Contradictions in Einstein s Special Relativity Theory: Amending the Lorentz Transformation

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1 Open Science Journal of Modern Physics 2015; 2(2): Published online April 20, 2015 ( Contradictions in Einstein s Special Relativity Theory: Amending the Lorentz Transformation Robert J. Buenker Fachbereich C-Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Wuppertal, Germany address buenker@uni-wuppertal.de, bobwtal@yahoo.de To cite this article Robert J. Buenker. Contradictions in Einstein s Special Relativity Theory: Amending the Lorentz Transformation. Open Science Journal of Modern Physics. Vol. 2, No. 2, 2015, pp Abstract Symmetric and asymmetric time dilation are defined. It is noted that the Lorentz transformation (LT) of Einstein's original theory of relativity predicts that time dilation is always symmetric, i.e. that two clocks can each be running slower than the other, whereas experimental studies have invariably found that it is always possible to determine which of two clocks is slower than the other. Moreover, consideration of Newton's law of inertia clearly indicates that two clocks in pure translation must have constant rates that are in fixed proportion to one another, thereby ruling out the occurrence of symmetric time dilation. It is concluded on this basis that the LT is invalid and needs to be replaced by another space-time transformation that is consistent with asymmetric time dilation. It is shown that there is another space-time transformation that also satisfies Einstein's two postulates of relativity, but one which assumes that clock rates in different rest frames are strictly proportional to one another. It is therefore in complete agreement with both Newton's First Law and the results of the above time-dilation experiments. It is also perfectly consistent with the clock-rate adjustment procedure applied to satellite clocks in the methodology of the Global Positioning System (GPS); hence the designation GPS-LT for this alternative space-time transformation. Unlike the original LT, the GPS-LT is consistent with the absolute remote simultaneity of events, and it eliminates the necessity of assuming that space and time are inextricably mixed. It also disagrees with the FitzGerald-Lorentz length-contraction predictions of the original theory, finding instead that isotropic length expansion always accompanies time dilation in a given rest frame. Keywords Newton's First Law of Kinematics, Clock-Rate Proportionality, Lorentz Transformation (LT), Relativistic Velocity Transformation (RVT), Global Positioning System-LT (GPS-LT), Transverse Doppler Effect, Hafele-Keating Experiment, Universal Time-Dilation Law (UTDL) 1. Introduction There are a number of inconsistencies in the Special Theory of Relativity (STR) introduced by Einstein in his landmark 1905 paper [1]. A good starting point for recognizing this is with its predictions of time dilation, i.e. the slowing down of clocks when they are in motion relative to an observer. There are two kinds of time dilation: symmetric and asymmetric. In the first case, which is based on the Lorentz transformation (LT) of STR, two clocks in motion can each be running slower than the other. By contrast, asymmetric time dilation implies that it is always possible to decide which of two clocks runs slower than the other. It will be seen that Newton's First Law of Kinematics leads to a definite conclusion as to which type of time dilation can actually occur in natural processes. 2. Clock-rate Proportionality Time dilation can most easily be understood in terms of two identical clocks that are in pure translation with relative speed v. According to Newton's First Law of Kinematics (law of inertia), the clocks cannot change their velocity (speed and direction) in the absence of unbalanced forces. A straightforward extension of this law leads to the conclusion that the respective rates of the two clocks also will not vary with time, in obvious disagreement with the contention of STR that there must be a symmetric relation between these clock rates. Instead, the rates of clocks at rest in different

2 15 Robert J. Buenker: Contradictions in Einstein s Special Relativity Theory: Amending the Lorentz Transformation inertial systems must satisfy a strict proportionality relationship to be consistent with Newton's First Law. Experiments to test which type of time dilation occurs in a given case have invariably found that the phenomenon is indeed asymmetric. This includes studies of the transverse Doppler effect using high-speed rotors [2-4] as well as of the elapsed times on atomic clocks carried onboard airplanes [5-6]. The overall conclusion from experiment is that elapsed times in different rest frames (S and S') are strictly proportional to one another: t' = t/q, (1) in obvious agreement with the above theoretical conclusion about the rates of such clocks. The constant Q in the above equation can be calculated if the speeds of the two clocks relative to a definite reference frame (objective rest system ORS [7]) such as the axis of the rotor in the transverse Doppler studies [2-4] are known. The Global Positioning System (GPS) makes direct use of eq. (1) for the rates of clocks located on a satellite and a station on the earth's surface [8], in which case the earth's center of mass (ECM) is the ORS. The rate of a given clock is then assumed to be inversely proportional to γ (v ir ), where v ir is the speed of the clock relative to the ECM and γ=(1-v ir 2 c -2 ) -0.5 (note that gravitational effects on clock rates are ignored in the present discussion). The theoretical procedure used by Hafele and Keating (HK) [5] is illustrative of how the above inverse proportionality for clock rates can be applied in practice. The elapsed time τ of an atomic clock on an airplane moving with speed v relative to the earth's surface is compared with the corresponding value τ 0 measured on an identical clock at rest at the airport of departure. The speed of the airport clock relative to the ECM is RΩ (in the easterly direction), while that of the airplane clock is v+ RΩ (R is the earth's radius and Ω is its rotational frequency). The sign convention used is v>0 if the airplane moves in the easterly direction and v<0 if it moves in the westerly direction. The ratio of these elapsed times is given below: τ/τ 0 = γ(rω)/γ(rω+v). (2) Since v is small compared to c, a Taylor series expansion is used to evaluate eq. (2): τ/τ 0 = ( R 2 Ω 2 c -2 )/[1+0.5 c -2 (R 2 Ω 2 + 2vRΩ + v 2 )] = c -2 (2vRΩ + v 2 ). (3) From eq. (3) one obtains eq. (1) of Ref. 5: τ-τ 0 = c -2 (2vRΩ + v 2 ) τ 0. (4) HK then evaluate eq. (4) for the typical case in their experiment in which v ~RΩ. They conclude correctly, i.e. in quantitative agreement with observation, that τ<τ 0 if v>0 (easterly direction) and τ>τ 0 if v<0 (westerly direction). In the above case the elapsed time τ i on a given clock satisfies the relation: τ 1 γ(v 10 )=τ 2 γ(v 20 ). (5) Exactly the same formula [9] applies to the high-speed rotor experiments [2-4], in which case the axis of the rotor serves as reference (ORS) for the speeds of the absorber and x-ray source that are to be inserted in the γ factors. In this case, setting v i0 = R i ω and τ i = ν i -1 leads directly to the empirical formula given in Ref. 2 (R i is the distance from the rotor axis of a given clock and ω is the corresponding rotational frequency; ν i is the measured frequency of the x- rays). Thus, eq. (5) can be called the Universal Time-dilation Law (UTDL). Accordingly, with the appropriate change in notation, the constant Q in eq. (1) is given by: Q= γ(v')/γ(v). (6) Note that Q>1 if the clock at rest in S runs more slowly, and Q<1 if it runs faster than that in S. In the typical case where the clock at rest in S has been accelerated to speed v' relative to S before returning to a state of pure translation, the value of Q is γ(v'). Eq. (6) is more general since it also accounts for the situation when both clocks being compared are moving relative to the original rest frame (ORS). It is clearly necessary in applying eq. (6) to first identify the ORS. The relative speed v of S and S is not directly involved in the UTDL, thereby eliminating the subjective character of the measurement process otherwise inherent in Einstein s LT. The high accuracy of the GPS technology constitutes strong empirical evidence that time dilation is fundamentally asymmetric, in agreement with eq. (5), and therefore that the LT is invalid. Furthermore, experiments with muon decay [10, 11] have shown that the degree of acceleration of clocks has no effect on their rates, i.e. the constant Q in eq. (1) only depends on their velocities relative to the pertinent ORS. The corresponding relation between elapsed times in the LT is: t' = γ ( t v xc -2 )= γ η -1 t. (7) In this equation v is the relative speed of the S and S' rest frames (defined to be moving along the mutual x,x' axis of the coordinate system), c is the speed of light in free space ( x10 8 ms -1 ), x is the spatial separation of the two events measured by a stationary observer in S, γ=(1-v 2 c -2 ) -0.5 and η= (1-vc -2 x/ t) -1. It is clearly incompatible with the proportionality relation in eq. (1). There is a key qualitative distinction between eq. (7) and eq. (1) with regard to whether the two events can occur simultaneously for both observers, i.e. so that t' = t = 0. According to eq. (7), if both v and x are not equal to zero, the two events are predicted to be non-simultaneous since t' t in this case. Thus, the situation in general is referred to as the remote non-simultaneity of events. Since the LT is invalid because of its erroneous prediction of symmetric time dilation, it follows that the latter result has no credibility. In the past, an example of a train being struck by lightning flashes [12] has been used to justify the STR conclusion of

3 Open Science Journal of Modern Physics 2015; 2(2): remote non-simultaneity. It is shown on the basis of various assumptions consistent with the LT that the lightning flashes from opposite ends of the train can arrive simultaneously at its midpoint for an observer on the station platform but at different times for his counterpart riding on the train itself. This rationale overlooks the fact that the light-speed postulate (LSP), which is assumed in deriving the LT [1], demands that the speed of light between each end of the train has the same value of c for both observers. As a result, since the distance travelled by the two light pulses is equal for both observers (according to the FitzGerald-Lorentz length-contraction (FLC) prediction of STR [13]), it follows that they must also agree on the corresponding elapsed times. It is therefore possible to use the LT to obtain opposite conclusions with regard to the simultaneity of the two events or lack thereof. This ambiguity is further proof that the LT is invalid. On the other hand, use of the proportionality relation in eq. (1), by contrast, is only consistent with the remote simultaneity of events since