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Physics & Astronomy International Journal

Research Article Volume 1 Issue 1

Testing Einstein’s second postulate with an experiment of the Sagnac type

Gianfranco Spavieri

Centro de Física Fundamental, Universidad de Los Andes, Venezuela

Correspondence: Gianfranco Spavieri, Centro de Física Fundamental, Universidad de Los Andes, 5101-Mérida, Venezuela

Received: July 13, 2017 | Published: July 28, 2017

Citation: Spavieri G. Testing Einstein’s second postulate with an experiment of the Sagnac type. Phys Astron Int J. 2017;1(1):14-24. DOI: 10.15406/paij.2017.01.00003

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Abstract

Einstein’s clock synchronization procedure has led to the contention of conventionalism that clock synchronization is arbitrary and, thus, the one-way speed of light cannot be measured even in principle. In the context of relativistic theories, we analyze the linear Sagnac effect and show that its interpretation implies the existence of superluminal light speeds. Furthermore, we consider an experiment of the Sagnac type that can test the second postulate of special relativity by discriminating absolute synchronization from Einstein synchronization. The mere existence in principle of such a test settles the nearly century-long controversy about the conventionality of the one-way speed of light. An immediate consequence is that the one-way speed is measurable and the Lorentz transformations maintain their unique physical meaning and, thus, cannot be substituted by transformations based on absolute synchronization. The outcome of the experiment, attainable with present technology, will either corroborate the postulate of a universal light speed in all inertial frames or identify the preferred frame of reference of relativistic theories. PACS: 03.30.+p; 03.65.-w; 45.50.-j

Keywords: sagnac effect, relativistic theories, one-way speed of light, Einstein synchronization

Introduction

A theory is physically meaningless unless its basic postulates can be tested, i.e., verified experimentally. This procedure is sometimes referred to by saying that the theory can be falsified. The second postulate of the special theory of relativity reflects Einstein’s assumption that the universal one-way speed of light c is constant. With the exception of experts, many physicists believe that the speed of light measured in experiments refers to the one-way speed,1 while what is measured is the round-trip value,2 in accordance with Einstein synchronization procedure using a mirror on a round-trip flight path. In fact, Einstein did not provide a way to test his basic assumption. Instead, by proposing his well-known clock synchronization procedure, spatially separated clocks are conventionally synchronized by assuming that the average round-trip speed and the one-way speed are the same. Considering that, in principle, the two speeds (round-trip and one-way) may not be the same, the question of whether or not the one-way speed of light is measurable, represents one of the prominent controversial debates of modern physics. The conventionality of the synchronization procedure of spatially separated clocks in special relativity (SR), and the related measurement of the one-way speed of light, has been thoroughly discussed by Poincaré3 and Einstein,4 and reinforced in the works of Reichenbach,5 and Grünbaum.6 Discussions on this fundamental issue were revived by the works of Möller7 and Mansouri and Sexl,8 who confirmed that synchronization by means of clock transport, is equivalent to Einstein’s procedure, favoring the conventionalist view.

According to Mansouri and Sexl,8 the intrinsic indetermination in the one-way speed c renders the validity of the second postulate of SR not testable. This has led to a renaissance of alternative relativistic theories that assume the existence of a preferred frame where space and light speed are isotropic while, in a relatively moving inertial frame, they are no longer isotropic. Thus, physicists have focused their attention to preferred frame theories that use transformations with the same rod-contraction and clock-retardation as the Lorentz transformation (LT), but differ from it by an arbitrary synchronization parameter. By their very construction, preferred frame theories interpret all the known experiments that support SR. For example, in the Michelson-Morley experiment, the observable turns out to be independent from synchronization. Therefore, besides the Lorentz transformations, also the transformations based on absolute synchronization, such as the Tangherlini transformations (TT),9 provide the same null result of SR. What matters in this and in other optical experiments is the average speed on the optical path, which turns out to be exactly c with transformations (such as the TT) that foresee the Lorentz-Fitzgerald contraction of moving rods and the time dilation of moving clocks. One important contribution to the subject was given by Bell,10 who stated that although there is a stringent “difference in philosophy” between the view of SR and that of a preferred frame theory, “The facts of physics do not oblige us to accept one philosophy rather than the other”. Bell’s assertion was used as a starting point in a series of works adopting the conventionalist thesis within relativistic theories,1014 where it has been argued that there is essentially one theory, Bell’s two philosophies corresponding rather to different aspects of the same theory.1119

If synchronization is arbitrary and the two synchronizations (Einstein’s and absolute) are equivalent, important consequences arise. Indeed, the Lorentz covariance concept exploits the symmetry of the Lorentz group and is usually applied by theoreticians to several branches of modern physics. If Einstein and absolute synchronizations are physically indistinguishable, the LT could be equivalently substituted by the TT, which possesses a lower symmetry, similar to that of the Galileo group. In this case, the equivalence of the two synchronizations implies that what is important is the contraction of moving rods and the slowing down of moving clocks but not the symmetry of the transformations. This fact is hardly acceptable for most physicists who have been relying for decades on the properties of the Lorentz group. There are instances in the literature where, instead of the LT, the TT transformations are equivalently used to describe physical reality. In the context of relativistic theories, an example is found in the interpretation of the Sagnac20 effect by Kassner,21 who proposes to solve Selleri’s22,23 paradox by replacing Einstein synchronization with the absolute synchronization. Kassner’s conclusion is then that the light speeds c+v and cv, pointed out by Selleri in his paradox, are to be interpreted in terms of the arbitrariness of synchronization and, thus, do not invalidate the second postulate of SR that requires the unique, universal value c for the speed of light.

Kassner’s paper is important for two reasons. First, he makes it evident that the approach of conventionalism has spread in the literature even among journals of didactic nature. Second and most importantly, because it supports the opinion shared by many physicists that within relativistic theories the principle of relativity remains valid even when we adopt a synchronization procedure different from that of Einstein, regardless of the validity or not of Einstein’s second postulate. In this complex scenario, the important issue consists in clarifying the physical equivalence — or lack thereof — of preferred frame theories with standard SR (i.e., special relativity with Einstein synchronization). The purpose of our paper is to show that, at least in principle and even on a kinematical basis,2427 there are experiments capable of testing the one-way speed of light, and the Sagnac effect is one of them. In Sects. 2 and 3 we consider the relativistic interpretation of the Sagnac effect, using a "linearized" version of it, and discuss the related controversial problem of the superluminal speeds of the light signals, highlighted by detractors of Einstein synchronization as disproving the validity of Einstein’s second postulate.

In Sect. 4 we show that an experiment of the Sagnac type can discriminate Einstein synchronization from absolute synchronization, i.e., standard SR from identifiable preferred frame theories. If the experiment is performed, its outcome will have significant consequences on modern physics. The first important result is that, from a theoretical perspective and in the description of physical theories, the LT are not equivalent to and cannot be arbitrarily replaced by the TT. In fact, if the one-way speed of light is observable, even only in principle, the validity of the second postulate of special relativity can be tested. Thus, standard SR maintains its unique meaning, physically different from that of theories that assume the existence of an identifiable preferred frame. Finally, we provide an indication of the sensitivity required by the experimental setup, capable of detecting in certain situations the velocity of the preferred frame if it were to exist.

The sagnac effect and Einstein’s second postulate

The usual circular Sagnac experiment is pictured in Figure 1. Consider a disk rotating at constant angular velocity ω and an observer with a clock O∗ stationary on a point at radius R, the origin of a co-rotating reference frame S∗. As indicated in Figure 1, this observer sends two light signals (as in the Sagnac experiment), or two particles (or two clocks, as in the Hafele-Keating experimen28) at velocity C around the disk circumference in opposite directions, respectively. Equivalently, we may consider the Sagnac experiment performed in a conveyor belt, as shown in Figure 2, which has been used to describe a modified, but equivalent, "linearized" version of the Sagnac effect.29

Figure 1 Clock O∗(originally at point A) is fixed on the circumference of the disk of radius Rrotating counter-clockwise at the angular velocity ωwith respect to the laboratory frame S. Two light signals are sent around the circumference of the disk in opposite directions starting from clock O∗, which measures the time span on the return of each signal to O∗.

We wish to know the time span observed by O∗ on the return of each signal on the disk. First, we consider the description given by an inertial observer O at the center of the disk stationary in the laboratory frame S. The observer O∗ at R is co-moving with the disk with angular velocity ω and possesses the tangential speed v=ωR MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG2bGaeyypa0JaeqyYdCNaamOuaaaa@3B49@ . If the laboratory frame S is chosen as the preferred frame, the description of the Sagnac effect and corresponding time span variation Δt= t t + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG0bGaeyypa0JaamiDa8aadaWgaaqcfasaa8qa cqGHsisla8aabeaajuaGpeGaeyOeI0IaamiDa8aadaWgaaqcfasaa8 qacqGHRaWka8aabeaaaaa@4050@ in such a frame is the same for Newtonian mechanics and relativistic theories with any synchronization. In general, with different speeds C + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWdamaaBaaajuaibaWdbiabgUcaRaqcfa4daeqaaaaa @395A@ and C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWdamaaBaaajuaibaWdbiabgkHiTaqcfa4daeqaaaaa @3965@ in the counter-rotating (+ clockwise) and co-rotating (- counterclockwise) directions, respectively, the result, for the propagation time t + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiDam aaBaaajuaibaGaey4kaScajuaGbeaaaaa@393D@ and t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiDam aaBaaajuaibaGaeyOeI0cajuaGbeaaaaa@3948@ of the signal in S, is

c ± = 2 π γ R t ± * c ± v , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGJbWdamaaBaaajuaibaWdbiabgglaXcWdaeqaaKqba+qa cqGH9aqpdaWcaaWdaeaapeGaaGOmaiabec8aWjabeo7aNjaadkfaa8 aabaWdbiaadshapaWaa0baaKqbGeaapeGaeyySaelapaqaa8qacaGG QaaaaaaajuaGcqWIdjYocaWGJbGaeyySaeRaamODaiaacYcaaaa@4B44@ (1)

where L=πR MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbGaeyypa0JaeqiWdaNaamOuaaaa@3B0F@ and Δt= t t + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG0bGaeyypa0JaamiDa8aadaWgaaqcfasaa8qa cqGHsisla8aabeaajuaGpeGaeyOeI0IaamiDa8aadaWgaaqcfasaa8 qacqGHRaWka8aabeaaaaa@4050@ The local time of O∗ runs slower than the time of O by the time dilation factor , so that for O∗ the light signals take the times t ± * = t ± /γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiabgglaXcWdaeaapeGaaiOk aaaajuaGcqGH9aqpcaWG0bWdamaaBaaajuaibaWdbiabgglaXcWdae qaaKqba+qacaGGVaGaeq4SdCgaaa@42B8@  and,

Δ t * = t * t + * = 2L( C + C +2v ) γ( C + +v)( C v ) 4γLv c 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG0bWdamaaCaaajuaibeqaa8qacaGGQaaaaKqb akabg2da9iaadshapaWaa0baaKqbGeaapeGaeyOeI0capaqaa8qaca GGQaaaaKqbakabgkHiTiaadshapaWaa0baaKqbGeaapeGaey4kaSca paqaa8qacaGGQaaaaKqbakabg2da9maalaaapaqaa8qacaaIYaGaam itamaabmaapaqaa8qacaWGdbWdamaaBaaajuaibaWdbiabgUcaRaqc fa4daeqaa8qacqGHsislcaWGdbWdamaaBaaajuaibaWdbiabgkHiTa WdaeqaaKqba+qacqGHRaWkcaaIYaGaamODaaGaayjkaiaawMcaaaWd aeaapeGaeq4SdCMaaiikaiaadoeapaWaaSbaaKqbGeaapeGaey4kaS capaqabaqcfa4dbiabgUcaRiaadAhacaGGPaWaaeWaa8aabaWdbiaa doeapaWaaSbaaKqbGeaapeGaeyOeI0capaqabaqcfa4dbiabgkHiTi aadAhaaiaawIcacaGLPaaaaaGaeyO0H49aaSaaa8aabaWdbiaaisda cqaHZoWzcaWGmbGaamODaaWdaeaapeGaam4ya8aadaahaaqcfasabe aapeGaaGOmaaaaaaqcfaOaaiilaaaa@69CA@ (2)

where Δ t * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG0bWdamaaCaaajuaibeqaa8qacaGGQaaaaaaa @3A21@ is the time span variation measured by the clock O∗ and the last term corresponds to the case C + = C =c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWdamaaBaaajuaibaWdbiabgUcaRaqcfa4daeqaa8qa cqGH9aqpcaWGdbWdamaaBaaajuaibaWdbiabgkHiTaqcfa4daeqaa8 qacqGH9aqpcaWGJbaaaa@3F2E@ .

With reference to Figure 1, for the observer O∗ on the disk at r=R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGYbGaeyypa0JaamOuaaaa@3978@ , the circumference length corresponds to the proper ground length of the path followed by the signals and is given by 2πγR MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIYaGaeqiWdaNaeq4SdCMaamOuaaaa@3B9B@ . For the usual case C + = C =c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWdamaaBaaajuaibaWdbiabgUcaRaqcfa4daeqaa8qa cqGH9aqpcaWGdbWdamaaBaaajuaibaWdbiabgkHiTaqcfa4daeqaa8 qacqGH9aqpcaWGJbaaaa@3F2E@  , the time spans are t ± =2πR/( c±v ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaBaaajuaibaWdbiabgglaXcWdaeqaaKqba+qa cqGH9aqpcaaIYaGaeqiWdaNaamOuaiaac+cadaqadaWdaeaapeGaam 4yaiabgglaXkaadAhaaiaawIcacaGLPaaaaaa@4529@ . Then, for the Sagnac effect, the average speeds of light in the counter- and co-rotating senses are

c ± = 2πγR t ± * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGJbWdamaaBaaajuaibaWdbiabgglaXcqcfa4daeqaa8qa cqGH9aqpdaWcaaWdaeaapeGaaGOmaiabec8aWjabeo7aNjaadkfaa8 aabaWdbiaadshapaWaa0baaKqbGeaapeGaeyySaelapaqaa8qacaGG Qaaaaaaaaaa@4504@ c±v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGJbGaeyySaeRaamODaaaa@3A76@ ,(3)

Where the results c±v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGJbGaeyySaeRaamODaaaa@3A76@ correspond to the speed of light in vacuum, C=c. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbGaeyypa0Jaam4yaOGaaiOlaaaa@3A16@

Different interpretations of the sagnac effect

Besides other properties, the Sagnac effect is widely recognized as an optical experiment capable of indicating the state of rotation of the interferometer.30 We mention here only some of the several contrasting interpretations of the Sagnac effect, originally thought to disprove special relativity.20 Landau and Lifshitz31 trace the physical cause of the existence of the Sagnac effect in the rotating reference system as due to a difference between the velocities of counter-propagating waves. This velocity difference, which refers to the average velocities in (3), leads to a paradox according to the interpretation of Selleri,22 who shows that when the radius of the rotating disk is increased to R → ∞ keeping constant ωR=v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHjpWDcaWGsbGaeyypa0JaamODaaaa@3B49@ , the local speed of light in an inertial frame would be c + vor c vdepending on the direction of propagation, in disagreement with the unique value c foreseen by the second postulate of special relativity. The existence of Selleri’s paradox has been supported by other papers32-37 where the authors claim that the phase difference of counter-propagating waves in the reference system co-rotating with a ring interferometer, calculated in the context of standard SR, is equal to zero. Instead, Malykin38 argues that we are rather in the presence of a Zeno paradox because, considering that the rotating disk is not an inertial frame, the interpretation of the Sagnac effect is reliant on the equivalence principle and the effect of time dilation in a gravitational field.

