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y of Engineering t k t, al ks at time o clo c e send a w A kat ery precise clo c ed. This statemen olv v ks not only in one p oin t notions are implied: oundation and the Swiss Academ 1 and comes bac 2 tial frames hronisation pro cedure are unable t of an inertial frame. W TED SYSTEMS Implications on v en in the case of a rotating platform. B t signals for the setting of t k sync ysical time: \time is what is measured and at time A a di Bari - Via G. Amendola, 173 - 70126 t and indep enden hronisation. Mathematically: ts of space. B t. tion in iner ts CCELERA 1 ersit + t p oin es at onisa 1 e can use ligh h p oin t hronous with the midtime of departure and arriv o p oin o di eren tional when inertial systems are in w = w 2 en t h arriv v View metadata,citationandsimilarpapersatcore.ac.uk TION IN A ws in eac to b e sync ts in di eren , whic 1 h are at t d en gr-qc/9607047 22 Jul 1996 t e see that t to di Fisica - Univ ONISA [email protected] ARIANT VELOCITY OF LIGHT AND CLOCK de ne h time o Financial supp ort of the Swiss National Science F y yofev t to apply this de nition for the setting of clo c is 1 at time hronisation is con an B (1.2) (1.1) e [1] and Einstein [2], w Sciences. A ks are inertial of ew elativity odin [8], elativity, k sync 2 d only to ultaneit SYNCHR - Dipartimen t oincar 1 NON-INV y Clo c ast one ory of r ks". If w estricte tical fabrication, whic alues of the one ory of r erywhere in space, w In , clo c The sim The rate at whic Conventionality of clock synchr act. wing P F. Go . This de nition is called the Einstein's sync hronisation on earth are discussed. tv t signal from at le ). A 0 e conclude that theories based on the Einstein's clo c . The time bstr ollo y clo c 3 ;t cial the in t ligh of iden F Let us b egin withb an op erationalistic de nition of ph but ev is no longer true in accelerated systems. A demonstration is giv W to explain, for example, the Sagnac e ect on the platform. sync 1 Bari - Italy - E-mail: go A By ound that this de nition cial the 0 e e ary rule r ;z according to (1.2) agreed, 0 " ;y 0 x 1 ) es so on the gr 1 <" < t 0 e false. If the sp ho ose freely corresp ond to di eren e assume that there is 3 ,itdo db " ) t 2 1 ( e call it the priviledged frame and denote t = 2 e to fol low an arbitr al le 1 ytoc etharaniam and Stedman [11], Anderson and y the letters: ( 3 .W t ( " alues of qual to + e tv 1 elop ed in the parametric test theory of sp ecial relativit t " ould not c baum [5], Jammer [6], Mansouri and Sexl [7], Sj = 2 un t as dev "This de nition is essential for the sp essary. If we wer wing these authors, w , di eren c es isotropically v " On the p ossibilit ollo quate and c ted [3]: al ly ne t b eha e ade gic t p osition w elations. t. h ligh h commen bac tly di eren provided byCERNDocumentServer hen alleri and Bernasconi [9], Ungar [10], V brought toyouby v y-sp eed of ligh efers the rst de nition, i.e., sets a ads to simpler r space and time co ordinates in this frame b frame in whic among others, Winnie [4], Gr w of Mansouri and Sexl. F A sligh le Ca Stedman [12 ], [13 ]. Clearly pr the form it would likewise b but not epistemolo Reic CORE

synchronised with Einstein's pro cedure. We consider also an other system S moving with

uniform velo city v

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:

x = (x vt )

0 0

y = y

z = z (1.3)

t = s ( x vt )+ 1 t ;

0 0 0

where = v=c. The parameter s, which determines the synchronisation in the S frame

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 ]:

1 1

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

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]. The Sagnac e ect is essentially the

observation of the phase shift b etween two coherent b eams travelling on opp osite paths in

an interferometer placed on a rotating platform. In 1925, Michelson and Gale [22 ], detected

the earth rotation rate with a giantinterferometer constructed with this aim. Michelson

explained the measured e ect with a velo city of light which is not constant. Nowdays the

Sagnac e ect is observed with light (in ring lasers and b er optics interferometers [23]) and in

interferometers built for electrons [24], neutrons [25 ], atoms [26] and sup erconducting Co op er

pairs [27]. The phase shift in interferometers is a consequence of the time delaybetween

the arrival of the two b eams, so a Sagnac e ect is also measured directly with atomic clo cks

timing light b eams sent around the earth via satellites [28 ].