Q is a non-zero proportionality constant, i.e. if t=0, then t =0 as well. Detailed examination of eq. (7) reveals another problem with the LT. Consider the case of an object moving relative to S and S. The component of the object s velocity in the x direction measured by the observer in S is u x = x/ t, i.e. x is the distance traveled in the x direction in time t. Since remote events must occur simultaneously based on the experimental finding of clock-rate proportionality discussed above, it follows that the ratio of elapsed times for the two observers is equal to the ratio of the clock rates in S and S'. Thus, according to eq. (7), the clock-rate ratio is: t / t = (1-v 2 c -2 ) -0.5 (1 - vu x c -2 ). (8) The conclusion from the LT is therefore that the ratio of clock rates depends on the velocity of the object being measured, which result therefore violates the principle of causality. From the point of view of fundamental theory, the most important consequence of the observations of clock-rate proportionality is that it rules out the concept of space-time mixing. Newton and other classical physicists were convinced that space and time are quite distinct from one another, exactly as one observes in everyday life. Einstein and his followers claimed to have overthrown this basic principle on the basis of the LT and its eq. (7). The judgment of the physics community over the past century has clearly been that Einstein was right and Newton was wrong. The only rational way to settle this issue is by experiment. The universal finding of clock-rate proportionality, reinforced by the straightforward argument given above for clocks at rest in inertial systems (Newton's law of inertia), leaves only one conclusion: eq. (1) is correct and eq. (7) is false. Time and space are distinct. Space-time mixing is a myth. It has given mathematicians a great opportunity to apply their trade and develop new theories that claim to have found solutions to important problems in cosmological science, e.g. string theory [14]. The observation of strictly proportional clock rates tells us that there must be a different way to explain what occurs in the universe that avoids any assumptions that are in conflict with eq. (1). 3. GPS-Compatible Lorentz Transformation The goal must therefore be to find a new version of Einstein's relativity theory that is not based on the LT. For this purpose it is important to review his original derivation [1]. One finds [15] that an additional assumption had to be made regarding a normalization function that appears in the general form of the Lorentz transformation. Einstein claimed without proof [1] that this function only depends on the relative speed v of the two inertial systems in his derivation. This decision leads directly to a specific value for the normalization function (φ=1), and with it, to the LT. The experiments with the transverse Doppler effect [2-4] and atomic clocks onboard airplanes [5-6] carried out over a halfcentury later show instead that the normalization function must be chosen to be compatible with the clock-rate proportionality relation in eq. (1). When this is done [16,17], the following alternative space-time transformation results, which will be referred to below as the GPS-LT: t = t/q x = (η/q) ( x v t) y = η y/γq z = η z/γq, (9a) (9b) (9c) (9d) Note that γ and η in these equations are the same as defined after eq. (7). Eq. (1) is simply repeated in eq. (9a). The GPS-LT satisfies both postulates of relativity [17,18] and is compatible with Einstein's relativistic velocity transformation (RVT) [1], in addition to agreeing with all experimental data on time dilation. The RVT is obtained by dividing each of the GPS-LT spatial equations with eq. (9a) for the elapsed times, exactly as has been done in Einstein's original derivation based on the LT. The result in each case is given below, with u x = x / t, u x = x/ t, etc.: u x ' = (1 vu x c -2 ) -1 (u x - v)= η (u x - v) (10a) u y ' = γ -1 (1-vu x c -2 ) -1 u y = ηγ -1 u y u z ' = γ -1 (1-vu x c -2 ) -1 u z = ηγ -1 u z (10b) (10c) Alternatively, one can start with the RVT and multiply each of its equations with eq. (1) to obtain eqs. (9b-d). The GPS-LT is not compatible with the FLC [13]. Distance L is obtained in the revised theory by multiplying the speed of an object such as a light pulse with the elapsed time t required for it to travel between the two corresponding end points, i.e. L=c t. On this basis, it is clear that the value of the distance is proportional to the elapsed time measured by a given observer, i.e. the faster one's clock, the larger will be the value of the distance measured in that rest frame. This prediction is therefore opposite to what one expects from the

