Scientific Examination of Relativistic Velocity and Acceleration Addition Copyright 2010 Joseph A. Rybczyk Abstract Direct comparison of the millennium relativity relativistic velocity and acceleration addition treatment to Einstein s velocity addition treatment reveals the hidden weakness of Einstein s formula while simultaneously demonstrating the superiority of the millennium relativity formulas on the basis of proper adherence to valid scientific principles. 1
Scientific Examination of Relativistic Velocity and Acceleration Addition Copyright 2010 Joseph A. Rybczyk 1. Introduction A long standing argument against the millennium relativity (MR) forms of relativistic velocity and acceleration addition 1 in defense of Einstein s special relativity (SR) velocity addition 2 is finally resolved. It will be shown in the simplest and most direct terms practical that the millennium relativity treatment holds up under the most stringent examination for adherence to valid evidence supported scientific principles. 2. Considerations Underlying the Examination Process When all of the factors involving the differences between Einstein s treatment of velocity addition and the millennium relativity treatments are known, it is apparent that the process of direct comparison of the underlying principles is extremely difficult to conduct. There are several reasons for this. For one, the fundamental nature of Einstein s formula is deceptively simple in appearance yet very difficult to fully understand. Subsequently, there are claims made in its defense that seem valid on the surface yet are scientifically difficult or, in at least one case, impossible to verify. The latter claim involves its symmetrical treatment of the two intermediate velocities that seems to be verifiable through experimental means when in reality it is impossible to verify at all. In fact there is no real direct evidence in support of velocity composition at all. All of the evidence is indirect. Subsequently, to use the mathematical results of Einstein s formula as a test with regard to the validity of the results of an alternative formula is scientifically invalid to begin with. As for the millennium relativity formulas, the velocity addition formula is simple in form similar to Einstein s formula, but allows for the theoretical possibility of light speeds in excess of c. The circumstances, by which this is predicted, however, might be impossible to test for anytime in the foreseeable future. Next, there is the millennium relativity acceleration addition formula. This formula, based on constant acceleration rather than uniform motion, actually gives results almost identical to Einstein s formula. Consequently, this finding challenges the very basis of Einstein s formula and strongly suggests that it too is an acceleration based formula, and not a uniform motion based formula as given in his theory. The problem here, however, is that although the millennium relativity acceleration based formula is very revealing insofar as the underlying behavior involved in the relativistic addition process, the formula is many times more complicated than either of the other two formulas being discussed. Thus, it is more difficult to analyze and evaluate by way of direct comparison to the other two formulas. Yet, it appears that this may be the most useful of the three formulas being discussed insofar as making the true nature of the relativistic addition process understandable. With these considerations in mind, let us begin the examination. 3. Einstein s Formula and the Symmetry Principle For simplicity purposes, assume there are three objects, A, B, and C in uniform motion relative to each other such that A serves as the stationary frame point of reference to which object B s motion is referenced, and B serves as the uniform motion frame point of reference relative to which the velocity of a faster object C moving in the same direction as B is referenced. According to the principles of relativistic physics, if the velocities are a significant 2
fraction of light speed c, the velocity of object C relative to stationary object A is not the direct sum of velocities of B-A + C-B. For that velocity we need Einstein s velocity addition formula as given next. Using the millennium relativity mathematical convention for comparison purposes Einstein s formula for velocity addition can be stated as 1 1 where v 1 is the uniform motion velocity of object B relative to stationary object A, while V 2 is the uniform motion velocity of object C relative to object B, and v is the resultant composition uniform motion velocity of object C relative to object A. (From this point forward is should be understood that A, B, and C refer to objects A, B, and C, and also to the inertial states that their motions represent.) One defense of the validity of Einstein s formula is that it adheres to the principle of symmetry that holds the laws of physics to be the same in all inertial systems. In accordance with this principle and the principle of relative motion, if v 1 is the velocity of B to A, then it is conversely the velocity of A to B. The same holds true for the velocity V 2 of C to B and the velocity v of C to A. It should be understood here, that this investigator is in agreement with these principles and in fact the millennium theory of relativity is founded on them. There is, however, one manner by which this argument is used in defense of Einstein s formula that is invalid as will be discussed next. One argument that claims Einstein s formula conforms to the principles of symmetry and relative motion is that it gives the same result for speed v when the values for speeds v 1 and V 2 are reversed. Since, a superficial analysis of both millennium relativity formulas seems to indicate that those formulas violate these principles; such principles are