Velocity Composition for Dummies. Copyright 2009 Joseph A. Rybczyk

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1 Velocity Composition for Dummies Copyright 29 Joseph A. Rybczyk Abstract Relativistic velocity addition is given in terms so simple that anyone should be able to understand the process. In so doing it becomes abundantly clear that since Einstein s treatment is different from that given here, he obviously got this part of his theory wrong. 1. Introduction The author has done extensive research analysis of the relativistic velocity addition process 1&2 and determined that the Special Relativity treatment is self contradicting and cannot be correct. Unfortunately, Einstein s treatment is so obscure that even those versed in Special Relativity are unable to understand it. They defend it purely on the basis that it must be correct if it is accepted by the general scientific community. Here we will show that absolute adherence to relativistic principles leads directly to the Millennium Relativity velocity addition formula and not to Einstein s formula. 2. What Relativistic Velocity Addition is all About Referring to Figure 1 we have a simple illustration that shows three objects located at points A, B and C, the outer two of which are moving away at uniform rates of speed in opposite directions from the one in the middle at location B. To avoid redundancy, it is understood that the term object is intended in the general sense and can be anything from a subatomic particle to a large body in outer space. In the case illustrated, the object at point B serves as the point of origin in the stationary frame of reference, relative to which the uniform motions of the objects at points A and C are referenced. A v 1 B v 2 C FIGURE 1 In classical Newtonian physics, if we wanted to know the speed of one of the moving objects relative to the other, we would simply add their speeds together as in v = v 1 + v 2 where v is the total speed of the one moving object relative to the other moving object. We can do this in Newtonian physics because time is the same in the three different frames of references; that is, in the frames of the moving objects at points A and C and also in the frame of the stationary object at point B relative to which the motions of the other two objects are referenced. In relativistic physics, this is not the case because the evidence shows that time itself, in the frame of a moving object, is affected by the speed of the object and is therefore different from the time in the 1

2 stationary frame or the frame of any other object that is moving at a different speed, or in a different direction. And since speed is a function of time, it too is affected by the difference in time rates between reference frames. This is not normally a problem in Newtonian physics because the speeds have to be a significant fraction of the speed of light in order for the time differences to be detectable and thereby cause a problem. Since light travels at meters per second, the effect the speed of an object has on time in its own frame and therefore on the speeds of other objects referenced to it only becomes an issue when dealing with the very high speeds that relativistic principles are based upon. Nonetheless, there are many fields of physics, such as particle physics for example, where such speeds are normal and therefore the differences in time rates must be taken into consideration or the computed results will be incorrect. The basic formula used to convert or transform the different time rates or intervals from one frame to another is given as 1 where t is the dilated (slower) time in the frame of the moving object, T is the corresponding (normal) time in the stationary frame to which speed v of the moving object is referenced, and c is the speed of light in the vacuum of space. Although the formula given is the straightforward Millennium Relativity formula for time transformation, it is simply another form of the Lorentz transformation formula used in Einstein s theory of relativity and gives identical results. 3. The Problem Referring now to Figure 2 we can discuss the problem in more specific detail. A v 1 B v 2 C A v 1 B v 2 C x 1 FIGURE 2 V 2 In reference to Figure 2-1, we state the problem in terms where speeds v 1 and v 2 are known relative to the point of origin at the location of object B in the stationary frame of reference. Then, using the time transformation formula (1) given previously, we can determine the corresponding times in the frames of objects A and C, for any time interval T that transpires in the stationary frame of object B. What we don t know, however, is the speed of object C relative to object A as indicated by speed x in view 1 of the illustration. Since this total speed is referenced to object A and not object B in the stationary frame, its value depends upon the time 2

3 in the frame of object A and not the time in the stationary frame of object B that speed v 2 is referenced to. In other words, since speed v 2 is referenced to the stationary frame of object B and depends on the time in frame B, it is not a valid speed relative to the time in the frame of object A and therefore cannot be added directly to speed v 1 to determine the total speed x. Conversely, if we know the total speed V of object C relative to object A as show in Figure 2-2, we find that not only is it not the sum of speeds v 1 and v 2, but that it has two different values depending on how the problem is viewed. Since the issue of multiple values comes up again in reference to speed v 2 of object C, let us defer discussion of it until we reach that part of the analysis. 4. The Solution Actually, the given problem has an easy solution by simply following the principles of relativity as will be done now. According to the principles of relativity the speed of one object relative to a second object is the same as the speed of the second object relative to the first object. That is, if the speed is a uniform speed as given here, either of the two objects can be considered the stationary frame point of reference relative to which the speed of the other object is referenced. In that case we can make the object at point A the point of origin in a stationary frame of reference relative to which the speed of the object at point B is referenced as shown in Figure 3. A v 1 B V 2 C x FIGURE 3 But now the speed of object C must be increased to compensate for the fact that the time in the frame of object B is no longer stationary frame time. Since the actual distance an object travels cannot change just because the speed of the object is referenced to a different time standard we can easily determine what the new speed V 2 of object C must be. The solution lies in converting the speed-to-time relationships into the distances that they represent as shown in Figure 4. A v 1 T B V 2 t 1 C xt FIGURE 4 3

