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**Correct Derivation Of Lorentz Transforms Eliminates Contradictions Of Einstein's Relativity **

By Harry H. Ricker III

1.0 Introduction

This paper presents a refutation of Einstein’s special theory of relativity by a new mathematical method and uses this result to derive a new theory which eliminates the inconsistency of Einstein relativity. The method relies on a procedure called evaluation which is essentially the same as used in the evaluation of an equation given a specified condition of evaluation. A crude and poorly defined nonrigorous method of evaluation was used by Einstein in his fundamental papers to determine the ‘’physical meaning" of the Lorentz transforms "in respect to moving rigid bodies and moving clocks". In this paper, the method of evaluation is rigorously applied to the same problem. The results obtained from the new approach to the evaluation of the Lorentz transforms are used to prove that the traditional solutions are inconsistent and contradictory.

Using the new method, it is demonstrated that the problem in Einstein relativity can be eliminated by a different mathematical interpretation, in which different inverse Lorentz transformations are derived that do not exhibit the paradoxes inconsistencies and contradictions of Einstein relativity. In the new theory, Einstein’s light velocity postulate is shown to be incorrect, and it is replaced with a different postulate that makes light velocity a constant but with different scales of measure. The new system is consistent with the postulate that the laws of physics, mechanics and electrodynamics, take the same form when transformed by the new Lorentz transformation equations. This is demonstrated by the fact that in the new theory, the Galilean transformation law remains valid, while in Einstein relativity it is invalid, a result which contradicts the postulate that the laws of mechanics keep the same form.

2.0 Background

In 1962 the late Herbert Dingle published his first refutation of Einstein’s special theory of relativity^{1}. Many years later, he published a book titled __Science At The Crossroads__^{2} which presents clearly why the special theory is untenable. The significance of Dingle’s results have been obscured by polemics launched by opponents who have obscured the issues and asserted without a really good proof that Dingle’s refutation is incorrect^{3,4,5,6}. However, these claims have never addressed the fundamental inconsistencies at the heart of the theory. Dingle’s refutation stands, although few scientists are aware of its existence.

This paper is the result of a analysis undertaken to study and understand the debate. It depends upon a new approach to demonstrating the inconsistencies and contradictions in the special theory of relativity. At the end, the reader will understand why it is successful. Dingle had already shown the theory was untenable in 1962. Therefore, as the years go by, it is not surprising that new refutations are produced. These will progressively become better, stronger, and more convincing as the faults inherent to the theory become better understood.

2.1 What Is Relativity?

The theory of relativity is a controversial topic largely because it is a diverse and confusing subject. The confusion is deepened by the fact that the theory as currently understood is not a single coherent theory, but a diverse collection of theories integrated into a package which is taught in textbooks. Different theories appear in different levels of textbooks. Introductory books try to keep the theory simple and closely follow the historical development and stress Einstein’s 1905 version of relativity^{7} while giving the student an introduction to the more challenging methods of the Minkowski space-time approach. Higher level books begin with the Minkowski^{8} approach and present Einstein’s 1907/1910 version^{9,10} in a more elegant form. The student is unaware that there are significant differences between these different interpretations that are inconsistent.

Most textbooks present the Einstein version of relativity which has largely superceded the earlier versions of relativity pioneered by FitzGerald, Lorentz, Poincare, and Ives^{11}. The theory is presented as if it is the creation of Einstein exclusively. But, Einstein incorporated these earlier versions into his theory^{9,10}. He changed the theory with each new publication of a paper^{7,9,10}. Following the introduction by Minkowski of a four dimensional geometry now called space-time, the theory was radically altered in form and content. The problem for the critic of relativity is to determine the correct accepted version of relativity that is to be criticized. However, no single accepted definitive theory is defined as the true theory of relativity.

2.2 Experiments and Relativity

The interpretation of the experimental evidence in the theory of relativity is another major problem. Proponents of the theory seem obvious to the fact that the experimental evidence is confusing and contradictory. This reinforces the problem of the different theories of relativity which exist under the umbrella concept of relativity. Einstein’s 1905 version of relativity, which is closer to the Lorentz theory of relativity than the modern version of relativity, asserts that moving clocks run slow and that distances appear to contract. The 1907 version, which was formulated to explain the redshift of moving canal rays, asserts that the both the Lorentz contraction and the slowing of clocks is apparent. The accepted explanation of the twins paradox is that the travelling twin ages slower than his stay at home twin because the slowing of time in his reference frame is real. In the explanation as to why the effect is not reciprocal as required by Einstein’s 1905 tand 1907 theories of relativity, the answer is given that the fast moving twin experiences acceleration relative to the inertial reference frame of space while the stay at home twin does not. Furthermore, it invokes general relativity. Here the explanation relies upon space as an absolute frame of reference, which contradicts the relativity postulate. The experiments which claim to support asymmetrical aging in the twins paradox, i.e. mu meson and travelling clocks, therefore refute the relativity postulate, and are inconsistent with Einstein’s 1907 version of relativity but consistent with Lorentz’s version.

All of the experiments are claimed to support Einstein relativity, but which version of relativity do they confirm? Since there are at least seven versions of Einstein relativity (the 1905 version, the 1907 version, the 1910 version, the 1911 version. the 1912 version, the 1916 version of general relativity, and the Minkowski version) we need to know which experiments confirm which version of the theory. Furthermore, some experiments are inconsistent with Einstein relativity, and consistent with Lorentz relativity, while others appear to contradict Lorentz relativity. But the answer is only that the experiments confirm relativity. Which relativity do they mean? It is clear that we can not rely upon the experiments to guide us towards the truth in relativity. Therefore, a careful examination of the mathematics at the foundation of the theory is required.

3.0 Method Of Approach

Here the method used is significantly different from the usual approach which follows the method of Einstein. That approach is based on philosophical postulates and attempts to derive physical consequences based on mathematical deductions from the postulates. Here we take a different route. We assume as axioms the truth of the Lorentz transforms and inquire into the validity of the resulting mathematical structure.

The particular problem which will be examined is the inconsistency of solutions obtained when using the Lorentz transformation. Einstein’s first innovation in the theory of relativity was the introduction of the idea that all inertial reference frames were equivalent, because there was no absolute rest frame. When he combined this idea with the postulate that the velocity of light is the same for all inertial reference frames, he derived the Lorentz transformations as a consequence of the two postulates. The purpose of this paper is to show that the Lorentz transformations are inconsistent with the two postulates that Einstein used to derive them. Hence there is an error in the mathematics. This error will be identified.

In mathematics a proof involves the demonstration of necessity and sufficiency. This means that we must demonstrate both sufficient and necessary conditions. This is generally done by showing that the conclusion follows from the assumptions (sufficiency), and then reversing the process, showing that the conclusion can only follow from the premises used and no others. Einstein achieved the proof of sufficiency in his 1905 paper, but the corresponding proof of necessity has never been demonstrated. This paper shows that this second requirement is not consistent with the theory of relativity as usually presented in textbooks.

