mirrored file at http://SaturnianCosmology.Org/ For complete access to all the files of this collection see http://SaturnianCosmology.org/search.php ========================================================== *Herbert Dingle Was Correct!* *By Harry H. Ricker III* *1.0 Introduction** * Herbert Dingle is well known for his claims of inconsistency in Einstein’s Special Theory of Relativity. One of the most interesting 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.” Here the thesis is presented that Dingle’s modest claim is based upon a mathematically correct derivation of the transformation of time from a moving frame into a rest frame following Einstein’s methods. It is concluded that Dingle’s assertion of an inconsistency in Einstein’s 1905 paper on relativity is correct. Furthermore, in addition to the inconsistency, there are errors in Einstein’s 1905 proof that he attempted to rectify in his 1907 and 1910 papers on relativity. This claim may seem outrageously absurd to a physics community which is convinced of the absolute correctness of Einstein’s special theory. The purpose here is to take up Dingle’s challenge, and to examine with mathematical rigor the mathematical derivations which lead to the famous conclusion in the special theory that “moving clocks run slow”. In taking up Dingle’s challenge, the focus will be on Einstein’s three fundamental papers which establish the theory^2,3,4 . An examination of the proofs of time dilation which appear in these papers indicates that three different methods are presented. Einstein presented a different proof in every one of the three papers. The final method of proof given in 1910 is completely different from the proof of 1905. The reason for these different proofs is interpreted here as an attempt to correct deficiencies in the 1905 paper in light of Stark’s discovery of spectral line shifts of canal rays. The 1907 method was further revised in 1910 which is taken as Einstein’s final version of the proof. The argument will be advanced that mistakes and inconsistencies in the 1905 paper contributed to the inconsistency arguments and confusion which has accompanied this subject for nearly 100 years. In the final analysis, it is determined that Dingle’s argument fails to prove that moving clocks run fast. Instead, his proof correctly demonstrates that clocks at rest in the moving frame run slow compared to clocks at rest in the rest frame. But, Dingle’s assertion that moving clocks run fast according to Einstein’s proofs is correct. Because of mathematical errors, Einstein incorrectly derives the transformation equations from a moving frame into a rest frame. He misinterprets them as indicating that moving clocks run slow, instead of the correct result, which is that they run fast. The correct conclusion which is given here is that moving clocks run slow, however, proofs based upon Einstein’s 1905 method are incorrect because the equations used actually indicate that moving clocks run fast. *2.0 Objective* The main objective of this paper is to take up Dingle’s challenge to examine Einstein’s proofs of time dilation and resolve the inconsistency that Dingle has discovered. This analysis has revealed mistakes in Einstein’s method of analysis that will be discussed and explained. This paper will demonstrate that Dingle’s criticism of Einstein’s 1905 derivation of time dilation is valid and that his 1905 paper contains mathematical errors. In the 1907 and 1910 papers, Einstein attempted to rectify the errors of the 1905 paper by presenting revised proofs. Einstein was not satisfied with the 1907 result and presented a further revision in 1910. The main purpose of this paper is to demonstrate that Dingle’s 1962 demonstration of an inconsistency in special relativity is an attempt to point out the mistakes in Einstein’s 1905 paper. While Dingle does not assert this, it is shown that his solution is the mathematically correct derivation which Einstein should have obtained. *3.0 Approach* The thesis advanced in this paper is that Dingle’s demonstration of an inconsistency in special relativity was a low key, or quiet urging to the physics community, to examine Einstein’s 1905 derivation of time dilation. Forewarned of the hostility to his views, Dingle did not assert that Einstein’s 1905 presentation was wrong, he merely pointed to an inconsistently, which he hoped would prod an examination of Einstein’s method and conclusions. The means by which this was to be achieved was to compare Einstein’s incorrect statements with his own correct statements. The essence of Dingle’s argument is that based upon the mathematical method of Einstein’s 1905 paper, the correct conclusion is that moving clocks run fast. However, Einstein reached the opposite conclusion that moving clocks run slow. Dingle shows that if we follow Einstein’s 1905 method using the inverse Lorentz transform instead of the Lorentz transform which Einstein uses, then we must conclude that moving clocks run fast. This argument will be analyzed further here. One result of the analysis is the discovery that Einstein’s 1905 paper contains a major blunder. He incorrectly concludes by his method that moving clocks run slow. This error is corrected in the 1907 and 1910 papers by a revision of the method of proof and by deleting all references to the conclusion that moving clocks run slow as a result of their motion. It is argued that Einstein recognized his errors and attempted to rectify them in his 1907 and 1910 papers. A further revision in the 1910 paper attempted to