The argument against Einstein synchronization can be resumed as follows. Because of the motion of O∗, for the observer O of the laboratory frame S, the paths ± of the signal have different lengths, 2πR/( 1 ±v/c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIYaGaeqiWdaNaamOuaiaac+capaWaaeWaaeaapeGaaGym aiaabccacqGHXcqScaWG2bGaai4laiaadogaa8aacaGLOaGaayzkaa aaaa@4240@ and, since the velocity c of the signal is the same in both directions, it appears obvious that we must have different propagation times t±. For O∗, instead, the propagation paths have the same length 2πγR MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIYaGaeqiWdaNaeq4SdCMaamOuaaaa@3B9B@ but still we have different propagation times, t ± * = t ± /γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiabgglaXcWdaeaapeGaaiOk aaaajuaGcqGH9aqpcaWG0bWdamaaBaaajuaibaWdbiabgglaXcqcfa 4daeqaa8qacaGGVaGaeq4SdCgaaa@42B8@ . Thus, for propagation on paths of the same lengths, if the respective propagation times, t ± * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiabgglaXcWdaeaapeGaaiOk aaaaaaa@3AC8@ , are different, different speeds c ± MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGJbWdamaaBaaajuaibaWdbiabgglaXcWdaeqaaaaa@39F8@ (and not the same speed c) are expected. This argument, which seems to cast doubts about the validity of a universal constant speed of light c, has been expanded by Selleri who claims that the paradox can be solved only if relativistic theories adopt the "natural" absolute synchronization that foresees the different speedsc + v and c - v of expression (3).

In this context, it is worth recalling the arguments of Kassner about Selleri’s paradox. In rebutting Selleri’s claim, Kassner21 attempts to refute the paradox by explaining the Sagnac effect in the frame of the co-moving observer O∗ in two ways: through Minkowsky’s analysis and by means of the absolute synchronization. The point is that, if Einstein synchronization is applied along the closed circular path, the resulting average speed in (3) is c, and not c + v or c - v. Thus, in this case, the Sagnac effect invalidates standard SR. After his Minkowsky analysis, Kassner concludes by acknowledging that “Einstein synchronization fails when performed along a path around a full circle”, i.e., on a closed path on the rotating disk, a failure that has also been observed by Weber39 and earlier by Anandan.40

Thus, in order to account for the resulting unphysical time discontinuity arising from the speeds c+v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGJbGaey4kaSIaamODaaaa@396A@ and cv MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGJbGaeyOeI0IaamODaaaa@3975@ and solve Selleri’s paradox, Kassner introduces the unusual concept of a “time gap” on the rotating disk and states, “the speed of light is c everywhere except at the point on the circle where we put the time gap. The position of this point is arbitrary but there must inevitably be such a point.”

The solution based on Minkowsky’s analysis presented by Kassner has been objected to by Gift41 who rejects Kassner’s adjustable “time gap” based on an unphysical time discontinuity claiming that is a theoretical construct that has no basis in reality. Similarly, Selleri has claimed that "it is not true that the synchronization procedure can be chosen freely because Einstein convention leads to an unacceptable discontinuity in the physical theory". Moreover, according to Selleri "Very probably the above discontinuity is the origin of the synchronization problems met with by the Global Positioning System". In relation to this, Gift emphasizes that the failure of Einstein synchronization is well known to GPS engineers who discovered41,42 that they cannot synchronize GPS clocks fixed on the rotating Earth using Einstein synchronization.43,44

The second argument presented by Kassner21 to solve Selleri’s paradox consists of replacing Einstein synchronization with the absolute synchronization. Thus, instead of the Lorentz transformations and without citing the related literature, Kassner makes use of the Tangherlini transformations. Considering that the one-way speed of light is synchronization dependent, for Kassner a difference from c does not constitute a problem because the observable two-way (average) speed of light remains c, as in the interpretation of the Michelson-Morley experiment. In fact, if synchronization is arbitrary (as also assumed by Kassner), the two synchronizations are equivalent and provide the same observable result, as they actually do within Kassner’s basic set up and hypotheses. Then, although acknowledging the anisotropy due to the different speeds c+v and cv in Selleri’s paradox, Kassner argues that the difference is to be interpreted in terms of the arbitrariness of synchronization and, thus, does not invalidate the second postulate of special relativity. Yet, in Sect. 4 we demonstrate that, in general, Einstein synchronization and the absolute synchronization are not physically equivalent. Therefore, the assumption of conventionalism is groundless, so that Kassner’s argument based on it is untenable. Moreover, while adhering to the conventionality of the speed of light and the equivalence of the two synchronizations, Kassner does not discuss the consequences of this assumption on the countless physical theories traditionally based on the LT. These physical theories would be substantially modified if they were based on and developed by means of the TT. In the next section, we review the arguments in favor and against Einstein synchronization and the problem of superluminal speeds within the context of the "linearized" Sagnac effect, shown in Figure 2.

Figure 2 The conveyor belt system represents a "linearized" version of the Sagnac effect. The belt stands for a flexible tube and the signals propagate in opposite directions along the closed path inside the tube, acting as a wave-guide and rotating counter-clockwise while leaning over the two pulleys separated by the arm AB of fixed length L. The inertial frame S’’ is co-moving with the upper part of the tube with velocity vwith respect to the arm AB, stationary in the laboratory frame S, while frame S’ and clock O*are co-moving with the lower part of the tube in the opposite direction.

Linear Sagnac effect, clock synchronization and superluminal "ground" speed

With reference to Figure 2, we assume that, if the distance L between the two pulleys of small radius r is long enough (r << L), the time τR/v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHepaDcqWIdjYocaWGsbGaai4laiaadAhaaaa@3C1F@ spent by the signals in the short portion of the tube in contact with the pulleys will be negligible with respect to the total time of flight and we may omit it. Furthermore, the acceleration a= v 2 /r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGHbGaeyypa0JaamODa8aadaahaaqcfasabeaapeGaaGOm aaaajuaGcaGGVaGaamOCaaaa@3CEE@ that could cause a change in the flow of time of clocks co-moving with the tube is constant and depends on vand r. Therefore, the change due to the effect of the acceleration does not depend on L and, if the distance L is long enough, becomes negligible with respect to the time variation Δ t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG0bWdamaaCaaajuaibeqaa8qacqGHxiIkaaaa aa@3A61@ , proportional to L, observed in the Sagnac effect.

If the tube where the light signals propagate is filled with a co-moving medium (for example, a fluid of refractive index n where the local speed of light is C 0 =c/n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWdamaaCaaabeqcfasaa8qacaaIWaaaaKqbakabg2da 9iaadogacaGGVaGaamOBaaaa@3CB7@ ) the speed of light is no longer c but C0. In this more general case, the two signals circumnavigate the tube in opposite directions with the same local (ground) speed C * = C 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWdamaaCaaajuaibeqaa8qacaGGQaaaaKqbakabg2da 9iaadoeapaWaaWbaaKqbGeqabaWdbiaaicdaaaaaaa@3C0F@ with respect to the tube.

  1. For special relativity with the LT.

For simplicity we consider here the case C 0 =c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWdamaaCaaajuaibeqaa8qacaaIWaaaaKqba+aacqGH 9aqpcaWGJbaaaa@3B20@ We have,

t + = 2L c+v ; t = 2L cv MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaBaaajuaibaWdbiabgUcaRaWdaeqaaKqba+qa cqGH9aqpdaWcaaWdaeaapeGaaGOmaiaadYeaa8aabaWdbiaadogacq GHRaWkcaWG2baaaiaacUdacaWG0bWdamaaBaaajuaibaWdbiabgkHi TaWdaeqaaKqba+qacqGH9aqpdaWcaaWdaeaapeGaaGOmaiaadYeaa8 aabaWdbiaadogacqGHsislcaWG2baaaaaa@48B2@ (4)

Δ t * = Δt γ = t t + γ =4γ vL c 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG0bWdamaaCaaabeqcfasaa8qacaGGQaaaaKqb akabg2da9maalaaapaqaa8qacqqHuoarcaWG0baapaqaa8qacqaHZo WzaaGaeyypa0ZaaSaaa8aabaWdbiaadshapaWaaSbaaKqbGeaapeGa eyOeI0cajuaGpaqabaWdbiabgkHiTiaadshapaWaaSbaaKqbGeaape Gaey4kaScajuaGpaqabaaabaWdbiabeo7aNbaacqGH9aqpcaaI0aGa eq4SdC2aaSaaa8aabaWdbiaadAhacaWGmbaapaqaa8qacaWGJbWdam aaCaaajuaibeqaa8qacaaIYaaaaaaaaaa@5161@  (5)

It can be shown that the Sagnac effect, related to Δ t * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG0bWdamaaCaaabeqcfasaa8qacaGGQaaaaaaa @3A21@ is foreseen to be independent of the signals ground speed C 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWdamaaCaaajuaibeqaa8qacaaIWaaaaaaa@3896@ , for both the circular and linear cases.

  1. For a preferred frame theory based on the TT.

The TT between the preferred frame Sand the moving frame S′ are: x'=γ( xvt );y'=y;z'=z;t'=t/γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bGaai4jaiabg2da9iabeo7aN9aadaqadaqaa8qacaWG 4bGaeyOeI0IaamODaiaadshaa8aacaGLOaGaayzkaaWdbiaacUdaca WG5bGaai4jaiabg2da9iaadMhacaGG7aGaamOEaiaacEcacqGH9aqp caGG6bGaai4oaiaacshacaGGNaGaeyypa0JaamiDaiaac+cacqaHZo Wzaaa@5032@ . Using the TT, with C + = C 0 / γ 2 v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWdamaaBaaajuaibaWdbiabgUcaRaqcfa4daeqaa8qa cqGH9aqpcaWGdbWdamaaCaaajuaibeqaa8qacaaIWaaaaKqbakaac+ cacqaHZoWzpaWaaWbaaeqajuaibaWdbiaaikdaaaqcfaOaeyOeI0Ia amODaaaa@42E9@ and C  = C 0 / γ 2 +v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWaaSbaaKqbGeaacqGHsislaeqaaKqbakaabccacqGH 9aqpcaWGdbWdamaaCaaajuaibeqaa8qacaaIWaaaaKqbakaac+cacq aHZoWzpaWaaWbaaKqbGeqabaWdbiaaikdaaaqcfaOaey4kaSIaamOD aaaa@434E@ , we have

t + = 2L C + +v = 2 γ 2 L C 0 ; t = 2L C v = 2 γ 2 L C 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaBaaajuaibaWdbiabgUcaRaqcfa4daeqaa8qa cqGH9aqpdaWcaaWdaeaapeGaaGOmaiaadYeaa8aabaWdbiaadoeapa WaaSbaaKqbGeaapeGaey4kaScapaqabaqcfa4dbiabgUcaRiaadAha aaGaeyypa0ZaaSaaa8aabaWdbiaaikdacqaHZoWzpaWaaWbaaKqbGe qabaWdbiaaikdaaaqcfaOaamitaaWdaeaapeGaam4qa8aadaahaaqa bKqbGeaapeGaaGimaaaaaaqcfaOaai4oaiaadshapaWaaSbaaeaape GaeyOeI0capaqabaWdbiabg2da9maalaaapaqaa8qacaaIYaGaamit aaWdaeaapeGaam4qa8aadaWgaaqcfasaa8qacqGHsisla8aabeaaju aGpeGaeyOeI0IaamODaaaacqGH9aqpdaWcaaWdaeaapeGaaGOmaiab eo7aN9aadaahaaqcfasabeaapeGaaGOmaaaajuaGcaWGmbaapaqaa8 qacaWGdbWdamaaCaaajuaibeqaa8qacaaIWaaaaaaaaaa@5CAD@

Δ t * = Δt γ = t   t + γ =0, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG0bWdamaaCaaajuaibeqaa8qacaGGQaaaaKqb akabg2da9maalaaapaqaa8qacqqHuoarcaWG0baapaqaa8qacqaHZo WzaaGaeyypa0ZaaSaaa8aabaWdbiaadshapaWaaSbaaKqbGeaapeGa eyOeI0capaqabaqcfa4dbiabgkHiTiaacckacaWG0bWdamaaBaaaju aibaWdbiabgUcaRaqcfa4daeqaaaqaa8qacqaHZoWzaaGaeyypa0Ja aGimaiaacYcaaaa@4D5D@

which implies no Sagnac effect independently of C 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWdamaaCaaajuaibeqaa8qacaaIWaaaaaaa@3895@ , the same result as in Newtonian mechanics.

From a mathematical perspective, the internal consistency of the LT is not questionable and not questioned here. In the following, we try to adopt an objective point of view, simply presenting the arguments in favor or against Einstein synchronization, with the aim of clarifying the issue of superluminal average speeds in the context of the Sagnac effect. The incompatibility of Einstein synchronization over a closed path has been discussed above in Sect. 2. The objection that could be made in the context of the circular Sagnac effect is that the rotating frame is a non-inertial frame and, thus, Einstein synchronization is not applicable in such a frame. Here, with the linear Sagnac effect, we may apply Einstein synchronization separately to the inertial frames S′′ and S′ co-moving respectively with the upper and lower part of the tube. Still, we find that the mentioned incompatibility subsists and we are met once more with superluminal speeds and the conflicting point of views, or "realities", of S’and S’’.

