Redefining Einstein s Velocity Addition Formula Copyright 2010 Joseph A. Rybczyk Abstract It will be shown in the clearest and briefest terms practical that Einstein s velocity addition formula actually involves acceleration and not uniform motion as given in his special theory of relativity. 1. Introduction It has been shown in previous works 1 by this investigator that Einstein s velocity addition formula 2 cannot be correct for its intended purpose involving the addition of uniform motion velocities. This finding was unexpectedly corroborated in subsequent research 3 that showed beyond doubt that the formula actually involved the addition of velocities resulting from constant acceleration. The latter finding involving acceleration will now be presented in the simplest and briefest terms practical. 2. Differentiating between Acceleration and Uniform Motion Velocities In the strictest scientific sense there can only be one kind of velocity and it is the relative speed between two objects in different inertial states. In such case either of two such objects can be considered stationary relative to which the other object is in uniform motion. Acceleration then, is really a change in such motion. Whereas in the first case we are defining the relative velocity between two objects, in the second case we are transitioning from one inertial state to another. The importance of this distinction between uniform motion and acceleration involves, among other things, the distances traveled during an interval of time. This distinction takes on greater importance when the principles of relativistic physics are involved. Such relativistic principles involve effects that are non-newtonian in nature and that increase asymptotically as light speed is approached. By way of emphasis, such considerations normally do not involve uniform motion speeds. For example, it is quite valid to claim that a speed of 0.8c is the twice the speed of 0.4c. But, to accelerate an object to 0.8c takes more than twice the energy required to accelerate the same object to 0.4c. Of even greater importance, however, is the different effects that uniform motion and constant acceleration have on distances traveled during an interval of time as will be discussed next. 3. The Nature of Relativistic Constant Acceleration Distances As in the case of the example speeds discussed previously it is quite reasonable to claim that the distance traveled at a uniform rate of speed of 0.8c in a given interval of time will be twice the distance traveled as that related to a uniform rate of speed of 0.4c for the same interval of time. That, however, is not the case involving relativistic constant acceleration. In that case the behavior is that of Newtonian constant acceleration for speeds that are not a significant fraction of light speed and then changes to Newtonian uniform motion behavior as the speed of light is approached. This means that at low acceleration speeds the travel distance will be half that obtained during uniform motion for the same speed v. As the acceleration continues and speed v approaches c, however, the distance traveled during constant acceleration approaches the distance traveled by uniform motion. This transverse Newtonian/relativistic behavior is easily demonstrated by simply analyzing the mathematical relationships of one of several previously 1
introduced Millennium Relativity constant acceleration distance formulas 4 as will be accomplished next. We begin with the basic form of the appropriate formula 1 where D is the stationary frame (SF) distance traveled by an object under constant acceleration that reaches speed v relative to the stationary frame at the end of stationary frame time interval T. The term c, of course, is the speed of light. Next, we rearrange the formula to the form 2 where the vt term to the right is the mathematical definition for the Newtonian uniform motion distance traveled at uniform motion speed v during time interval T. If we now set speed v to 0, we obtain 0 0 3 giving a distance of zero as is expected. Now let us set speed v to a speed that is just at the border of relativistic speeds, such as v = 0.01c. We then obtain 0.01 4 that reduces to approximately 1 5 2 that is, of course, the Newtonian formula for distances traveled under constant acceleration. For the final step of the analysis we set speed v to v = c. This gives 6 that reduces to 7 0 that, as can be seen, is the Newtonian distance for uniform motion and not constant acceleration. 4. Relativistic Constant Acceleration Relationships From the analysis just completed it is apparent that the distances traveled during relativistic constant acceleration emulate the behavior of Newtonian constant acceleration 2