correct the deficiencies of the backward method used previously by substituting the inverse Lorentz transform in place of the Lorentz transform used in 1905 and 1907. However, he continues to conclude that moving clocks run slow. The reason that Einstein continues to insist that moving clocks run slow is a mathematical error that will be discussed. The analysis method used here presents a comparative analysis of the two derivations and identifies two main mistakes in Einstein’s 1905 paper; the incorrect use of the Lorentz transform and the incorrect interpretation of the result. Dingle’s presentation corrects these errors while maintaining the other aspects of Einstein’s method. His result is therefore interpreted as the correct result that Einstein was unable to obtain. Dingle correctly uses the inverse Lorentz transform and correctly recognizes that the resulting contraction of time interval indicates that moving clocks run fast. The conclusion is that, based on Einstein’s principles enunciated in 1905, the correct conclusion should have been that moving clocks run fast, and not slow as Einstein mistakenly concluded. The evidence to support this argument is divided into two main lines of approach. One is historical. It proceeds by an analysis of Einstein’s three papers on special relativity and demonstrates that Einstein recognized that the 1905 derivation was incorrect. But he never recognized that his 1905 conclusion regarding moving clocks running slow might be in error. The second approach investigates the rigor of the mathematical method and points out mathematical errors in the 1905 paper. *4.0 Background* Here the brief note presented by Dingle is described^1 . 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 demonstration consists of four parts. The first part defines the reference frames and the coordinates in accordance with Einstein’s original terminology. Dingle makes an addition by defining the inverse Lorentz transform which Einstein does not use. The second part presents a statement of the desired solution in Einstein’s exact words by quotation. The third part states Einstein’s solution as a direct quotation. In the fourth part, Dingle presents his alternate solution, exactly as Einstein presents it with the appropriate changes in mathematical symbols and words to make the necessary corrections. After demonstrating that the two solutions are inconsistent, Dingle poses the following problem: “...it must be explained not why the two cases are different-that is obvious-but why, consistently with the theory, the former result must be accepted as true while the latter must be rejected as false.” He then sent copies of the reprints to relativity experts with the enclosed handwritten remark: “With kindest regards. Test case for the integrity of scientists.” This goaded Max Born to write a response which dismissed Dingle’s argument with a snide and curt refutation. While Born’s argument has been accepted as a correct refutation of Dingle, it has not been generally regarded as completely satisfactory. Dingle’s method and his approach is curious. At first glance, the problem he poses appears to be merely a semantic gambit; not worthy of notice. This is apparently why Born responded by saying: “Dingles objections are just a matter of superficial formulation and confusion.” Born attributes the confusion to Dingle and not to the theory which he is being urged to critically examine. Dingle does not assert that Einstein’s statements are incorrect. Instead, Dingle merely asserts an inconsistency. But the inconsistency can only be resolved in one of two ways. Admit that Einstein’s derivation is wrong, or assert that Dingle’s derivation is wrong. Born, and all other commentators, choose the latter course of action. Dingle’s approach was to goad Born and others to critically examine Einstein’s solution. He expected them to discover that Einstein’s answer was wrong. This is clear from his admonition to explain, not why the answers were different, but to justify why Einstein’s answer is considered the correct one. The response was not as required. Born evaded the injunction to explain, and others have asserted simply that Dingle’s paradox is a fallacy. *5.0 Comparison of Dingle And Einstein* The method used by Dingle, which permits a word-for-word step-by-step comparison of Einstein’s solution with the solution by Dingle, is analyzed here. This comparison reveals the following significant differences. First, Dingle uses the inverse Lorentz transformation equations of coordinates from the moving system into the stationary system, while Einstein uses the Lorentz transformation equations. Dingle exchanges the moving and stationary system coordinates where necessary in the solution. Dingle follows every step of the solution exactly as Einstein, so that the principles used in the solution are the same. Dingle reduces the solution in the same manner as Einstein. At the conclusion, Dingle states that the solution indicates that time marked by the clock in the moving system runs fast when viewed from the stationary system, while Einstein says that it runs slow. This summarizes the only differences in the two proofs. So we see that the main differences are the use of the inverse Lorentz transform and the conclusion that the result indicates that moving clocks run fast. Dingle defines the problem exactly as Einstein; a transformation from the moving frame into the rest frame. Dingle uses the inverse Lorentz transform; Einstein uses the Lorentz transform. Both Dingle and Einstein use