The Lorentz transformations from S’’ to S’ are,

x =  γ ( x  w t );  t =  γ ( t     w x c 2 ), MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG4bWdayaafaWdbiabg2da9iaabckacuaHZoWzpaGbauaa peWaaeWaa8aabaWdbiqadIhapaGbayaapeGaeyOeI0IaaeiOaiaadE haceWG0bWdayaagaaapeGaayjkaiaawMcaaiaacUdacaqGGcGabmiD a8aagaqba8qacqGH9aqpcaqGGcGafq4SdC2dayaafaWdbmaabmaapa qaa8qaceWG0bWdayaagaWdbiaacckacqGHsislcaGGGcGaaiiOamaa laaapaqaa8qacaWG3bGabmiEa8aagaGbaaqaa8qacaWGJbWdamaaCa aajuaibeqaa8qacaaIYaaaaaaaaKqbakaawIcacaGLPaaacaGGSaaa aa@5696@ (6)

with w=2v/( 1+ v 2 / c 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG3bGaeyypa0JaaGOmaiaadAhacaGGVaWaaeWaa8aabaWd biaaigdacqGHRaWkcaWG2bWdamaaCaaabeqcfasaa8qacaaIYaaaaK qbakaac+cacaWGJbWdamaaCaaajuaibeqaa8qacaaIYaaaaaqcfaOa ayjkaiaawMcaaaaa@445E@  and γ'= ( 1 w 2 / c 2 ) 1/2 = γ 2 ( 1+ v 2 / c 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHZoWzcaqGNaGaeyypa0ZaaeWaa8aabaWdbiaaigdacqGH sislcaWG3bWdamaaCaaajuaibeqaa8qacaaIYaaaaKqbakaac+caca WGJbWdamaaCaaabeqaa8qacaaIYaaaaaGaayjkaiaawMcaa8aadaah aaqabKqbGeaapeGaeyOeI0IaaGymaiaac+cacaaIYaaaaKqbakabg2 da9iabeo7aN9aadaahaaqcfasabeaapeGaaGOmaaaajuaGdaqadaWd aeaapeGaaGymaiabgUcaRiaadAhapaWaaWbaaeqajuaibaWdbiaaik daaaqcfaOaai4laiaadogapaWaaWbaaeqajuaibaWdbiaaikdaaaaa juaGcaGLOaGaayzkaaaaaa@5460@ .

The crucial question for standard SR is: How does the observer O∗ manage to explain the observed time span difference for two signals counter-propagating along a path of the same length and with the same ground speed C 0 =c? MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWdamaaCaaajuaibeqaa8qacaaIWaaaaKqbakabg2da 9iaadogacaGG=aaaaa@3BD4@

For our example, we choose a special case where the effect of non-conservation of simultaneity is made apparent. For theexperimental set-up shown in Figure 3, we assume that the three origins O, O′, O′′ of frames S, S’, S’’, coincide at the initial time t= t = t = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bGaeyypa0JaamiDa8aadaahaaqcfasabeaapeGaeyOm GikaaKqbakabg2da9iaadshapaWaaWbaaKqbGeqabaWdbiabgkdiIk abgkdiIcaajuaGcqGH9aqpcaqGGaGaaGimaaaa@4478@ , while the pulley at A is positioned at the distance L o =vL/ C 0 =vL/c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbWdamaaBaaajuaibaWdbiaad+gaa8aabeaajuaGpeGa eyypa0JaamODaiaadYeacaGGVaGaam4qa8aadaahaaqcfasabeaape GaaGimaaaajuaGcqGH9aqpcaWG2bGaamitaiaac+cacaWGJbaaaa@43F5@ to the left with respect to O, and the pulley at B at the distance L L o =L( 1 v/ C 0 ) =L( 1 v/c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbGaeyOeI0Iaamita8aadaWgaaqcfasaa8qacaWGVbaa juaGpaqabaWdbiabg2da9iaadYeapaWaaeWaaeaapeGaaGymaiaabc cacqGHsislcaWG2bGaai4laiaadoeapaWaaWbaaeqajuaibaWdbiaa icdaaaaajuaGpaGaayjkaiaawMcaa8qacaqGGaGaeyypa0Jaamita8 aadaqadaqaa8qacaaIXaGaaeiiaiabgkHiTiaadAhacaGGVaGaam4y aaWdaiaawIcacaGLPaaaaaa@4E6A@ to the right. For our purposes, it is sufficient to consider the clockwise propagation of a light signal and determine its time of flight starting from the clock O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaajuaibeqaa8qacqGHxiIkaaaaaa@38D6@ , co-moving with the tube and coincident with the origin O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaajuaibeqaa8qacqGHYaIOcqGHYaIOaaaa aa@3AE7@ at t = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaCaaabeqcfasaa8qacqGHYaIOcqGHYaIOaaqc faOaeyypa0Jaaeiiaiaaicdaaaa@3DFD@ , and returning to O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaajuaibeqaa8qacqGHxiIkaaaaaa@38D6@ after a complete cycle. The single clock O∗ alone is used to determine the time interval of the closed trip and, therefore, no synchronization procedure is involved. The two physical realities of S’’ and S’ will be confronted.

Figure 3 The linear Sagnac effect described from the perspective of frame S′′. The arm AB of length L, fixed in the laboratory frame S, moves with velocity vwith respect to S’’, stationary with the upper part of the tube. Frame S’, co-moving with the lower part of the tube, moves with velocity w with respect to S′′ and velocity v with respect to S. The origins O′′, O′ coincide with the origin O of frame S at t = t′ = t′′=0, when

the light signal is sent counter-propagating at the velocity C 0 =c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWdamaaCaaabeqcfasaa8qacaaIWaaaaKqbakabg2da 9iaadogaaaa@3B11@ . At the time t B '' = t A '' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiaadkeaa8aabaWdbiaacEca caGGNaaaaKqbakabg2da9iaadshapaWaa0baaKqbGeaapeGaamyqaa WdaeaapeGaai4jaiaacEcaaaaaaa@3F80@ , the signal has reached point B while clock O∗ has simultaneously reached point A.

Physical reality and total time for the clockwise trip according to S’’

Clocks at rest in some inertial reference frame may be “internally” synchronized according to Einstein’s procedure. Without reference to other inertial frames in relative motion, to say that the clocks at rest along the x′′ of the inertial frame S’’ are synchronized, it implies, for both Newton and Einstein, that in S’’ the clocks show the same readings simultaneously. Within a given inertial frame, “internal” simultaneity or, simply, simultaneity, is a reflection of internally synchronized clocks. Then, physical phenomena are described in S’’ as a function of the time t′′, the same time simultaneously displayed by all the synchronized clocks. If a hypothetical signal is sent at infinite speed along the x′′ axis of S’’, all the synchronized clocks on the x′′ axis will display the same time simultaneously when the signal passes by. At least in principle, clocks can be properly (internally) synchronized by signals with either finite or infinite speed, if the latter were available. The same conclusions apply for S’. For both Newton and Einstein, the concept of (internal) simultaneity applies when keeping within the physical reality of any inertial frame. The difference between Newton and Einstein shows up only when we relate two inertial frames in relative motion by means of the space-time coordinates transformations. In fact, as is well-known, if the hypothetical signal sent along the x′′ axis of S’’ had infinite speed also along the x′ axis of S’, there would be a conflict with the Lorentz transformations that assume the relativity of time and foresee different, non-simultaneous time readings between the two sets of clocks of S’’and S’. Supporters of special relativity overcome this problem by arguing that a signal with infinite speed is not operationally acceptable within the theory, which requires c to be the maximum speed permitted. However, given that the concept of simultaneity holds (internally) for both Newton and Einstein when keeping within the physical reality of a given frame, we may say that physical phenomena are portrayed in that frame as a function of the same "absolute" time. Thus, the restriction imposed by the finite speed c does not impede to conceive that, if spatially separated clocks are synchronized, they will display the same time simultaneously when a hypothetical infinite-speed signal passes by. The same is true for the synchronized clocks of any other inertial frame.

What is to expect in any case, is that the related physical realities of the two inertial frames in relative motion are compatible and reflect the same objective physical reality. In our analysis of the linear Sagnac effect, we shall consider the compatibility of the internal simultaneity of an inertial frame S’’ with that of another generic inertial frame S’ in relative motion with respect to S’’. As shown below, we find that internal simultaneity is not compatible with Einstein synchronization and the Lorentz transformations, because the latter imply, for S’’ and S’, two contrasting and irreconcilable physical realities.

We calculate now the time taken by the signal to reach point B after leaving the clock O∗. Since, at t = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaCaaajuaibeqaa8qacqGHYaIOcqGHYaIOaaqc faOaeyypa0Jaaeiiaiaaicdaaaa@3DFD@ point B is at the distance L(1−v/c)/γ to the right from the origin O = O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaajuaibeqaa8qacqGHYaIOcqGHYaIOaaqc faOaeyypa0Jaam4ta8aadaahaaqcfasabeaapeGaey4fIOcaaaaa@3EAD@ and point A is at vL/γc to the left, the signal reaches B at the time

t B '' = L γc = t A '' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiaadkeaa8aabaWdbiaacEca caGGNaaaaKqbakabg2da9maalaaapaqaa8qacaWGmbaapaqaa8qacq aHZoWzcaWGJbaaaiabg2da9iaadshapaWaa0baaKqbGeaapeGaamyq aaWdaeaapeGaai4jaiaacEcaaaaaaa@4434@ (7)

and at the same time t A '' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiaadgeaa8aabaWdbiaacEca caGGNaaaaaaa@3A48@  clock O = O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaqcfaOaeyyp a0Jaam4ta8aadaahaaqcfasabeaapeGaeyOmGiQaeyOmGikaaaaa@3EAD@ reaches point A, the two events at A and B being simultaneous. In the time span t B '' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiaadkeaa8aabaWdbiaacEca caGGNaaaaaaa@3A49@  the signal has covered the ground length L/γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbGaai4laiabeo7aNbaa@39CF@ of the upper part AB of the tube at the local speed c. As shown in Figure 3, after the clock O∗, initially co-moving with the origin O′′, reaches point A, it turns around the pulley and goes toward point B, co-moving now with the lower part AB of the tube and the x′ axis of S′.

Frame S’’and S′ agree on and share the common value for the total invariant proper length 2γL of the tube where propagation takes place. It follows that both S’’ and S′ agree that, after changing direction at B, the signal is bound to cover the remaining proper length L'=2γLL/γ=γ'L/γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbGaae4jaiabg2da9iaaikdacqaHZoWzcaWGmbGaeyOe I0Iaamitaiaac+cacqaHZoWzcqGH9aqpcqaHZoWzcaqGNaGaamitai aac+cacqaHZoWzaaa@46F3@  from B to O∗ by traveling entirely in the lower part of the tube until clock O∗ is met, as apparent from Figures 3, 4, and 5. For S’’, the ground length of the lower part of the tube is γ L/γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHZoWzpaWaaWbaaKqbGeqabaWdbiabgkdiIcaajuaGcaWG mbGaai4laiabeo7aNbaa@3DF3@ , which (because of its relative motion) is seen as being contracted by the factor γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHZoWzpaWaaWbaaKqbGeqabaWdbiabgkdiIcaaaaa@3A3A@ to L/γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbGaai4laiabeo7aNbaa@39CF@ and, thus, it fits exactly the length L/γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbGaai4laiabeo7aNbaa@39CF@ of the segment AB. Conversely and in contrast with the reality of S’’, frame S′ estimates that it is the upper part of the tube that has ground length γ L/γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHZoWzpaWaaWbaaKqbGeqabaWdbiabgkdiIcaajuaGcaWG mbGaai4laiabeo7aNbaa@3DF3@ , seen by S′ as being contracted by the factor γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHZoWzpaWaaWbaaKqbGeqabaWdbiabgkdiIcaaaaa@3A3A@ to the value L/γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbGaai4laiabeo7aNbaa@39CF@ . At this point, detractors of Einstein synchronization may point out the following two arguments against the validity of Einstein synchronization.

  1. They may first argue that the time interval Δ t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG0bWdamaaCaaajuaibeqaa8qacqGHYaIOaaaa aa@3AF2@ , corresponding to propagation in the lower part of the tube, can be inferred by S’’, knowing that the ground speed is c also in that lower section of the tube. In fact, the ground length of the lower part of the tube, L = γ L/γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbWdamaaCaaajuaibeqaa8qacqGHYaIOaaqcfaOaeyyp a0Jaeq4SdC2damaaCaaajuaibeqaa8qacqGHYaIOaaqcfaOaamitai aac+cacqaHZoWzaaa@4247@ , to be covered by the signal, and the corresponding ground speed c of the signal are known. Thus, an observer of S’’ may say that, whatever may be the time displayed by a clock of S′ when the signal starts propagating along the x′ axis, this clock will have advanced by the time interval Δ t = L /c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG0bWdamaaCaaajuaibeqaa8qacqGHYaIOaaqc faOaeyypa0Jaamita8aadaahaaqcfasabeaapeGaeyOmGikaaKqbak aac+cacaWGJbaaaa@416F@ , after the signal has completely covered the ground distance L′. About the corresponding elapsed time of clock O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaaaaa@38D6@ , we need to consider that, when co-moving in the lower part of the tube, clock O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaaaaa@38D6@ is sharing the same properties of the clocks fixed to the x′ axis. Therefore, since for S’’clock O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaaaaa@38D6@ is always co-moving with the lower part of the tube while the signal is propagating along this lower part, the time interval by which O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaaaaa@38D6@ advances must be the same as that of any other clock of S′. It follows that, as inferred by S’’, the proper length L = γ L/γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbWdamaaCaaajuaibeqaa8qacqGHYaIOaaqcfaOaeyyp a0Jaeq4SdC2damaaCaaajuaibeqaa8qacqGHYaIOaaqcfaOaamitai aac+cacqaHZoWzaaa@4247@ must be covered by the signal in the time interval

    Δ t =Δ t * = L c = γ L γc , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarceWG0bWdayaafaWdbiabg2da9iabfs5aejaadsha paWaaWbaaKqbGeqabaWdbiaacQcaaaqcfaOaeyypa0ZaaSaaa8aaba WdbiqadYeapaGbauaaaeaapeGaam4yaaaacqGH9aqpcuaHZoWzpaGb auaapeWaaSaaa8aabaWdbiaadYeaa8aabaWdbiabeo7aNjaadogaaa Gaaiilaaaa@488E@ (8)

as measured by clock O∗ or any other clock of S′.

