View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server NON-INVARIANT VELOCITY OF LIGHT AND CLOCK SYNCHRONISATION IN ACCELERATED SYSTEMS 1 By F. Goy - Dipartimento di Fisica - Universita di Bari - Via G. Amendola, 173 - 70126 Bari - Italy - E-mail: [email protected] Abstract. Clo ck synchronisation is conventional when inertial systems are involved. This statement is no longer true in accelerated systems. A demonstration is given in the case of a rotating platform. We conclude that theories based on the Einstein's clo ck synchronisation pro cedure are unable to explain, for example, the Sagnac e ect on the platform. Implications on very precise clo ck synchronisation on earth are discussed. 1 Conventionality of clock synchronisation in inertial frames Let us b egin with an op erationalistic de nition of physical time: \time is what is measured by clo cks". If wewant to apply this de nition for the setting of clo cks not only in one p oint, but everywhere in space, we see that two di erent and indep endent notions are implied: The rate at which time ows in each p oint. The simultaneityofevents in di erent p oints of space. Following Poincare [1] and Einstein [2], we can use light signals for the setting of two clo cks of identical fabrication, which are at two p oints A and B of an inertial frame. We send a light signal from A at time t , which arrives at B at time t and comes backatAat time 1 2 t . The time t of B is de ned to b e synchronous with the midtime of departure and arrival 3 2 in A. This de nition is called the Einstein's synchronisation. Mathematically: 1 t = t + (t t ) (1.1) 2 1 3 1 2 Reichenbach commented [3]:"This de nition is essential for the special theory of relativity, but not epistemological ly necessary. If we were to fol low an arbitrary rule restricted only to gr-qc/9607047 22 Jul 1996 the form t = t + "(t t ) 0 <" < 1 (1.2) 2 1 3 1 it would likewise be adequate and could not cal ledbe false. If the special theory of relativity prefers the rst de nition, i.e., sets " equal to 1=2,itdoes so on the ground that this de nition leads to simpler relations." On the p ossibilitytocho ose freely " according to (1.2) agreed, among others, Winnie [4], Grun baum [5], Jammer [6], Mansouri and Sexl [7], Sjodin [8], Cavalleri and Bernasconi [9], Ungar [10], Vetharaniam and Stedman [11], Anderson and Stedman [12 ], [13 ]. Clearly, di erentvalues of " corresp ond to di erentvalues of the one way-sp eed of light. A slightly di erent p osition was develop ed in the parametric test theory of sp ecial relativity of Mansouri and Sexl. Following these authors, we assume that there is at least one inertial frame in which light b ehaves isotropically.We call it the priviledged frame and denote space and time co ordinates in this frame by the letters: (x ;y ;z ;t ). In , clo cks are 0 0 0 0 1 Financial supp ort of the Swiss National Science Foundation and the Swiss Academy of Engineering Sciences. 1 synchronised with Einstein's pro cedure. We consider also an other system S moving with uniform velo city v<calong the x -axis in the p ositive direction. In S , the co ordinates 0 are written with lower case letters (x; y; z; t). Under rather general assumptions, symmetry conditions on the two systems, the assumption that the two-wayvelo city of lightis c and furthermore that the time dilation factor has its relativistic value, one can derive the following transformation: 1 p x = (x vt ) 0 0 2 1 y = y 0 z = z (1.3) 0 q 2 t = s ( x vt )+ 1 t ; 0 0 0 where = v=c. The parameter s, which determines the synchronisation in the S frame p 2 2 remains unknown. Einstein's synchronisation in S involves: s = v=(c 1 ) and (1.3) b ecomes a Lorentz b o ost. For a general s, the inverse one-wayvelo city of lightisgiven by [14 ]: ! q 1 1 2 cos ; (1.4) = + + s 1 c () c c ! where is the angle b etween the x-axis and the lightrayinS. c () is in general dep endent ! on the direction. A simple case is s = 0. This means from (1.3), that at T =0 ofwe set all clo cks of S at t = 0 (external synchronisation), or that we synchronise the clo cks by means of lightrays with velo city c () = c=(1 + cos ) (internal synchronisation). It should b e ! stressed that, unlike to the parameters of length contraction and time dilation, the parameter scannot be tested, but its value must b e assigned in accordance with the synchronisation cho osen in the exp erimental setup. It means, as