The disproof of the second requirement (necessity) follows from the demonstration that the Lorentz transforms are not reciprocally equivalent as assumed in the relativity postulate, and that the Lorentz transforms do not require that the distance travelled by light in different reference frames be invariant.

4.0 Mathematical Analysis

The mathematical method used here is to first solve the system of Lorentz and inverse Lorentz equations for space and time simultaneously using a specified condition of evaluation. Here the term evaluation is used in the same sense as it is used when a polynomial equation is solved for its roots by setting the equation to equal zero and solving for the indeterminates. The procedure used here is similar. A selected variable is set to zero, and the resulting solutions are obtained. Solutions are obtained by setting one of the following four variables equal to zero, and then solving for the remaining three. The following variables are set equal to zero and the resulting solutions obtained by evaluation: x=x’=t=t’=0, each taken in turn.

The Lorentz transformation equations in a simplified form are assumed as follows:

x’=ß(x-vt)

t’=ß(t-vx/c^{2})

x=ß(x’+vt’)

t=ß(t’+vx/c^{2})

ß=(1-v^{2}/c^{2})^{-1/2}

Here there are four equations which express the simultaneous solutions for the transformation of coordinates. These equations are defined in the usual way in terms of two relatively moving reference frames S and S’. Where the origin of frame S’ is in motion with velocity v in the positive x direction of S.

Notice that ß is greater than unity when v is greater than zero, and that ß^{-1 }is less than unity when v is greater than zero. An equation of the form t’=ßt results in a dilation of the variable t’ with respect to t because t’ is greater than t The equation t=ß^{-1}t’ results in a contraction of the variable t with respect to t’ because t is less than t’. The definition of ß implies that it is always equal to or greater than unity, and can never be less than unity.

The coordinate frames S and S’ are assumed to be orthogonal coordinate systems with the requirement that time is defined such that t=t’=0 occurs when the origins coincide; i.e. x=x’=y=y’=z=z’=0 at t=t’=0. The axes for the x, y, and z directions are assumed to be parallel, and the y and z coordinates are assumed to be identical and coincide when the origins coincide at t=t’=0. The purpose of the solutions is to determine the relations governing the transformation of the x and t coordinates according to the Lorentz transform equations.

4.1 Results for x=0 (Specification of an evaluation in space )

To consider the role of evaluation in space, we determine the simultaneous solution of the four equations when we specify the condition that x=0. The results are as follows:

Equation 1: x’=ß(x-vt)=-ßvt

Equation 2: t’=ß(t-vx/c^{2})=ßt

Equation 3: x=ß(x’+vt’)^{=}0 , Therefore x’=-vt’

Equation 4: t=ß(t’+vx’/c^{2})=ßt’(1-v^{2}/c^{2})=ß^{-1}t’

Notice that equation 4 is the inverse of equation 2, so they are the same solution. Equation 4 is solved by substitution with the result from equation 3. Therefore, from equations 2 and 4 we have the following solution for the condition x=0: t’=ßt. The solutions for equations 1 and 3 give the results x’=-ßvt=-vt’, from which we conclude that t’=ßt. A result which is the same as obtained from equation 2 which is the primary result for the condition x=0.

4.2 Results for x’=0 (Specification of an evaluation in space )

To consider the role of evaluation in space, we determine the simultaneous solution of the four equations when we specify the condition that x’=0. The results are obtained as follows:

Equation 5: x’=ß(x-vt)=0, Hence x=vt

Equation 6: t’=ß(t-vx/c^{2})=ßt(1-v^{2}/c^{2})=ß^{-1}t

Equation 7: x=ß(x’+vt’)=ßvt’

Equation 8: t=ß(t’+vx’/c^{2})=ßt’.

Notice that equation 6 is the inverse of equation 8, so they are the same solution. Equation 6 is solved by substitution with the result from equation 5. Therefore, from equations 6 and 8 we have the following solution for the condition x’=0: t=ßt’. The solutions for equations 5 and 7 give the results x=vt =ßvt’, from which we conclude that t=ßt’. A result which is the same as obtained from equation 8 which is the primary result for the condition x’=0.

4.3 Results for t=0 (Specification of an evaluation in time )

To complete the analysis of evaluation, we now consider the role of evaluation in time. We determine the simultaneous solution of the four equations when we specify the condition that t=0. The results are as follows:

Equation 9: x’=ß(x-vt)=ßx

Equation 10: t’=ß(t-vx/c^{2})= -ßvx/c^{2}

Equation 11: x=ß(x’+vt’)^{=}ßx’(1-v^{2/}c^{2})=ß^{-1}x’

Equation 12: t=ß(t’+vx’/c^{2})=0, therefore t’=-vx’/c^{2.}

Notice that equation 11 is the inverse of equation 9, so they are the same solution. Equation 11 is solved by substitution with the result from equation 12. From equations 9 and 11, we have the following solution for the condition that t=0: x’=ßx. The solutions for equations 10 and 12 give the results t’=-ßvx/c^{2}=-vx’/c^{2} from which we conclude that x’=ßx. A result which is the same as obtained from equation 9 which is the primary result for the condition t=0.

4.4 Results for t’=0 (Specification of an evaluation in time)

To consider the role of evaluation with the opposite condition, we determine the simultaneous solution of the four equations when we specify the condition that t’=0. The results are as follows:

Equation 13: x’=ß(x-vt)=ßx(1-v^{2/}c^{2})=ß^{-1}x

Equation 14: t’=ß(t-vx/c^{2})=0, therefore t= vx/c^{2}.

Equation 15: x=ß(x’+vt’)^{=}ßx’

Equation 16: t=ß(t’+vx’/c^{2})= ßvx’/c^{2}.

Notice that equation 13 is the inverse of equation 15, so they are the same solution. Equation 13 is solved by substitution with the result from equation 14. From equations 13 and 15 we have the following solution for the condition that t’=0: x=ßx’. The solutions for equations 14 and 16 give the results t=ßvx’/c^{2}=vx/c^{2} from which we conclude that x=ßx’. A result which is the same as obtained from equation 15 which is the primary result for the condition t’=0.

4.5 Comments on Above Results

The solutions presented as Equations 2 and 8 obtained with the evaluations x=0, and x’=0 correspond to the usually accepted reciprocally related equations which demonstrate time dilation for moving clocks in the special theory of relativity. These results lead to the commonly used description that moving clocks run slow. Equation 5 was used by Einstein in 1905 to deduce the result in equation 6 as time dilation. In 1910 Einstein revised his method and obtained the equation for time dilation as equation 8. Later authors cited equation 8 as the equation for time dilation. During the 1930s, some textbooks cited equation 2, and since then either equation 2 or 8, and sometimes both, have been used as the definition of time dilation. However, most textbooks continue to specify equation 8 as time dilation. Sometimes the inverse solutions given by equations 4 and 6 are referred to as equations for time dilation, but they are not generally or widely accepted as defining this concept. Notice that Einstein deduces time dilation using evaluation in frame S’ by setting x’=0.