S’’ can claim the above result to be a necessary consequence of the observed physical reality shown in Fig. 3, reflecting the simultaneity of the two events: signal at point B and clock O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaaaaa@38D6@ at point A. For S’’, this reality requires that the signal has still to travel the ground distance L = γ L/γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbWdamaaCaaajuaibeqaa8qacqGHYaIOaaqcfaOaeyyp a0Jaeq4SdC2damaaCaaajuaibeqaa8qacqGHYaIOaaqcfaOaamitai aac+cacqaHZoWzaaa@4247@ at the ground speed c after the clock L = γ L/γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbWdamaaCaaajuaibeqaa8qacqGHYaIOaaqcfaOaeyyp a0Jaeq4SdC2damaaCaaajuaibeqaa8qacqGHYaIOaaqcfaOaamitai aac+cacqaHZoWzaaa@4247@ reaches point A and, therefore, result (8) must inevitably hold. Then, for S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWGtbWdayaagaaaaa@3799@ , the total time t + ' = t + * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiabgUcaRaWdaeaapeGaai4j aaaajuaGcqGH9aqpcaWG0bWdamaaDaaajuaibaWdbiabgUcaRaWdae aapeGaaiOkaaaaaaa@3E64@  displayed by clock O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaaaaa@38D6@ for the clockwise signal propagation must be,

t + * = t B '' +Δ t =2γ L c . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiabgUcaRaWdaeaapeGaaiOk aaaajuaGcqGH9aqpcaWG0bWdamaaDaaajuaibaWdbiaadkeaa8aaba WdbiaacEcacaGGNaaaaKqbakabgUcaRiabfs5aejqadshapaGbauaa peGaeyypa0JaaGOmaiabeo7aNnaalaaapaqaa8qacaWGmbaapaqaa8 qacaWGJbaaaiaac6caaaa@4910@ (9)

For the counter-clockwise propagation, the result is the same. Therefore, the propagation time difference for the light signals is Δ t * = t * t + * =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG0bWdamaaCaaajuaibeqaa8qacaGGQaaaaKqb akabg2da9iaadshapaWaa0baaKqbGeaapeGaeyOeI0capaqaa8qaca GGQaaaaKqbakabgkHiTiaadshapaWaa0baaKqbGeaapeGaey4kaSca paqaa8qacaGGQaaaaKqbakabg2da9iaaicdaaaa@45B7@ , and no Sagnac effect is foreseen if the local speed of the signal is C 0 =c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWdamaaCaaajuaibeqaa8qacaaIWaaaaKqbakabg2da 9iaadogaaaa@3B11@ everywhere along the tube.

b) The second argument that detractors of standard special relativity may point out are the following.

Let us suppose that in (9) we have Δ t Δ t SR ' =γ ( 1v/c ) 2 L/c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarceWG0bWdayaafaWdbiabgkziUkabfs5aejaadsha paWaa0baaKqbGeaapeGaam4uaiaadkfaaKqba+aabaWdbiaacEcaaa Gaeyypa0Jaeq4SdC2aaeWaa8aabaWdbiaaigdacqGHsislcaWG2bGa ai4laiaadogaaiaawIcacaGLPaaapaWaaWbaaKqbGeqabaWdbiaaik daaaqcfaOaamitaiaac+cacaGGJbaaaa@4DA8@ , in agreement with special relativity (see (4) and (5)) and experimental observation. Then, we may write

t + * = t B '' +Δ t SR ' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiabgUcaRaWdaeaapeGaaiOk aaaajuaGcqGH9aqpcaWG0bWdamaaDaaajuaibaWdbiaadkeaa8aaba WdbiaacEcacaGGNaaaaKqbakabgUcaRiabfs5aejaadshapaWaa0ba aKqbGeaapeGaam4uaiaadkfaa8aabaWdbiaacEcaaaaaaa@45AB@ (10)

and, correspondingly,

2γL( 1v/c ) c = L/γ c + γ L/γ c  , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaaGOmaiabeo7aNjaadYeadaqadaWdaeaa peGaaGymaiabgkHiTiaadAhacaGGVaGaam4yaaGaayjkaiaawMcaaa WdaeaapeGaam4yaaaacqGH9aqpdaWcaaWdaeaapeGaamitaiaac+ca cqaHZoWza8aabaWdbiaadogaaaGaey4kaSYaaSaaa8aabaWdbiqbeo 7aN9aagaqba8qacaWGmbGaai4laiabeo7aNbWdaeaaceWGJbGbamba aaWdbiaacckacaGGSaaaaa@4F5F@  (11)

where, on the lhs of (11), 2γL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIYaGaeq4SdCMaamitaaaa@39D8@ is the total length covered by the signal, and c/(1−v/c) ≃c+v is the corresponding superluminal average speed. On the rhs of (11), the length L/γ of the first term represents the length covered by the signal in the upper part of the tube and c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabm4yay aataaaaa@3786@ the corresponding local speed. The length γ L/γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHZoWzpaWaaWbaaKqbGeqabaWdbiabgkdiIcaajuaGcaWG mbGaai4laiabeo7aNbaa@3DF3@ in the second term represents the length covered by the signal in the lower part of the tube and c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqcfayaamaayeaaba Gaam4yaaqaaaGaay5n+daaaa@394D@  the corresponding local speed. Simple algebra indicates that the local speed c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4yaa aa@376D@ , in the lower part of the tube, must be c c+2v , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaagbaqaaiaadogaaeaaaiaawEJ=aKqzGeGaeS4qISJaae4y aiabgUcaRiaaikdacaqG2bGaaiiOaiaacYcaaaa@4089@ , i.e., superluminal, in contradiction with Einstein’s second postulate, as pointed out by Selleri. It follows that internal simultaneity, as assumed by both Newton and Einstein, is not compatible with the Lorentz transformations based on Einstein synchronization.

Physical reality and total time for the trip according to S′

We have shown in the previous section that, according to the physical reality of S’’, the local speed of light in the lower part of the tube, co-moving with S’, must be superluminal. In order to restore the local speed of light to c in this lower section of the tube, frame S’ has to synchronize clocks ("de-synchronize clocks", would argue a detractor) by means of Einstein synchronization, so that the LT can be used.

In this way, the physical reality of S’corresponds now to the reality of S’’changed, or "transformed", by means of the LT (6) as shown in Figure 4. It follows that, according to S’, simultaneity no longer holds because the signal reaches point B at the time

t B ' = γ t B '' ( 1    w c )=γ L c ( 1    v c ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiaadkeaa8aabaWdbiaacEca aaqcfaOaeyypa0Jafq4SdC2dayaafaWdbiaadshapaWaa0baaKqbGe aapeGaamOqaaWdaeaapeGaai4jaiaacEcaaaqcfa4aaeWaa8aabaWd biaaigdacaGGGcGaeyOeI0IaaiiOaiaacckadaWcaaWdaeaapeGaam 4DaaWdaeaapeGaam4yaaaaaiaawIcacaGLPaaacqGH9aqpcqaHZoWz daWcaaWdaeaapeGaamitaaWdaeaapeGaam4yaaaadaqadaWdaeaape GaaGymaiaacckacqGHsislcaGGGcGaaiiOamaalaaapaqaa8qacaWG 2baapaqaa8qacaWGJbaaaaGaayjkaiaawMcaa8aadaahaaqabKqbGe aapeGaaGOmaaaaaaa@58F0@

while clock O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaaaaa@38D6@ reaches point A at the later time

t A ' = t B ' +δ t = γ t B '' =γ L c ( 1 +  v 2 c 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWcpaWaa0baaKqbGeaajugWa8qacaWGbbaajuaipaqa aKqzadWdbiaacEcaaaqcfaOaeyypa0JaamiDa8aadaqhaaqcfasaa8 qacaWGcbaapaqaa8qacaGGNaaaaKqbakabgUcaRiabes7aKjqadsha paGbauaapeGaeyypa0Jafq4SdC2dayaafaWdbiaadshapaWaa0baaK qbGeaapeGaamOqaaWdaeaapeGaai4jaiaacEcaaaqcfaOaeyypa0Ja eq4SdC2aaSaaa8aabaWdbiaadYeaa8aabaWdbiaadogaaaWaaeWaa8 aabaWdbiaaigdacaGGGcGaey4kaSIaaiiOamaalaaapaqaa8qacaWG 2bWdamaaCaaabeqcfasaa8qacaaIYaaaaaqcfa4daeaapeGaam4ya8 aadaahaaqcfasabeaapeGaaGOmaaaaaaaajuaGcaGLOaGaayzkaaaa aa@5BE6@

being

δ t = t A ' t B ' = γ t B '' w c =2γL v c 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH0oazceWG0bWdayaafaWdbiabg2da9iaadshapaWaa0ba aKqbGeaapeGaamyqaaWdaeaapeGaai4jaaaajuaGcqGHsislcaWG0b WdamaaDaaajuaibaWdbiaadkeaa8aabaWdbiaacEcaaaqcfaOaeyyp a0Jafq4SdC2dayaafaWdbiaadshapaWaa0baaKqbGeaapeGaamOqaa WdaeaapeGaai4jaiaacEcaaaqcfa4aaSaaa8aabaWdbiaadEhaa8aa baWdbiaadogaaaGaeyypa0JaaGOmaiabeo7aNjaadYeadaWcaaWdae aapeGaamODaaWdaeaapeGaam4ya8aadaahaaqcfasabeaapeGaaGOm aaaaaaqcfaOaaiilaaaa@547E@ (12)

Figure 4 The linear Sagnac effect described from the perspective of frame S′ at the time t B ' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWcpaWaa0baaKqbGeaajugWa8qacaWGcbaajuaipaqa aKqzadWdbiaacEcaaaaaaa@3C33@ . The origins O′′ and O′ were coinciding with the origin O at t = t′ = t′′= 0. At the time t B ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiaadkeaa8aabaWdbiaacEca aaaaaa@399D@ the signal has reached point B while clock O* has not yet reached point A. At time t B ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiaadkeaa8aabaWdbiaacEca aaaaaa@399D@ , the reading ofclock O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaaaaa@38D6@ is t B '' = t B ' / γ < t B '' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiaadkeaa8aabaWdbiaacEca caGGNaaaaKqbakabg2da9iaadshapaWaa0baaKqbGeaapeGaamOqaa WdaeaapeGaai4jaaaajuaGcaGGVaGafq4SdC2dayaafaWdbiabgYda 8iaadshapaWaa0baaKqbGeaapeGaamOqaaWdaeaapeGaai4jaiaacE caaaaaaa@4691@ , while the proper ground length O B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaqcfaOaamOq aaaa@3A2B@ is L/γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbGaai4laiabeo7aNbaa@39CF@ . Hence, the corresponding ground speed is c c+2v>c. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaagbaqaaiaadogaaeaaaiaawEJ=aiabloKi7iaadogacqGH RaWkcaaIYaGaamODaiabg6da+iaadogacaGGUaaaaa@40CC@

the time interval (Kassner’s time gap) introduced by non-conservation of simultaneity. Fig. 4 pictures the physical reality of S’ at the time indicating that O∗ is at the distance L/γ γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbGaai4laiabeo7aNjabeo7aN9aadaahaaqcfasabeaa peGaeyOmGikaaaaa@3D65@ from B and at the distance Δ x = 2( v 2 / c 2 )γL/ γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG4bWdamaaCaaajuaibeqaa8qacqGHYaIOaaqc faOaeyypa0JaaeiiaiaaikdapaWaaeWaaeaapeGaamODa8aadaahaa qabKqbGeaapeGaaGOmaaaajuaGcaGGVaGaam4ya8aadaahaaqcfasa beaapeGaaGOmaaaaaKqba+aacaGLOaGaayzkaaWdbiabeo7aNjaadY eacaGGVaGaeq4SdC2damaaCaaajuaibeqaa8qacqGHYaIOaaaaaa@4C79@ from A. The physical reality of S’ at the later time t A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaBaaajuaibaWdbiaadgeaa8aabeaajuaGdaah aaqcfasabeaapeGaeyOmGikaaaaa@3B4E@ is pictured by Figure 5, indicating that when clock O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaaaaa@38D6@ reaches A the signal has already traveled from B toward point A in the lower part of the tube during the time interval δt′ and is now at the shorter distance L/γcδ t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbGaai4laiabeo7aNjabgkHiTiaadogacqaH0oazcaWG 0bWdamaaCaaajuaibeqaa8qacqGHYaIOaaaaaa@4031@ from A and clock O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaaaaa@38D6@ .  According to S’, as a consequence of

non-conservation of simultaneity the signal is not located at point B at the time t A ' , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiaadgeaa8aabaWdbiaacEca aaqcfa4daiaacYcaaaa@3AEA@ , but has already moved in the lower part of the

tube from B toward O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaaaaa@38D6@ by the amount cδ t = 2γL( v/c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGJbGaeqiTdqMaamiDa8aadaahaaqcfasabeaapeGaeyOm GikaaKqbakabg2da9iaabccacaaIYaGaeq4SdCMaamita8aadaqada qaa8qacaWG2bGaai4laiaadogaa8aacaGLOaGaayzkaaaaaa@45D1@ , as shown in Figure 5. Meanwhile, in the same time interval δ t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH0oazcaWG0bWdamaaCaaajuaibeqaa8qacqGHYaIOaaaa aa@3B31@ point A has moved with respect to O' by vδ t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHsislcaWG2bGaeqiTdqMaamiDa8aadaahaaqcfasabeaa peGaeyOmGikaaaaa@3D19@ . Thus, at the time t A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaBaaajuaibaWdbiaadgeaa8aabeaajuaGdaah aaqcfasabeaapeGaeyOmGikaaaaa@3B4E@ , the remaining shorter distance to be covered by the signal is L/γ ( cv )δt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbGaai4laiabeo7aNjabgkHiTiaabccapaWaaeWaaeaa peGaam4yaiabgkHiTiaadAhaa8aacaGLOaGaayzkaaWdbiabes7aKj aadshaaaa@4294@ ′, so that when the signal finally reaches O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaaaaa@38D6@ this clock must have advanced by the time interval

Δ t SR ' = L/γcδ t +vδ t ' c = γL c ( 1   v c ) 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG0bWdamaaDaaajuaibaWdbiaadofacaWGsbaa paqaa8qacaGGNaaaaKqbakabg2da9maalaaapaqaa8qacaWGmbGaai 4laiabeo7aNjabgkHiTiaadogacqaH0oazceWG0bWdayaafaWdbiab gUcaRiaadAhacqaH0oazcaWG0bWdamaaCaaabeqaa8qacaGGNaaaaa WdaeaapeGaam4yaaaacqGH9aqpdaWcaaWdaeaapeGaeq4SdCMaamit aaWdaeaapeGaam4yaaaadaqadaWdaeaapeGaaGymaiaacckacqGHsi slcaGGGcWaaSaaa8aabaWdbiaadAhaa8aabaWdbiaadogaaaaacaGL OaGaayzkaaWdamaaCaaabeqcfasaa8qacaaIYaaaaKqba+aacaGGUa aaaa@5ADD@  (13)

Then, according to S’, the total time displayed by clock O∗ at the end of the clockwise cycle is

t + * = t B '' +Δ t SR ' = 2γL c 2γLv c 2 = 2L γ( c+v ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiabgUcaRaWdaeaapeGaaiOk aaaajuaGcqGH9aqpcaWG0bWdamaaDaaajuaibaWdbiaadkeaa8aaba WdbiaacEcacaGGNaaaaKqbakabgUcaRiabfs5aejaadshapaWaa0ba aKqbGeaapeGaam4uaiaadkfaa8aabaWdbiaacEcaaaqcfaOaeyypa0 ZaaSaaa8aabaWdbiaaikdacqaHZoWzcaWGmbaapaqaa8qacaWGJbaa aiabgkHiTmaalaaapaqaa8qacaaIYaGaeq4SdCMaamitaiaadAhaa8 aabaWdbiaadogapaWaaWbaaKqbGeqabaWdbiaaikdaaaaaaKqbakab g2da9maalaaapaqaa8qacaaIYaGaamitaaWdaeaapeGaeq4SdC2aae Waa8aabaWdbiaadogacqGHRaWkcaWG2baacaGLOaGaayzkaaaaaiaa cYcaaaa@5D59@  (14)

Figure 5 The linear Sagnac effect described from the perspective of frame S’ at the time t A ' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiaadgeaa8aabaWdbiaacEca aaaaaa@399D@ . Clock O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaaaaa@38D6@ has reached point A and, after turning around the pulley, starts co-moving with the lower part of the tube and frame S’. Because of the swifter ground speed C >c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaagbaqaaiaadoeaaeaaaiaawEJ=aiabg6da+iaadogaaaa@3B48@ in the upper part of the tube, the signal has already moved along the lower part of the tube advancing by cδ t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGJbGaeqiTdqMaamiDa8aadaahaaqcfasabeaapeGaeyOm Gikaaaaa@3C19@ from B toward O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaaaaa@38D6@ .