4 Referring to Figure 4, if T is the stationary frame time interval that transpires in the frame of object A during which distance v 1 T is traveled, then using time transformation formula (1) we obtain 2 where t 1 is the corresponding time interval that takes place in the frame of object B. Obviously, since time interval t 1 is a smaller value than stationary frame time interval T, this time interval will be less than that (i.e. interval T) that would have been used in conjunction with speed v 2 as originally given in Figures 1 and 2 where object B was the origin of the stationary frame to which speed v 2 was referenced. Subsequently, speed V 2 in Figures 3 and 4 must be a greater speed than the original speed v 2 in order that the distance traveled by object C is unchanged from what it was previously. The actual conversion formula involving the changes in reference frames is derived from 3 giving 4 where V 2 is the speed used in conjunction with moving frame time interval t 1 and time interval t 1 is related to stationary frame time interval T as defined by the time transformation formula (2) above. For ease of reference, the convention used is simply that the upper case characters represent the larger of two corresponding values and the lower case characters represent the smaller of the two values. To summarize, in reference to Figure 4, we can now equate distances v 1 T and V 2 t 1 to distance xt as in 5 where xt is the total distance traveled, v 1 T is the distance traveled by object B and V 2 t 1 is the distance traveled by object C. Then by substituting the right side of time transformation formula (2) in place of t 1 in equation (5) we obtain 6 4

5 that simplifies to 7 where the unknown value x has been replaced by the known value for speed v. This is the Millennium Relativity velocity composition formula for speed v as illustrated in Figure 5 below. A v 1 B V 2 C v FIGURE 5 5. The Alternate Solution Now that we understand the time, distance, and speed relationships we can directly solve the problem as originally given in Figures 1 and 2. By converting the original speeds into distances as given in Figure 6 below, we can solve for speed v as will be shown next. A v 1 T B v 2 T C Vt 1 FIGURE 6 First we state the distance relationships given in Figure 6 as 8 where time t 1 is defined by time transformation formula (2). Then we define the relationship between speed V and its alternative value v as given by 9 where we can immediately substitute the right side of this equation in place of Vt 1 in equation (8) to derive 1 5

6 into which we substitute the left side of equation (3) in place of v 2 T to derive 11 into which we substitute the right side of equation (2) in place of t 1 to obtain 12 that simplifies into 13 that is the same as the velocity composition formula (7) derived earlier. 6. Conclusion With this final paper on the subject of relativistic velocity composition the author has made his last effort to demonstrate the validity of the Millennium Relativity velocity composition formula. Eventually the scientific community will be forced to deal with this issue and give it the attention it deserves. In the meantime the author will go forth with his research into new areas of relativistic physics on and beyond the fringes of what is known today. REFERENCES 1 Joseph A. Rybczyk, Millennium Relativity Velocity Composition, (22); Millennium Relativity Acceleration Composition, (23); Integrated Relativistic Velocity and Acceleration Composition, (24); Complete Relativistic Velocity and Acceleration Composition, (28); (These are the four main papers on the subject of velocity composition and acceleration composition by this author. They are all available from the home page of the Millennium Relativity web site at 2 Joseph A. Rybczyk, Velocity Composition Completely Deciphered, (23); Einstein s Velocity Composition Proven Wrong, (23); More Proof that Einstein s Velocity Composition Law is Wrong, (24); Velocity Composition the Scientific Establishment s Way, (24); Millennium Relativity Velocity Composition in Eight Easy Steps, (28); The Failure of Einstein s Velocity Composition in the Reverse Direction, (28); Einstein s Biggest Mistake, Acceleration Composition, Finally Proven, (28); Alternative Versions of the Relativistic Acceleration Composition Formula, (28); Einstein s Erroneous Derivation of Velocity Composition, (28); The Enigma of Einstein s Velocity Composition, (28); The Special Relativity Velocity Composition Paradox, (28); (These are the 11 short papers on the subject of velocity composition and acceleration composition by this author. They are all available from the Millennium Briefs Section of the Millennium Relativity web site at 6

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