In the typical exp eriment for the study of the e ect, a mono chromatic light source placed on

a disk emits two coherent light b eams in opp osite directions along the circumference until

they reunite after a 2 propagation. The p ositionning of the interference gure dep ends on

the disk rotational velo city.Textb o oks deduce the Sagnac formula in the lab oratory, but

say nothing ab out the description of the phenomenon on the rotating platform. Exception

to this trend are Michelson [22], Langevin [29], Anandan [30 ], Dieks and Nienhuis [31 ] and

Post [32 ], but dissatisfaction remains widespread, b ecause none of these treatments is free

of ambiguities. Michelson uses a noninvariantvelo city of light, but with a galilean addition

of velo cities, whichisincontradiction with his exp eriment of 1887. Langevin, in his 1937, 3

pap er recognized the p ossibility of a nonstandard velo city of light on the rotating platform,

but gives a formula only valid to the rst order. Post's relativistic formula is not generally

valid, but limited to the case where the origin of the tangential inertial frame coincides with

the center of the rotating disk.

So let us deduce here the Sagnac formula for a circular Sagnac device rotating in the an-

ticlo ckwise sense in the priviledged frame , for simplicity. We rst calculate the time

di erence b etween a clo ckwise and an anticlo ckwise b eam of light constrained to followa

circular path of radius R as seen by an observer in the lab oratory and secondly we will make

the calculation on the rim of the disk. The two b eams leave the b eamsplitter at time t =0

when it is in p osition C (see gure 1). The clo ckwise circulation is opp osite to the direction

of rotation and meets the b eamsplitter when it is in p osition C' shifted bys with resp ect

to C, at time t . The anticlo ckwise b eam, travelling in the same direction as the direction

of rotation meets the b eamsplitter in the later p osition C", shifted bys , with resp ect to

C at time t . The geometry is given in the following gure. ! is the rate of rotation of the

interferometer.

Figure 1: Simpli ed Sagnac con guration: the light b eams are drawn with dashed lines

Wehave:

0 0 0 0

s = !Rt ; L s = ct (2.1)

0 0

and

00 00 00 00

; (2.2) ; L +s = ct s = !Rt

0 0

where L is the circumference of the rotating disk measured in the lab oratory. The length

of this circumference is reduced relative to the length L =2R measured directly on the

disk (see section 3 for a discussion of the geometry of the rotating disk). Wehave: L =

0 00

2 2 2

L 1 ! R =c . Eliminating s and s from equations (2.1) and (2.2), we obtain the

00 0

time di erence t = t t in the lab oratory

0 0

4!A

t = ; (2.3)

2 2 2 2

c 1 ! R =c

. Because of the time dilation, an observer on the platform must where A stands for R

nd a time di erence t, so that:

t = ; (2.4)

2 2 2

1 ! R =c 4

which gives using (2.3):

4!A

t = (2.5)

Let us now directly calculate the result on the platform, as done by Selleri [17]. Every

in nitesimally small p ortion of the rim of the disk can b e considered to b e at rest in a

comoving inertial frame tangential to the disk. The Clock Hypothesis of sp ecial relativity

tells us that the rate of ideal clo cks on the rim is the same as in the comoving inertial

frame. Similary ideal unit measuring ro ds b ehaves in the same way on the rim and in the

comoving inertial frame [33, p. 254]. For reasons of continuity,wemust cho ose the same

synchronisation on a small p ortion of the rim as in the tangeantial inertial frame. The one-

wayvelo city of light is given by (1.4). The time di erence b etween the arrival of the two

light b eams as calculated on the disk is given by:

0 1

2 2 2

sc 1 ! R =c

1 1 4!A

00 0

@ A

t = t t = L = 1+ (2.6)

c (0) c ( ) c !R

! !

Comparing this last formula to (2.5), we see that the only value giving a correct prediction

of the Sagnac e ect on the platform is s =0. In all other cases, the formula (2.5) is

not recovered as it should b e, b ecause (2.3) is predicted by all theories having the general

transformation (1.3) since they all use an Einstein's synchronisation in the lab oratory.In

particular, the theory of sp ecial relativity gives t = 0, when the value of s given after (1.3)

is substitued in (2.6).