The solutions for t=0 and t’=0, given in equations 9 and 15, do not lead to the usually accepted reciprocally related solutions for FitzGerald-Lorentz contraction in the special theory of relativity. Instead, the inverse solutions given in equations 11 and 13 are given as the traditional solutions for the Lorentz-FitzGerald contraction.This leads to the commonly used description that moving objects physically contract in the direction of motion. Einstein obtained the result given by equation 11 in his 1905 paper. The primary solutions given by equations 9 and 15, do not appear as solutions in the special theory of relativity. Notice that Einstein deduces the FitzGerald -Lorentz contraction using evaluation in frame S by setting t=0. An evaluation procedure that is opposite to the evaluation used for the derivation of time dilation. In other words, time is evaluated in frame S’ while space is evaluated in frame S. These results lead to the commonly used description in special relativity that in a moving frame of reference space contracts and time dilates.

The method used here provides the traditional results in a very simple and efficient manner. But notice that there are additional equations which the traditional theory does not obtain. These results show that solutions obtained by the traditional methods are not complete. Equations 1, 7, 12, and 16 appear here for the first time. In addition, this is the first time that a fully complete solution set of all possible solutions has been obtained.

To summarize, when an evaluation condition is imposed upon the system of Lorentz transformation equations, two simultaneous solutions result. One solution, which we call the primary result, and its inverse, which we call the secondary result. The primary results appear in equations 2, 8, 9, and 15, with the secondary results given by equations 4, 6, 11, and 13. The primary results always give a dilation of the transformed variable while the secondary result is always a contraction. The primary result is distinguished from the secondary because it is also obtained as a solution of the remaining two equations which are redundant with the result of the primary solution. The primary equation is considered the solution because the secondary solutions are inversely related to the primary.

The following primary solutions are therefore interpreted as the solutions for the four evaluation conditions. They are as follows:

For evaluation x=0, the solution is equation 2: t’=ßt.

For evaluation x’=0, the solution is equation 8: t=ßt’.

For evaluation t=0, the solution is equation 9: x’=ßx.

For evaluation t’=0, the solution is equation 15: x=ßx’.

Notice that none of these solutions is a contraction, they are all dilations of coordinates.

5.0 Discussion Regarding the Interpretation and Meaning of The Results

Consider the case of evaluation with x=0. There are four resulting equations which have the following interpretation. Equation 3 gives the equation for the motion of the origin of the coordinate system S, the coordinate x=0, in terms of the time and space coordinates of S’. It represents the equation of motion obtained by an observer in S’ in terms of his coordinates. Equation 1 gives the motion of the origin of S relative to space in S’ in terms of time defined in system S. Thus, an observer in S can calculate his position in S’ using this equation and a clock at rest in S. An observer in S’ measures the motion of the origin of S in terms of his space coordinates using a clock at rest in S’ that records time in terms of t’. From the point of view of the observer in S using time t, the coordinates measured by an observer in S’ using time t’ are dilated in terms of the time measured in frame S. This is the result indicated by equation 2, the primary solution. The secondary solution given by equation 4 specifies how measurements of time performed in frame S’ are transformed back to the time standard of frame S.

Consider the case of evaluation with x’=0. There are four resulting equations which have the following interpretation. Equation 7 gives the equation for the motion of the origin of the coordinate system S’, the coordinate x’=0, in terms of the time and space coordinates of S. It represents the equation of motion obtained by an observer in S in terms of his coordinates. Equation 5 gives the motion of the origin of S’ relative to space in S in terms of time defined in system S’. Thus, an observer in S’ can calculate his position in S using this equation and a clock at rest in S’. An observer in S measures the motion of the origin of S’ in terms of his space coordinates using a clock at rest in S that records time in terms of t. From the point of view of the observer in S’ using time t’, the coordinates measured by an observer in S using time t are dilated in terms of the time measured in frame S’. This is the result indicated by equation 8, the primary solution. The secondary solution given by equation 6 specifies how measurements of time performed in frame S are transformed back to the time standard of frame S’.

When we consider the meaning of the system of equations for evaluation specified by a time, we are considering the concept dual to the meaning of the motion of the origin in space. This dual concept is the equation of synchronization of time. The equation gives the time lead or lag of a clock at a space coordinate relative to a clock located at the origin of space coordinates. The equation specifies the time measured at a space coordinate when the clock at the origin reads zero time.

Consider the case of evaluation with t=0. There are four resulting equations which have the following interpretation. Equation 12 gives the equation for the synchronization of the clocks in coordinate system S’ in terms of the time and space coordinates of S’. It represents the equation of synchronization for an observer in S’ in terms of his space coordinates. Equation 10 gives the equation of synchronization of time in S’ relative to space coordinates defined in S. Thus, an observer in S can calculate his time lag relative to the clocks in S’ using this equation and his location in S. From the point of view of the observer in S using distance x, the coordinates measured by an observer in S’ using distance x’ are dilated in terms of the distances measured in frame S. This is the result indicated by equation 9, the primary solution. The secondary solution, given by equation 11 and usually called the Lorentz-FitzGerald contraction, specifies how measurements of distance performed in frame S’ are transformed back to the distance standard of frame S.

Consider the case of evaluation with t’=0. There are four resulting equations which have the following interpretation. Equation 14 gives the equation for the synchronization of the clocks in coordinate system S in terms of the time and space coordinates of S. It represents the equation of synchronization for an observer in S in terms of his space coordinates. Equation 16 gives the equation of synchronization of time in S relative to space coordinates defined in S’. Thus, an observer in S’ can calculate his time lag relative to the clocks in S using this equation and his location in S’. From the point of view of the observer in S’ using distance x’, the coordinates measured by an observer in S using distance x are dilated in terms of the distances measured in frame S’. This is the result indicated by equation 15, the primary solution. The secondary solution, given by equation 13 and usually called the Lorentz-FitzGerald contraction, specifies how measurements of distance performed in frame S are transformed back to the distance standard of frame S’.

5.3 The Clock Paradox - Contradiction Of Clock Rates

The essence of the clock paradox is contained in the problem of determining which set of equations should be used to calculate the rates of identical clocks at rest in relatively moving reference frames. The paradox results because the solution sets for the different evaluation conditions are inconsistent. To see what this means within the context of the above definitions, we attempt to solve the famous problem posed by Herbert Dingle. We seek to answer the question: When we have two identical clocks at rest in relatively moving reference frames, which clock runs slow? At first the answer seems easy; the moving clock runs slow. But the paradox arises when we try to identify the moving clock, because both clocks are moving with respect to the other one, so which one runs slow?

