Fresnel light-drag formula, Einstein s dual theories of time dilation and the amended Lorentz transformation

Transcription

1 Open Science Journal of Modern Physics 2015; 2(1): 1-9 Published online January 30, 2015 ( Fresnel light-drag formula, Einstein s dual theories of time dilation and the amended Lorentz transformation Robert J. Buenker Fachbereich C-Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaussstr. 20, Wuppertal, Germany address bobwtal@yahoo.de, buenker@uni-wuppertal.de To cite this article Robert J. Buenker. Fresnel Light-Drag Formula, Einstein s Dual Theories of Time Dilation and the Amended Lorentz Transformation. Open Science Journal of Modern Physics. Vol. 2, No. 1, 2015, pp Abstract The history of Fresnel s light-drag formula is reviewed and its impact on relativity theory is assessed. Fizeau s experimental results reported in 1851 not only demonstrated that light drag is a real effect, they also provided the first concrete indication that the speed of light in free space is independent of the state of motion of the source. This ultimately became one of the main justifications for Einstein s second postulate of relativity and his ether-free description of light propagation. However, it is a little known fact that Einstein s original work on relativity presented two distinct theories of time dilation. One is based on the Lorentz transformation (LT) and claims that a moving clock always runs slower than the observer s stationary clock (symmetric theory). The other version assumes instead that accelerated clocks always run slower than their identical counterparts which remain at rest in the original position (asymmetric theory). It is pointed out that there have been numerous confirmations of the asymmetric theory, such as by using the transverse Doppler effect or comparing elapsed times on atomic clocks in various states of motion. The LT is only valid for uniformly translating systems and is therefore contradicted by these results because of its prediction of exclusively symmetric outcomes. It is shown that a different Lorentz-type transformation, referred to as the ALT or GPS-LT, exists which is compatible with asymmetric time dilation, and therefore with all known experiments. A degree of freedom in the definition of the general Lorentz transformation allows this goal to be readily achieved, while still satisfying Einstein s light-speed postulate and the relativity principle (RP). Keywords Fresnel Light-Drag Formula, Postulates of Special Relativity, Degree of Freedom in the Lorentz Transformation, Relativistic Velocity Transformation (RVT), Alternative Lorentz Transformation (ALT), Global Positioning System-Lorentz Transformation (GPS-LT), Amended Relativity Principle (ARP) 1. Introduction One of the most influential theories of light in the early 19 th century was put forward by Fresnel. It was based on the ether concept and one of its main predictions was that light traveling through a medium which itself was in motion in the laboratory would be subject to a drag effect. Experiments carried out by Arago in 1810 indicated that the resultant speed of the light could not be obtained in the standard way by simply adding the speeds of the medium and that of light in free space. Fizeau carried out accurate studies of this effect in the 1850s and his results confirmed the light-drag effect, albeit of a smaller magnitude than had originally been thought. On this basis it was widely concluded that the only way to explain the slowing down of the light was by accepting as fact that it moves through a stationary ether. When Einstein introduced his ether-free relativity theory in 1905 [1], Fizeau s light-drag verification represented a potentially serious obstacle to establishing its validity. Although Einstein did not mention this experiment in his original paper, von Laue was able to show a few years later [2] that its results could be explained quantitatively on the

2 2 Robert J. Buenker: Fresnel Light-Drag Formula, Einstein s Dual Theories of Time Dilation and the Amended Lorentz Transformation basis of the new theory. His argument was based on Einstein s relativistic velocity transformation (RVT) [1], which was a direct consequence of the light-speed postulate (LSP) that was crucial to the theory as a whole. As a result, Fresnel s conjecture about the light-drag effect has ironically turned into one of the more stunning indications that the properties of light do not require the existence of an ether for their explanation. In the following discussion attention will be focused on the derivation of the RVT from Einstein s relativity postulates. It will be seen that the Lorentz transformation (LT), which is the cornerstone of Einstein s theory, is not essential in arriving at this goal, thereby raising questions about some of its other predictions that, unlike the light-drag effect, have not yet received direct experimental confirmation. 