raised in an attempt to refute the validity of the latter formulas. Given the higher degree of complexity by which these principles relate to the millennium relativity formulas, those analyses will be dealt with on an individual basis as we progress in this paper. Of immediate interest here is to show that the manner by which the argument of symmetry and reciprocity involving the interchanging of speeds v 1 and V 2 are used in support of Einstein s formula is without any scientific credibility at all. By simply viewing the mathematical relationships of speeds v 1 and V 2 in Einstein s formula, it is apparent that the formula cannot distinguish between the two different speeds. Thus, to claim that the fact that the formula gives the same result when the values for v 1 and V 2 are switched, proves that the formula adheres to the principles of relative motion and symmetry is in and of itself scientifically invalid. We are not referring to Newtonian velocity addition where the order in which the sum or product of two velocities appears in the formula has no bearing on the mathematical result. We are referring to relativistic velocity addition and acceleration addition in which case the velocity of the second object relative to the first, directly affects the physical properties upon which the velocity of the third object is dependent. Specifically, the time in the frame of the second object is dilated in direct relation to its velocity relative to the first object. And subsequently, if the total distance traveled by the two objects is held constant, then the velocity of the third object relative to the inertial frame of the second object is affected. Conversely, if the sum of the two velocities v 1 and V 2 is held constant while their individual values are varied then the total distance traveled by the two objects is affected. 3
This then means that the velocity v of object C relative to object A is affected since it is a function of time and distance also and only the distance changed but not time. That is, since the stationary frame time is unaffected it cannot compensate for the change in distance as would be required to hold velocity v constant. By way of contrast, the millennium relativity formulas do distinguish between the speed of the closer object and the speed of the farther object, but that, as will be shown, is an independent consideration that has nothing directly to do with the principles of symmetry and relative motion. 4. Millennium Relativity Velocity Addition We will begin the examination process with the millennium relativity velocity addition formula since it is far less complicated than the millennium relativity acceleration addition formula. Then, since the acceleration addition formula is derived in the same manner, it will be easier to understand how the same principles apply to it. Referring now to Figure 1 assume that the distances D 1, D 2, and D are traveled at the uniform motion speeds v 1, V 2, and v, as shown, during stationary frame time interval T in the frame of object A and the corresponding time interval t 1 in the uniform motion frame of object B. A B C D 1 = v 1 T t 1 D 2 = V 2 t 1 D = vt T Figure 1 Millennium Relativity Velocity Addition By simply relating these distances algebraically while applying the Lorentz time transformation 3 formula that relates time interval t 1 to time interval T, we can derive the millennium relativity velocity addition formula. That is, given 2 for the distance relationship, and 3 for the Lorentz time transformation, by way of substitution of the right side of equation (3) in place of t 1 in equation (2) we derive 4 that simplifies to 5 4
that is the millennium relativity velocity addition formula. Still referring to Figure 1, assume that object B has a speed of v 1 = 0.8c relative to object A. Further assume that object C has a speed of V 2 = 0.6c relative to object B. If we then compute the speed of object C relative to object A using the millennium relativity velocity addition formula (5) we will obtain the result, v = 1.16c. Disregarding for the moment that light speed c is being exceeded by object C relative to object A (This aspect of velocity addition will be discussed later in the paper.) what we have then, is two different speeds for object C; one relative to object B and one relative to object A. Now let us discuss why the velocity obtained for object C relative to object A is in disagreement with the principles of Newtonian physics whereupon speed v would be determined by simply adding speeds v 1 and V 2. According to the principles of relativity, time slows down in the frame of a moving object. If we apply the millennium relativity version of Lorentz time transformation as given by equation (3) to the uniform motion frame of object B and compute the time interval t 1 that takes place in that frame relative to the stationary frame time interval T in the frame of object A during which all of the motion takes place, we will find that less time has passed in the frame of object B then in the frame of object A. Thus, when the speed of object C is determined in the frame of object B using the slower time in Bs frame, a higher speed will be determined for object C than would be in the frame of object A using the faster time in that frame. The reason is simple. If the same distance (between C and B) is traveled relative to the frames of both A and B during the same corresponding time period, then it will be traveled faster in the moving frame of B than in the stationary frame of A, since less time has passed in Bs frame then in As frame. It s really that simple. In consideration of the fact that the millennium relativity formula appears to violate the evidence supported principle that light speed c cannot be exceeded, however, we are left to question its validity and assume that Einstein s formula must therefore be correct on that basis alone since it does not violate that principle. This would seem to imply that the entire basis upon which the millennium relativity formula is founded must be in error somehow. Before accepting that conclusion, let us first examine the millennium relativity acceleration addition formula. 