Herbert Dingle is well known for his claims of inconsistency in Einstein’s Special Theory of Relativity. One of Dingle’s arguments appeared in the September 8, 1962 issue of Nature under the title "Special Theory of Relativity"^{1}. This short note by Herbert Dingle points out "what appears to be an inconsistency in the kinematical part of Einstein’s special theory of relativity." Dingle states that "The alleged inconsistency lies in the fact that the argument used to prove that ‘moving clocks run slow’ (with which all the kinematical implications of the theory are bound up) proves with exactly the same validity, that moving clocks run fast. Both cannot be right, so the basis of the theory must be faulty." Dingle uses Einstein’s notation and copies his exact words in a comparison of Einstein’s solution for the transformation of time with Dingle’s alternative solution based on the inverse Lorentz transformation.

Dingle’s argument was simple. He derived equations 4 and 6 using the same method as used above, and showed that the result obtained by Einstein in 1905 as equation 6 is contradicted by the result of equation 4. In the first, the moving clock runs slow and in the second it runs fast. Here we experience a major problem. These conclusions are wrong because Einstein’s definition of the rate of a clock assumes that the clock runs slow when the Lorentz transform is contracted, and fast when it is dilated. These results are contradicted by the calculation of the clock frequency as Einstein discovered in 1907. This discrepancy resulted in a major revision of the theory in 1907 so that the theory would be consistent with the Lorentz transformation of clock frequency. The 1905 and 1907 theories give different interpretations of the Lorentz transform, in the following we will use the 1905 interpretation. The correct result is that when a Lorentz transformation of time is a contraction, the clock runs fast, and when dilated, it runs slow. In the following we will use this definition for clock rate.

The primary solutions given by equations 2 and 8 are also inconsistent. If we choose S as the rest frame, then equation 2 indicates that the clock in S’ runs slow, but equation 8 gives the solution that it also runs fast. Why? Because the correct physical situation is given by the inverse of equation 8 or equation 6 which leads to the interpretation that the clock at rest in S’ runs fast. This contradicts the previous result, hence the paradox and the contradiction. The equations are inconsistent.

The solution is to permit only one evaluation condition. This means using equation 2 and its inverse from the same solution set; i.e., equation 4. We see that equation 2 indicates that the clock in S’ runs slow, and that the interpretation of equation 4 is also that the clock in S’ also runs slow. Hence the paradox is avoided and the problem solved. Dingle’s question can now be answered with confidence, the moving clock runs slow relative to the reference clock in an absolute reference frame. Why? Because now there is only one rest frame. We eliminated the other one because its solutions were inconsistent with the existence of two equivalent rest frames.

The paradox of the twins can now be answered in the following traditional way. The stay-at-home-twin resides in the earth rest frame. Then according to equation 2 the travelling twin’s clock runs slow. The twins appear to age asymmetrically. When we use earth time only, there is no paradox. Notice the important result that we only encounter a paradox when we attempt to combine conclusions based on two different conditions of evaluation, one for evaluation in S and another one for S’. But this conclusion violates the relativity postulate. A fact that has been pointed out consistently and often. The purpose of the following sections is to make this conclusion more precise and mathematically rigorous.

6.0 Inconsistency of The Mathematical Solutions

This section will present a proof that the usual conclusions in relativity are incorrect because they do not form a bijective; i.e. one-to -one and onto set of transformations. This proof strikes at the heart of traditional relativity and shows why the theory has drawn considerable criticism and controversy. The essence of the proof that necessary and sufficient conditions are not fulfilled in Einstein relativity is that the Lorentz transformations are not mathematically bijective transformations.

The usual understanding is that transformations between reference frames are symmetrical in the sense that when transforming from S into S’ and from S’ into S the equations are equivalent or symmetrical. This means that when we obtain the transformation of length from S into S’ we obtain x’=ß^{-1}x. This is the result given in equation 11 for evaluation t=0. The equivalent or symmetrical transformation from S’ to S is x=ß^{-1}x’. This is the result given in equation 13 for evaluation t’=0. A result that can be obtained merely by exchanging x for x’ and x’ for x. This is why the transformations are called symmetrical. The unfortunate fact is that these equations do not form a bijective set. The proof of this is that they do not give an identity upon substitution. Hence given x=ß^{-1}x’, substitute the result x’=ß^{-1}x. The result is: x=ß^{-1}x’=ß^{-1}(ß^{-1}x)=ß^{-2}x not x=x as we should obtain for a true bijective transformation. Reversing this does not correct the fault. To see this, substitute the result of the second equation into the first: x’=ß^{-1}x=ß^{-1}(ß^{-1}x’)=ß^{-2}x’, and not x’=x’. This result is the source of many paradoxes. The solution is that the theory is incorrect.

The fault is not corrected by the use of the primary solutions of equations 9 and 15. For the transformation of length from S into S’ we obtain x’=ßx when we evaluate in S’. The equivalent or symmetrical transformation from S’ to S is x=ßx’. A result that can be obtained merely by exchanging x for x’ and x’ for x. Here again we see that the transformations are symmetrical. These equations also do not form a bijective set. The proof of this is that they do not give an identity upon substitution. Hence given x=ßx’, substitute the result x’=ßx. The result is: x=ßx’=ß(ßx)=ß^{2}x not x=x as we should obtain for a true bijective transformation. Reversing this does not correct the fault. To see this, substitute the result of the second equation into the first: x’=ßx=ß(ßx’)=ß^{2}x’, and not x’=x’.

The theory presented here can be used to define the correct solution for a bijective transformation. It is simple to do this. We define a transformation from S into S’ with an evaluation condition defined in either S or S’. Suppose we specify evaluation in S and transform length using equations 9 and 11. Hence we obtain the result that x’=ßx from 9. We now seek the transformation from S’ into S, but because we want this transformation to be bijective, we specify that evaluation is in S, which leads to equation 11, x=ß^{-1}x’. Now substituting equation 9 into that of equation 11 we obtain that x=ß^{-1}x’= ß^{-1}(ßx)= x. A result which shows that the transformations are bijective inverses. The trick or key ingredient to obtaining this result is to specify that the evaluation condition when going back into the original frame from the opposite one is to use the same evaluation condition as we used in the first transformation.

To see this is true, we do it again with the opposite evaluation. Transforming from S into S’ with evaluation in S’. The result is equation 13, x’=ß^{-1}x. Now transforming from S’ into S with evaluation specified in S’; notice it is the same for both. The result is 15, x=ßx’. Now substitute the first into the second as before, x=ßx’=ß(ß^{-1}x)=x. Again the result is an identity and the transformations are bijective. The reason they are bijective is that when we transform from S into S’ and then back into S, we have been careful to require that the transformation maps into the same conditions, i.e. events, from which the process originated. Hence the process transforms events in S into events in S’ and then transforms back into S at the same events from which the process originated.