2. Fizeau s Light-Drag Experiment and the Light-Speed Postulate The dependence of the speed of light c on the speed v of a transparent medium characterized by refractive index n through which it is transmitted was determined by Fizeau to be [3]: c = cn -1 + v(1 n -2 ), (1) where c= x10 8 ms -1 is the speed of light in free space. According to the classical Galilean transformation, c should simply be equal to cn -1 + v, i.e the sum of the speed of light in the refractive medium and that of the medium itself. The (1-n -2 ) factor of v in eq. (1) is less than unity for normally refractive media (with n>1) such as water, hence the coining of the term light drag to describe the effect. The true speed of light in the medium at rest in the laboratory is not inversely proportional to n but rather to the group index of refraction n g, whereby the latter is defined by the expression [4,5]: n g = n + ωdn/dω, (2) where ω is the circular frequency of light. The refractive index n varies with ω due to what is conventionally described as a dispersion effect [6]. As a result, the experimental value [7] for the light speed in the medium at rest is given as c (v=0) = cn g -1 = cn -1 kcn -2 dn/dk, (3) where k = ωnc -1. A more accurate value for the variation of c with v in eq. (1) is therefore obtained by substituting n g for n in eq. (1), as was subsequently verified experimentally by Zeeman [8-9]. The clear result of Fizeau s successful execution of the Fresnel light-drag experiment was to demonstrate that the classical (Galilean) velocity transformation is not directly applicable in this case. The reaction of most physicists of the time was not to reject the law per se but rather to invoke a third participant in the interaction between light and water in the experiment, namely an ether. A similar situation occurred nearly a century later in discussions of neutron decay, in which case Pauli [10] eventually provided the correct solution when he postulated the existence of the neutrino to explain the fact that the momentum conservation laws appeared to be violated by the finding of only an electron and a nucleus in the observed dissociation products. There is another piece of information to be gleaned from eq. (1) that seems to have eluded the attention of 19 th century physicists, however, and continues to be largely ignored to the present day. If one examines the limiting situation for light traveling in free space (n=1), it is seen that the speed v of the medium disappears entirely, with the simple result: c (v) = c. (4) This is an experimental verification, or at least a straightforward and unavoidable deduction based on experiment, of Einstein s LSP [1]. It shows that Fizeau s experiment could just as well have been taken as motivation to confirm eq. (4) in different empirical investigations as it was to prove the existence of an ether. As it turned out, as for example with the Michelson-Morley experiment [11], attempts to do the latter kept coming back to the same, thoroughly surprising to the great majority of contemporaries, result that the speed of light in free space actually does not depend on the relative speed of the source to the observer, exactly as Fizeau demonstrated with his light-drag experiment. Contemporary experiments [12-14] with the Sagnac effect [15] have tended to cloud the issue, however. The point of contention is perhaps best illustrated by the operation of the Global Positioning System (GPS) in computing distances on the earth s surface. For this purpose the speed of light must be adjusted to take account of the receiver speed v [16], e.g. it is taken to be c + v if the receiver is moving towards the satellite and c v if it moves in the opposite direction. The object is to measure the distance D between the satellite and the receiver at the time that the light signal is emitted. If the receiver is moving away from the satellite with speed v, however, the signal must travel farther than D in order to be received. If the time to reach the receiver is t, this means the actual distance traveled by the light is D + v t, not simply D. According to Einstein s LSP [1], the signal traverses this distance with speed c, so that the elapsed time t is equal to (D + v t)c -1. The value of the required distance between the satellite and the original position of the receiver is thus D = (c v) t. The adjustment of the speed of light in the GPS procedure is thus seen to be a consequence of the fact that the light must travel a different distance than D to reach the receiver, and not because the speed of light is dependent on the speed of the receiver. The GPS effect is actually a confirmation of the LSP and a further demonstration that the corresponding Galileantransformation result is contradicted by experiment. A similar argument applies to the original Sagnac effect [15] observed for the interference of two light waves moving in opposite directions on a rotor. There is a time discrepancy t caused by the fact that the waves must travel different distances

3 Open Science Journal of Modern Physics 2015; 2(1): before arriving at the same point on the rotor. The observed value of t is obtained by assuming that the light speed is c for both waves. One perfectly straightforward way of addressing the novel results of Fizeau s study was to come up with a different velocity transformation than the classical additivity law, something akin to c+v=c. The seeming absurdity of this approach certainly was a hindrance to giving the idea serious consideration, but this is ultimately what Einstein accomplished with his theory of relativity some 50 years later [1]. As the discussion in the next section will show, however, he was not so much interested in obtaining a suitably relativistic velocity transformation (RVT) as he was in finding a space-time transformation that leaves Maxwell s equations invariant. This conclusion is supported by the title of his paper (translated from the German): On the Electrodynamics of Moving Bodies. His lack of emphasis on the velocity transformation itself is perhaps best illustrated by the fact that he did not use it to derive eq. (4) in his paper [3]. Instead, it fell to von Laue [2] to point out the connection two years later. All of the above might simply be material for history books were it not for a number of conclusions Einstein ended up drawing that are not specifically related to his RVT but rather to the Lorentz space-time transformation (LT) that is the actual centerpiece of his theory. 