Synchronization of the clocks in the revolving reference marks within the framework of the restricted Relativity.

Synchronization in restricted relativity

That is to say an inertial reference mark R with an origin O. One lays out in each point of the reference mark a motionless clock in R (what is checked easily while placing rules standards and by checking that the space coordinates of the clock do not vary during time).

One chooses then a clock of reference, for example that located out of O. One sends a signal then has known speed then $V$ at the moment $t\_0$ indicated by the clock out of O. If this signal reaches the clock located at the distance $L$, one adjusts this clock so that it indicates to the moment reception time $t\_0+L/V$.

It is checked easily that this procedure of synchronization is consistent (absence of contradiction by changing clock of reference or speed of the signal) as long as one is in a flat space time.

Can imports the speed of the signal as long as it is known. And it is determined easily by measuring it on a return ticket.

The speed of the signal could despite everything be affected of an awkward and undetectable anisotropy to a return ticket measure. To be sure, it is necessary to carry out a great number of measures to all the directions. There remains possibly still a residual anisotropy which could prove to be undetectable whatever measurements. But in this case it does not have a physical consequence since precisely it is undetectable.

It is for these reasons that one employs usually an electromagnetic signal (of the light, for example) in the vacuum, being propagated at the speed $c$. Its constancy, its invariance and its isotropy are guaranteed by:

• the Principle of relativity.
• many experimental data (checking its constancy and its invariance under all the possible conditions).

It is also of a very practical use as well in the real experiments as in the experiments of thoughts.

This procedure using a signal at speed $c$ is also called synchronization of Einstein.

Synchronization on the motionless disc

Now let us look at the problem of synchronization on the disc a little more closely.

We have a procedure to synchronize the clocks of the same reference mark. But this procedure exploited we it in the case of the inertial reference marks.

Knowing restricted relativity and knowing this procedure, how to apply it in the reference mark in R' rotation?

Simultaneity is determined by the time indicated by the clocks. Simultaneous is synonymous with " identical time indicated by two horloges".

Consequently the procedure of synchronization amounts making sure that two motionless clocks in a reference mark indicating same time correspond to simultaneous events.

The concept of reference mark in relativity will be well defined if one can strew the reference mark with synchronized clocks, i.e. if one can make all the points of the reference mark simultaneous (surface of simultaneity or place where all the points are simultaneous). If it were not the case, there would be then a serious problem to allot a temporal coordinate to certain events in this reference mark.

Let us take 4 motionless events and 4 observers in a reference mark R inertial given.

There are 4 observers 1,2,3 and 4, all motionless in inertial reference mark R. And 4 events has, B, C and D taking place at the places indicated.

Attention, it is not because we drew a circle that it there rotating. That indicates only the provision of the events and observers.

The events are selected so as to be able to carry out synchronizations. With and B are simultaneous for 1. B and C simultaneous for 2. C and D simultaneous for 3. And D and has simultaneous for 4.

1 and 2 being motionless in the inertial reference mark. Then, has and B will be also simultaneous for 2. And thus C and A. Of the same, C and B and have and B will be simultaneous for 3, and thus D and A. Enfin, for 4, have, B, C and D all are well simultaneous. All is coherent.

It is even rather commonplace since the inertial reference mark is spatially Euclidean in relativity restricted by assumption. The distances are thus perfectly defined and the guaranteed relative immobility. And thus also simultaneities.

We already announced that a flat space time, and thus a flat space and the Euclidean geometry, guarantees that the procedure of usual synchronization is consistent.

But even if that seems commonplace, we will use this procedure further in a way definitely more delicate.

That is to say a class of accelerated reference marks (with all same acceleration compared to R). With a constant and uniform acceleration. Then, which we have just seen valid remainder. Indeed, that is to say 1 and 2 two observers defining two reference marks R' and R" accelerated and initially at rest (for example, or in any case with same speed at the initial moment). Then, the constant acceleration guarantees that the speed of 1 and 2 will remain identical at any moment in R and the distance between 1 and 2 constant in R. Consequently, 1 and 2 are motionless one compared to the other and one can apply the preceding procedure to synchronize all the clocks in, for example, R'.

From a dynamic point of view, in this class of accelerated reference marks, one must add a virtual force. But from a kinematic point of view, any valid remainder within this class of accelerated reference marks.

Let us announce however that accelerations can be vicious. And there is a defect with the reasoning above. Can you see which?

If accelerations are identical in R, then in R' the distance which joint 1 and 2 must increase (by simple application of the contraction lengths). Consequently they are not motionless one compared to the other in R'! To guarantee that the observers are motionless in R' it is necessary that 1 and 2 has slightly different accelerations in R. It is not very difficult to calculate and to realize (only the distance is needed which separates them in R decreases exactly in the same way as the contraction lengths). Then R' will have a whole of motionless observers which can carry out a synchronization.

It is seen that there are already difficulties with simple linear accelerations (we will not deepen that here, but space is not already more Euclidean there and the reference mark is called reference mark of Rindler, to see the references).

