Special Relativity velocity composition

 

 

In the previous article I've shown that there is a major difference between the transformation of the Millennium Relativity and Lorenz transformation of Special Relativity. While MR transformation only scales time intervals and distances, Lorenz transformation also adds difference in simultaneity. I've also shown that there is a serious difficulty in describing the constancy of light speed in the relative motion direction using the MR transformation.

 

As I see it, scientific theory can be considered correct and satisfactory as long as:

 

  1. There is no internal logical contradiction in it.
  2. There is no reliable experiment or observation that is incompatible with the laws it suggests.

 

Since we are not analyzing the results of experiments here, the only way to show that SR velocity composition is wrong is to show that there is an internal contradiction in it. That is, we must show that this equation is in conflict with the assumption of SR itself. The Millennium Relativity argues that "a conflict exists between Einstein's velocity composition law and the second postulate of special relativity". This looks like a claim of such contradiction. But in this article I will show that what is really done in all but one of the attempts to prove it, is relying on the MR transformation to show that the velocity composition equation is invalid. This doesn't show an internal contradiction and thus does not refute the velocity composition equation. All it does is showing that if we accept the MR transformation then the equation cannot be right.

 

Let's start with the article titled "Millennium Relativity Velocity Composition". The following sentences are taken from section 2 of this article:

"Consider for example velocity U. It is quite apparent that it is the difference between velocities v and u even though such definition is in direct conflict with the relativistic premise that velocities are not additive"

There is no such premise in special relativity. The reason for saying that velocity cannot be linearly added is the law of velocity composition that is derived from Lorenz transformation. This assumption is also not a result of logical sequence from the premises of the Millennium Relativity.

What is done here is accepting one claim of SR while rejecting other that led to it (Lorenz transformation and velocity composition law). This ungrounded assumption is then used to show that the very claim that led to it is wrong. This is a wrong logic.

 

In section 4 of the same article a one spatial axis model is used for proving the argument. I agree with the author that it is hard to show anything related to velocity composition on this model, but I can't agree with the following statement: "… although Einstein’s formula appears to be a very elegant mathematical solution for relating moving frame velocities to their equivalent stationary frame values, the model upon which it is based is nonetheless in violation of rigorously applied relativistic principles and thus the formula cannot be valid for its intended purpose." This is just turning things upside down. The model is just a way to understand the logic better. The logic is not based on the model. The SR velocity composition equation can be mathematically derived directly from Lorenz transformation. If we accept Lorenz transformation, then the velocity composition must be correct too. If the model is unclear or do not match the mathematics, the problem is with the model, not with the equation.

The problem with the x only model is similar to the problem with the MR basic model – it doesn't show the time relationship between things. With such model we can only show a view from a single reference frame if we want to keep a correct geometric relationship between objects. A much better model for SR is the space-time model. In this case we can use 2 dimensions – one for x and one for t as shown in Figure 1.

 

 

The figure show an x-t diagram of reference frame S in which an object A is at rest at x = 0. The axes of a second reference frame S' are x' and t' and a second object B is at rest at this frame at x' = 0. A third object C is moving relative to S faster than B. In this model the relative velocity between A and B is x1/ t1 the relative velocity between B and C is x'1/ t'1 and the relative velocity between A and C is x2/ t1.

This model gives us an idea of how things are related to each other according to SR. We can see that time and lengths cannot be just scaled from one frame to another. The mathematical development of the velocity composition equation from Lorenz transformation is shown here.

Section 4 of the article uses t as "the time interval in the stationary frame during which all of the motion between A, B and C occure". T1 "is the time interval in the moving frame". This is good for the Millennium Relativity scaling transformation, but not for Lorenz transformation. The interval here is different for different points. Figure 2 shows the lines of points A, B and C on a time-space diagram. The line EF represents all the points along the x axis after a time interval t at the stationary frame. The object A is at E, the object B is at I, and the object C is at F. In the moving frame, the time at point I is t/γ. At this time in the moving frame A is at G (a little earlier than E) and C is at H (a little later than F). Since both A and C are moving relative to this frame, also their position will be different. The relative position of the bodies in both frames when the time that is measured by B is t/γ is shown in Figure 3. I've added points m and n where n is fixed in the stationary frame and m is fixed in the moving frame.

 

 

We can see that unlike the drawing in the article, here ut + u1t1/γ is a little longer than vt (in the stationary frame n at t is at the left of F). Also vt/γ is not equal to  ut1 + u1t1 () though it is hard to see it on this drawing.

So, if we use Lorenz transformation, the equations that are shown in this section are incorrect. Lengths that are considered equal are not the same if we use this transformation. The equations are only correct if we use the MR transformation. What this section show is that Einstein velocity composition is not compatible with the MR transformation. This is quite sensible, since it is derived form Lorenz transformation. There is nothing here that can prove anything about how correct or incorrect is the equation.

 

 

I will not go into the details, but section 5 of the above article as well as the article "Velocity composition the Scientific Establishment way" do similar thing. They use the MR transformation to show that Einstein formula is wrong.

An exception is the article "More Proof that Einstein's Velocity Composition is Wrong". Here, graphs that show velocities calculated by SR and MR velocity composition equations are plotted, along with a claim that the graph that shows the MR velocities is what is expected. Well, this depends on one's expectations. Special relativity predicts non linear behavior of velocity composition. This is exactly what is shown in the graph. The behavior of the MR velocities is nearly linear. No reason is given why one would expect such behavior. If these were results of experiment it would be a proof that Einstein's equation is wrong. But just showing a graph of different equation and saying that one of them is what expected proves nothing.

 

So, no logical contradiction in special relativity is shown in any of these articles. The question goes back to the transformation equation. Which is correct - the one of MR or Lorenz transformation? If we assume that there is no logical contradiction in the Millennium Relativity too, we are left with two theories that might be correct. The only thing to that can judge which (if any) of them is correct is the results of experiments.

 

Originally I meant to write another article that would show that the velocity composition equation of the Millennium Relativity was wrong. But as I looked at it again I realized that what I was going to do is to argue again that simultaneity difference between reference frames is necessary. This I've already done before.

The fundamental disagreement between special relativity and the Millennium Relativity is about the transformation equation. I still can't see how the simultaneity difference can be avoided. Any other solution that is logically valid, leads to differences in the physical reality observed from different reference frame. Accepting such idea actually pulls the rug out under the whole concept of relativity. How can we talk about length position and velocity of objects as measured from different reference frames while these objects might not exist at all in some reference frames or might be in a completely different part of space in others?