the same method of evaluation, but-this is significant-with different conditions. Einstein’s condition specifies the coordinate of the clock at rest in the moving frame; x=0 hence x=vt, and Dingle uses the condition that measurement is in the rest frame at x=0, hence x=-vt. Dingle uses the condition appropriate to the inverse Lorentz transformation. Dingle and Einstein obtain the same basic equation with the variables t and t exactly reversed as required. Dingle interprets the result as showing that the moving clock runs fast, while Einstein says that it runs slow. We will see that Dingle’s solution can be viewed as the correct one. He has exactly corrected all of the mistakes in Einstein 1905 paper, but as noted before, he does not make this claim. Parsing Dingle’s method, we see why his method of solution is correct and Einstein’s is in error. The problem is defined as a transformation from the moving frame as domain into the stationary frame as co-domain. The first step defines the domain as the moving frame with time coordinate t and co-domain as coordinate t. Dingle writes the inverse Lorentz transform with the domain coordinates on the right and co-domain on the left. Einstein writes a Lorentz transform with domain coordinates on the left and co-domain on the right. Dingle performs evaluation in the co-domain using a substitution of co-domain coordinates on the right. Einstein performs evaluation in the domain using a substitution of domain coordinates. Dingle’s result is a transformation with the co-domain on the left side expressed as a function of the domain coordinates on the right: t=tp(1-v^2 /c^2 ). Einstein’s result expresses the domain coordinates as a function of the co-domain coordinates: t=tp(1-v^2 /c^2 ) . Clearly Einstein’s result is in the improper format, because the domain and co-domain are reversed. Dingle’s equation expresses time-as indicated by seconds “marked” on the clock in the moving system- as it is measured in the stationary frame in terms of a function relating time in the moving frame to measured time in the stationary frame. Einstein’s equation expresses time in the moving frame in terms of time in the rest frame. This is the mistake that has caused nearly 100 years of confusion and controversy. Einstein concludes that the moving clock runs slow. But this seems wrong because the equation which Einstein presents does not support this conclusion. Correctly understood, it states that the rest frame clock runs “slow” when viewed from the moving frame. A result which partly explains the confusion. Einstein’s method of analysis is the following. It is characterized by the peculiar method in which he performs the analysis backwards, going from the moving system into the stationary system. Time is defined in the moving system as being the same as marked by a clock in the stationary system. This is a strange approach, and it is the source of considerable confusion and argument. The definition appears to establish that the clocks at rest in both frames run at the same rate. But there is sufficient ambiguity to reach the opposite conclusion. Another difficulty is that the analysis proceeds backwards, there does not appear to be any reason why this is required, it seems to only deepen the confusion. Einstein solves the Lorentz transform -which goes backwards relative to the desired transformation- and obtains an equation for the transformation of time going from the rest frame into the moving frame. He then solves for time in the moving frame when the rest frame clock dial indicates one unit of time. The result is that when the rest clock reads one time unit, the moving clock reads p(1-v^2 /c^2 ) units of time. Since this has a value less than one for any velocity v greater than zero, Einstein concludes that the moving frame clock is slow. Dingle uses the same procedure on his resulting equation. Setting the rest frame time equal to one unit of time, he solves for the time on the moving frame clock as 1/ p(1-v^2 /c^2 ); the inverse of Einstein’s result. Hence, the moving clock runs fast in Dingle’s solution. Dingle’s challenge is to explain why the solution obtained by Einstein should be considered the correct one. When we consider the strange method of solution-going backwards-, the ambiguous definition of time marked by the moving frame clock, and the awkward method of clock comparison, Dingle’s demand that the validity of Einstein’s solution be explained is amply justified. *6.0 Analysis of Mathematical Method Of Solution* Before we can determine the mistakes or errors in either Einstein’s or Dingle’s methods of solution, we must have some method or approach to determine what constitutes a correct solution. This seems to consist of at least two basic considerations. A method of clock comparison and a method to implement this comparison of clocks mathematically. The first involves a theory of how we compare the clocks, and the second addresses the mathematical procedure to accomplish the comparison. The theory of clock comparison is addressed first. *6.1 Theory Of Clock Comparison In Special Relativity* A clock is generally defined today as a system which produces an output without an input. What this means is that a clock measures time by counting the vibrations or beats of a self sustaining oscillatory system. The type of self sustained oscillation used in 1900 was a mechanically vibrating system which performed repetitive harmonic motions. The specific requirement of the vibrations, which was required to make a good clock, was the exact repetitive