In agreement with (10) and (11) as foreseen by special relativity. For the counterclockwise propagation, we find t * =2γL( 1v/c )/c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiabgkHiTaWdaeaapeGaaiOk aaaajuaGcqGH9aqpcaaIYaGaeq4SdCMaamitamaabmaapaqaa8qaca aIXaGaeyOeI0IaamODaiaac+cacaWGJbaacaGLOaGaayzkaaGaai4l aiaadogaaaa@4610@  and thus, being Δ t * = t * t + * =4γLv/ c 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG0bWdamaaCaaajuaibeqaa8qacaGGQaaaaKqb akabg2da9iaadshal8aadaqhaaqcfayaaKqzadWdbiabgkHiTaqcfa 4daeaajugWa8qacaGGQaaaaKqbakabgkHiTiaadshapaWaa0baaKqb GeaapeGaey4kaScapaqaa8qacaGGQaaaaKqbakabg2da9iaaisdacq aHZoWzcaWGmbGaamODaiaac+cacaWGJbWdamaaCaaajuaibeqaa8qa caaIYaaaaaaa@4F49@  as in (5), the LT applied in frame S′ correctly foresee the Sagnac effect. It should be pointed out that the difference between (8) and (13) is precisely the time interval (12), δt=2 γLv/ c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH0oazcaWG0bGaae4jaiaabckacqGH9aqpcaaIYaGaaeiO aiabeo7aNjaadYeacaWG2bGaai4laiaadogapaWaaWbaaKqbGeqaba Wdbiaaikdaaaaaaa@442D@  , introduced by non-conservation of simultaneity. As pointed out above, according to S′ and in agreement with standard special relativity, result (14) indicates that, for C 0 c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWdamaaCaaajuaibeqaa8qacaaIWaaaaKqbakabgkDi Elaadogaaaa@3C68@ , the average counter-propagation speed, along the invariant ground length 2γL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIYaGaeq4SdCMaamitaaaa@39D8@ of the closed path, is superluminal and given by c+v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqWIdjYocaWGJbGaey4kaSIaamODaaaa@3A9A@ .

Supporters of special relativity may claim that, although the average speed is superluminal, after Einstein synchronization the local speed in the lower part of the tube co-moving with S’is c, as it should be. Nevertheless, detractors of Einstein synchronization may argue that the local speed in the lower part of the tube has been achieved by re-setting the clocks in that part of the tube in such a way that the speed be c. For doing this, frame S’ has to pay a price because now the average superluminal speed c+v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqWIdjYocaWGJbGaey4kaSIaamODaaaa@3A9A@ along the closed path 2γL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIYaGaeq4SdCMaamitaaaa@39D8@ cannot be justified by S’, unless the ground speed in the upper part of the tube is not only superluminal, but greater than the average value c+v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqWIdjYocaWGJbGaey4kaSIaamODaaaa@3A9A@ and of the order c+ 2v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqWIdjYocaWGJbGaey4kaSIaaeiiaiaaikdacaWG2baaaa@3BF9@ . In fact, by synchronizing clocks with Einstein procedure, S’ has introduced the kinematical mechanism of non-conservation of simultaneity and the physical reality of S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGtbGaaiygGiaacMbiaaa@38F6@ has been changed. The resulting effect consists of modifying the time interval (8), making it shorter by a first order contribution in v/c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG2bGaai4laiaadogaaaa@393A@ in order to equal the value (13). Because of non-conservation of simultaneity, when seen from the snapshot perspective of S’shown in Fig. 4, the clocks of S’’ in the upper part of the tube display readings predicted by the Lorentz time transformations. If the perspective of S’ is taken realistically (and is not a mere mathematical makeup), the reading of O*, at the time t B ' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiaadkeaa8aabaWdbiaacEca aaaaaa@399E@ , can be inferred by means of the inverse Lorentz transformations, so that the time reading of clock O*

(at x′′ = 0) turns out to be

t B '' = t B ' γ = t B '' ( 1     w c )=  L γc ( 1     w c ). MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaGraae aacaWG0baabaaacaGL34padaqhaaqcfasaaabaaaaaaaaapeGaamOq aaWdaeaapeGaai4jaiaacEcaaaqcfaOaeyypa0ZaaSaaa8aabaWdbi aadshapaWaa0baaKqbGeaapeGaamOqaaWdaeaapeGaai4jaaaaaKqb a+aabaWdbiqbeo7aN9aagaqbaaaapeGaeyypa0JaamiDa8aadaqhaa qcfasaa8qacaWGcbaapaqaa8qacaGGNaGaai4jaaaajuaGdaqadaWd aeaapeGaaGymaiaacckacaGGGcGaeyOeI0IaaiiOaiaacckadaWcaa WdaeaapeGaam4DaaWdaeaapeGaam4yaaaaaiaawIcacaGLPaaacqGH 9aqpcaGGGcWaaSaaa8aabaWdbiaadYeaa8aabaWdbiabeo7aNjaado gaaaWaaeWaa8aabaWdbiaaigdacaGGGcGaaiiOaiabgkHiTiaaccka caGGGcWaaSaaa8aabaWdbiaadEhaa8aabaWdbiaadogaaaaacaGLOa GaayzkaaGaaiOlaaaa@6336@ (15)

Obviously, the clock reading t B '' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWaa0baaKqbGeaacaWGcbaabaGaai4jaiaacEcaaaaa aa@3A0A@  represents the time of flight
of the signal from O* to the point of the tube reached by the signal at that time. According to S’’, at the time t B '' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaagbaqaaiaadshaaeaaaiaawEJ=amaaDaaajuaibaGaamOq aaqaaiaacEcacaGGNaaaaaaa@3BF5@ the light signal has not yet reached point B but has covered only shorter ground length L B =c t B '' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbWdamaaBaaajuaibaWdbiqadkeapaGbayaaaeqaaKqb a+qacqGH9aqpcaWGJbGabmiDayaataWdamaaDaaajuaibaWdbiaadk eaa8aabaWdbiaacEcacaGGNaaaaaaa@3F10@  to the point B′′ shown in Fig. 4. For the Newtonian mindset, the fact that, at the displayed time t B '' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaagbaqaaiaadshaaeaaaiaawEJ=a8aadaqhaaqcfasaaKqz adWdbiaadkeaaKqbG8aabaWdbiaacEcacaGGNaaaaaaa@3D90@ , the signal can be in two different places, at B or B′′, depending on the point of view of S’ or S’’, hints at an inconsistency between the two physical realities. However, if we admit non-conservation of simultaneity and focus on the reality of S’, it is a fact that, at the time t B ' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bWdamaaDaaajuaibaWdbiaadkeaa8aabaWdbiaacEca aaaaaa@399E@ , the signal has already reached point B, past B′′.

Thus, we may conclude that the signal has been sped up by non-conservation of simultaneity, while propagating in the upper section of the tube from clock O∗ to point B, past B′′. Considering that the ground length O*B is L/γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbGaai4laiabeo7aNbaa@39CF@ and that t B '' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaagbaqaaiaadshaaeaaaiaawEJ=a8aadaqhaaqcfasaaKqz adWdbiaadkeaaKqbG8aabaWdbiaacEcacaGGNaaaaaaa@3D90@  is the time of flight of the signal, we are compelled to infer that the local ground speedof the signal along O∗B is,

c =( L/γ )/ t B '' =c/( 1w/c )c+2v , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaagbaqaaiaadogaaeaaaiaawEJ=aiabg2da9maabmaapaqa a8qacaWGmbGaai4laiabeo7aNbGaayjkaiaawMcaaiaac+cadaagba qaaiaacshaaeaaaiaawEJ=a8aadaqhaaqcfasaa8qacaWGcbaapaqa a8qacaGGNaGaai4jaaaajuaGcqGH9aqpcaWGJbGaai4lamaabmaapa qaa8qacaaIXaGaeyOeI0Iaam4Daiaac+cacaWGJbaacaGLOaGaayzk aaGaeS4qISJaam4yaiabgUcaRiaaikdacaWG2bGaaiiOaiaacYcaaa a@552D@  (16)

i.e., superluminal for C 0 =c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbWdamaaCaaabeqcfasaa8qacaaIWaaaaKqbakabg2da 9iaadogaaaa@3B11@ , in conflict with the postulate of the constancy of c
In short, the argument discussed in b) in the above subsection, holds here also. If the average speed over the total length  2γL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGGcGaaGOmaiabeo7aNjaadYeaaaa@3AFC@ is c+v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqWIdjYocaWGJbGaey4kaSIaamODaaaa@3A9A@ , and, over the partial length γL ( 1v/c ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHZoWzcaWGmbWdamaabmaabaWdbiaaigdacqGHsislcaWG 2bGaai4laiaadogaa8aacaGLOaGaayzkaaWaaWbaaKqbGeqabaWdbi aaikdaaaaaaa@402D@ of (13) in the lower part of the tube, the local speed is c, it follows that the local speed in the remaining part must be c+ 2v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqWIdjYocaWGJbGaey4kaSIaaeiiaiaaikdacaWG2baaaa@3BF9@ , i.e., superluminal.

Preliminary conclusions

The two physical realities of Figure 3 (for S’’) and Figure 4 (for S’) are physically incompatible and lead to different results and interpretations of the Sagnac effect. For both Newton and Einstein, in an inertial reference frame the physical reality is described synchronously, i.e., at the same simultaneous time in every point of space of the inertial frame. The contrasting results obtained by S’’ and S’((9) and (14), respectively) are simply the reflection of the incompatibility between the concept of simultaneity and Einstein’s concept of relative time.

Non-conservation of simultaneity and the Lorentz transformations are direct consequences of Einstein synchronization and the validity of the second postulate of special relativity. Detractors of Einstein synchronization may claim that the resulting average propagation speed c+v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqWIdjYocaWGJbGaey4kaSIaamODaaaa@3A9A@ of (14) implies signal propagation at a superluminal local ground speed along some section of the closed path, in line with Selleri’s paradox for the circular Sagnac effect. We have shown that, for the equivalent linear Sagnac effect, if in one of the inertial frames, S’’or S’, the local ground speed is c, the local speed in the other inertial frame must be superluminal and of the order of c+2v, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqWIdjYocaWGJbGaey4kaSIaaGOmaiaadAhacaGGSaaaaa@3C07@ , in contrast with Einstein’s second postulate. This result is seen as reflecting the incompatibility of Einstein synchronization applied to the closed path of the Sagnac effect, even when applied separately to the upper and lower parts of the tube. Therefore, since Einstein synchronization implies the existence of superluminal local speeds, the second postulate of SR is invalid.

Nevertheless, thinking beyond the controversy about the validity of the postulate of the universal light speed c in all inertial systems, supporters of SR may shift the debate to a more pragmatic scenario by highlighting the correctness of the predictions of the theory. They could argue that, even though the second postulate of SR is questionable because of the superluminal speeds involved in the interpretation of the Sagnac effect, from an operational standpoint, Einstein synchronization can still be applied to the upper and lower parts of the tube. Therefore, from a pragmatic point of view, followers of standard special relativity may claim that the theory is sound because, by means of non-conservation of simultaneity, it predicts for the Sagnac effect the observed results, derived either in the laboratory frame S or in the tube co-moving frame S’’or S’. Along these lines, assuming the physical equivalence between absolute and Einstein synchronization, Kassner argues that the difference between a local speed c and a superluminal average or local speed c+v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqWIdjYocaWGJbGaey4kaSIaamODaaaa@3A9A@ is to be interpreted in terms of the arbitrariness of synchronization and, thus, does not invalidate the second postulate of special relativity.

Thus, the debate of relative versus absolute time is at a standstill and it is unlikely that historically long debates such as this can be settled through mere theoretical arguments. Fortunately, physicists agree that, if there is a meaningful difference between opposing points of view, it must be ultimately observable, i.e., experimentally testable. Therefore, the best clear-cut way to solve the controversy is by means of a test capable of discriminating absolute versus Einstein synchronization. As shown in the next section, such a test exists and is precisely the Sagnac effect.

Regardless of the feasibility or not of experimental verification, the mere existence in principle of such a test proves that the Lorentz transformations are not physically equivalent to transformations based on absolute synchronization. An immediate consequence is that it is conceptually unfeasible to use the Tangherlini transformations and absolute synchronization in trying to conciliate, as claimed by Kassner, the superluminal average speed of the Sagnac effect with the postulate of the constancy of the speed of light. In principle, the two synchronizations are physically distinguishable and the arbitrariness of synchronization can no longer be invoked to substitute the LT with the TT in order to explain Selleri’s paradox, which remains unsolved.