3. Degree of Freedom in the Derivation of the Lorentz Space-Time Transformation The general attitude of physicists in the second half of the 19 th century has been summarized in Pais s biography of Einstein as follows [17]: The most important question for all theses authors of ethers and makers of Maxwell theories was to find a dynamic understanding of the aberration of light, of Fresnel drag, and later, of the Michelson-Morley experiment. In the minds of many of these scientists, this goal was best to be achieved within the context of the RP by finding a spacetime transformation that, unlike the Galilean transformation, leaves Maxwell s equations of electricity and magnetism invariant. Voigt [18] was the first to obtain the desired result, but the set of equations he found differ by the same factor in each case from the LT equations that Einstein later derived [1], but which do not satisfy the RP. The LT itself was first reported by Larmor in 1898 [19-20]. At about the same time, Lorentz [21] recognized that the condition for the space-time transformation of leaving Maxwell s equations invariant can be accomplished in an infinite number of ways. He gave the general form for these equations as follows: t = γ ε (t vxc -2 ) (5a) x = γ ε (x vt) (5b) y = ε y (5c) z = ε z. (5d) In these equations, x, y, z and t are the space-time coordinates of an object as measured by an observer who is at rest in inertial system S, whereas the corresponding primed symbols correspond to the measured values for the same object obtained by a second observer who is stationary in another inertial system S which is moving along the common x,x axis at constant speed v relative to S [γ = (1- v 2 c -2 ) -0.5 ]. The factor ε corresponds to an undefined degree of freedom in this definition. Voigt s value for this quantity was γ -1 whereas Larmor had ε = 1 [20] in his version of the transformation. At the time that Lorentz reported the above equations [21], there seemed to be no consensus as to how best to determine the value of ε in an unequivocal manner. Einstein [1] was aware of the degree of freedom in the general version of the Lorentz transformation. He derived exactly the same equations as above but simply used different notation than Lorentz had. For example, he referred to the aforementioned factor as φ instead of ε in eqs. (5a-d). However, Einstein based his derivation on the LSP, for which eq. (4) from the light-drag effect is simply a special case. The idea was to find a space-time transformation that satisfies the condition of equal values of the speed of a given light pulse measured by the different observers in S and S. It was already a novel result to show that the resulting equations are the same using this condition as those obtained earlier by other authors [18-19, 21] who had instead assumed the invariance condition of the Maxwell equations in their respective derivations. There was something else that distinguished Einstein from these other colleagues. He asserted (see p. 900 of ref. 1) that φ is a temporarily unknown function of v. This allowed him to make use of a symmetry condition to prove that the only physically acceptable value for the common factor φ/ε in the above transformation equations should be unity. However, he gave no justification for this assumption about the functional dependence of φ. Indeed, he did not acknowledge it as an assumption at all. Over the ensuing century many authors have accepted this declaration as fact and have concluded along with Einstein that substitution of ε=1 in eqs. (5a-d) leads to the only viable relativistic spacetime transformation. The result is the same as Larmor s version [19] and has been referred to since Einstein s paper appeared as simply the Lorentz transformation (LT in the notation introduced above). This choice of φ = 1 has very significant consequences, as will be discussed in detail below. It is just as essential for the successful derivation of the LT as Einstein s two declared postulates, and therefore many of the most famous predictions of his theory rest squarely on its validity. This does not include the explanation for the Fresnel light-drag formula, however. The reason is because all that is necessary for this application is Einstein s RVT. The latter follows directly from Lorentz s general transformation in eqs. (5a-d) since it is obtained by division of the x,y,z spatial quantities by the corresponding value of the time t of a given event. The