nature of the harmonic motion so that the cyclic motion always repeated itself within the same interval of time. This time interval is called the period of oscillation, and the reciprocal of the period of oscillation is called the frequency of oscillation. Time measurement circa 1900 was performed by counting the oscillations or beats of the clock using a dial mechanism which converts the vibration counts into dial readings, calibrated in terms of measures of time intervals; usually days, hours, minutes and seconds. The beats measure tiny intervals of time, these are counted and related to the dial movement or ticks, so that a certain number of beats corresponds to a standard interval of time. Since the counting of beats is fixed in relation to the dial reading, the main error that effects the time measure is the change in period of the harmonic vibrations. The method used to display time related the clock beats to time intervals inversely via the dial readings. Clocks are calibrated in terms of time by comparison to a standard of time. This standard is a clock maintained by the government at an astronomical observatory. A clock is said to run slow when it reads time behind the reference standard of time. It runs fast when it reads time ahead of the reference. This process requires that the readings of two clocks be obtained and compared when the standard clock has run for a defined interval of time by its own measure. The terms fast and slow therefore refer to the rate of beats of the clock compared with the reference. Rate is a measure of the frequency of vibration. When the clock runs fast, its vibration frequency is higher than the reference clock frequency, and when it runs slow its frequency is lower than the reference clock frequency. The reverse is true for the time interval or period of the vibrations. The clock runs fast when the time period is smaller or less than the reference time period, and it runs slow when the time period is longer or greater than the reference time period. The reader should notice the following important result. With the definitions used here, the dial readings of clocks are inversely related to the changes in time interval. (The dial of a slow clock reads less than the reference because its period of oscillation is longer than the reference.) A slow clock reads behind the reference because its period is longer. The clock is slow because there are fewer beats or ticks of the clock during the reference time period. The frequency is lower, and the time required to complete one period of oscillation is longer than the reference. A clock reads fast because more ticks are recorded during the reference time period. The frequency is higher, and the time to complete one period of oscillation is less than the reference period. Hence while the dial readings of clocks are directly related to the clock rate, they are inversely related to the transformation of the time intervals. When discussing the transformation of clock rate between reference frames, it is not necessary to consider dial readings. We can compare the clocks based upon the transformation of time intervals directly. Because the period is inversely related to clock rate, the direct transformation of time intervals gives the necessary result. A clock runs fast when the transformation contracts the time interval; numerically a smaller measure, and it runs slow when the time interval is dilated; a numerically larger measure. In the following sections, we will see that Einstein makes a serious blunder when he confuses the time as defined by clock dial readings with the transformation of real time intervals. The procedure for clock comparison is as follows. A reference clock is defined, and it runs for a unit of time by its own measure. The comparison clock is defined and it is started or adjusted so that its start time coincides with the start of the reference clock. The dial readings which represent time intervals are compared at the end of a unit of time run on the reference. But, we can not use this procedure unless we assume that the time that is measured is the same for both clocks. When we are trying to determine if there is an actual change in clock rate due to possible differences in time scale, the comparison of time intervals is preferred. Since the supposed relativistic transformations change the time scale, the preferred method is to directly compare the frequencies of oscillation as the measure of clock rate. *6.2 The Theory of Mathematical Transformation* The deceptively simple procedure for clock comparison is complicated by the problem of transformation of time in the theory of relativity. The question arises: What is to be compared when we do the measurement? Mathematically we need to define the symbols that represent the intervals used in the comparison of clock rate measurements. There are two symbols used in the transformations; one for the domain of the transformation, and one for the co-domain. But they don’t always represent the same concept in terms of clock time. In some cases, the symbols represent the time determined by the reference clock in the rest frame, in others the time generated by the moving clock and measured in the rest frame, and in others the time obtained from the clock situated at rest in the moving frame. The different cases depend on how the Lorentz transform is interpreted. Three specific cases can be identified: projection as a contraction, projection as a dilation, and a change of basis or time scale change. The interpretation of the