This solves the famous pole vaulter and barn paradox. Starting from the viewpoint of the barn, the observer in the barn measures the length of the vaulter’s pole. According to the traditional theory, the measurement result is x=ß^{-1}x’ where x’ is the rest length in the vaulter’s frame of reference. The observer at rest finds the true length of the vaulter’s pole by application of a measurement scale change law to the measured length, i.e. x’=ßx. Now by the above result x’=ßx= ß(ß^{-1}x’)= x’, it is clear that the observer in the barn obtains the true rest length of the pole so that it can be compared with the width of the barn. Hence both observers can agree on the rest length of the pole without encountering a paradox. The reason is that they are using a bijective pair of transformations. The paradox in traditional theory arises because the transformations used are not bijective. This arises because the Lorentz and inverse Lorentz transformations are mathematically inconsistent.

We now turn to the consideration of the consistency of the solutions of the Lorentz transforms. An examination of the mathematical consistency of the solutions given above shows that the solutions obtained using evaluation in frame S are inconsistent with solutions obtained using evaluation in frame S’. This inconsistency of solutions is the cause of the numerous paradoxes in the special theory of relativity. These paradoxes have been discussed previously where it was suggested that the inconsistencies can be eliminated by only considering solutions obtained using evaluation in only one reference frame.

The inconsistency of the solution sets for evaluation specified in frames S and S’ simultaneously (in the mathematical sense of the word) can be easily seen by direct comparison of the primary solutions given in equations 2, 8, 9, and 15. Comparing equations 2 and 8 for time and 9 and 15 for space, we see that there are only three possible simultaneous sets of solution. The trivial one with all solutions equal to zero; in which case we have no solutions; the relative rest solution where ß=1 and v=0 hence no motion, and the alias solution, in which case all the primed variables or indeterminates are identical to the unprimed variables upon the exchange of variable names. The last case being essentially that the two different solution sets for evaluation in frames S and S’ result in the same solution, but with the symbol names (which represent the same physical entities) exchanged. So the solutions represent the same solution with the names of the variables or indeterminates exchanged. Physically the two different solutions are meaningless, there is only one solution, either the solution given by evaluation in frame S or the solution given by evaluation in frame S’ but not both simultaneously. The clarification of this difference is one of the main objectives of this paper.

Given the above analysis of the inconsistency of the solutions obtained in the special theory of relativity, the procedure for eliminating all paradoxes can be easily defined. The correct procedure is to only use, consistently throughout all the derivations and calculations, an evaluation that is defined for only one frame of reference. The best way to do this is to discard the solution set obtained for evaluation in frame S’. Hence, we discard equations 5 through 8, and 13 through 16. All derivations of mathematical results using these should be avoided since they will inevitably lead to paradoxical results or incorrect conclusions. In a later section we will see that the inverse Lorentz transforms must also be discarded and replaced by different ones.

Before ending this topic, it is necessary to consider the arguments used to prove that Dingle’s paradoxes and inconsistencies are fallacious. Clearly if these arguments are correct, then the proof of inconsistency given here fails. The curious result is that an examination of the arguments used to refute Dingle actually prove the argument given here. The only solution that avoids the inconsistency is the "alias solution". This can mean either, the two solutions are the same solution with the names changed to appear to be different solutions, or they apply to two physically different situations and are meaningless. Dingle’s critics have used the second method, and tried to prove that Dingle’s inconsistencies were fallacious because they applied to different physical situations. This approach seems peculiar because they never make it clear what exactly the other different physical situations are. The skeptical reader wants to know what are these different physical situations for which the solutions apply?

Born tries to make this argument by saying that the symbols t and t’ "have not the same meaning" in equations 2 and 6. The only way this can occur is if we exchange the symbol names as in the alias solution. Otherwise, the solutions remain contradictory. Born goes on to say that, the symbols t and t’ refer to different physical situations, and "these are inverse and must of course correspond to an exchange of the symbols". But these solutions are not inverses as shown previously. Hence they must either represent an inconsistency or the same physical situation as in the alias solution. In either case the refutation remains valid.

At this point, the reader can see that the proof of inconsistency does not rely as much on the contradictory nature of the solutions as it does upon the failure of the solutions to be bijective. If the solutions are bijective, then we can be sure that the same physical situation is being represented mathematically. If they are not bijective, we can not be sure what the equations actually represent physically. Therefore, the arguments against Dingle’s inconsistencies leave the problem as it stood. The solutions have no physically consistent meaning and the contradiction stands unresolved.

7.0 Interpretation Of Lorentz Transform Equations In Light Of Evaluation

This section has the objective of determining the correct interpretation to be applied to the previous results. To summarize, we rejected the solutions obtained using evaluation conditions x’=t’=0 because they represent solutions inconsistent with the solutions obtained using the evaluation conditions x=t=0. The objective is now to place an interpretation upon the remaining equations.

7.1 The Equations Define The Measurement Scale Change Laws

The most obvious and clearly fruitful interpretation is that the equations resulting from the process of evaluation give the coordinate transformation laws between the reference frames S and S’. This interpretation is different from the traditional interpretation given to the equations in Einstein’s theory, where the solutions are interpreted as real changes in the physical state of space and time between reference frames. The new interpretation views the transformation equations as coordinate scale changes between reference frames. Hence, the transformation of the measurement scale for time in S into the time measurement scale in S’ is given by equation 2. The transformation from S’ into S is given by the inverse of equation 2 as in equation 4. These are bijective transformations that are one-to-one and onto between the reference frames. Since the transformations of Einstein relativity are not bijective, paradoxes, inconsistencies, and contradictions are eliminated.

The transformation of the measurement scale for distance in S into the distance scale in S’ is given by equation 9. The transformation of the measurement scale for distance from S’ into S is given by the inverse of equation 9 as in equation 11. These are bijective transformations that are one-to-one and onto between the reference frames. Notice that this interpretation is nicely confirmed by the equations for the motion of the origin of S as viewed from S’, equations 1 and 3. Similarly, the synchronization lag equations 10 and 12 are consistent with the scale change interpretation.

7.2 The Scale Change Laws Transform Velocity Invarantly

This section proves the surprising result that application of the measurement scale change interpretation leads to the conclusion that velocity is transformed invarantly between reference frames S and S’. This new approach requires some preparation. Consider the result of the scale change laws. In frame S we measure time and space in terms of units of time and distance measure so that the law x=ct applies. This means that distance and time measurement scales are defined in terms of the light velocity c.