4 4 Robert J. Buenker: Fresnel Light-Drag Formula, Einstein s Dual Theories of Time Dilation and the Amended Lorentz Transformation common factor ε is eliminated in the process, which relationship clearly explains the existence of Lorentz s degree of freedom in the first place. If all one knows is a relationship between two velocities of an object, as is the case with the LSP, it is not possible to further specify either how far or for how long that object has traveled during the course of the corresponding measurement. One needs an additional piece of information for this to be done, and Einstein merely asserted that this was uniquely provided by the sole dependence on v of Lorentz s undetermined function ε/φ. It is important to see that one can forego any assignment for the latter function, however, if the goal is simply to obtain a transformation for the velocity components of the object. The required division gives a unique result for the RVT without making this choice, namely as: u x = (1 vu x c -2 ) -1 (u x - v) = η (u x - v) u y = γ -1 (1 vu x c -2 ) -1 u y = η γ -1 u y u z = γ -1 (1 vu x c -2 ) -1 u z = η γ -1 u z, (6a) (6b) (6c) where u x =x /t, etc. and η= (1 vu x c -2 ) -1 = (1 vxt -1 c -2 ) -1. As discussed in Sect. 2, von Laue [2] was able to successfully derive Fizeau s experimental result simply by using the inverse of eq. (6a) (obtained by exchanging the primed and unprimed symbols and changing v to v): u x = (1 + vu x c -2 ) -1 (u x + v) = η (u x + v). (6a ) One obtains the desired result in eq. (1) by setting u x =c/n and neglecting higher-order terms in the expansion of η : u x = c = (1 + vn -1 c -1 ) -1 (cn -1 + v) cn -1 + v vn -2. (7) The expression for the aberration of starlight at the zenith, which was discussed in Einstein s original work [1], can also be obtained directly from the RVT. In this case one assumes that the starlight approaches the earth from a position in the neighboring sky with speed c along the y direction. Hence, u y = c and u x = 0 (η=1) in eq. (6b), so that the corresponding component in the earth s rest frame is u y = γ -1 c. The angle of aberration [3] is thus given as tan -1 (u x u y -1 ) = tan -1 (γ vc -1 ), where u x = v is the earth s orbital speed relative to the position of the star in the neighboring sky. Note that in both of the above cases, the value chosen for the normalization factor ε/φ in Lorentz s general transformation of eqs. (5a-d) has no effect whatsoever on the result because the RVT has been used exclusively to obtain it. This fact is often obscured by lengthy derivations of the same effects that begin with the LT, such as, for example, Kilmeister s theoretical discussion of starlight aberration [22]. It requires many more steps to obtain the same result as when the RVT is used directly, and in the process one can easily get the impression that relativity theory is far more complicated than need be. At the same time, it is commonplace to see claims that the successful derivation of a given effect is confirmation of the validity of the LT. Pais s treatment [3] of the light-drag effect provides a good example for this. He concludes that the agreement with Fizeau s empirical formula that von Laue [2] obtained starting from the RVT is a direct consequence of Lorentz invariance. However, the latter condition is only satisfied for the LT, i.e for the particular value of ε = 1 in eqs. (5a-d). On the contrary, the exclusive use of the RVT in von Laue s derivation proves that any other choice for this quantity would do just as well. More generally, it is clear from the conditions under which it is derived that the RVT satisfies the LSP exactly. The direction of the velocity of a light beam is usually different for the respective observers in S and S, but the speed is always the same for both. Again, there is a wide-spread mistaken belief among authors of textbooks dealing with relativity theory that the LT is the only space-time transformation that has this characteristic. 4. Asymmetric Time-Dilation Measurements The discussion in the previous section makes clear that the only way to make a definitive test of the LT s validity is to carry out an experiment in which either time or spatial relationships are being probed separately and not just some function of the ratios of these quantities. This state of affairs effectively narrows the choice down to studies of either time dilation or Fitzgerald-Lorentz length contraction, whereby the former have received far more attention because of the availability of atomic clocks and accurate measurements of the transverse Doppler effect and decay lifetimes of elementary particles. None of these experimental results were known at the time that Einstein wrote his famous paper [1], so he was forced to speculate about very basic details of the predicted phenomenon. A key aspect of his discussion centers around the question of whether time dilation is subject to a symmetry principle whereby two observers in relative motion would each find that it was the other s proper clocks that were running slower. This is a key result of the LT since according to the version of the RP on which it is based, all inertial systems are equivalent. When ε = 1 in eqs. (5a-d), which it will be recalled is assumed in the LT definition, it can be shown that both clock-rate comparisons are possible in the same experimental situation, i.e. by suitable manipulation of these equations, one obtains both t =γt and t= γt. This result is unusual to say the least, and many authors have concluded that it amounts to a contradiction that proves the LT is invalid. The situation is not quite so simple, however, because the clocks under comparison are in relative motion and therefore cannot be compared by simply putting them side by side, so there is at least the theoretical possibility that such a situation does exist in nature. After all, when two observers are in relative motion, they each think it is the other who is actually moving, so there would seem to be some precedent for the predicted symmetry of clock rates. The challenge according to that view is to devise an experiment to verify this theoretical result.

