P′)

If point P′ remains outside the circle, (c + v) is realized. O′S < O′P′ (the case OP < O′P′)

The yellow arrow connecting points S and P′ indicates how much, according to A Frame, the signal deviates from speed c. That is, it represents the magnitude of v in the expression (c+v)(c-v). The magnitude of v is calculated using v = SP′ / t. If the arrow points toward O′ (OP > O′P′), v takes a minus sign; if it points outward (OP < O′P′), v takes a plus sign.

The signal speed according to A Frame:

If O′S < O′P′ c′ = O′P′ / t c′ = O′S / t + SP′ / t c′ = c·t / t + v·t / t c′ = c + v

If O′S > O′P′ c′ = O′P′ / t c′ = O′S / t - SP′ / t c′ = c·t / t - v·t / t c′ = c - v

Using the equalities above, we can write for (c+v)(c-v) mathematics: OP / O′P′ = c / (c ± v). Since it is a very important equality, I call it the Alice Equality.

 (c+v)(c-v) mathematics is a dynamic mathematics. In this example, the case of a single signal is considered. When continuous motions are considered, the motions of the frames change the positions of points O and P, and accordingly the distances OP, OO′,

PP′ and O′P′ change continuously.

Therefore, in (c+v)(c-v) mathematics, the calculation must be repeated for each new position of the frames.