5. Millennium Relativity Acceleration Addition Using the exact same method that was used to derive the millennium relativity velocity addition formula, but using distances traveled by constant acceleration instead of uniform motion we derive the millennium relativity acceleration addition formula as will be shown next. Referring to Figure 2, assume that the distances shown were traveled during constant acceleration from 0 to the speeds previously given for v 1 and V 2. More specifically, assume that speed v 1 = 0.8c was reached while object B accelerated away from object A at a constant rate of acceleration a 1 during time interval T in A s stationary frame of reference. Similarly assume that object C reached speed V 2 = 0.6c while accelerating away from object B at a constant rated of acceleration A 2 during time interval t 1 of the uniform motion frame reached by object B at the end of time interval T. Thus, time interval t 1 is given by the same time transformation formula (3) used in the preceding problem. Distance D 1 is then the stationary frame distance traveled by object B relative to object A during time interval T in the stationary frame of object A. Distance D 2 is the distance simultaneously traveled by object C relative to object B during time interval t 1 of the inertial frame reached by object B during interval T. And last of all, distance D is the stationary frame distance traveled by object C relative to object A during time interval T. 5
A D 1 = cv 1 T c + c 2 2 v 1 B t 1 D 2 = C cv 2 t 1 c + c 2 2 V 2 T cvt D = c + c 2 v 2 Figure 2 Millennium Relativity Acceleration Addition In a manner similar to that in which the millennium relativity velocity addition formula (5) was originally derived from the uniform motion distances traveled in the stationary frame relative to objects A and B, the constant acceleration distances giving in Figure 2 can be used to derive the millennium relativity acceleration addition formula. We begin by taking the formula for distance D, (that was first introduced in the earlier work on energy 4 by this investigator), 6 and solving for speed v 7 8 9 2 2 10 11 to obtain 2 2 2 2 12 13 14 15 2 16 for the basic formula for acceleration speed v. Then given the relationship 6
17 as shown in Figure 2, we obtain by way of substitution with equation (16), the simplified version of the millennium relativity formula 2 18 for the constant acceleration speed v of object C relative to object A. For the full version of the formula we simply substitute the right sides of the corresponding distance equations 19 and 20 in place of D 1 and D 2 of equation (18) to derive 2 21 where v is the composition velocity of speeds v 1 and V 2. If we then substitute the right side of the time transformation equation (3) in place of t 1 in equation (21) we derive 2 22 that reduces to 7
2 23 upon simplification. Unlike the millennium relativity velocity addition formula (5), based on uniform motion distances, this formula based on constant acceleration distances gives results almost identical to Einstein s velocity addition formula (1). This finding not only gives the formula credibility but offers convincing evidence that Einstein s formula is actually an acceleration based formula and not a uniform motion based formula as given in his theory. We will discuss this finding further in the next Section. 6. Initial Findings Involving Acceleration Addition Upon comparing the results of the millennium relativity acceleration addition formula just derived to the results of Einstein s velocity addition formula, it is found that if the two velocities, v 1 and V 2, are the same, both formulas give identical results for velocity v out to 16 decimal places. If the two velocities are not the same, the difference in the results for v is no more than approximately 1.5 percent in the worst case. On this basis alone there can be no doubt that both formulas exhibit the same basic behavior and must therefore have a common fundamental nature to their underlying relationship with the principles of relativity. This view is further substantiated by reformatting the formula in a manner that makes its relationship to Einstein s formula even more apparent as will be done next. By taking the contents inside the brackets of equation (23) and placing them equal to a variable such as 24 and then substituting the variable back into equation (23) in place of the bracketed contents, we obtain 2 25 that upon dividing the numerator and denominator of the fraction on the right by c 2 gives 2 1 26 where the similarity to Einstein s equation (1) is undeniable. An even more important consideration has to do with how the millennium relativity formula was arrived at to begin with. The fact of the matter is that the acceleration formulas for 8