Consider the result of the scale change laws. Equation 9 defines the distance scale in S’ in terms of distance measure in S as x’=ßx, and time in S’ as t’=ßt. The light velocity law transforms as ßx=cßt , which is the same as the law in frame S after dividing both sides by the factor ß. Hence the Lorentz transformations leave the coordinate measure of light velocity invariant. This is not the same as in Einstein’s relativity. In Einstein’s relativity, the distances x=ct and x’=ct’ are equal; i.e. x=x’, because the time and distance measurement scales in S and S’ are the same. In this interpretation, the time and distance scales are transformed and therefore represent different times and distances while c is numerically invariant after the transformation. This result applies to all transformations of velocity between S and S’. Hence velocities transform invarantly.

This result shows that the Einstein light velocity postulate is inconsistent with the interpretation that Lorentz transforms change the scale of measurement to keep the coordinate measure of light velocity invariant. The inconsistency in the Lorentz and inverse Lorentz transforms occurs because light velocity in an absolute sense, i,e, that the same measurement scale is used in both frame S and S’, is inconsistent with the fact that the Lorentz transform changes the measurement scales. Hence the physical standards of measure in frames S and S’ are different. But Einstein relativity asserts that they are the same. Hence the contradictions and paradoxes. This is illustrated by the first and most basic paradox, the paradox that arises when we assert that the distance travelled by light is the same in frames S and S’. The light spheres do not represent the same distance in both cases, but only that when the coordinate measures of distance in S and S’ are compared, they are equal. Since the measurement standards for distance and time are not the same in both frames, there is no paradox.

The mathematical explaination follows. Relativity assumes the two-way velocity of light is the same in frames S and S’ or that c=c’. But here it is assumed c does not equal c’. Instead it is assumed that the following holds, c’=ß^{-1}c. Now in frame S’, we change the units in S’ to make light velocity transform invariently relative to the units in S. The reason for this is that in frame S’ the distance travelled in unit time t measured in S’ is longer than travelled in S because we have the equation x’=c’t’. Substituting using the equations x’=ßx and t’=ßt gives: ßß^{-1}ct=ßx or ct=ßx. So in terms of unit measure of S, in S’ the distance light travels is greater than in S. Hence, there is no light paradox at all. In order to make the laws of physics defined in frame S transform into S’ with the same form, we transform the units of S into new units in S’ which make the light velocity ratio c as in frame S’. When this is done as defined above, the same laws are valid in S and S’, and the requirement is satisfied.

7.3 The Scale Change Laws Preserve The Galilean Transformations

Here the fact that the Galilean transformations are preserved by the Lorentz transformations is proved. A corollary, the velocity addition law is also proved and shown to contradict Einstein relativity. The proof relies upon converting the Lorentz transforms and the evaluation solutions to differential form. This is easily performed by replacing x with dx, x’ with dx’, t with dt, and t’ with dt’. The Lorentz transform becomes: dx’=ß(dx-vdt). A velocity defined relative to the frame S’ is defined as: U_{x}’=dx’/dt’. Here U_{x}’ means the component of velocity defined in S’ which is parallel to the direction of the x’ axis of the S’ reference frame. A velocity defined relative to the frame S is defined as U_{x}=dx/dt, where this is the velocity component parallel to the x axis of S. Here the velocity addition law is proved by showing that a velocity U_{x}’ defined in S’ is given by: U_{x}’=U_{x}+v , where v is the velocity of the origin of S’ relative to S.

The proof is as follows. Given dx’=ß(dx-vdt), substitute dx’=U_{x}’dt’, and obtain, U_{x}’dt’=U_{x}’ßdt=ß(dx-vdt). Divide both sides by ß : U_{x}’dt=(dx-v)dt. Rearrange to obtain the result that: (U_{x}’+v)dt=dx, which upon dividing by dt gives: U_{x}=U_{x}’+v. Hence the velocity addition law holds. The opposite result, the subtraction law, is obtained as follows. Here we are given a velocity defined relative to frame S as U_{x}. We require the velocity relative to S’.

The proof is as follows. Given dx’=ß(dx-vdt), substitute dx=U_{x}dt to obtain: dx’=ß(U_{x}dt-vdt)=(U_{x}-v)ßdt. Substitute dt’=ßdt and divide by dt’ to obtain the following: dx’/dt’=U_{x}-v=U_{x}’, which is the required transformation law. Hence we see that the velocity addition law of the Galilean transformation law is preserved.

The significance of this result is that in Einstein relativity, the velocity addition law does not hold in the same form in frames S and S’. But here in this new theory, it is clear that the velocity addition law is the same as the Galilean transform. So the laws of mechanics retain the same form in the new theory but not in Einstein’s theory.

7.4 Lorentz Transforms Are Galilean Transforms Incognito

The results in the previous section suggest a new line of investigation. This leads to the surprising conclusion that Lorentz transforms are disguised Galilean transforms. This idea, however, is a little obscure at first glance, but it becomes much clearer if we reconsider our concept of a Galilean transformation. In a Galilean transformation, time in frame S is considered as a parameter in the calculation of events in transforming from S into S’. Einstein uses a flaw in the definition of the Galilean transform to establish a new physics. He disposes of absolute time, claiming that it is a false concept at the foundation of physics. He replaces this absolute time with a new concept. A simpler approach might have been to modify the definition of the Galilean transform as we proceed to do here as follows.

The idea is extremely simple. Taking the Lorentz transformation for time as a guide, we see that transformation of time in a Galilean transform should change from t=t’ to the following form: t’= t-vx/c^{2}. Here a very small correction is applied to time in S’ relative to time defined in S. Going the other way we have the result t=t’+vx’/c^{2}. The reader should notice that for all practical purposes, the factor vx/c^{2} is negligible at low velocities. The point here is that once this idea is fixed in the mind, the additional change in the measurement scale that is involved in the Lorentz transformation removes all the mystery of the appearance of the term vx/c^{2} in the transformations. Recapping, we have the following Galilean transformation equations: x’=x-vt, t’=t-vx/c^{2}, y=y’, and z=z’. Going the other way we have: x=x’+vt, t=t’+vx’/c^{2}, y’=y, and z’=z. By closely examining these, we see that we are only one small step away from the Lorentz transforms, viz: x’=ß(x-vt), t’=ß(t-vx/c^{2}), x=ß(x’+vt’), and t=ß(t’+vx’/c^{2}). So fundamentally there is only the scale change factor ß, applied at high velocity, that makes Galilean transformations different from Lorentz transformations.

8.0 Vindicating The Inverse Lorentz Transformations

In a previous section, it was demonstrated that the results deduced from inverse Lorentz transformations derived under the same assumptions and procedures as Lorentz transforms lead to inconsistencies and contradictions. These inconsistencies demonstrated that Einstein relativity is untenable. As a consequence, deductions based upon evaluation in frame S’ were eliminated. Deductions which were based upon the inverse Lorentz transformations were also avoided. The reason is that when we attempt to form conclusions based on the inverse Lorentz transforms we obtain contradictions with the deductions based upon Lortentz transforms. Here the object will be to deduce consistent inverse Lorentz transformations from the Lorentz transformations.

