The weakness of the basic model of the Millennium Relativity

 

 

 

The basic model of the Millennium Relativity consists of two circles and a triangle as shown in Figure 1:

 

 

 

The line A-B shows the displacement of the moving point B relative to some stationary reference frame S, during a time interval t measured in this stationary frame. The line A-C represents the motion of a light point in the same reference frame during the same interval of time. The circle S1 represents a light sphere that is made of all the light points that left point A at the instant that point B passed it. This sphere is also in reference frame S. Line B-C is connecting two points in S, but is also a line in reference frame S' in which point B is stationary. It is the course of the same light point C as seen from reference frame S'. Less clear is the circle S2. This circle represent the idea that in reference frame S' there is also a light sphere around point B. This is true of course if we accept the constancy of light speed, but the drawn circle has very little meaning. Except for the line B-C it doesn't show the real geometric relationship with other parts of the drawing that are drawn as viewed from reference frame S. It can be easily seen that in the example in Figure 1 point A is out of this circle. This cannot be true if we accept the assumption that the relative motion between A and B is less then light speed in both reference frames. What's the use of drawing this circle at all if it doesn't show the correct relationship with other parts of the drawing?

With agreement with SR, the ratio between the line A-C and B-C as calculated by Pythagoras law is used as the factor for scaling lengths and time intervals between the reference frames. But is scaling enough as a complete transformation formula?

Let's put two objects at points D and E as shown in Figure 2. It can be seen that the distance from point A to each of these points is the same. As measured in reference frame S, a light pulse generated at point A will hit both objects at the same time.

 

Assuming the light pulse occurred at the instant   when point B passed point A, the same light pulse in frame S' propagates equally at light speed around point B. Since point B is at point A only when the pulse is generated, point B cannot be at the same distance from D and E at any later time. No matter how we scale time and distance, a light sphere around B cannot hit both objects at the same time. So, in frame S' the same light pulse hits object E before it hits object D. This fact is not reflected by the MR transformation equations:

 

x = γ x'                        (1)

ct = γ ct'

 

How does special relativity handle all this?

 

First, Lorenz transformation does reflect the simultaneity difference by adding components to the equations for x and t:

 

x = γ( x' + β ct')                      (2)

ct = γ( t' + β x')

 

With this transformation x is not only a function of x' but also of time. This is obvious, because due to the relative motion a fixed point in S' changes its position in S with time. Also, t is not only a function of t' but also a function of x'. That is, for a fixed t' we get different values of t for different points along the x axis. This is just that simultaneity difference.

But we don't want only equations; we want an explanation; we want to understand. Special relativity adds a time axis to its model. Since we are able to visualize no more than 3 dimensions, I will use only 2 spatial dimensions and talk about light circles and not light spheres. Anyhow, MR doesn't use the third dimension and always shows a plane that is a cross section of the light sphere.

The notion of light sphere or light circle is strongly related to time and simultaneity. A light sphere is the collection of all the point to which a single pulse of light reaches simultaneously. The light generated at a point creates a constantly growing light circle. In the time-space model of SR it forms a cone around the time axis as shown in Figure 3. Any cross section of this cone that is parallel to the X-Y plane forms a light circle in the stationary reference frame S. Point A on this model is moving only in time but not in space, so it is always on the red time axis and in the center of the circle.

 

Point B, on the other hand, moves in this reference frame and is always on the blue line. This point is stationary in frame S', so the blue line also forms the time axis for this frame. A light circle in frame S' must have the same distance from point B to any two opposite points on it. A sloped cross section of the light cone can form an ellipse that satisfies this condition. Yet, this is an ellipse and not a circle, even if we project it on an X-Y plane. But this is how it is in reference frame S, not in S'. As we remember, distances in frame S are shorter in frame S' (and vice versa) along the axis of motion but not perpendicular to it. So, in frame S' this ellipse forms a perfect circle and light circles of S form ellipses as shown in Figure 4.

 

So, here are two basic differences between the Millennium Relativity and Special Relativity:

 

 

 

 

MR does not explain how two different points in space can be the center of the same light sphere. SR solve the problem by saying that though there is only one light cone, there are different light spheres in different reference frames due to a rotation of the space axes in time space, or in other words, a difference in the simultaneity between reference frames. There are other possible ways to solve the logical problem by saying one of the two:

 

  1. Distances along the motion direction that are equal in one frame are not equal in another frame.

 

  1. The order of events in one frame is different from that of another frame.

 

But both of these assumptions allow a different physical history in different frames and this is unacceptable - as explained here.

So I don't see how the scaling-only transformation of MR can avoid logical contradiction.

 

This apparently incorrect transformation (together with some more strange assumptions) leads to conclusions with internal logical contradictions. In the next article(s) I will show why all the attempts to prove that SR velocity composition is wrong are incorrect, and how the MR equation for velocity composition contradicts the basic assumption of MR.