Nothing prohibits, a priori, to study the movement of an accelerated object, even in rotation. But nothing guarantees that the formulas of Lorentz keep their form in a reference mark in rotation because acceleration is not uniform in all the reference mark. In the same way, synchronization becomes noncommonplace as we will see it.

Synchronization on the disc in rotation

We now will try to synchronize the clocks on the disc in rotation.

For that, it is also necessary to return a series of simultaneous events. Let us take again the procedure of the preceding section.

The same assumption of choice of the events is made: With and B simultaneous for an observer located into 1 motionless in R, B and C for an observer located into 2 motionless in R, C and D for 3, D and has for 4.

Now let us consider the observers located into 1,2,3 and 4 but in the R' reference mark in rotation.

Speeds in the inertial reference mark R of the various observers are indicated in the figure above. Speeds of 1 and 2, which are different, show (by using the transformations of Lorentz) that, for 2, occurs before B. It has is simply what we know in restricted relativity: simultaneity is relative.

In the same way, for 3, B occurs before C, for 4, C before D and, for 1, D before A.

If we try to synchronize our clocks, we will thus have, in total a R' reference mark, that has occurs before B which occurs before C which occurs before D which occurs before has

This contradiction shows that it is impossible to synchronize, in R', the 4 clocks of observers 1 to 4 in a coherent way by using the procedure of restricted relativity. It is simply impossible to synchronize the clocks in a reference mark turning with the procedures of the relativity restricted, at least overall on all the circumference and by respecting the principle of relativity.

That with two important implications:

• the reference marks, such as we used them up to now, are not applicable to the disc in rotation. I.e. it is always possible to define an inertial reference mark around a vicinity of an observer in rotation during a short time, but it is impossible to define a reference mark which would be valid for the whole disc since we are not able to synchronize its clocks and thus to define valid temporal coordinates on all the disc.
• the usual formulas of restricted relativity lose their validity in the turning reference mark (dilation of time, transformations of Lorentz).

Moreover, the reasoning preceding watch that by making the turn of the disc in rotation, one observes a " Time Gap". I.e. a discontinuity in time (in the synchronization of the clocks). If, on the turning disc, one slowly moves a clock (at very low speed in front of $V$ and $c$) and that one makes the full rotation, one should observe a shift between the clock remained with the starting point and that having made the turn. It is a direct consequence of the successive shifts between synchronizations of has to B, of B with C, etc as the clock traverses the whole of the observers of the disc.

The calculations can be made rigorously, but that gives an index on the origin of the shift of the signals in the Effet Sagnac. It could be a question of this Time Gap which would have in more the advantage of being universal (the reasoning above does not utilize the speed of a possible signal around the disc, the procedures of synchronization in restricted relativity being able to be done with any signal at known speed, as we saw above).

Let us note that, contrary to the case of linear accelerations, it is not possible to be left there by a similar easy way (to have a nonuniform acceleration). For two reasons:

• Si in periphery of the disc each observer had a centripetal acceleration (i.e. an angular velocity and thus a tangential speed) different, they would not be long in meeting! Therefore, that cannot guarantee a relative immobility and certainly not symmetry of the disc.
• the phenomenon of shift above remains even if speeds are different. It is enough that there is rotation in a given direction.

The case of the reference marks in rotation is thus particularly thorny.

Various synchronizations

There exists an infinity in manner of synchronizing the clocks. In fact, it is possible to carry out unspecified transformations of the coordinates without changing physics.

It is right a mathematical transformation, it should not have consequence on the physical phenomena. It is a simple reparametrisation of the coordinates. If they are the coordinates of a reference mark " physique" given (i.e. a reference mark attached to a physical object), the transformations are known as interns with the reference mark. It is also said that it is a transformation of gauge.

An observable physics is a quantity which depends on the reference mark considered but its description must be independent of the selected reparametrisation. I.e. the quantity must be invariant of gauge. Cattaneo developed such techniques and you will find further information on its method in the references.

Here, we only will interest we in the synchronization of the clocks.

The question which arises here is " which is the synchronization most adapted to the description of the Sagnac effect? "

An unspecified change of the frame of reference is given by the arbitrary functions:

$\backslash \; bar\; x^\; \backslash \; mu=\; \backslash \; bar\; x^\; \backslash \; driven\; \backslash \; left\; (x^\; \backslash \; driven\; \backslash \; right)$

With the additional condition $\backslash \; partial\; \backslash \; partial\; bar\; x^0/\backslash \; x^0>0$ not to change the arrow of time and thus to ensure causality.

The transformations are still too general. We here only will interest we in a transformation introduced by Selleri.

One chooses a reference mark of reference, for example the inertial reference mark R, and one modifies the coordinates in R1 according to speed compared to R.