The procedure used to deduce the inverse Lorentz transformation for distance in S is to first assume the Lorentz transform from S into S’ and solve for the opposite transformation. This turns out to be a simple procedure as follows:

Given x’=ß(x-vt)=ßx-vßt=ßx-vt’. Rearranging and writing ßx=x’+vt’, we obtain the required result: x=ß^{-1}( x’+vt’). This equation is in the same form as the traditional result except for the fact that the scale factor is ß^{-1} in place of ß.

The procedure used to deduce the inverse Lorentz transform for time in S is to assume the Lorentz transform from S into S’ and solve for the opposite transformation. Given t’=ß(t-vx/c^{2})=ßt-vßx/c^{2}=ßt-vx’/c^{2}. Rearranging and writing

ßt=t’+vx’/c^{2}, we obtain the required result: t=ß^{-1}(t’+vx’/c^{2}). This equation is in the same form as the traditional result except for the fact that the scale factor is ß^{-1} in place of ß. As before we also have y’=y, and z’=z.

To prove that the new inverse transforms are consistent, the procedure applied in section 4.0 is applied to the new equations. The results are as follows.

8.1 Results for x=0 (Specification of an evaluation in space )

To consider the role of evaluation in space, we determine the simultaneous solution of the four equations when we specify the condition that x=0. The results are as follows:

Equation 17: x’=ß(x-vt)=-ßvt

Equation 18: t’=ß(t-vx/c^{2})=ßt

Equation 19: x=ß^{-1}(x’+vt’)^{=}0 , Therefore x’=-vt’

Equation 20: t=ß^{-1}(t’+vx’/c^{2})=ß^{-1}t’+vß^{-1}ßx/c^{2}=ß^{-1}t’, since x=0.

Notice that equation 20 is solved by substitution with the result from equation 17. Therefore, from equations 18 and 20 we have the following two solutions for the condition x=0: t’=ßt and t=ß^{-1}t’. The solutions for equations 17 and 19 give the results x’=-ßvt=-vt’, from which we conclude that t’=ßt. A result which is the same as obtained from equation 18 which is the primary result for the condition x=0. All these results and conclusions are the same as obtained in section 4.1.

Although traditionally the solutions to equations 18 and 20 have been interpreted as the solutions, it is a conclusion of this analysis that it is the solutions 17 and 19 which give the definitive proof that the primary solution ought to be equation 18. In fact, this solution alone is sufficient and solutions 18 and 20 can be viewed as redundant. Interpreted as the basis change law, this solution shows that the scale of measure in S’ is dilated by the factor ß relative to the scale of measure in S.

8.2 Results for x’=0 (Specification of an evaluation in space )

To consider the role of evaluation in space, we determine the simultaneous solution of the four equations when we specify the condition that x’=0. The results are obtained as follows:

Equation 21: x’=ß(x-vt)=0, Hence x=vt

Equation 22: t’=ß(t-vx/c^{2})=ßt-ßvx/c^{2} =ßt-vx’/c^{2}=ßt, since x’=0.

Equation 23: x=ß^{-1}(x’+vt’)=ß^{-1}vt’

Equation 24: t=ß^{-1}(t’+vx’/c^{2})=ß^{-1}t’.

Notice that equation 22 is solved by substitution with the result from equation 25. From equations 22 and 24 we have the following two solutions for the condition x’=0: t’=ß^{-1}t and t=ßt’. The solutions for equations 21 and 23 give the results x=ß^{-1}vt’=-vt, from which we conclude that t=ß^{-1}t’. A result which is the same as obtained from equation 24 which is the primary result for the condition x’=0. Here the results and conclusions are not the same as obtained in section 4.2. Here the primary solution, t=ß^{-1}t’, is inverse to the primary solution obtained in section 8.1, t’=ßt.

8.3 Results for t=0 (Specification of an evaluation in time )

To complete the analysis of evaluation, we now consider the role of evaluation in time. We determine the simultaneous solution of the four equations when we specify the condition that t=0. The results are as follows:

Equation 25: x’=ß(x-vt)=ßx

Equation 26: t’=ß(t-vx/c^{2})= -ßvx/c^{2}

Equation 27: x=ß^{-1}(x’+vt’)^{=}ß^{-1}x’+ß^{-1}vt’=ß^{-1}x’+vt=ß^{-1}x’, since t=0.

Equation 28: t=ß^{-1}(t’+vx’/c^{2})=0, therefore t’=-vx’/c^{2}.

Notice that equation 27 is solved by substitution with the result from equation 24. From equations 25 and 27, we have the following two solutions for the condition that t=0: x’=ßx and x=ß^{-1}x’. The solutions for equations 26 and 28 give the results t’=-ßvx/c^{2}=-vx’/c^{2}, from which we conclude that x’=ßx. A result which is the same as obtained from equation 25 which is the primary result for the condition t=0. All these results and conclusions are the same as obtained in section 4.3.

8.4 Results for t’=0 (Specification of an evaluation in time)

To consider the role of evaluation with the opposite condition, we determine the simultaneous solution of the four equations when we specify the condition that t’=0. The results are as follows:

Equation 29: x’=ß(x-vt)=ßx-ßvt=ßx-vt’=ßx, since t’=0

Equation 30: t’=ß(t-vx/c^{2})=0, therefore t= vx/c^{2}.

Equation 31: x=ß^{-1}(x’+vt’)^{=}ß^{-1}x’

Equation 32: t=ß^{-1}(t’+vx’/c^{2})= ß^{-1}vx’/c^{2}.

Notice that equation 29 is solved by substitution with the result from equation 18. From equations 29 and 31 we have the following two solutions for the condition that t’=0: x’=ß^{-1}x and x=ßx’. The solutions for equations 30 and 32 give the results t=ß^{-1}vx’/c^{2}=vx/c^{2}, from which we conclude that x=ß^{-1}x’. A result which is the same as obtained from equation 31 which is the primary result for the condition t=0. Here the results and conclusions are not the same as obtained in section 4.4. Here the primary solution, x=ß^{-1}x’, is inverse to the primary solution obtained in section 8.1, x’=ßx.

8.5 Comments On The Above Results

The solutions presented as the primary solutions for sections 8.1, 8.2, 8.3, and 8.4 are as follows:

For evaluation x=0, the solution is x’=ßx

For evaluation x’=0, the solution is x=ß^{-1}x’

For evaluation t=0, the silution is t’=ßt

For evaluation t’=0, the solution is t=ß^{-1}t’

When these solutions are compared with the primary solutions presented in section 4.5, we see that the solutions given here are inversely related, rather than symmetrically related as in section 4.5. These equations represent a consistent solution set compared with the inconsistent solution set of section 4.5. From this it is concluded that the correct inverse Lorentz transforms have been deduced in section 8.0.