Non-equivalence of Einstein and absolute synchronization

In relativistic theories, the coordinate transformations between inertial frames of reference in relative motion may take into account the synchronization procedure adopted (for example, internal and external ) through the dependence of the time ton a convenient synchronization parameter ε. The contention of conventionalism is that clock synchronization is arbitrary and, thus, the one-way speed of light, that can be expressed as c(ε) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGJbWdaiaacIcapeGaeqyTdu2daiaacMcaaaa@3ABA@ , is conventional and not measurable in principle. It follows that, within this scenario, an observable quantity must be synchronization-independent in order to be physically meaningful. If the observable quantities of relativistic theories are independent of the chosen synchronization procedure, all the experiments supporting standard special relativity also support a preferred frame theory with absolute synchronization. Then, non-conservation of simultaneity, which is a consequence of Einstein synchronization procedure, should not be observable per se. If, instead, synchronization is not arbitrary, it is feasible that some observable quantities are dependent on synchronization and, thus, their measurement would allow for the determination of the "natural" synchronization that fits with observation. In the case of the Sagnac effect, the propagation times over a cycle are measured by means of a single clock and there is no need to perform synchronization of distant, spatially separated clocks. If the laboratory frame S is chosen as the preferred frame where space is isotropic, the time relation t*= t/γ between clock O in motion with relative velocity v and a clock stationary in Sis the same for TT and LT. Therefore, both theories foresee the same results, independent of ε. However, it is unlikely that the laboratory where the Sagnac experiment is performed, located somewhere on the rotating Earth, should coincide with the preferred frame of some ether theory. Thus, it is justified to consider the hypothetical case when, in frame S, space is no longer isotropic, as a consequence of its possible motion with respect to some preferred frame. Let us then envision the existence of the preferred inertial frame Σ ( X , T ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHJoWupaWaaeWaaeaapeGaamiwaiaacYcacaWGubaapaGa ayjkaiaawMcaaaaa@3C45@ where space is isotropic and the one-way speed of light is c. In the Figure 2 we have to suppose that the laboratory frame S, where the set-up (arm AB) of the linear Sagnac effect is stationary, moves with velocity Win the Xdirection with respect to Σ. The clock O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaajuaibeqaa8qacqGHxiIkaaaaaa@38D6@ , co-moving with frame S′, has velocity W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWGxbWdayaafaaaaa@379C@ with respect to Σ. The following transformations8 apply to relativistic theories (special relativity with arbitrary synchronization parameter ε):

x= γ W ( XWT );y=Y;z=Z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bGaeyypa0Jaeq4SdC2damaaBaaajuaibaWdbiaadEfa aKqba+aabeaapeWaaeWaa8aabaWdbiaadIfacqGHsislcaWGxbGaam ivaaGaayjkaiaawMcaaiaacUdacaWG5bGaeyypa0JaamywaiaacUda caWG6bGaeyypa0JaamOwaaaa@48B1@

t= γ W [ T( 1+ W 2 c 2 ( ε1 ) )ε WX c 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bGaeyypa0Jaeq4SdC2damaaBaaajuaibaWdbiaadEfa aKqba+aabeaapeWaamWaa8aabaWdbiaadsfadaqadaWdaeaapeGaaG ymaiabgUcaRmaalaaapaqaa8qacaWGxbWdamaaCaaajuaibeqaa8qa caaIYaaaaaqcfa4daeaapeGaam4ya8aadaahaaqcfasabeaapeGaaG Omaaaaaaqcfa4aaeWaa8aabaWdbiabew7aLjabgkHiTiaaigdaaiaa wIcacaGLPaaaaiaawIcacaGLPaaacqGHsislcqaH1oqzdaWcaaWdae aapeGaam4vaiaadIfaa8aabaWdbiaadogapaWaaWbaaeqajuaibaWd biaaikdaaaaaaaqcfaOaay5waiaaw2faaaaa@5428@  (17)

In Equation (17), when ε= 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH1oqzcqGH9aqpcaqGGaGaaGymaaaa@3AAF@ we recover the time relation t= γ W ( TWX/ c 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bGaeyypa0Jaeq4SdC2damaaBaaajuaibaWdbiaadEfa aKqba+aabeaadaqadaqaa8qacaWGubGaeyOeI0Iaam4vaiaadIfaca GGVaGaam4ya8aadaahaaqcfasabeaapeGaaGOmaaaaaKqba+aacaGL OaGaayzkaaaaaa@44AC@ of the LT and, when ε= 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH1oqzcqGH9aqpcaqGGaGaaGimaaaa@3AAE@ , the relation t=T/ γ W MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bGaeyypa0Jaamivaiaac+cacqaHZoWzpaWaaSbaaKqb GeaapeGaam4vaaqcfa4daeqaaaaa@3DBD@ of the TT. After taking the derivative dx/dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGKbGaamiEaiaac+cacaWGKbGaamiDaaaa@3B1F@ , for the components of the velocity in the x, X direction, we obtain from (17),

u= UW ( 1+ W 2 c 2 ( ε1 ) )ε WU c 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG1bGaeyypa0ZaaSaaa8aabaWdbiaadwfacqGHsislcaWG xbaapaqaa8qadaqadaWdaeaapeGaaGymaiabgUcaRmaalaaapaqaa8 qacaWGxbWdamaaCaaabeqcfasaa8qacaaIYaaaaaqcfa4daeaapeGa am4ya8aadaahaaqabKqbGeaapeGaaGOmaaaaaaqcfa4aaeWaa8aaba Wdbiabew7aLjabgkHiTiaaigdaaiaawIcacaGLPaaaaiaawIcacaGL PaaacqGHsislcqaH1oqzdaWcaaWdaeaapeGaam4vaiaadwfaa8aaba WdbiaadogapaWaaWbaaKqbGeqabaWdbiaaikdaaaaaaaaaaaa@5001@ (18)

,p>Where u=dx/dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG1bGaeyypa0JaamizaiaadIhacaGGVaGaamizaiaadsha aaa@3D1F@ is the velocity with respect to the lab frame Sand U=dX/dT MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGvbGaeyypa0JaamizaiaadIfacaGGVaGaamizaiaadsfa aaa@3CBF@ is the velocity with respect to the preferred frame Σ. If U= W MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGvbGaeyypa0Jaam4va8aadaahaaqcfasabeaapeGaeyOm Gikaaaaa@3B4F@ is the velocity with respect to Σof the clock O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaaaaa@38D6@ (or interferometer) co-moving with S′, from (18) we find that the corresponding relative velocity u=v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG1bGaeyypa0JaamODaaaa@399F@ of the clock with respect to Sis,

v= W W 1+ W 2 c 2 ( ε1 )ε W W c 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG2bGaeyypa0ZaaSaaa8aabaWdbiqadEfapaGbauaapeGa eyOeI0Iaam4vaaWdaeaapeGaaGymaiabgUcaRmaalaaapaqaa8qaca WGxbWdamaaCaaajuaibeqaa8qacaaIYaaaaaqcfa4daeaapeGaam4y a8aadaahaaqcfasabeaapeGaaGOmaaaaaaqcfa4aaeWaa8aabaWdbi abew7aLjabgkHiTiaaigdaaiaawIcacaGLPaaacqGHsislcqaH1oqz daWcaaWdaeaapeGaam4vaiqadEfapaGbauaaaeaapeGaam4ya8aada ahaaqabKqbGeaapeGaaGOmaaaaaaaaaaaa@4E95@  (19)

We can see that for the LT (ε= 1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbGaamivaiaabccapaGaaiika8qacqaH1oqzcqGH9aqp caqGGaGaaGyma8aacaGGPaaaaa@3E83@ the velocity composition (19) gives, as expected, v= ( W W )/( 1  W W/ c 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG2bGaeyypa0Jaaeiia8aadaqadaqaa8qacaWGxbWdamaa Caaajuaibeqaa8qacqGHYaIOaaqcfaOaeyOeI0Iaam4vaaWdaiaawI cacaGLPaaapeGaai4la8aadaqadaqaa8qacaaIXaGaaeiiaiabgkHi TiaadEfapaWaaWbaaKqbGeqabaWdbiabgkdiIcaajuaGcaWGxbGaai 4laiaadogapaWaaWbaaeqajuaibaWdbiaaikdaaaaajuaGpaGaayjk aiaawMcaaaaa@4C6F@ , while for the TT (ε= 0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGubGaamivaiaabccapaGaaiika8qacqaH1oqzcqGH9aqp caqGGaGaaGima8aacaGGPaaaaa@3E8A@ we have v= γ W 2 ( W W ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG2bGaeyypa0Jaeq4SdC2damaaDaaajuaibaWdbiaadEfa a8aabaWdbiaaikdaaaqcfa4aaeWaa8aabaWdbiqadEfapaGbauaape GaeyOeI0Iaam4vaaGaayjkaiaawMcaaaaa@4179@ . Expressed in terms of Wand v, the velocities of S′ and S′′ with respect to Σ are respectively,

W = W+v[ 1+ W 2 c 2 ( ε1 ) ] 1+ε vW c 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWGxbWdayaafaWdbiabg2da9maalaaapaqaa8qacaWGxbGa ey4kaSIaamODamaadmaapaqaa8qacaaIXaGaey4kaSYaaSaaa8aaba WdbiaadEfapaWaaWbaaKqbGeqabaWdbiaaikdaaaaajuaGpaqaa8qa caWGJbWdamaaCaaajuaibeqaa8qacaaIYaaaaaaajuaGdaqadaWdae aapeGaeqyTduMaeyOeI0IaaGymaaGaayjkaiaawMcaaaGaay5waiaa w2faaaWdaeaapeGaaGymaiabgUcaRiabew7aLnaalaaapaqaa8qaca WG2bGaam4vaaWdaeaapeGaam4ya8aadaahaaqabKqbGeaapeGaaGOm aaaaaaaaaaaa@515E@ ; W '' = Wv[ 1+ W 2 c 2 ( ε1 ) ] 1ε vW c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGxbqcfa4damaaCaaajuaibeqaa8qacaGGNaGaai4jaaaajuaG cqGH9aqpdaWcaaWdaeaapeGaam4vaiabgkHiTiaadAhadaWadaWdae aapeGaaGymaiabgUcaRmaalaaapaqaa8qacaWGxbWdamaaCaaabeqc fasaa8qacaaIYaaaaaqcfa4daeaapeGaam4ya8aadaahaaqcfasabe aapeGaaGOmaaaaaaqcfa4aaeWaa8aabaWdbiabew7aLjabgkHiTiaa igdaaiaawIcacaGLPaaaaiaawUfacaGLDbaaa8aabaWdbiaaigdacq GHsislcqaH1oqzdaWcaaWdaeaapeGaamODaiaadEfaa8aabaWdbiaa dogapaWaaWbaaKqbGeqabaWdbiaaikdaaaaaaaaaaaa@539B@  (20)

The following results, which depend on ε, are obtained for the linear Sagnac effect. As measured in the preferred frame Σ, the time intervals for the counter-propagating signal (clockwise propagation in Figure 2) with one-way speed c from A to B and from B to O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaaaaa@38D6@ are respectively:

T AB = L ( cW ) γ W ; T B O * = L ( c+ W ) γ W W W c+ W T AB MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGubWdamaaBaaajuaibaWdbiaadgeacaWGcbaajuaGpaqa baWdbiabg2da9maalaaapaqaa8qacaWGmbaapaqaa8qadaqadaWdae aapeGaam4yaiabgkHiTiaadEfaaiaawIcacaGLPaaacqaHZoWzpaWa aSbaaKqbGeaapeGaam4vaaqcfa4daeqaaaaapeGaai4oaiaadsfapa WaaSbaaeaajuaipeGaamOqaiaad+eajuaGpaWaaWbaaKqbGeqabaWd biaacQcaaaaajuaGpaqabaWdbiabg2da9maalaaapaqaa8qacaWGmb aapaqaa8qadaqadaWdaeaapeGaam4yaiabgUcaRiqadEfapaGbauaa a8qacaGLOaGaayzkaaGaeq4SdC2damaaBaaajuaibaWdbiaadEfaaK qba+aabeaaaaWdbiabgkHiTmaalaaapaqaa8qaceWGxbWdayaafaWd biabgkHiTiaadEfaa8aabaWdbiaadogacqGHRaWkceWGxbWdayaafa aaa8qacaWGubWdamaaBaaajuaibaWdbiaadgeacaWGcbaajuaGpaqa baaaaa@5EC4@ .

The time T+ for counter-propagation and T− for co-propagation turns out to be, respectively,

T + = T AB + T B O * = 2L c( 1+ W /c )( 1W/c ) γ W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGubWdamaaBaaajuaibaWdbiabgUcaRaqcfa4daeqaa8qa cqGH9aqpcaWGubWdamaaBaaajuaibaWdbiaadgeacaWGcbaajuaGpa qabaWdbiabgUcaRiaadsfapaWaaSbaaKqbGeaapeGaamOqaiaad+ea juaGpaWaaWbaaKqbGeqabaWdbiaacQcaaaaapaqabaqcfa4dbiabg2 da9maalaaapaqaa8qacaaIYaGaamitaaWdaeaapeGaam4yamaabmaa paqaa8qacaaIXaGaey4kaSIabm4va8aagaqba8qacaGGVaGaam4yaa GaayjkaiaawMcaamaabmaapaqaa8qacaaIXaGaeyOeI0Iaam4vaiaa c+cacaWGJbaacaGLOaGaayzkaaGaeq4SdC2damaaBaaajuaibaWdbi aadEfaaKqba+aabeaaaaaaaa@5723@

T = 2L c( 1 W /c )( 1+W/c ) γ W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGubWdamaaBaaajuaibaWdbiabgkHiTaqcfa4daeqaa8qa cqGH9aqpdaWcaaWdaeaapeGaaGOmaiaadYeaa8aabaWdbiaadogada qadaWdaeaapeGaaGymaiabgkHiTiqadEfapaGbauaapeGaai4laiaa dogaaiaawIcacaGLPaaadaqadaWdaeaapeGaaGymaiabgUcaRiaadE facaGGVaGaam4yaaGaayjkaiaawMcaaiabeo7aN9aadaWgaaqcfasa a8qacaWGxbaapaqabaaaaaaa@4BFD@

ΔT= T T + = 4L c 2 γ W 2 γ W ( W W ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWGubGaeyypa0Jaamiva8aadaWgaaqcfasaa8qa cqGHsislaKqba+aabeaapeGaeyOeI0Iaamiva8aadaWgaaqcfasaa8 qacqGHRaWkaKqba+aabeaapeGaeyypa0ZaaSaaa8aabaWdbiaaisda caWGmbaapaqaa8qacaWGJbWdamaaCaaajuaibeqaa8qacaaIYaaaaa aajuaGcqaHZoWzpaWaa0baaKqbGeaapeGabm4va8aagaqbaaqaa8qa caaIYaaaaKqbakabeo7aN9aadaWgaaqcfasaa8qacaWGxbaajuaGpa qabaWdbmaabmaapaqaa8qaceWGxbWdayaafaWdbiabgkHiTiaadEfa aiaawIcacaGLPaaaaaa@528F@ (21)

where ΔT MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWGubaaaa@38E3@ is the time span difference. The corresponding time difference Δ t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG0bWdamaaCaaajuaibeqaa8qacqGHYaIOaaaa aa@3AF2@ measured by the clock O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaajuaibeqaa8qacqGHxiIkaaaaaa@38D6@  for the Sagnac effect is,