distances at Newtonian speeds and asymptotically transition to Newtonian uniform motion distances as the speed of light is approached. This difference in behavior involving relativistic constant acceleration distances as opposed to uniform motion distances will now be exploited to evaluate the true nature of Einstein s velocity addition formula. We will accomplish that by first deriving the Millennium Relativity acceleration addition formula in a simplified easy to understand form and then show that it gives the same results as Einstein s velocity addition formula. It should be understood here that the employed method of derivation is identical to that used in derivation of the Millennium Relativity velocity addition formula with the exception being that constant acceleration distances will replace the uniform motion distances previously used. Referring now to Figure 1, the curve labeled a 1 shows the rise in velocity v 1 for an object accelerating 5 at a constant rate of acceleration a 1 relative to the stationary frame point of origin 0 SF during stationary frame time interval T. In a similar manner the curve labeled a shows the rise in velocity v for an object accelerating at the higher constant rate of acceleration a relative to the same stationary frame point of origin during the same time interval T. D 1 is the stationary a t 1 A 2 T a 1 0 MF 0 SF Figure 1 Relativistic Acceleration Distances frame distance traveled at acceleration rate a 1 and D is the stationary frame distance traveled at acceleration rate a during the time interval T just mentioned. An object accelerating at acceleration rate A 2 relative to the uniform motion frame v 1 transitioned to by the object accelerating at rate a 1 at the end of interval T, reaches speed V 2 relative to the uniform motion frame v 1 during the corresponding moving frame (MF) v 1 time interval t 1. D 2 is the stationary frame distance traveled at rate A 2 during time interval t 1. The relationship between the two time intervals is given by the Lorentz 6 time transformation formula 1 8Lorentz Time Transformation where t 1 is the uniform motion frame v 1 time interval that corresponds to stationary frame time interval T. 3
5. Deriving the Acceleration Addition Formula The first step in the derivation of the acceleration addition formula is to take the constant acceleration distance formula (1) and solve it for speed variable v. Beginning then with 1 and solving for v, we obtain 9 10 11 2 13 12 2 2 2 2 2 14 15 16 17 18 2 19 that can be rearranged to 2 20 for the initial version of the formula for acceleration speed v. Then, on the condition that 21 we obtain by way of substitution with equation (20) 4
2 22 for the simplified version of the Millennium Relativity acceleration addition formula. When used in conjunction with stationary frame distance 23 and moving frame distance 24 formula (22) gives the same results as Einstein s velocity addition formula throughout the entire range of input velocities v 1 and V 2 from 0 to c. For the full version of the formula we simply complete the implied substitutions to arrive at 2 25 where v is the resultant stationary frame velocity from the addition of stationary frame velocity v 1 and moving frame velocity V 2. For a complete version of the formula without the T and t 1 time variables we first convert the Lorentz time transformation formula (8) into its millennium relativity form given by 26 and then substitute the right side this version in place of the variable t 1 in equation (25). This then gives 2 27 5
that by way of simplification gives 2 2 28 29 2 30 ending with equation (30) for the final form without the time variables. Unfortunately, equation (30) does not appear to simplify beyond what is given here. 6. Comparison to Einstein s Velocity Addition Formula Using the same variable definitions given previously for the acceleration addition formula, we can state Einstein s velocity addition formula as 1 31 where v is supposedly the stationary frame uniform motion velocity that results from the addition of stationary frame uniform motion velocity v 1 and moving frame uniform motion velocity V 2 relative to moving frame v 1. Comparison of the results obtained from this velocity addition formula to those of acceleration addition formula (30) shows that both formulas give virtually identical results for speed v for the same input values of speeds v 1 and V 2. More specifically, if speeds v 1 and V 2 are given the exact same values from 0 to c, the results for both speeds v will be identical out to 16 decimal places in the computer based math program used. And speed c will not be exceeded. Only when speeds v 1 and V 2 are given different values from each other, will there be any difference at all for speeds v between the two different formulas. And even then, in the worse case the difference will be no more than approximately 1.5% between the two different formulas. Moreover, neither formula varies in the same direction. Einstein s formula will give a slightly higher result for v if v 1 is higher than V 2, and the Millennium formula will give a slightly higher result for v if V 2 is higher than v 1. But in no case does the difference exceed approximately 1.5% and it does not happen at all in the Newtonian range of values. Nor does it 6