As a final note, the reader should consider the ironic situation which results if the traditional Lorentz transforms are defended. The method of derivation given in section 8.0 proceeds directly from the Lorentz transform and obtains the inverse Lorentz transforms by simple algebraic manipulation. A procedure that should have been discovered and used by Einstein to deduce the correct equations. The irony is that defending the traditional equations means that we will have two mutually inconsistent sets of inverse Lorentz transforms unless the simple derivation of section 8.0 can be shown to be incorrect. But this is not possible, because it is correct.

9.0 Summary and Conclusions

9.1 Summary of Mathematical Results

The results of the evaluation of the Lorentz and inverse Lorentz transforms in section 4.0 indicated that the primary solutions are all dilations, as shown at the end of section 4.5. These primary solutions for transformation of distance and time between the systems S and S’ are inconsistent. They have no solutions other than the trivial ones and the solution ß=1, which means there is no relative motion. These traditional solutions are not reciprocal in the sense of bijective inverses as was shown in section 6.0. This means that the Lorentz and inverse Lorentz transformations are contradictory and inconsistent. Therefore, it is not surprising that Einstein relativity is plagued by contradictions, paradoxical results, inconsistency, confusion, and acrimonius argument.

In section 7.0 it was shown that the inconsistency can be removed by the measurement scale change interpretation of Lorentz transforms. This interpretation was used to show that velocities measured in frames S and S’ transform invarantly in terms of different units of measure defined in S and S’. This result was used to prove that the Einstein velocity addition law is incorrect. Next it was shown that Lorentz transforms are Galilean transforms with a transformation of measurement scales. When the different scales are used as determined by the evaluation conditions, the transformations become Galilean. This means that the laws of physics are the same in frames S and S’ when we transform the measurement scales as required. Finally, a new set of inverse Lorentz transforms were derived which are bijective and are consistent with the requirements of a Galilean transformation. The new equations are consistent, as shown in section 8.0, with reinterpretations of the relativity and light velocity postulates which are physically different in interpretation from the interpretations given in physics textbooks.

9.2 Discussion and Conclusions

In his 1905 relativity paper Einstein did not derive the inverse Lorentz transformations. However, the equivalence of the equations x^{2}+y^{2}+z^{2 }= c^{2}t^{2} and x’^{2}+y’^{2}+z’^{2 }= c^{2}t’^{2} was essential to his derivation of the Lorentz transforms. These equations assert that the velocity of light is an invariant constant when measured in two different frames of reference, which use the same standards of measurement and identical systems of coordinates. This proof of sufficiency showed that a sufficient condition for the deduction of the Lorentz transformation was the constancy of light velocity assumption. However, there was no proof of necessity. Here we have shown that the Lorentz transform implies that the coordinate measures in S and S’ are different when the Lorentz transformations are imposed. Hence, although the coordinate measure of c is the same in both frames, the measurement scales can not be the same. The problem is to assess what this implies in relation to the theory of relativity.

It is clear that these results refute the version of Einstein relativity that appears in the traditional textbooks. But can we infer from this that relativity is refuted? This depends on which theory of relativity we mean. Here the problem becomes complicated. In the 1905 paper, Einstein refers to the principle of relativity as suggesting that there is no concept of absolute rest. Then he redefines this as meaning that "the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good." So it appears that relativity can be maintained even though the transformations turn out to be Galilean.

In his 1907 relativity paper, Einstein made a very clear statement regarding the reciprocal relation between frames S and S’. "In general, according to the principal of relativity each correct relation between "primed" (defined with respect to S’) and "unprimed" (defined with respect to S) quantities or between quantities of only one of these kinds yields again a correct relation if the unprimed symbols are replaced by the corresponding primed symbols, or vice versa, and if v is replaced by -v." This assertion is the source of the contradictions and inconsistencies. This statement is trivially true, when interpreted to mean a redefinition by renaming of coordinates, but it is false as shown above when applied to the derivation of the relation between reference frames. As shown in section 8, the inverse Lorentz transforms we require are not the ones obtained using the procedure described by Einstein in 1907. The problem is that the bijective requirement for the Lorentz transformations was never demonstrated mathematically. It was simply assumed to be valid without actually proving it. This turned out to be a mistake, because the transformations are not bijective inverses.

From this it was concluded that there are two different relativity postulates, or perhaps different interpretations applied to the same postulate. It appears that the postulate that denies the existence of an absolute rest frame is the cause of the problem, and dropping this assertion removes the inconsistencies and paradoxes. This procedure was performed in this paper, with the result that different inverse Lorentz transformations, which satisfy the requirements of the postulate that the laws of physics transform with the same form, were derived. Hence, the traditional inverse transforms have been replaced with new ones that have the advantage that they do not cause the solutions to be contradictory or inconsistent.

The light velocity postulate was also revised. When combined, the two postulates as usually interpreted produce the incorrect inverse Lorentz transform equations. These are derived from the equations x^{2}+y^{2}+z^{2 }= c^{2}t^{2} and x’^{2}+y’^{2}+z’^{2 }= c^{2}t’^{2}, and make the reference frames S and S’ equivalent. This equivalence is the basis for asserting that no reference frame has the right to be considered as the absolute rest frame. However, these equations are really the same equation with the symbol names changed. But it was demonstrated in section 7.2 that a different interpretation of the light velocity postulate leads to the scale change interpretation which produces consistent Lorentz transformations. The error in traditional relativity was corrected by deriving new inverse Lorentz transformations. These equations required the concept of an absolute rest frame.

It can be seen that this new viewpoint is consistent with experiment while the old theory of relativity is not. When we measure the real, not the apparent, lifetime of a mu meson particle, which is what the experiments actually measure, then an asymmetry in the lifetimes is established between the mesons relative to the earth as an absolute rest frame. The mu mesons live longer at higher velocities relative to the earth based reference frame. A result consistent with the Lorentz and 1905 Einstein versions of relativity. But, we can not reverse reference frames and expect that in the fast moving reference frame, the slow mu mesons live longer. That is absurd as pointed out by Dingle and others many times. That prediction is a mathematically inconsistent or contradictory result. So the theory of Einstein relativity as it is traditionally taught is incorrect. This theory removes this difficulty and many others that have perpetuated confusion in relativity for 100 years.

The fault lies in the way that the inverse Lorentz transforms are derived so that they apparently support the postulate that there is no absolute rest frame. But when derived this way the system of Lorentz transforms becomes inconsistent. This can be avoided by asserting one frame as reference, and then deriving inverse Lorentz transforms consistent with this hypothesis. The resulting system of equations transforms to make velocity in the direction of motion appear invariant. The relativity and light velocity postulates in a revised form are reinstated but the interpretation of the revised theory is quite different from the traditional approach.

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