5 Open Science Journal of Modern Physics 2015; 2(1): Einstein described such a possibility in his paper, namely by deriving the transverse Doppler effect from the LT [1]. Accordingly, the frequency ν r of light emitted from a source moving in a perpendicular direction should always be measured to be smaller than the standard frequency ν e the observer finds when the identical source is stationary in his laboratory [23]: ν r = ν e (1 v 2 /c 2 ) 0.5 = γ -1 ν e. (8) As a result, when light signals are exchanged between two observers moving in a transverse direction to one another, each will find it is the period of the other s radiation that is always greater. This in turn would mean that all naturally occurring periodic events in another inertial system would appear to be running slower than the identical processes taking place in one s own rest frame. Einstein gave another example of time dilation in his paper, however. He discussed the case of two identical clocks, one of which is accelerated in a curved path away from the other. He concluded that the latter clock would return to its original position with less elapsed time for the flight than is recorded on the clock that was left behind. On this basis he also argued that a clock located at the Equator must run slower than an identical clock located at one of the Poles. There is a key difference between these examples than in those discussed first, however. Einstein was saying in these cases that there was no question which of the two clocks runs slower, namely the one that is accelerated. There has been a widespread tendency to look upon these different examples as just being two sides of the same coin, but this position overlooks the basic fact that time dilation is assumed to be an asymmetric phenomenon in the latter case. Von Laue [24] explained the distinction by noting that the RP is not relevant to the case when one of the clocks has been accelerated while the other continues on in uniform translation [25]. That is certainly a convincing argument, but it has important consequences to relativity theory as a whole. The fact is that the LT predicts that time dilation and length contraction are completely symmetric for two moving observers, never asymmetric. There are thus two fundamentally distinct theories that must be contended with. In order to test the symmetry properties of time dilation, it is necessary to have a two-way experiment [26]. The first successful demonstration of the transverse Doppler effect was performed by Ives and Stilwell in 1938 [27], but it was only a one-way experiment. It found that there was time dilation in the rest frame of a light source moving at high speed in the laboratory. The results were in quantitative agreement with the symmetric theory of eq. (8) but they were also in agreement with Einstein s argument about clocks losing time at the Equator because of their acceleration relative to the Earth s polar axis. This situation was remedied with the high-speed rotor experiments carried out by Hay et al. in 1960 using the Mössbauer technique [28]. In this case it was the absorber/detector rather than the light (x-ray) source that was subject to acceleration since it was mounted on the rim of the rotor. The empirical findings for the shift in frequency ν/ν are summarized by the formula: ν/ν = (R a 2 R s 2 ) ω 2 /2c 2, (9) where R a and R s are the respective distances of the absorber and light source from the rotor axis (ω is the circular frequency of the rotor). It shows that a shift to higher frequency (blue shift) is observed when R a is greater than R s, as in the present case. The corresponding result expected from eq. (8) would be: ν/ν = γ -1 ( R a R s ω) 1 -(R a R s ) 2 ω 2 /2c 2, (10) i.e. a red shift should be observed in all cases in accordance with the symmetric interpretation of the time dilation. However, the results shown in eq. (9) indicate on the contrary that the effect is anti-symmetric, in clear contradiction to both eq. (8) and the LT. Hay et al. [28] nonetheless declared that their results were consistent with Einstein s theory [1] without mentioning the difficulty with the LT s prediction of symmetry and the sign of the shift. They also noted that eq. (9) can be derived from Einstein s equivalence principle [29], which equates centrifugal force and the effects of gravity. Subsequent experiments by Kündig [30] and Champeney et al. [31] also found that their results were summarized by eq. (9). Kündig stated explicitly that the results confirmed the position that it is the accelerated clock that is slowed by time dilation. In response to Hay et al. s work, Sherwin [32] pointed out that their key result was to show that it was unambiguous which clock was running slower in the rotor experiments. He attributed this to the fact that acceleration of the clocks was involved, thereby agreeing with Einstein s conclusion for the example of a clock returning to its original location over a circular path [1]. Sherwin also stated that the situation was qualitatively different when both clocks were in uniform translation, in which case it was completely ambiguous which one had the slower rate. He gave no reference for such an experiment, however, merely referring to his conclusion as an established fact. There was an important aspect of the overall theory that was left out of both Einstein s and Sherwin s discussions, however. If there are two different theories of time dilation, and only one of them is consistent with the symmetry predicted by the LT, is it not essential to find a different Lorentz-type space-time transformation to cover the other cases where one of the clocks is accelerated and the phenomenon is asymmetric in character? This question becomes all the more pertinent when it is realized that all subsequent time-dilation tests have also failed to find an example where the expected symmetry has been verified. For example, the Hafele-Keating experiments with atomic clocks located on circumnavigating airplanes [33, 34] show clearly that it is the speed v i0 relative to the earth s center of mass that ultimately determines such rates. The elapsed time τ i on a given clock satisfies the relation: τ 1 γ(v 10 ) = τ 2 γ(v 20 ). (11)