distance and rate of acceleration that are used to derive the acceleration addition formula were themselves derived early on in the millennium relativity theory involving relativistic kinetic energy and total energy 5 that have nothing to do with acceleration addition. The resulting formulas for kinetic energy and total energy have been shown to be different forms of Einstein s formulas and subsequently give the associated formulas for acceleration the same degree of credibility as Einstein s formulas involving energy. It was not until two years and eight additional papers later 6, including a paper on velocity addition, that the work on acceleration addition came about. And it was based on the same principles as the earlier work on velocity addition involving uniform motion; not acceleration. The point being, that the formula for acceleration addition followed directly from the evidence that supported the earlier works and was in no way contrived based on this investigator s personal beliefs or preferences. This should be rather apparent from the fact that the earlier formula for velocity addition appears to violate the principle of light speed constancy. Yet, when the same derivation process was applied using the distances related to constant acceleration, and not uniform motion, the acceleration addition formula (23) directly followed. It should be rather apparent that all of these considerations combined give credibility not only to the millennium relativity acceleration addition formula, but also to the velocity addition formula that preceded it. Now let us continue with the examination. 7. Symmetry and the Principle of Relative Motion Whereas Einstein s formula does not distinguish between the two intermediate velocities v 1 and V 2, the millennium relativity formula for acceleration addition not only does, but gives a different result for velocity v when the values are interchanged. This result appears to be normal and has no effect on the principles of symmetry and relative motion as will be shown next. When evaluating a formula for compliance with the principle of relative motion it is important to understand the role played by reversing the direction of motion. This point is brought up to dispel a false belief that Einstein s formula conforms to the principle of symmetry because it does not reverse directions in producing the same result for v when the values for v 1 and V 2 are interchanged. Actually, the very basis of the principle of relative motion involves a reversal of direction between objects in motion. More specifically, if B has a certain speed relative to A, then according to the principle of relative motion, A has that same speed relative to B. The fact that a change of direction is involved is obvious. In the first case A is at rest relative to which B is moving, and in the second case B is at rest relative to which A is moving in the opposite direction. The same principle holds true for the motion of C relative to both B and A. With that understood, let us now do a reverse analysis of the acceleration addition example just completed, but this time with the speeds for B and C interchanged. Referring to Figure 3 it should be noted that as a result of reversing the direction of motion, the speed of object B is now referenced to the stationary frame of object C, and not object A as in the previous case involving the forward motion illustrated in Figure 2. Subsequently, the speed of object A that is now in motion is referenced to the moving frame of object B. This means that the distance traveled at speed V 2 by object B relative to object C is now based on stationary frame time T and not moving frame time t 1 as was the case for object C when the motion was in the forward direction. And also that the distance traveled at speed v 1 by object A relative to object B is dependent on uniform motion time t 2 as given by the new time transformation formula 9
27 where t 2 is the time interval in the moving frame of object B based on its speed relative to object C, and not object A, as in the former case involving forward motion. A D 1 = cv 1 t 2 c + c 2 2 v 1 t 2 B D 2 = C cv 2 T c + c 2 2 V 2 cvt D = c + c 2 v 2 T Figure 3 MR Acceleration Addition in the Reverse Direction The distance traveled by object A is unaffected by the change in direction of motion since it is still based on the same speed v and the same stationary frame time interval T as in the previous case. Therefore, using the same distance formula (6) 6 and solving it for speed v as done previously in Section 5, leads to the same formula (16) 2 16 for speed v as previously obtained. Substituting the right side of the new equation for the reverse order of distances, D 1 and D 2 28 in place of D in equation (16) then gives 2 29 in place of the original equation (18). We then substitute the right sides of the new equations (30) and (31) 30 31 10
in place of D 2 and D 1 in equation (29) to derive 2 32 for the full version of the reverse direction formula. Substituting the right side of the reverse direction time transformation formula (27) in place of t 2 in equation (32) then gives 2 33 for the final version of the reverse direction formula upon simplification. For the same velocities, v 1 and V 2, these reverse direction formulas, (29), (32) and (33) give identical results for velocity v as obtained from the forward direction formulas, (18), (21) and (23) previously derived. This proves beyond any doubt that the millennium relativity acceleration addition formula is in full agreement with the principles of relative motion and symmetry. For that matter, even the reverse velocity addition formula 34 derived in a similar manner to the reverse acceleration addition formula just completed, gives identical results for velocity v for the same input velocities, v 1 and V 2, as the forward direction velocity addition formula (5) previously derived. This provides additional theoretical evidence as to the validity of the derivation principles used for both types of formulas. (Note: An earlier reverse analysis 7 between the millennium relativity velocity addition formula and Einstein s formula in which Einstein s formula gives different results for the two different directions is recommended for addition reading.) 8. The Absence of Evidence Prohibiting Speeds in Excess of Light Speed To this investigator s knowledge there is no scientific evidence to support the belief that light speed cannot be exceeded. All that the evidence really shows is that nothing can be made to exceed the speed of light relative to the frame of reference that provides the energy or force required to do so. When viewed from the normal perspective of any point of existence in the universe, this restriction has no noticeable effect on anything we do. That, however, is not the case involving relativistic velocity addition. Unlike the conditions that exist in our local environment, including those involving our exploration of the universe, relativistic velocity addition spans across two or more such environments, or points in the universe that can be 11