Indep endently of the ab ove considerations, wewould draw the attention of the reader to

our formula (2.3), which di ers in the second order in ! R=c, from the formula given byPost

[32 ] and followers. In the same physical situation, he gives in the lab oratory:

4!A

P ost

t = ; (2.7)

2 2 2 2

c (1 ! R =c )

The discrepancy comes from the fact that Post do es not take account of the Lorentz con-

traction of the circumference of the disk. From our p oint of view, this is erroneous. Unfor-

tunately, the precision of measures of the Sagnac e ect do es not enable to decide b etween

the two formulas on an exp erimental level, so that wehave no empirical information on the

physics of a relativistic rotating disk.

3 Clock synchronisation in the formalism of general relativity

The same disk as in section 2 is considered. We can obtain the metric on the disk as follow:

in the inertial system the squared line element ds in cartesian co ordinates is:

2 2 2 2 2 2

(3.1) dz dy dx ds = c dt

0 0 0 0

In the formalism of general relativity one has a large lib erty in the choice of the co ordinate

system useful for solving a given problem. Since we are doing physics and not only di erential

geometry, once given co ordinates are cho osen, the problem of their interpretation in terms

of measurable quantities has still to b e solved. In the case of the rotating disk it is simpler

to use the co ordinates in the right-hand side of the following transformations, as done for 5

examples by Langevin [29] and by some textb o oks [33, p. 253], [34, p. 107], [35, p. 281].

t = t

x = r cos (' + !t)

y = r sin(' + !t) (3.2)

z = z

As stressed byMoller, in his treatment of the problem of the rotating disk, the rst equation

of (3.2) do es not mean that we are dealing with Newtonian physics, where time dilation

e ects are missing, but that the co ordinate t is measured by a clo ck on the disk, that runs

2 2 2

1= 1 ! r =c faster than the clo cks of the lab oratory, when compared at rest. Once this

co ordinate clo ck will b e put on a disk rotating with angular velo city ! at radius r , it will

have the same rate as the clo cks in the lab oratory, b ecause of the time dilation. Similary

the second and the third equation of (3.2) havea physical meaning, taking account of e ects

of longitudinal length contraction, only if the co ordinate ' is measured with tangential

2 2 2

co ordinates-ro ds that are 1= 1 ! r =c longer than the measuring ro ds of the lab oratory

when compared at rest. Substituting (3.2) in (3.1) one can easily obtain:

2 2 2 2 2 2 2 2

d' ( cdt) dr r d' dz (3.3) (cdt) 2 ds = 1 ! r =c

0 1 2

Eq. (3.3) de nes a metric g which is stationary, but not static. If x = ct, x = r , x = ',

x = z , its element are:

2 2 2

g =1! r =c ; g = g = 1

00 11 33

2 2

g = g = !r =c ; g = r ; (3.4)

02 20 22

all other elements b eing zero. Note that the space-time describ ed by (3.4) is at b ecause

R (t ;x ;y ;z )=0)R (t; r;';z)=0 [i; j; k ; l =0;1;2;3], where R is the Riemann

ij k l 0 0 0 0 ij k l ij k l

tensor. For the same reason of covariance, the metric de ned in (3.4) is necessarly a solution

of Einsteins equations in empty space R =0 [i; j =0;1;2;3], where R is the Ricci tensor.

ij ij

As is well known, in the case of a non-static space-time metric, the spatial part of the metric

is not only given by the space-space co ecients of the four dimensional metric, but by:

2 2

g g r d'

0 0

2 2 2

dl = g + dx dx = dr + dz + ; (3.5)

2 2 2

g 1 ! r =c

where 0 is the time index, and ; represent the space indices and can take the values 1; 2; 3.

The right-hand-side of (3.5), with dz =0,isinterpreted by most textb o oks as the standard

result, showing that the spatial part of the metric is not at on the rotating disk. The opinion

of the author is that it demonstrates exactly the contrary, namely that space is at on the

rotating disk. Wehave seen that the angle ' was measured with tangeantial co ordinates-

ro ds, which, at rest, are longer than those of the lab oratory, the radial measuring ro ds on the

disk having the same rest length as those of the lab oratory. If in contrary we had cho osen

tangential unit ro ds which, at rest, have the same length as the ro ds of the lab oratory the

co ordinate ' measured with these ro ds on the disk would satisfy the following equation:

2 2 2

d' = d' 1 ! r =c ; (3.6) 6

so that measured with these \normal" ro ds (3.5) b ecomes:

2 2 2 2 02

dl = dr + dz + r d' ; (3.7)

which is a at metric in p olar co ordinates.