4 17 Robert J. Buenker: Contradictions in Einstein s Special Relativity Theory: Amending the Lorentz Transformation FLC of STR: length expansion accompanies time dilation in a given rest frame, not length contraction. Moreover, the increase in length must be the same in all directions in order to be consistent with the LSP, since it requires that the light speed must be independent of orientation. The above discussion of relativistic distance variations has been purely theoretical. What does experiment have to say about whether the lengths of objects expand or contract? Previous claims of length-contraction observations [19, 20] involve distributions of a large ensemble of particles such as electrons. As such, they ignore the effects of de Broglie wave-particle duality [21], which is known to produce a decrease in the wavelength of the distribution in inverse proportion to the momentum of the particles ( p = hλ 1 ) [16,22]. It should be noted that the FLC has a substantially different dependence on the speed of particles than does the above de Broglie relation. For example, doubling v in the latter case leads to a reduction in the de Broglie wavelength of the particles by 50%, whereas if the FLC is assumed, a much smaller decrease is expected, namely by a maximum γ ( 2v) 2 2 factor of v c. Since vc γ ( v) -1 is never greater than 10-6 in these experiments, it is seen that the actual effect of the FLC would be scarcely noticeable. A better place to study length variations is the Ives-Stilwell study of the transverse Doppler effect [23]. A light source with a standard wavelength λ 0 is accelerated, and the wavelength λ of the radiation is measured in the laboratory. Two values are obtained for opposite directions of the light source s motion. Averaging of these two values therefore eliminates the first-order Doppler effect caused by the motion of the light source to and from the observer, respectively. It is found that the average wavelength is larger than the standard value (λ>λ 0 ). Einstein s LSP is then assumed, from which is concluded that the average frequency ν measured in the laboratory is inversely proportional to the average wavelength and therefore that ν<v 0. This result was considered by the authors [23] to be experimental proof that clocks in the rest frame of the light source run slower than their identical counterparts in the rest frame of the laboratory, in quantitative agreement with STR. Yet, the experiment actually measures wavelengths directly and finds that they are larger in the laboratory than in the rest frame of the light source. The analogous conclusion to time dilation, namely that lengths expand instead of contracting, is never made in textbooks discussing this experiment. Sometimes, the argument is made that the observed result can be ignored because length contraction only refers to material objects. This conclusion overlooks the consequence of Einstein s first postulate of relativity, however, the relativity principle (RP). It states that the observer co-moving with the light source will measure the standard wavelength value for the light source, i.e. λ =λ 0, even though his colleague in the laboratory measures a larger value for the same radiation. The only rational conclusion from the RP is that the diffraction grating (or comparable measuring device) in the rest frame of the light source has increased in all directions by the same fraction as the wavelength, so that no change is noticeable. The observer himself must also have experienced the same amount of length expansion in all directions, since otherwise he would be able to distinguish between the two rest frames, in direct contradiction to the RP. Analysis of another experiment leads to the same conclusion. Rossi et al. [24] showed that the range of decay of meta-stable particles such as muons increases when they are accelerated in the upper atmosphere. Because of the RP, the corresponding range must be smaller for observers moving with the particles. Although the original authors did not mention it, their results have been hailed as a confirmation of the FLC [20,25,26]. The truth is that this experiment tells us just the opposite. The reason the observer co-moving with the muons measures smaller distances is because the length of his meter stick has increased as a result of the acceleration, and in exactly the same proportion as everything else that is stationary in the rest frame of the muons. The numerical value of a measurement is inversely proportional to the unit in which it is expressed. When the meta-stable particles are produced in collisions, the rates of all clocks in their rest frame slow down and the lengths of all objects increase in the same proportion so that measured speeds of other objects are unaffected by these changes, i.e., both smaller distance values and shorter elapsed times by the same factor are measured in all cases. The Rossi et al. experiment [24] is therefore another confirmation of isotropic length expansion accompanying time dilation, not anisotropic length contraction as the FLC and LT predict. 4. Conclusion In summary, the LT fails to predict the proportionality of clock rates observed in different rest frames that is a key ingredient of the GPS methodology and also is the inevitable consequence of Newton's First Law and the causality principle for stationary clocks in inertial systems. It also is contradicted by the observation of isotropic length expansion in rest frames where time dilation occurs. It needs to be recognized that the strict proportionality of clock rates obviates the possibility of remote non-simultaneity of events. At the same time, it unequivocally supports the view that space and time are indeed distinct physical quantities, exactly as Newton and his contemporaries concluded over three centuries previously. The GPS-LT on the other hand assumes clock-rate proportionality explicitly in its formulation while also satisfying both of Einstein's postulates of relativity and being consistent with the RVT. It therefore deserves the recognition of the physics community as the true relativistic space-time transformation. References [1] A. Einstein, Ann. Physik 17, 891 (1905). [2] H. J. Hay, J. P. Schiffer, T. E. Cranshaw and P. A. Egelstaff, Phys. Rev. Letters 4, 165 (1960).

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