symbols used is different in each of these cases. The solution to the problem of transformation of time intervals is based on the use of the Lorentz transform. This is a relation that expresses coordinates defined in what Einstein calls the stationary frame into coordinates of the moving frame. Mathematically it is a relation that transforms the rest frame space-time events taken as the domain into the moving frame space-time events taken as co-domain. In order to express the transformation in terms of time coordinates only, a procedure termed “evaluation” is used to eliminate the spatial coordinates from the solution. Einstein does this by substitution of the equation x=vt, into the Lorentz transform. Einstein’s evaluation is mathematically equivalent to evaluation by solving the Lorentz transform for space-time events located at the coordinate x’=0 in the co-domain or the moving frame. The resulting equation is then reduced to its simplest form. Einstein obtained the result: t=tp(1-v^2 /c^2 ) . We see that this equation expresses the moving frame coordinates in terms of the rest frame coordinates. But, the solution requested the reverse of this; rest frame time in terms of moving frame time. An unrecognized complication is that there is an alternative method of evaluation of the Lorentz transform. This procedure is to evaluate the Lorentz transform for the space-time events located at x=0 in the domain. The first procedure results in a contraction of time in the co-domain and the second in a dilation in the co-domain. Unfortunately these solutions are not what is required if we are asked to determine what is the rate of the clock in the moving frame when it is viewed from the rest frame. This problem is solved by inverting the results of the previous evaluations. Then the contraction becomes a dilation and the dilation a contraction. The third alternative is that the Lorentz transform is a scale change or basis transformation between reference frames. The result is either a contraction or dilation as determined by the evaluation, but the procedure for comparison of the time intervals is different in this case. The result of the first two cases is a comparison of the apparent transformation of time , while in the last case, the results represent a real change in the rate of the moving frame clock. With this brief introduction to the issues that must be confronted, we proceed to the examination of Einstein’s method of solution. *6.3 Einstein’ s Method Of 1905* This section presents an analysis of Einstein’s derivation of time dilation in his first paper on relativity published in 1905. Einstein’s 1905 derivation of the transformation of time is analyzed and the errors in his derivation are discussed. Einstein’s derivation of time dilation is presented in section 4 of the 1905 paper. After discussing the transformation of space which leads to a contraction, Einstein takes up the transformation of time. The first step is to define the domain and co-domain coordinates for the transformation. Einstein does this as follows. A clock which marks the time t at rest in the moving frame is located at the origin of this system. Einstein asks for the transformed rate of this moving clock when viewed in the stationary or rest frame. The problem is defined as a transformation from the moving frame as domain into the rest frame as co-domain. Therefore, we expect the result to be in the form of a functional relation which expresses the measured or observed rest frame time as a function of the moving frame time. The second step is to select a Lorentz transformation which expresses the equivalence of moving frame coordinates as domain with rest frame coordinates as co-domain. Einstein uses the Lorentz transformation going from the rest frame as domain into the moving frame as co-domain. This is an error that leads him astray. It is ironic that the solution is in the correct form for transforming time from the rest frame into the moving frame, but that is not the solution which was specified. So it must be judged as incorrect. The proof of this error is that the result which Einstein gives, is an equation which expresses the moving frame time as a function of the rest frame time. Hence it is expressed incorrectly, because the required solution is an equation expressing the rest frame time as a function of moving frame time. Einstein proceeds to use this incorrect result to support the conclusion that the moving clock runs slow. This approach is wrong because it uses an incorrect derivation of the Lorentz transform. A second problem is the question of the correct method of evaluation. There are two possibilities; evaluation for x=0 and evaluation for x’=0. The answer requires a knowledge of which evaluation condition is correct. The correct evaluation occurs when we transform from the moving frame into the rest frame, with rest frame measurement performed at the same place --this is where the reference clock is located in the rest frame-- but at different times. This condition is only achieved by evaluation at x=0 as correctly specified in Dingle’s method of solution. Einstein uses an incorrect method which performs evaluation in the moving frame of reference, and not the rest frame as required. Einstein’s use of an incorrect evaluation method is apparently motivated by his desire to show that the rate of a moving clock runs slow. This being the reason for transforming from the moving frame into the rest frame and his use of the evaluation method which employs substitution of the condition x=vt