6 6 Robert J. Buenker: Fresnel Light-Drag Formula, Einstein s Dual Theories of Time Dilation and the Amended Lorentz Transformation The same formula applies to the Hay et al. experiments [28], in which case the axis of the rotor serves as reference for the speeds of the absorber and x-ray source that are to be inserted in the γ factors. Expansion of eq. (11) with v i0 = R i ω and τ i = ν i -1 leads directly to the empirical formula given in eq. (9) [26]. The lack of symmetry in the time-dilation experiments does not contradict in any way the LSP employed by Einstein to derive the LT. What it suggests instead is that his assumption regarding the degree of freedom in the general form of the Lorentz transformation needs to be reexamined [35-36]. There is no clear justification for his assertion that φ can only be a function of the relative speed v of the two rest frames and therefore for his conclusion that it can only have a constant value of unity. Given the experimental facts, it makes more sense to insist that φ be chosen so as to maintain consistency with the empirical formula given by eq. (11). This can easily be brought into a form which is consistent with the notation in eqs. (5a-d) by assuming the following proportionality: t = tq -1, (12) where Q is a ratio of the two elapsed times in eq. (11). This relation allows one to obtain a different value for the undetermined function φ/ε by comparing with eq. (5a): Solving for ε yields: t = γ ε (t vxc -2 ) = tq -1. (13) ε = [(1-vxc -2 )γq] -1 = η(γq) -1, (14) where η is the same function that appears in each of the RVT relations in eqs. (6a-c). Inserting the above value for ε in eqs. (5a-d) therefore produces a new space-time transformation, hereafter referred to as the alternative Lorentz transformation (ALT [35, 36]) that is consistent with the LSP as well as with the empirical time-dilation formula of eq. (11), namely: t = Q -1 t (15a) x = η Q -1 (x vt) (15b) y = η (γ Q) -1 y (15c) z = η (γ Q) -1 z, (15d) whereby the first of these equations is identical to eq. (12). Note that the proportionality factor Q also appears in eqs. (15b-d). The above result is clearly necessary [30-31] in order to ensure that the new transformation is consistent with the RVT of eqs. (6a-c). However, it also has important consequences for the other postulate of relativity, the RP. 5. Lorentz Invariance and the Amended Version of the RP One of the most attractive features of Einstein s theory is its Lorentz invariance condition. It follows directly from the general form of the Lorentz transformation by squaring and adding eqs. (5a-d): x 2 + y 2 + z 2 - c 2 t 2 = ε 2 (x 2 + y 2 + z 2 - c 2 t 2 ). (16) Lorentz invariance is the special case for this equation when the value of ε = 1 for the LT is substituted therein. When both S and S are inertial systems, the RP demands that respective observers who are stationary in either of these rest frames find the same laws of physics to be valid. That implies a definite symmetry that is far from obvious from the general form of eq. (16). As noted above, Einstein derived the LT from eqs. (5a-d) by claiming that ε = 1 is the only acceptable value for this quantity [1]. It is clear that this choice produces a symmetric appearance for eq. (16) as well as the condition of Lorentz invariance. When any other value for ε is assumed, such as that used to derive the ALT in eqs. (15a-d), it is critical to show that the RP is also satisfied by this choice. One might try to circumvent this question by insisting that the latter transformation is only valid for non-inertial systems, but such a position does not stand up to further scrutiny for the reason that an accelerated clock can assume a state of uniform translation at any time. When this happens, there is no evidence to indicate that its clock rate changes as a consequence. On the contrary, the experimental data for muon decays in storage rings [37, 38] indicate that only the speed of the particles affects their lifetime, and not the level of acceleration to which they are subjected. The question thus arises as to whether the choice of ε/φ = η (γ Q) -1 that leads to the ALT is also consistent with the RP. Voigt s value [18] of γ -1 does not satisfy the RP, for example, as will be shown below. Substitution in eq. (16) gives the following alternative condition of invariance: x 2 + y 2 + z 2 - c 2 t 2 = η 2 (γ Q) -2 (x 2 + y 2 + z 2 - c 2 t 2 ). (17) To satisfy the RP, it is necessary for the inverse of eq. (17) to have the same form from the vantage point of the observer in S, i.e x 2 + y 2 + z 2 - c 2 t 2 = η 2 (γ Q ) -2 (x 2 + y 2 + z 2 - c 2 t 2 ). (18) In this equation η must be obtained from η= (1 vu x c -2 ) -1 = (1 vxt -1 c -2 ) -1 in the standard way by exchanging corresponding primed and unprimed values and changing v to v, i.e. η = (1 + vu x c -2 ) -1 = (1 + vx t -1 c -2 ) -1. The value of γ remains the same because it is a function of v 2. The value of Q = Q -1 is fixed by forming the inverse of eq. (15a), i.e. t = Q -1 t = Q t. There is another way to invert eq. (17), namely to simply divide both sides by ε 2 = η 2 (γ Q) -2, with the result: x 2 + y 2 + z 2 - c 2 t 2 = η -2 (γ Q) 2 (x 2 + y 2 + z 2 - c 2 t 2 ). (19) Both equations must be equivalent in order to satisfy the RP, hence η -2 (γ Q) 2 in eq. (19) must be equal to η 2 (γ Q ) -2 in eq. (18). The primed variables in the definition of η can be eliminated by using the RVT [39], whereupon the following identity is obtained:

7 Open Science Journal of Modern Physics 2015; 2(1): ηη = γ 2. (20) Consequently, eqs. (18) and (19) are seen to be equivalent since Q = Q -1. The choice of ε = η (γ Q) -1 in the general Lorentz transformation of eqs. (5a-d) to define the ALT therefore satisfies both of the relativity postulates just as well as Einstein s φ/ε= 1 value does for the original LT. On the other hand, Voigt s value [18] of γ -1 does not satisfy the RP because εε 1 in this case. Yet, unlike the LT, the ALT is consistent with the asymmetry that is always observed in two-way time-dilation tests. There is another topic that needs careful attention when proposing a new version of the Lorentz transformation to remove the experimental contradictions that result from reliance on Einstein s LT. The usual argument in textbooks is that the laws of physics are satisfied in all inertial systems because of the Lorentz invariance condition that is a characteristic of the LT. The discussion in the preceding paragraph notwithstanding, it is clearly necessary to show explicitly how a different space-time transformation guarantees satisfaction of the RP. According to eq. (15a) of the ALT, elapsed times measured in different inertial systems always differ by a definite proportionality factor Q, and this guarantees that time dilation is asymmetric (i.e. when Q 1, in agreement with all experiments. Unlike the case for the LT, however, timings are not subject to space-time mixing, and this fact greatly simplifies matters as far as satisfaction of the RP is concerned. It allows one to look upon Q in the above equation as a conversion factor between the respective units of time in S and S, respectively [40]. The laws of physics are ordinary equations that are independent of the units chosen for the various quantities that appear therein. For example, the laws are not affected in any material way when the unit of length is changed from meters to yards or the unit of force from newtons to pounds. All changes in the numerical values of the quantities contained in the laws that are caused by such changes in units are simply cancelled out in the overall equations so that the laws themselves are unaffected. This is exactly the situation when eq. (15a) of the ALT is used to determine the conversion factor between the different units of time in the two inertial systems. The RP is therefore satisfied in all cases, as the RP demands. Moreover, we also know that the unit of speed/velocity must be the same in all inertial systems because otherwise it is impossible, as required by the LSP, to explain how the speed of light has the same value in each of them. Therefore, no conversion factor for velocities is required to change from one inertial system to another. The constancy of the unit of velocity has another important consequence, however. The only way the values of speeds can be the same for observers in different inertial systems even though their respective timing measurements differ is for their length measurements to differ by exactly the same factor. Since the unit of time varies with Q according to eq. (15a) of the ALT while the unit of speed does not change at all, it therefore follows that the unit of distance must vary in exactly the same way as that of time. One of the consequences of this state of affairs is that length measurements are also asymmetric. There is never any ambiguity as to which of two lengths is greater, just as there is no question about which clock rate is slower. The RP is automatically satisfied for distances because any law that is valid in one initial system which involves distances must also be valid in any other inertial system. The above result for distance variations is quite different than what Einstein predicted using the LT [1]. He concluded (Fitzgerald-Lorentz contraction) that lengths contract by the same fraction that clocks slow down and that the effect is anisotropic, i.e. they decrease along the parallel direction of motion relative to the observer but they do not change at all in perpendicular directions. Consistent with Lorentz invariance, the effect should be symmetric for two observers, the same characteristic as he found for time dilation. The Ives-Stilwell transverse Doppler experiments provide a straightforward test for these predictions [27]. It was found that the wavelength of light emitted in a perpendicular direction from a moving source increases by a factor of γ (v). Moreover, the form of the general relativistic Doppler equation indicates that the same increase in wavelength occurs in all directions once the non-relativistic contribution due to relative motion of the source and detector is accounted for [26]. In accordance with the RP, an observer co-moving with the light source does not notice any change in the standard wavelength of the light source. This conclusion indicates that the dimensions of the diffraction grating used in the local measurement have changed by exactly the same fraction in all directions as the wavelength of the standard light source. The result is perfectly consistent with the isotropic length expansion expected from the asymmetric theory of time dilation, but it stands in direct contradiction to the Fitzgerald-Lorentz length contraction phenomenon predicted by the LT. The latter prediction of Einstein s relativity theory has sometimes been claimed [41] to have received experimental confirmation, but such conclusions confuse relativistic length contraction with the decrease in the de Broglie wavelength [42] of the distribution of the particles that is known to occur with their increasing momentum in the laboratory. The general treatment of changes in clock rates and dimensions of objects discussed above can be concisely summarized in terms of the uniform scaling of physical properties that is caused by application of force. The conversion factor for energy and inertial mass is the same as for time and distance, i.e. the value of Q in the ALT connecting the two inertial systems. More details of this aspect of the asymmetric version of the theory are given in a companion publication [43]. In general, one simply needs to know the composition of a given property in terms of the units of length time and inertial mass to determine its conversion factor. The value is always a power of the proportionality factor that appears explicitly in the ALT for a given pair of rest frames. This guarantees that the measurement process is objective, without there being any ambiguity as to which observer will find a larger/smaller