cause speed c to be exceeded by either formula. It is a condition most likely resulting from the mathematical rounding process based on the difference in complexity between the two formulas and not a result of a difference in theoretical definition between the formulas. 7. Moving Frame to Stationary Frame Velocity Conversion It should be rather obvious from the relationships shown in Figure 1, that if equation (1) gives the correct stationary frame constant acceleration distance for speed v, and if equation (23) gives the correct stationary frame constant acceleration distance for speed v 1, then distance D 2 for speed V 2 presently defined by equation (24) using moving frame values could alternately be defined as 32 where stationary frame values v 2 and T for speed and time replace the moving frame values V 2 and t 1 used in equation (24). Then, using the same procedure previously used in Section 5 for equation (1) to derive equation (20) for speed v, we can similarly derive 2 33 and 2 34 for stationary frame constant acceleration speed v 2 and the corresponding moving frame constant acceleration speed V 2. The right side of equation (24) can then be substituted for D 2 in equation (33) to derive 2 35 for speed v 2, and the right side of equation (32) can be substituted for D 2 in equation (34) to derive 2 36 7
for speed V 2, where v 2 is the stationary frame value and V 2 is the corresponding moving frame value for the constant acceleration speeds responsible for distance D 2. In accordance with the millennium relativity mathematical convention, lower case and upper case characters can be used to distinguish between the respective smaller and larger of two corresponding values for the same function. Thus, equations (35) and (36) provide a convenient means for converting between speed V 2 relative to moving frame v 1, and its corresponding value v 2 relative to the stationary frame. 8. Conclusion Upon verification that Einstein s velocity composition formula gives the same results as the Millennium Relativity acceleration composition formula and the fact that the latter formula clearly involves the distances associated with constant acceleration, there can be no doubt that Einstein s formula is actually an acceleration based formula and not a uniform motion related formula as given in his theory. The failure of the scientific community to acknowledge this finding and correct the meaning of Einstein s formula in the literature is not only unscientific and lacking in ethics but most assuredly has and will continue to lead to false assumptions about the nature of relativistic mechanics and the universe in general. We can only wonder what new and different discoveries in relativistic physics and astrophysics await us upon complete acceptance within the scientific community of the findings presented here. REFERENCES 1 Joseph A. Rybczyk, Millennium Relativity Velocity Composition, (2002); More Proof that Einstein s velocity Composition Law is Wrong, (2004); Velocity Composition the Scientific Establishment s Way, (2004); Millennium Relativity Velocity Composition in Eight Easy Steps, (2008); The Failure of Einstein s Velocity Composition in the Reverse Direction, (2008); Einstein s Erroneous Derivation of Velocity Composition, (2008); Velocity Composition for Dummies, (2009); Einstein s Special Relativity Velocity Addition Invalidated, (2010) 2 Albert Einstein, On the Electrodynamics of Moving Bodies, now known as the Special Theory of Relativity, Annalen der Physik, (1905) 3 Joseph A. Rybczyk, Millennium Relativity Acceleration Composition, (2003); Velocity Composition Completely Deciphered, (2003); Einstein s Velocity Composition Proven Wrong, (2003); Integrated Relativistic Velocity and Acceleration Composition, (2004); Einstein s Biggest Mistake, Acceleration Composition, Finally Proven, (2008); Alternative Versions of the Relativistic Acceleration Composition Formula, (2008); The Enigma of Einstein s Velocity Composition, (2008); Complete Relativistic Velocity and Acceleration Composition, (2008); The Special Relativity Velocity Composition Paradox, (2008) 4 Joseph A. Rybczyk, Time and Energy, (2001); Time and Energy, Inertia and Gravity, (2002); Relativistic Motion Perspective, (2003); Directly Deriving the Millennium Relativity Acceleration Distance Formula, (2003); Corroboration Research Letter, (2003); Most Direct Derivation of Relativistic Constant Acceleration Distance Formula, (2008); Using Integral Calculus for Relativistic Acceleration, (2009) 5 Joseph A. Rybczyk, The Laws of Acceleration, (2001) 6 H. A. Lorentz, Dutch physicist, discovered a new way to transform distance and time measurements between two moving observers so the Maxwell s equations would give the same results for both, (1895); The use of the formula was expanded upon by Einstein in his special theory of relativity. 8
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