into the Lorentz transformation formula. Unfortunately, this method is not properly matched to the use of the Lorentz transformation. But, at this stage, Einstein has not developed the inverse transformation formula that would have permitted the use of the evaluation procedure which he uses. The correct procedure would have been to use the Lorentz transform evaluated at x=0. Instead he uses a confusing and incorrect procedure that leads him into error. The first mistake is now clear, for the mathematical solution that is presented, the problem is defined incorrectly. The correct definition would have been to ask, what is the rate of the rest frame clock when viewed from the moving frame. Presented this way, the mathematical solution given is correct. The second mistake cited notes that the Lorentz transform is used incorrectly relative to the problem definition. The inverse Lorentz transform should have been used instead of the Lorentz transform. With this change, and the appropriate change in the evaluation condition, a correct derivation results as demonstrated by Dingle. An analysis of the method used by Einstein reveals the following points: • The problem is defined by asking for the rate of the moving system clock when viewed from the stationary system • Einstein uses the Lorentz transform formulation of the problem which provides the transformation of time defined in the stationary frame into the moving frame • The solution is written with the moving frame time written as a function of the stationary frame time • Einstein interprets the solution, which shows a time interval in the moving frame is contracted relative to the stationary frame, as demonstrating that the moving frame clock runs slow • Einstein uses this result to assert that when a clock at rest in the stationary system is moved with a velocity v from A to B in the stationary system, it will lag behind the clock at B when it arrives because of this motion • Einstein then asserts that a clock at the equator “must go more slowly ...than a similar clock situated at one of the poles” The first two points addresses the problem mentioned before that the resulting equation used to compare the clock rates is backwards. Hence, it can not be used to represent a projection of the time marked on the moving frame clock into a measurement of time in the stationary frame. So it must represent a direct comparison of the rate of a moving clock with a clock at rest. Points four, five and six can now be seen as following directly from the demonstration that a moving clock runs slow. The comparison assumes a time scale or basis change from the moving frame into the rest frame with time in the rest frame defined as reference time. The result that the moving frame clock runs slow is then used to support the conclusions given in the last two points. *6.4 Einstein’s Big Blunder* Einstein’s blunder is as follows. He says that when we have a Lorentz transformation between two time intervals, representing time marked by clocks, when we set the time indicated on the reference clock to 1 unit and calculate time on the opposite clock, if the result is less than 1 unit, the subject clock is slow. This is incorrect. The equation for the transformation of time intervals is in the form of a function as follows: t_s =L_t (t_r ). Here L_t is the transform function, t_s is the subject time interval, and t_r is the reference time interval. The subject time interval is a representation of the clock time subject to the transformation. We calculate the comparison of the clock rates as follows. When f_s >f_r , the subject clock is fast. Solving for the times we obtain: f_s >f_r , hence, 1/t_s >1/t_r , so t_r > t_s , and t_s f_r . The subject clock is slow when we have the result t_s = bt_r. Because in this case, b is greater than unity , and we have the result that f_s >f_r , because t_r < t_s. These results are exactly opposed, i.e. inversely related, to the method Einstein uses. Einstein’s blunder is confusing dial readings with time intervals. He assumes a direct 1 to 1 relation such that clock dial or display readings represent time. This is not true in the theory of Lorentz transforms, which act directly upon the time itself, and not dial readings as assumed in the operational method adopted by Einstein. Dial readings are counts of clock beats. Beats are oscillations with a fundamental cyclic frequency of occurrence. They are usually assumed uniform of fixed period. Clocks keep time by counting these periods of oscillation. Oscillations with shorter periods, ie, contracted time intervals, mean the clock runs fast because more beats are counted during a given reference period as measured by the reference clock. Einstein’s mistake, interpreting a time interval contraction as indicating that a clock runs slow, must be recognized as a blunder of major significance for the following reason. It led to an incorrect interpretation of the theory and the wrong equations. To explain this lets take up an analysis of the 1905 paper from a different viewpoint. In 1905, Einstein performed an analysis of the Fitzgerald-Lorentz contraction going backwards. The method can be summarized in modernized form as follows. Starting with the Lorentz transform for distance as x’=b(x-vt), where x’ is the distance defined in the moving frame and x the distance in the rest frame, and b=(1-v^2 /c^2 )^- 1 .We have for t=0 the resulting solution, x’=bx, which is solved for x=b^-1 x’. The result being the required Lorentz contraction going from the moving into the rest