8 8 Robert J. Buenker: Fresnel Light-Drag Formula, Einstein s Dual Theories of Time Dilation and the Amended Lorentz Transformation value for a given property and by what ratio. It is also important to note that the proportionality relationship in eq. (15a) rules out both remote non-simultaneity of events and the inexorable mixing of space and time coordinates that is such a staple of modern cosmology (e.g. string theory [44]). The strict proportionality of clock rates in different inertial rest frames is a key assumption [45] in the methodology of the Global Positioning System (GPS). Hence, it is appropriate to refer to the ALT as the GPS version of the Lorentz transformation (GPS-LT). 6. Conclusion Einstein presented two distinct theories in his original paper on relativity theory, one symmetric and one asymmetric. The former is based on the Lorentz transformation (LT) and it predicts that a moving clock always runs slower than an identical stationary one. The other claims instead that it is always the accelerated clock that runs slower. In this case, it is completely unambiguous which clock rate is slower because it is possible to distinguish a clock which has been subject to a force from one that has remained stationary at the original position. By contrast, the fact that inertial systems are all equivalent is said to preclude a similar distinction between clocks that are in uniform motion. All experimental tests such as the Hay et al. rotor measurements [28, 30, 31] and the observations of rates of atomic clocks located on circumnavigating airplanes [33, 34] have thus far verified the asymmetric theory. Since the LT irrevocably predicts that there must be complete symmetry between these clocks, it is clear that it is contradicted by these results. Although the search continues for a test example that does confirm the LT s time-dilation predictions, it needs to be recognized that another space-time transformation is required to successfully describe the asymmetric cases that are well characterized in the existing experiments. The results of Fizeau s realization of the Fresnel light-drag experiment in 1851 already demonstrated that the speed of light in free space has the same value for all observers, consistent with the LSP [1]. The RP is also well established experimentally. Unlike the conventional view, however, the LT is not the only such transformation that satisfies both the LSP and RP. Lorentz showed in 1898 [21] that there is a degree of freedom which also needs to be removed before a unique space-time transformation that leaves Maxwell s equations invariant and still satisfies the RP can be specified. Einstein s attempt to resolve this issue [1] was by making an undeclared assumption about the velocity dependence of a normalization function in Lorentz s general transformation equations. This leads directly to the LT and the aforementioned symmetry characteristics. A simple argument (clock riddle [43]) shows that this assumption is contradicted by Einstein s two postulates, however. The theory leads to opposite predictions for the same experiment depending on how it is applied to clocks that are both in uniform translation. A satisfactory alternative to the LT can be obtained instead by replacing Einstein s assumption with a well established result from experimental tests of time dilation, namely t = Q -1 t. This choice makes relativity theory suitably asymmetric and unambiguous. The new set of space-time equations is referred to as the Alternative Lorentz Transformation (ALT) or also as the GPS-LT. It satisfies both of Einstein s postulates but eliminates the space-time mixing feature of the LT [35, 36, 44]. It is perfectly consistent with the remote simultaneity of events, a principle long disputed because of the LT. It also eliminates Fitzgerald-Lorentz length contraction and replaces it with an asymmetric theory that allows distances to be measured with a clock. It is consistent with the modern definition of the meter as the distance travelled by light in free space in c -1 s. It forces one to the conclusion that the slowing down of proper clocks upon acceleration is accompanied by a corresponding increase in the lengths of co-moving objects in all directions; isotropic length expansion instead of the anisotropic length contraction implied by the LT. In addition, the asymmetric theory provides a simpler means of ensuring satisfaction of the RP. The physical characteristics of objects generally change when they are subjected to an outside force. This is tantamount to changing the standard units employed in a given rest frame. No change in the laws of physics is observable because all the changes are uniform within a given rest frame. This leads to a restatement of Galileo s RP: The laws of physics are the same in all inertial systems but the units in which they are expressed can and do vary in a systematic manner from one system to another. The variables are subject to a uniform scaling, whereby the conversion factors from one system of units to another are found to be powers of the proportionality factor Q in eqs. (15 a-d) of the ALT. More details about the general scaling procedure are given elsewhere [46, 47], including corresponding prescriptions for the changes in physical units that occur when an object moves to a different gravitational potential. 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