Wenow try to set clo cks with an Einstein's synchronisation on the disk. Generally,ifwe

send a light signal from p oint A with co ordinates x ; =1;2;3 to an in nitesimally near

p oint B with co ordinates x + dx ; =1;2;3 and back, the co ordinate time di erence dt

(dt ) for the \there" (back) trip is obtained by solving the equation ds =0.We obtain:

q 2

1 !r d' dl

dt = g dx + (g g g g ) dx dx = +

1;2 0 0 0 00

2 2 2 2

2 2 2

cg c (1 ! r =c )

c 1 ! r =c

(3.8)

By de nition the time t ,atA, which is synchronous with the arrival time t at B is

A B

the midtime of departure and arrival at A. So, two Einstein-synchronous events are not

co ordinate-time-synchronous (see gure 2) and have a di erence t such that:

1 g dx !r d'

t = t +t = t (3.9) = t +

B A A A

2 2 2 2

c g c (1 ! r =c )

If we generalise this pro cedure, not only in an in nitesimal domain, but along a curve, we

Figure 2: Illustration of equations (3.8) and (3.9). Einstein's synchronous events t and t

A B

are joined with a dashed line. Co ordinate-time synchronous events are joined with a doted

line.

obtain that generally it is path dep endant, b ecause t is not a total di erential in ' and r .

As consequence, time can not b e de ned globally on the disk with Einstein's pro cedure. This

is what in fact happ ens on earth: if one synchronises atomic clo cks all around the earth with

Einstein's pro cedure and comes back to the p oint of departure after a whole round trip, a

time lag will result. This means that a clo ck is not synchronisable with itself, which is clearly

absurd. Moreover, we will generally not obtain the same result, when synchronising clo ckA

with clo ck B using two di erent paths. This mean also that if a clo ck B is synchronised with

A and a clo ck C is synchronised with B, C will generally not b e synchronised with A. Hence

the physicist Ashby from Boulder has said: \Thus one discards Einstein's synchronisation

in the rotating frame"[36]. 7

The existence of synchronisation problem is physically strange, b ecause if the whole disk is

initially at rest in the lab oratory, with clo cks near its rim synchronised with the Einstein's

pro cedure, since we are in , then when the disks moves accelerates and attains a constant

angular velo city, the clo cks must slow their rate, but not desynchronise, for symmetry rea-

sons, since they had at all time the same sp eed. From suchapoint of view, it is dicult to

see why there should b e any diculty in de ning the time on the rotating platform. In fact,

we see that we can easily de ne a global time since the co ordinate time t is already global.

Rememb ering that the co ordinate time t is measured with clo cks that run faster than clo cks

2 2 2

at rest in , we de ne a global time t = 1 ! r =c t. The one-wayvelo city of lighton

the rim is now given by:

dl dl c

AB AB

c ()= = = ; (3.10)

2 2 2

dt 1 !r=c

1;2

dt 1 ! r =c

1;2

where the last step comes from (3.5) and (3.8) with dr = dz = 0 and + () stands for

the anticlo ckwise (clo ckwise) propagation of light. It means that the velo city of lightinthe

tangential inertial frame is also equal to: c ()= c=(1 ), with = !r=c , corresp onding

to a parameter s = 0 of (1.4) and an angle of 0 ( ). From here, the Sagnac e ect can

easily b e calculated, in the same way as in section 2 and the result is found to b e identical.

So we see that the formalism of general relativity is able to describ e without contradictions

the synchronisation of clo cks on the rim of a rotating disk, but implies a velo city of lightin

the comoving inertial frame tangeantial to the rim which is noninvariant and so contradict

the clo ck synchronisation of sp ecial relativity.

4 Conclusion

In inertial systems the synchronisation of clo cks is conventional and a set of theories equiva-

lent to sp ecial relativity, as regards exp erimental results, can b e derived. When extended to

accelerated motion, the synchronisation is no longer conventional and only the theory using

s = 0 from (1.3) is consistent with the Clock Hypothesis. This conclusion is con rmed in

the case of the Sagnac e ect. An elementary calculation shows that only s = 0 enable to

explain the Sagnac e ect on the platform. Since one could say that accelerated motions have

to b e calculated with the formalism of general relativity,wehave done it and shown that

in this case also, only a noninvariantvelo city of light enables to synchronise clo cks globally

on the platform. So, the clo ck synchronisation used in sp ecial relativity is inconsistent with

the global de nition of time used in general relativity. Moreover wehave shown that, in our

view, the geometry of the rotating platform is at.

5 Acknowledgement

Iwant to thank the Physics Departement of Bari University for hospitality.

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