Δ t =Δ t * = ΔT γ W = 4L c 2 γ W γ W ( W W ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarceWG0bWdayaafaWdbiabg2da9iabfs5aejaadsha paWaaWbaaeqajuaibaWdbiaacQcaaaqcfaOaeyypa0ZaaSaaa8aaba Wdbiabfs5aejaadsfaa8aabaWdbiabeo7aN9aadaWgaaqcfasaa8qa ceWGxbWdayaafaaajuaGbeaaaaWdbiabg2da9maalaaapaqaa8qaca aI0aGaamitaaWdaeaapeGaam4ya8aadaahaaqcfasabeaapeGaaGOm aaaaaaqcfaOaeq4SdC2damaaBaaajuaibaWdbiqadEfapaGbauaaae qaaKqba+qacqaHZoWzpaWaaSbaaKqbGeaapeGaam4vaaqcfa4daeqa a8qadaqadaWdaeaapeGabm4va8aagaqba8qacqGHsislcaWGxbaaca GLOaGaayzkaaaaaa@56C0@ (22)

By means of (20), we may substitute in (22) W ′ expressed in terms of Wand v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG2baaaa@37A0@ and obtain,

Δ t = 4Lv c 2 γ W { ( 1+ε vW c 2 ) 2 ( W c + v c [ 1+ W 2 ( ε1 ) c 2 ] ) 2 } 1/2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarceWG0bWdayaafaWdbiabg2da9maalaaapaqaa8qa caaI0aGaamitaiaadAhaa8aabaWdbiaadogapaWaaWbaaeqajuaiba WdbiaaikdaaaqcfaOaeq4SdC2damaaBaaajuaibaWdbiaadEfaaKqb a+aabeaaaaWdbmaacmaapaqaa8qadaqadaWdaeaapeGaaGymaiabgU caRiabew7aLnaalaaapaqaa8qacaWG2bGaam4vaaWdaeaapeGaam4y a8aadaahaaqabKqbGeaapeGaaGOmaaaaaaaajuaGcaGLOaGaayzkaa WdamaaCaaajuaibeqaa8qacaaIYaaaaKqbakabgkHiTmaabmaapaqa a8qadaWcaaWdaeaapeGaam4vaaWdaeaapeGaam4yaaaacqGHRaWkda WcaaWdaeaapeGaamODaaWdaeaapeGaam4yaaaadaWadaWdaeaapeGa aGymaiabgUcaRmaalaaapaqaa8qacaWGxbWdamaaCaaajuaibeqaa8 qacaaIYaaaaKqbaoaabmaapaqaa8qacqaH1oqzcqGHsislcaaIXaaa caGLOaGaayzkaaaapaqaa8qacaWGJbWdamaaCaaabeqcfasaa8qaca aIYaaaaaaaaKqbakaawUfacaGLDbaaaiaawIcacaGLPaaapaWaaWba aeqajuaibaWdbiaaikdaaaaajuaGcaGL7bGaayzFaaWdamaaCaaaju aibeqaa8qacqGHsislcaaIXaGaai4laiaaikdaaaqcfa4daiaac6ca aaa@6D77@ (23)

Result (23) indicates that the optical path difference for two counter-propagating electromagnetic waves depends on the synchronization parameter ε in the Sagnac effect. If the procedure

used above is applied to other optical tests, for example, to the Michelson-Morley experiment, we find instead that ∆t′ is independent of ε, as also shown by Mansouri and Sexl.

If we set ε= 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH1oqzcqGH9aqpcaqGGaGaaGymaaaa@3AAF@ in (23), the term { ... } 1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiWaae aaqaaaaaaaaaWdbiaac6cacaGGUaGaaiOlaaWdaiaawUhacaGL9baa daahaaqabKqbGeaapeGaeyOeI0IaaGymaiaac+cacaaIYaaaaaaa@3E71@ becomes γWγ and as foreseen by the LT, we obtain Δ t = 4γLv/ c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG0bWdamaaCaaabeqaa8qacqGHYaIOaaGaeyyp a0JaaeiiaiaaisdacqaHZoWzcaWGmbGaamODaiaac+cacaWGJbWdam aaCaaabeqcfasaa8qacaaIYaaaaaaa@4364@ independent of W. Withε=0 , the coupled term vW/c2 does not vanish and, being attributable to the absolute synchronization, is present also in the Newtonian case. Thus, for the TT, from (23) we obtain,

Δ t =Δ t * = 4Lv c 2 [ ( 1     vW c 2 ) 2    v 2 c 2 ] 1/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarceWG0bWdayaafaWdbiabg2da9iabfs5aejaadsha paWaaWbaaeqajuaibaWdbiaacQcaaaqcfaOaeyypa0ZaaSaaa8aaba WdbiaaisdacaWGmbGaamODaaWdaeaapeGaam4ya8aadaahaaqabKqb GeaapeGaaGOmaaaaaaqcfa4aamWaa8aabaWdbmaabmaapaqaa8qaca aIXaGaaiiOaiaacckacqGHsislcaGGGcGaaiiOamaalaaapaqaa8qa caWG2bGaam4vaaWdaeaapeGaam4ya8aadaahaaqabKqbGeaapeGaaG OmaaaaaaaajuaGcaGLOaGaayzkaaWdamaaCaaabeqcfasaa8qacaaI YaaaaKqbakabgkHiTiaacckacaGGGcWaaSaaa8aabaWdbiaadAhapa WaaWbaaKqbGeqabaWdbiaaikdaaaaajuaGpaqaa8qacaWGJbWdamaa Caaabeqcfasaa8qacaaIYaaaaaaaaKqbakaawUfacaGLDbaapaWaaW baaeqajuaibaWdbiabgkHiTiaaigdacaGGVaGaaGOmaaaaaaa@618C@ (24)

To the order vW/ c 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG2bGaam4vaiaac+cacaWGJbWdamaaCaaajuaibeqaa8qa caaIYaaaaaaa@3B42@ and with γ 2 =1 v 2 / c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHZoWzpaWaaWbaaKqbGeqabaWdbiabgkHiTiaaikdaaaqc faOaeyypa0JaaGymaiabgkHiTiaadAhapaWaaWbaaKqbGeqabaWdbi aaikdaaaqcfaOaai4laiaadogapaWaaWbaaeqajuaibaWdbiaaikda aaaaaa@4319@ , the term [...]−1/2 in(24) becomes,

[ ] 1/2 [ γ 2 2 vW c 2 ] 1/2 γ( 1+ vW c 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWadaWdaeaapeGaeyOjGWlacaGLBbGaayzxaaWdamaaCaaa juaibeqaa8qacqGHsislcaaIXaGaai4laiaaikdaaaqcfaOaeS4qIS Jaai4waiabeo7aN9aadaahaaqabKqbGeaapeGaeyOeI0IaaGOmaaaa juaGcqGHsislcaaIYaWaaSaaa8aabaWdbiaadAhacaWGxbaapaqaa8 qacaWGJbWdamaaCaaajuaibeqaa8qacaaIYaaaaaaajuaGcaGGDbWd amaaCaaajuaibeqaa8qacqGHsislcaaIXaGaai4laiaaikdaaaqcfa OaeS4qISJaeq4SdC2aaeWaa8aabaWdbiaaigdacqGHRaWkdaWcaaWd aeaapeGaamODaiaadEfaa8aabaWdbiaadogapaWaaWbaaeqajuaiba WdbiaaikdaaaaaaaqcfaOaayjkaiaawMcaaaaa@5AFC@ (25)

When, after reaching point B, the clock O∗ changes direction and moves with velocity −v with respect to S, the relation W =W+v/ γ W 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWGxbWdayaafaWdbiabg2da9iaadEfacqGHRaWkcaWG2bGa ai4laiabeo7aN9aadaqhaaqcfasaa8qacaWGxbaapaqaa8qacaaIYa aaaaaa@3FEB@ has to be replaced by W =Wv/ γ W 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWGxbWdayaagaWdbiabg2da9iaadEfacqGHsislcaWG2bGa ai4laiabeo7aN9aadaqhaaqcfasaa8qacaWGxbaapaqaa8qacaaIYa aaaaaa@3FF6@ , while the factor γ W MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHZoWzpaWaaSbaaKqbGeaapeGabm4va8aagaqbaaqcfaya baaaaa@3A3E@ has to be replaced by γ W MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHZoWzpaWaaSbaaKqbGeaapeGabm4va8aagaGbaaqabaaa aa@39B1@ in the

calculations. Furthermore, instead of the time interval ∆t′ = ∆t∗, we have now ∆t′′ = ∆t∗, which is the same as the expression ∆t′ of (24) with the term −vW/c2 replaced by +vW/c2.

Result (24) can be immediately extended to the circular Sagnac effect, where the clock moves with tangential speed v=ωR MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG2bGaeyypa0JaeqyYdCNaamOuaaaa@3B49@ and its x-component is v x =vcos(ωt) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG2bWdamaaBaaajuaibaWdbiaadIhaaKqba+aabeaapeGa eyypa0JaamODaiaadogacaWGVbGaam4Ca8aacaGGOaWdbiabeM8a3j aadshapaGaaiykaaaa@42D9@ , by writing ±vW/ c 2 ( vW/ c 2 )cos(ωt). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHXcqScaWG2bGaam4vaiaac+cacaWGJbWdamaaCaaabeqc fasaa8qacaaIYaaaaKqbakabgkDiE=aadaqadaqaa8qacaWG2bGaam 4vaiaac+cacaWGJbWdamaaCaaabeqcfasaa8qacaaIYaaaaaqcfa4d aiaawIcacaGLPaaapeGaam4yaiaad+gacaWGZbWdaiaacIcapeGaeq yYdCNaamiDa8aacaGGPaWdbiaac6caaaa@4EEF@ Then, with the help of (25), the time span (24) measured by clock O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGpbWdamaaCaaabeqcfasaa8qacqGHxiIkaaaaaa@38D6@ in the Sagnac effect can be expressed as

Δ t * 4γLv c 2 [ 1+ vW c 2 cos( ωt ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG0bWdamaaCaaajuaibeqaa8qacaGGQaaaaKqb akabloKi7maalaaapaqaa8qacaaI0aGaeq4SdCMaamitaiaadAhaa8 aabaWdbiaadogapaWaaWbaaKqbGeqabaWdbiaaikdaaaaaaKqbaoaa dmaapaqaa8qacaaIXaGaey4kaSYaaSaaa8aabaWdbiaadAhacaWGxb aapaqaa8qacaWGJbWdamaaCaaabeqcfasaa8qacaaIYaaaaaaajuaG ciGGJbGaai4BaiaacohadaqadaWdaeaapeGaeqyYdCNaamiDaaGaay jkaiaawMcaaaGaay5waiaaw2faaaaa@52B5@ (26)

Precision measurements of (26) can provide the value of the component W of the absolute velocity of the laboratory frame with respect to Σ.

Concerning the smallest measurable time interval, there are techniques capable of resolving femtosecond ( 10 15 ( s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIXaGaaGima8aadaahaaqabKqbGeaapeGaeyOeI0IaaGym aiaaiwdaaaqcfa4damaabmaabaWdbiaadohaa8aacaGLOaGaayzkaa aaaa@3E2C@ ) [45] or even attosecond ( 10 18 ( s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIXaGaaGima8aadaahaaqcfasabeaapeGaeyOeI0IaaGym aiaaiIdaaaqcfa4damaabmaabaWdbiaadohaa8aacaGLOaGaayzkaa aaaa@3E2F@ )46,47 pulses of laser light while better limits may be achieved by means of advanced interferometer. In terms of phase shift, we may write Δφ=cΔ t /(γλ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcqaHgpGAcqGH9aqpcaWGJbGaeuiLdqKaamiDa8aa daahaaqabKqbGeaapeGaey4fIOcaaKqbakaac+capaGaaiika8qacq aHZoWzcqaH7oaBpaGaaiykaaaa@4595@ where λis the vacuum wavelength and c is the free-space velocity of light [30]. Then, result (26) can be expressed as

Δφ 4Lv λc [ 1+ vW c 2 coscos( ωt ) ]=K[ 1+Acos( ωt ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcqaHgpGAcqWIdjYodaWcaaWdaeaapeGaaGinaiaa dYeacaWG2baapaqaa8qacqaH7oaBcaWGJbaaamaadmaapaqaa8qaca aIXaGaey4kaSYaaSaaa8aabaWdbiaadAhacaWGxbaapaqaa8qacaWG JbWdamaaCaaabeWcbaqcLbmapeGaaGOmaaaaaaqcfaOaci4yaiaac+ gacaGGZbGaci4yaiaac+gacaGGZbWaaeWaa8aabaWdbiabeM8a3jaa dshaaiaawIcacaGLPaaaaiaawUfacaGLDbaacqGH9aqpcaWGlbWaam Waa8aabaWdbiaaigdacqGHRaWkcaWGbbGaci4yaiaac+gacaGGZbWa aeWaa8aabaWdbiabeM8a3jaadshaaiaawIcacaGLPaaaaiaawUfaca GLDbaaaaa@618B@ (27)

The amplitude Ain (27) could be more easily measured if interferometer techniques were available where the constant amplitude K is locked and fixed and we could estimate Wby resolving the relative fluctuation proportional to A=vW/c2 and alternating with frequency ω. In any case, the stability of the amplitude K, is probably the limiting factor in any such possible experiment because the inevitable vibrations of the table or pillar on which the experiment is mounted will be hard to cancel out altogether. Therefore, the "constant" Kwill have its own fluctuations which can exceed A. Hopefully, we can transfer some of the advanced techniques developed for gravitational wave detection,49 to the scenario of our experiment. There are vibration cancellation schemes available for wave interferometers, although very expensive. The phase measuring gravity wave detectors (which are the best in this category) keep Knear zero and, in special conditions by various enhancement techniques, one can theoretically hope for 10−9−10−11 of a fringe. Sometimes adding an optical cavity multiplies the signal by factor of 102.