regards exp erimental results, that theories using di erent s are equivalent. Of course, they may predict di erentvalues of physical quantities (for example the one-way sp eed of light). This di erence resides not in nature itself but in the convention used for the synchronisation of clo cks. For a recent and comprehensive discussion of this sub ject, see [15]. A striking consequence of (1.4) is that the negative result of the Michelson-Morley exp eriment do es not rule out an ether. Only an ether with galilean transformations is excluded, b ecause the galilean transformations do not lead to an invariant two-wayvelo city of lightinamoving system. Strictly sp eaking, the conventionality of clo ck synchronisation was only shown to hold in inertial frames. The derivation of equation (1.3) is done in inertial frames and is based on the assumption that the two-wayvelo city of light is constant in all directions. This last assumption is no longer true in accelerated systems. But sp ecial relativity is not only used in inertial frames. A lot of textb o oks bring examples of calculations done in accelerated systems, using in nitesimal Lorentz transformations. Such calculations use an additional assumption: the so-called Clock Hypothesis, which states that the rate of an accelerated ideal clo ckis identical to that of the instantaneously comoving inertial frame. This hyp othesis rst used implicitelyby Einstein in his article of 1905 was sup erbly con rmed in the famous timedecay 18 exp erimentofmuons in the CERN, where the muons had an acceleration of 10 g , but where their timedecaywas only due to their velo city [16]. We stress here the logical indep endence of this assumption from the structure of sp ecial relativityaswell as from the assumptions necessary to derive (1.3). The opinion of the author is that the Clock Hypothesis, added to sp ecial relativity in order to extend it to accelerated systems leads to logical contradictions when the question of synchronisation is brought up. This idea was also expressed by Selleri 2 [17 ]. The following example (see [18]) shows it: imagine that two distant clo cks are screwed on an inertial frame (say a train) and synchronised with an Einstein's synchronisation. The train accelerates during a certain time. After that, the acceleration stops and the train has again an inertial mouvement. It is easy to show that the clo cks are no more Einstein's synchronous. So the Clock Hypothesis is inconsistent with the clock setting of relativity.On the other hand, the Clock Hypothesis is tested with a high degree of accuracy [19] and cannot b e rejected, so one has to reject the clo ck setting of sp ecial relativity. The only theory which is consistent with the Clock Hypothesis is based on transformations (1.3) with s = 0. This is an ether theory. The fact that only an ether theory is consistent with accelerated motion give strong evidences that an ether exist, but do es not involve inevitably that our velo city relative to the ether is measurable. It remains an imp ortant op en question whichisbeyond the scop e of this pap er. The opinion of the author is that it cannot b e measured in the ab ove example, in spite of the logical diculties of sp ecial relativity. There are strong evidences that when a temp orary acceleration is used to go from one inertial frame to an other, an \absolute" motion cannot b e measured, as it has b een shown by the author in the case of stellar ab erration [20]. The situation could b e di erent when a p ermanent acceleration acts on a system. The logical diculties of sp ecial relativity indicates that the relativity principle (The lawofphysics are the same in all inertial frames) do es not only express an ob jective prop erty of nature, but also a human choice ab out the setting of clo cks. The ob jective part of the relativity principle is expressed in the statement: an uniform motion through the ether is not detectable. Equations (1.3) ob eys to this second statement. 2 The Sagnac effect An uniform motion through the ether was never detected, what is expressed in the negative result of the Michelson-Morley exp eriment. In contrary, rotational motion can b e detected by mechanical metho ds like the Foucault p endulum as well as by luminuferous metho ds like in the Sagnac e ect. Eightyears after 1905, Sagnac wrote an article whose title shows that he thoughttohave proved the reality of ether [21].