frame. Suppose we utilize an analogous method for the transformation of time. The Lorentz transform is t’=b(t-vx/c^2 ). Now solving with the condition that x=0, the solution t’=bt is obtained. Apparently Einstein rejected this solution because it leads to the conclusion that the clock in the moving frame runs fast -remember he thinks that a dilation of the time means that the clock runs fast because he interprets a contraction of the interval as a slow clock. But, this is the correct solution for the dilation of time experienced by a moving clock. It is the solution that he was searching for. However, because he misunderstood the correct interpretation of time intervals, he obtained the incorrect result of 1905. Hence Einstein missed the correct solution and the consequence was that he introduced a different interpretation which explained the slowing of moving clocks in his 1907 paper. The failure to clearly retract his errors in the 1905 paper has resulted in confusion, misinterpretation, bitter debate, argument, and dissent for nearly 100 years. *6.5 Einstein’s Blunder Vindicated - The 1907 Method Of Solution* In 1907 the blunder of the 1905 paper is erased with the presentation of a new interpretation and method of proof. The 1907 method of proof vindicates the blunder of 1905 by showing that moving clocks run slow just as concluded in 1905. This revised proof was motivated by new experimental results obtained by J. Stark. This suggested to Einstein that moving atomic clocks run slow because “ ...the oscillation process that corresponds to a spectral line is to be considered an intra-atomic process whose frequency is determined by the ion alone, we may consider such an ion as a clock...the effect of motion on the light frequency...reduces the (apparent) proper frequency of the emitting ions...” The revised theory presented in 1907 demonstrated the mathematical proof of this supposition by showing that the Lorentz transformation of the frequency of a moving clock observed from the stationary frame was less than the observed frequency of the same ions at rest. Hence the clocks of the moving ions appeared to run slow. Another revision was that the assertions given in the last two points of the summary of section 6.3 - the conclusions that moving clocks actually run slow- are dropped. We do not see any reference to them after this. This suggests a change in viewpoint is undertaken in the 1907 revision. This new viewpoint stresses the interpretation that the rate of a moving clock only appears slow when viewed from the rest frame. Hence in the 1907 revision, the interpretation that the slowing of moving clocks is a real change in clock rate gives way to the interpretation that the change is in the apparent rate of the clock observed from the moving frame. In the following, this altogether different interpretation will become apparent. It is clear that by 1907 Einstein believed his derivation of time dilation in the 1905 paper was unsound. The new interpretation required a different approach. For the new interpretation, it was necessary to show that the time marked by the clock in the moving frame was shifted in frequency. This led to a more elegant and convincing method of proof than that used in 1905. Hence, in 1907, the derivation of the transformation of time intervals was dramatically revised and appeared in a completely different form. The new form posed a completely different problem. It addressed the transformation of clock rate, as opposed to the previous approach based only on the transformation of time interval. Einstein’s method for demonstrating the transformation of clock rate begins by defining a clock in the moving frame which runs or beats ?_0 times faster than the other reference clocks used to measure time in the moving and rest frames. Hence this clock beats ?_0 times while every other clock beats only once. The method is to calculate the transformation of this clock rate from the moving frame to the rest frame. Here we see that the problem is defined backwards just as in 1905. Hence he again defines the transformation going from the moving frame to a rest frame, but the 1907 method proceeds differently after obtaining the same result as in 1905; t’=b^-1 t, where new notation is used (primed variables refer to the moving frame and unprimed to the rest frame). Although Einstein demonstrates that the clock rate measured in the rest frame appears slow compared with the rate defined in the moving frame, he does not present a complete mathematical derivation. He skips the steps which calculate the transformation of the time interval. He states this procedure as follows: “Using the first two transformation equations, one obtains”. He then proceeds to write the results: t=bt’. The reader is left with the task of filling in the intermediate steps. The implication here is that the same method that is used in the 1905 paper is to be applied, because he refers to the two transformation formulas, instead of only one which is needed for the transformation. This involves solution of the spatial equation with the condition x’=0, and following that with the substitution of x=vt into the Lorentz transformation equation for time. The proof now proceeds by inverting the result of the 1905 solution, and the solution becomes a dilation of clock rate transformed from the moving frame into the rest frame where it is measured by comparison with a rest frame clock. Hence he obtains: t=bt’. But t now represents the observed time interval, as viewed from the rest frame, marked by the clock in the moving frame, and not the time marked by the rest frame clock. The symbol t’ now represents the reference time which is assumed the same in both the moving frame and the rest frame. This change in the meaning of the symbols is crucial in understanding the difference between the transformation of reference clock rates -used to interpret a real change in clock rates- as opposed to the transformation of apparent clock rates. The proof follows immediately from the transformation formula obtained for the transformation of time by converting the time intervals into frequency. The method used however is awkward and difficult to follow. Defining frequency observed in the rest frame as ? and the moving frame frequency as ?