Already at the time of Sagnac, for similar experiments [30, 49-50] the fringe shift was ΔφK1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcqaHgpGAcqWIdjYocaWGlbGaeS4qISJaaGymaaaa @3DB4@ . With this value for Kin (27) and with an optimal resolution of 10−13 of a fringe, requiring a lot of work and money, we can get an idea of the value of W that can be detected in some experimental conditions. Assuming that the smallest detectable value of A=vW/ c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbGaeyypa0JaamODaiaadEfacaGGVaGaam4ya8aadaah aaqabKqbGeaapeGaaGOmaaaaaaa@3D0D@ is 10 13 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIXaGaaGima8aadaahaaqabKqbGeaapeGaeyOeI0IaaGym aiaaiodaaaaaaa@3AED@ , with a velocity v450( m/s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG2bGaeS4qISJaaGinaiaaiwdacaaIWaWdamaabmaabaWd biaad2gacaGGVaGaam4CaaWdaiaawIcacaGLPaaaaaa@3F5B@ (a value of the order of the tangential speed of the Earth at the equator), we have W=  10 13 c 2 /v20( m/s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGxbGaeyypa0JaaeiiaiaaigdacaaIWaWdamaaCaaabeqc fasaa8qacqGHsislcaaIXaGaaG4maaaajuaGcaWGJbWdamaaCaaabe qcfasaa8qacaaIYaaaaKqbakaac+cacaWG2bGaeS4qISJaaGOmaiaa icdapaWaaeWaaeaapeGaamyBaiaac+cacaWGZbaapaGaayjkaiaawM caaaaa@494A@ as the minimum absolute velocity W detectable. Historically, the velocity W with respect to the preferred frame has been expected to be that of the Earth’s orbital speed (W30( km/s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikaa baaaaaaaaapeGaam4vaiabloKi7iaaiodacaaIWaWdamaabmaabaWd biaadUgacaWGTbGaai4laiaadohaa8aacaGLOaGaayzkaaaaaa@4018@ , in relation to the Michelson-Morley experiment) or the absolute speed of the Earth through space (W300( km/s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikaa baaaaaaaaapeGaam4vaiabloKi7iaaiodacaaIWaGaaGima8aadaqa daqaa8qacaWGRbGaamyBaiaac+cacaWGZbaapaGaayjkaiaawMcaaa aa@40D2@ , in relation to the observed cosmic background-radiation anisotropy). Therefore, we could hope to verify these two assumptions and even check if the hypothetical preferred frame is identifiable with the inertial rest frame of the center of the Earth (W450( m/s )) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikaa baaaaaaaaapeGaam4vaiabloKi7iaaisdacaaI1aGaaGima8aadaqa daqaa8qacaWGTbGaai4laiaadohaa8aacaGLOaGaayzkaaGaaiykaa aa@4095@ . Although the realization and the cost of the experiment are certainly quite challenging factors, we believe this test not to be completely outside the reach of present technology. We may conclude that the two synchronizations (Einstein and absolute) are not physically equivalent in principle, but almost equivalent in practice and, should experimental evidence favor the absolute synchronization, it is understandable why experimental discrepancies with Einstein synchronization have not been evidenced so far. In this case, we may state that, from an operational perspective, Einstein synchronization represents a useful simple approach to the procedure of clock synchronization, capable of reproducing approximately the results of relativistic theories based on absolute synchronization.

Nevertheless, we will show in a future contribution that, with a modified experiment of the Sagnac type, the variation in Δ t * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqqHuoarcaWG0bWdamaaCaaabeqcfasaa8qacaGGQaaaaaaa @3A21@ of expression (26) can be enhanced to the order < (L/c)(vW/ c 2 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGOaGaamitaiaac+cacaWGJbGaaiykaiaacIcacaWG2bGa am4vaiaac+cacaWGJbWdamaaCaaabeqcfasaa8qacaaIYaaaaKqbak aacMcacaGGSaaaaa@419D@ which should be more easily measurable.

Consequences of the non-equivalence of Einstein and absolute synchronization

In the case of the Michelson-Morley experiment, the observed null result rules out the Galileo transformations, which foresee a non-null outcome. Instead, the TT and LT are confirmed because both predict the observed null result. In the case of the Sagnac effect, the non-null result W= 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGxbGaeyypa0Jaaeiiaiaaicdaaaa@39E3@ would confirm absolute synchronization, i.e., Galileo transformation and the TT, while disproving standard SR. If the experimental result indicates that W= 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGxbGaeyypa0Jaaeiiaiaaicdaaaa@39E3@ , the conclusion that can be drawn is the same as that of the Michelson-Morley experiment: the preferred frame coincides with the laboratory frame Sand both LT and TT are confirmed. However, it would be a conceptual error to conclude that they are equivalent, after having shown above that in principle they are physically distinguishable. A conclusive experimental proof consists of repeating the experiment having the Sagnac apparatus in motion with respect to S, identified as the preferred frame by the first experiment. Regardless of experimental verification, the mere existence in principle of such a test proves that absolute and Einstein synchronization are not physically equivalent. Even if the outcome of the experiment indicates the existence of a preferred frame of reference, the principle of relativity of physical laws holds. However, in this case, the principle has to reflect the invariance under transformations adopting absolute synchronization.

In conclusion, the countless theoretical approaches to modern physics based on the Lorentz covariance are substantially different from, and cannot be equivalently substituted by, approaches based on the invariance of physical laws under the TT. Therefore, it is conceptually unfeasible to use the Tangherlini transformations in trying to conciliate, as claimed by Kassner, the superluminal average speed of the Sagnac effect with the postulate of the constancy of the speed of light by invoking the arbitrariness of synchronization.

Conclusions

The thesis of conventionalism is based on the assumption that clock synchronization is arbitrary, so that the one-way speed of light is conventional and not measurable. Accordingly, Einstein’s second postulate of special relativity, requiring a universal light speed c in all inertial frames of reference, is not falsifiable. Consequently, all the experiments supporting special relativity can be interpreted in terms of the Lorentz transformations based on Einstein synchronization, as well as the physically equivalent Tangherlini transformations based on absolute synchronization. This scenario implies that the Lorentz transformations can be substituted by the Tangherlini transformations, a substitution hardly acceptable by physicists who have been relying on the symmetry of the Lorentz group for decades.

For the Sagnac effect, where light signals propagate in a closed path, for counter-clockwise light propagation, relativistic theories foresee the superluminal average speed c+v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqWIdjYocaWGJbGaey4kaSIaamODaaaa@3A9A@ , a well-known circumstance that leads to Selleri’s paradox and, according to detractors of SR, invalidates Einstein’s second postulate. The analysis of the "linear" Sagnac effect points out the role of non-conservation of simultaneity in keeping the local speed c in different sections of the tube when Einstein synchronization is used. Since the superluminal average speed is c+v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqWIdjYocaWGJbGaey4kaSIaamODaaaa@3A9A@ over the total length 2γL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIYaGaeq4SdCMaamitaaaa@39D8@ of the tube, if in the upper part of the tube clocks are synchronized in such a way that the local speed is c, it follows that in the lower part of the tube the local speed must be c+ 2v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqWIdjYocaWGJbGaey4kaSIaaeiiaiaaikdacaWG2baaaa@3BF9@ , and vice versa. Thus, detractors of SR may claim once more that the existence of superluminal light speeds invalidates Einstein’s second postulate. Despite Selleri’s paradox and superluminal average speeds, supporters of Einstein synchronization may argue that special relativity provides the correct observable result for the Sagnac effect and, thus, is still valid from a pragmatic operational point of view.

The controversy about the conventionality of the one-way speed of light and the validity or not of Einstein synchronization, reaches a turning point when it is shown that the second postulate of SR can be tested with an experiment of the Sagnac type, capable of discriminating absolute from Einstein synchronization. Hence, Einstein’s second postulate is falsifiable and the one-way speed of light is measurable in principle. From a theoretical point of view, an immediate consequence is that the symmetry of the Lorentz group maintains its unique physical meaning and cannot be equivalently substituted by the group properties of transformations based on absolute synchronization. Moreover, the claim that the existence of superluminal speeds in the Sagnac effect can be justified in terms of the arbitrariness of synchronization becomes groundless.

The experiment is realizable with present technology and the required sensitivity of the measuring apparatus is evaluated in some scenarios. The outcome of the test will have a significant impact on modern physics because, by testing Einstein’s postulate of a universal speed of light, it identifies, or rules out, the preferred frame of reference.

Acknowledgments

This work was supported in part by the CDCHTA of the Universidad de Los Andes, Mérida, Venezuela.

Conflicts of interest

Authors declare that there are no conflicts of interests.

References

  1. Krisher TP, Lute Maleki, George F Lutes, et al. (1990) Test of the isotropy of the one-way speed of light using hydrogen-maser frequency standards. Phys Rev D. 1990;42:731.
  2. Spavieri G, Quintero J, CS Unnikrishnan, et al. Can the one-way speed of light be used for detection of violations of the relativity principle? Phys Lett A. 2012;376:795–797.
  3. H Poincare. Létat Actuel Et l'avenir de la physique mathématique; Parm Henri Poincaré. Bulletin des Sciences Mathématiques. 1994;28:302–324.
  4. A Einstein. Einstein’s Miraculous Year. Princeton University Press, New Jersey, USA. 1998.
  5. R Reichenbach. Axiomatization of the Theory of Relativity. University of California Press 1st edn. (1924) & The philosophy of Space and Time. Dover (1st German edn.) (1928). 1969.
  6. Grunbaum. Philosophical Problems in Space and Time 1st edn. Cambridge Univ. Press, Cambridge, USA. 1973.
  7. C Moller. The Theory of Relativity. Suppl Nuovo Cimento. 1957;(6):381.
  8. R Mansouri, RU Sexl. A test theory of special relativity: I. Simultaneity and clock synchronization. Gen Rel Grav. 1977;8(7): 497–513.
  9. FR Tangherlini. The Tangherlini transformations are known in the literature also as the IST 20(1) transformations and (2015) The Selleri transformations and the one-way speed of light by Stephan JG Gift. 1961;28:474–481.
  10. JS Bell. Speakable and Unspeakable in Quantum Mechanics 2nd edn. Cambridge Univ. Press, Cambridge, USA. 1988.
  11. R de Abreu, V Guerra. Thought experiment discriminating special relativity from preferred frame theories. Eur J Phys. 2005;26:117–123.
  12. Vasco Guerra, Rodrigo de Abreu. On the Consistency between the Assumption of a Special System of Reference and Special Relativity. Foundation Physics. 2006;36(12):1826–1845.
  13. R de Abreu, V Guerra. Special relativity as a simple geometry problem. Eur J Phys. 2008;29:33–52.
  14. F Selleri. The Inertial Transformations and the Relativity Principle. Foundations of Physics Letters. 2005;18(4):325–339.
  15. B Ellisv, P Bowman. Conventionality in Distant Simultaneity. Philosophy of Science. 1967;34:116–136.
  16. RW Brehme. The role of dynamics in the synchronization problem. American Journal of Physics. 2004;72(2):56–59.
  17. R Anderson, I Vetharaniam, GE Stedman. Conventionality of synchronisation, gauge dependence and test theories of relativity. Physics Reports. 1988;295(3):93–180.
  18. MM Capria. On the Conventionality of Simultaneity in Special Relativity. Foundation Physics. 2001;31(5):775–818.
  19. HC Ohanian. The role of dynamics in the synchronization problem. American Journal of Physics. 2004;72:141–148.
  20. G Spavieri. On measuring the one-way speed of light. The European Physical Journal D. 2012;66:76.
  21. G Sagnac.Sagnac effect in an off-center rotating ring frame of reference. CR Acad Sci English translation in Relativity in rotating frames, chap 2nd ed. Rizzi G & Ruggiero ML Kluwer Academic Publishers, Dordrecht, Netherlands. 2004;1913; p. 5–7.
  22. K Kassner. Ways to resolve Selleri's paradox. American Journal of Physics. 2012;80(12):1061.
  23. F Selleri. Noninvariant one-way speed of light and locally equivalent reference frames. Foundations of Physics Letters. 1997;10(1):73–83.
  24. F Selleri. Recovering the Lorentz Ether. Apeiron. 2004;11(1):246–281.
  25. KR Atkins. Lorentz ether theory versus relativity — The possibility of new experimental evidence. Phys Lett A. 1980;80(4):243–245.
  26. M Consoli, E Costanzo. From classical to modern ether-drift experiments: the narrow window for a preferred frame. Phys Lett A. 2004;361(6):509–512.
  27. M Consoli, E Costanzo. Reply to:Comment on: From classical to modern ether-drift experiments: the narrow window for a preferred frame. Phys Lett A. 2007;361:513–514.
  28. G Spavieri. On measuring the one-way speed of light. 2012;66:76.
  29. Hafele C, Keating RE. Around-the-World Atomic Clocks: Predicted Relativistic Time Gains. Science. 1972;177(4044):166–170.
  30. R Wang, Y Zhengb, A Yaob, et al. Modified Sagnac experiment for measuring travel-time difference between counter-propagating light beams in a uniformly moving fiber. Physics Letters A. 2012;312:7–10.
  31. EJ Post. Sagnac Effect. Review of Modern Physics. 1967;39(2):475–493.
  32. LD Landau, EM Lifshitz. Teoriya Polya. The Classical Theory of Fields. Oxford: Pergamon Press. 1967.
  33. VI Bashkov, V Sintsova Yu. Connections in Principal Bundles and the Sagnac Effect , Grav. Cosmology. 1999;5:319–320.
  34. VI Bashkov, V Sintsova Yu. Rep Math Phys. 2001;48:353.
  35. VI Bashkov, M Malakhaltsev. In Intern Conf “Geometrization of Physics V”. 2001. p. 24–28.
  36. NV Kupryaev. Izv Vyssh Uchebn. Zave Fiz. 1999;42(7):8.
  37. NV Kupryaev. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika. 2001;44(8):63.
  38. NV Kupryaev. The Sagnac Vortex Optical Effect. Russian Physics Jornal. 2001;44(8):858–862.
  39. GB Malykin. Sagnac effect in a rotating frame of reference. Relativistic Zeno paradox. Physics-Uspekhi. 2002;45(8):907–909.
  40. TA Weber. Measurements on a rotating frame in relativity, and the Wilson and Wilson experiment. American Journal of Physics. 1997;65: 946–953.
  41. J Anandan. Sagnac effect in relativistic and nonrelativistic physics. Physical Review D. 1981;24:338–346.
  42. SJG Gift. On the Selleri Transformations: Analysis of Recent Attempts by Kassner to Resolve Selleri’s Paradox. Applied Physics Research. 2015;7(2):112–120.
  43. N Ashby. Relativity in the Global Positioning System. Living Reviews in Relativity. 2003;6(1):1–42.
  44. F Selleri. Relativity in Rotating Frames. In: G Rizzi & ML Ruggiero editors. Kluwer Academic Publishers, London. 2010.
  45. SJG Gift. One-Way Speed of Light Relative to a Moving Observer. Applied Physics Research. 2013;5(1):93–106.
  46. DA Ludlow, MM Boyd, J Ye, et al. Optical atomic clocks. Reviews Modern Physics. 2015;87:637.
  47. Kim J, Chen J, Cox J, et al. Attosecond-resolution timing jitter characterization of free-running mode-locked lasers. Optics Letters. 2007;32(24):3519–3521.
  48. Kwon D, Jeon CG, J Shin, et al. BMP2-modified injectable hydrogel for osteogenic differentiation of human periodontal ligament stem cells. Scientific Reports. 2007.
  49.  Abbott BP. Observation of Gravitational Waves from a Binary Black Hole Merger. Physical Review Letters. 2016;116(6):061102.
  50. Pogany B. Über die Wiederholung des Harress‐Sagnacschen Versuches. Ann Physik. 1928;385(11):217–231.
  51. Pogany B. Über die Wiederholung des Harressschen Versuches. Ann Physik. 1928;390(2):244–256.
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