moving at significant velocities relative to each other. The question then becomes, if one object or body in space is moving at near light speed relative to another object or body that in turn is moving in the same direction at near light speed relative to yet another object or body, is it possible for the faster object or body to exceed the speed of light relative to the slowest of the three objects or bodies? According to the principles of Einstein s theory, the answer is no. But, there is no real evidence to support that conclusion. It is simply one of the principles upon which Einstein postulated his theory. When Einstein s theory is properly understood, it is a theory that contains the universe as a separate entity in space. According to the real evidence, however, there is nothing that prevents the universe from going on to infinity in which case we might only be seeing that part of it that is within light speed c relative to our point in space. Subsequently, as we change our motion relative to something else, we might be experiencing a slight change in our universe relative to the universe experienced by that other entity. And the faster we move, the more we enter a different universe relative to the one we previously existed in. Unfortunately, there are many issues involved in this open view of the universe and it cannot be adequately treated in a brief research paper such as this. It is a subject more suited to a book and will most likely be dealt with by the author in that manner sometime in the near future. 9. Conclusion It is the investigator s claim that the principles applied in the derivation of the millennium relativity formulas for velocity addition and acceleration addition are in complete agreement with the properly interpreted principles of relativity. We are then left to explain how Einstein s formula can involve the addition of uniform motion velocities when the evidence presented in this work clearly shows it is an acceleration based formula. Moreover, there is also the question as to why Einstein s formula does not distinguish between forward motion and reverse motion as in the case of both millennium relativity formulas whereupon a different result is obtained for the two different conditions unless the proper formula is used in each case. And, finally, there is the elevated credibility of the millennium relativity velocity addition formula based on the fact that it is derived in the same manner as the acceleration addition formula. This implies that under proper conditions, light speed can be exceeded without violating the principles of relativity. On the basis of these latest findings one would reasonably expect the scientific authorities to openly acknowledge that Einstein s treatment of velocity addition is in need of further examination and possible revision. Acknowledgments The investigator expresses his gratitude to cincirob (cyber name), Johannes Valks, and Robert Freeland for their efforts in critiquing his findings on the subject of velocity and acceleration addition. It was a direct result of responding to their critiquing that led to the findings presented in this work. A special note of appreciation is extended to cincirob who first made the connection between the millennium relativity acceleration addition formula, (23) and Einstein s formula as exhibited in equation, (26). Appendixes The following appendixes are available at the end of this paper showing two of the Mathcad models used in its preparation: Appendix A Forward to Reverse Comparison Acceleration and Velocity Addition 12
Appendix B Forward to Forward Comparison Acceleration and Velocity Addition REFERENCES 1 Joseph A. Rybczyk, Millennium Relativity Velocity Composition, (2002); Millennium Relativity Acceleration Composition, (2003) 2 Albert Einstein, On the Electrodynamics of Moving Bodies, now known as the Special Theory of Relativity, Annalen der Physik, (1905) 3 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. 4 Joseph A. Rybczyk, Time and Energy, (2001), Section 5, equation (20) 5 Joseph A. Rybczyk, Time and Energy, (2001); The Laws of Acceleration, (2001); Time and Energy, Inertia and Gravity, (2002) 6 Joseph A. Rybczyk, The Laws of Acceleration, (2001); Time and Energy, Inertia and Gravity, (2002); Millennium Relativity Velocity Composition, (2002); A Summary Evaluation of the Evidence that Supports Relativity, (2002); Proof that Matter and Energy Are Not Interchangeable, (2002); Using Special Relativity s Gamma to Derive the Millennium Relativity Kinetic Energy Formula, (2003); Relativistic Motion Perspective, (2003); How Time Transformation Reconciles the Constancy of Light Speed, (2003) 7 Joseph A. Rybczyk, The Failure of Einstein s Velocity Composition in the Reverse Direction, (2008) (Note: All of the Millennium Relativity papers are available at the Millennium Relativity web site at http://www.mrelativity.net) 13
Appendix A Forward to Reverse Comparison 14
Appendix B Forward to Forward Comparison 15
Scientific Examination of Relativistic Velocity and Acceleration Addition Copyright 2010 Joseph A. Rybczyk All rights reserved including the right of reproduction in whole or in part in any form without permission. Note: If this document was accessed directly during a search, you can visit the Millennium Relativity web site by clicking on the Home link below: Home 16