_0 the transformation of time intervals can be converted into a transformation of frequency as follows: t=bt’, hence 1/? =b 1/?_0, and the resulting solution for frequency observed in the rest frame is ?_ =b^- 1 ?_0 . The result shows that the frequency generated in the moving frame is greater than the observed frequency in the rest frame. Therefore, the apparent clock rate is slow as viewed from the rest frame. As Einstein puts it “ The motion... reduces the (apparent) proper frequency of the emitting ions...” *6.6 Einstein’s 1910 Relativity Paper* Einstein was not satisfied with his solution of 1907, because a new but not significantly different solution is presented in his 1910 paper. The main change provided a justification for the equation, t=bt’, which was not presented in the 1907 paper. The method used reveals a completely different approach to the problem. The new feature of the 1910 derivation is the use of the inverse Lorentz transform equation for time. This, however, is written incorrectly as: t=b(t’-vx’/c^2 ). (The error is using a minus instead of a plus sign.) However, this is clearly intended to be the inverse Lorentz transform for time, despite the mistake. Einstein proceeds by saying: “and since clock H’ is at rest at the origin of S’, we must always have x’=0, which yields t=bt’.” Here evaluation is performed by substitution of x’=0 into the inverse Lorentz transform equation. This procedure is equivalent to the evaluation method used in 1905, so the same result is obtained. Using this evaluation method gives the same results as in 1905 and 1907, but it is another blunder. He misses another opportunity to discover the correct equation that proves moving clocks run slow. To see why, we return to the 1905 derivation. There the substitution, x=vt was used. This equation results when we solve the spatial Lorentz transform for the condition x’=0 (x=0 in the 1905 notation). But the inverse Lorentz transform is the correct equation, so the correct condition is actually x=0. Analysis of the 1910 derivation of time dilation brings Einstein’s erroneous misconception regarding evaluation into focus. He uses the specification x=vt in both the 1905 and 1907 papers. This results from the solution of the Lorentz transform for space when the condition x’=0 is imposed in the domain instead the co-domain. This same incorrect specification is also used in 1910, but in a different way. This requirement is wrong, because the solution asks for the transformation of time going from the moving frame into the rest frame. Einstein’s evaluation condition specifies the location of the clock in the moving frame but does not specify where the observer is to be located in the rest frame. This specification is necessary to correctly transform the clock rates. Einstein does this incorrectly. His specification that x’=0 is only valid for the transformation from the rest frame into the moving one. *6.7 All Of Einstein’s Derivations Of Time Dilation Are Incorrect* The purpose of the section is to show that Einstein’s derivations of time dilation in all three of the fundamental papers on relativity are incorrect. The proof follows when the equations used for transformation of time intervals are converted to frequency. To do this we define the clock rate in terms of frequency as follows: the clock at rest in S has the rate f=1/t, and the clock at rest in S’ has the rate f’=1/t’. These equations relate unit time intervals defined in S and S’ into clock rate or frequency to permit comparasion of clock rates. To calculate the rates of the clocks we use equations t’=b^- 1 t and t=b^-1 t’and convert time to frequency using f=1/t and f’=1/t’. The results for the 1905 result given in the first equation are as follows: t’=b^-1 t or 1/f’=b^-1 1/f. Therefore f=b^-1 f’ or ff. This result shows that the clock at rest in S’, the moving system, is fast relative to the clock at rest in S, the stationary system. These results prove that the conclusions given in the foundamental papers on relativity are incorrect. The contradiction of the 1905 result is obvious, but that the 1907 result is also wrong is not as clear and requires an explaination. In the 1907 and 1910 papers the equation t=bt’ was used. Einstein obtained the following result, which was calculated above based upon the 1905 equation: t’=b^-1 t or 1/f’=b^-1 1/f. Therefore f=b^-1 f’ or ff’ . Which says that the clock at rest in S, the stationary system, is fast relative to the clock at rest in S’, the moving system. The second equation, which is Dingle’s result, gives: t=b^-1 t’ or 1/f=b^-1 1/f’. Therefore, f’=b^-1 f, or f’ *Note: * If you entered this page directly during a search, you can visit the* Millennium Relativity *site by clicking on the* Home *link below: Home <../../Default.htm> [AddFreeStats.com Free Web Stats]