If one limits oneself to the synchronization of the clocks, one limits oneself then to the transformations

$\backslash \; begin\; \{matrix\}\; \backslash \; bar\; t=\; \backslash \; bar\; T\; \backslash \; left\; (T,\; x^1,\; x^2,\; x^3\; \backslash \; right)\; \backslash \; \backslash \; \backslash \; bar\; x^i=x^i\; \backslash \; end\; \{matrix\}$

where the coordinates $\backslash \; left\; (T,\; x^1,\; x^2,\; x^3\; \backslash \; right)$ are those obtained with the usual synchronization of Einstein which we saw. And the surlignées coordinates are those of the new frame of reference.

Attention, it is not a question here of a change of reference mark and transformations in the style of the transformations of Lorentz. Here they are two frames of reference attached to the same reference mark.

The transformation of Selleri is given by:

$\backslash \; begin\; \{matrix\}\; \backslash \; bar\; t\text{'}=t\text{'}+\; \backslash \; frac\; \{\backslash \; Phi\; \backslash \; left\; (\backslash \; beta\; \backslash \; right)\}\{c^2\}\; x\text{'}\; 1\; \backslash \; \backslash \; \backslash \; bar\; x\text{'}^i=x^i\; \backslash \; end\; \{matrix\}$

One arbitrarily chooses an absolute reference mark (for example the inertial reference mark R in the Sagnac effect) in which the coordinates are identical to the coordinates obtained by the synchronization of Einstein. And in another R1 reference mark moving in a given direction $x^1$ with speed $\backslash \; beta=V/c$, one uses the relations above where $\backslash \; Phi$ is an arbitrary function. Attention $\backslash \; bar\; t\text{'}$ and $t\text{'}$ (respectively $\backslash \; bar\; x\text{'}^i$ and $x\text{'}^i$) are not, again, the coordinates in the reference marks R and R1 but only the coordinates in R1 according to two methods of synchronization, while $\backslash \; beta$ refers to R well!

One can write this function like

$\backslash \; Phi\; \backslash \; left\; (\backslash \; beta\; \backslash \; right)\; =\; \backslash \; beta+\; \backslash \; frac\; \{e\_1\; \backslash \; left\; (\backslash \; beta\; \backslash \; right)\; C\}\; \{\backslash \; Gamma\}$

$e\_1$ is called the parameter of Selleri. When $e\_1=-\; \backslash \; beta\; \backslash \; Gamma/c$, one finds usual synchronization and when $e\_1=0$, one finds the synchronization of Selleri or gauge of Selleri. This synchronization is known as " absolue" because it uses an absolute reference mark as bases for the synchronization of all the reference marks.

In the gauge of Selleri, the change of coordinates is worth simply

$\backslash \; begin\; \{matrix\}\; \backslash \; bar\; t\text{'}=t\text{'}+\; \backslash \; frac\; \{\backslash \; beta\}\; \{c^2\}\; x\text{'}^1\; \backslash \; \backslash \; \backslash \; bar\; x\text{'}^i=x\text{'}^i\; \backslash \; end\; \{matrix\}$

Let us note that in the gauge of Selleri, the transformations of Lorentz of time take a particularly simple form:

$\backslash \; bar\; t\text{'}=\; \backslash \; Gamma\; \backslash \; bar\; t$

(where here $\backslash \; bar\; t$ and $\backslash \; bar\; t\text{'}$ refers to the reference marks R and R1, in the gauge of Selleri).

In this gauge the transformation of the time from one reference mark to another is particularly simple and related on the dilation of time and not to the position, which justifies its name of absolute synchronization better (it does not depend on the position of the reference marks).

In the gauge of Selleri simultaneity is absolute but, since one uses a different synchronization (and thus different hours indicated by the clocks), it is not a question of the same simultaneity as that marked relative with the synchronization of Einstein.

On the other hand, in the gauge of Selleri, speed of light is really anisotropic in R1.

In fact, the gauge of Selleri to the advantage of defining a space time division on the space surfaces which is independent of the reference mark considered.

But that does not have anything magic. This cutting on the space surface is right that obtained with usual synchronization in reference mark R. And as this one is regarded as absolute in the gauge of Selleri, to say that this flaky preparation is independent of the reference mark returns in fact to say that the flaky preparation of R is chosen and that one regards it as the single reference.

Selleri, like the majority of the authors, is of agreement to say that the choice of the gauge is right a convention. But he affirms that in the case of rotation it is different, his gauge being essential.

What is it actually? Let us recapitulate the advantages and disadvantages of the gauges of Selleri and Einstein.

• Physically, the use of one or the other gauge is equivalent. Physics must be independent of the gauge and one can always pass from the numerical results of a gauge to another gauge by the use of the transformation, purely mathematical, adequate.
• If one wishes to have a total synchronization for the disc, it is necessary to use the gauge of Selleri which consists in fact to use the synchronization of Einstein in reference mark R.
• If one wishes a complete description of the relations between local speeds, times clean, etc It is easier to use the gauge of Einstein.
• the synchronization of Einstein is based, as we saw on light signals (local if one synchronizes gradually). It guarantees an optical isotropy thus: it corresponds so that one " voit" taking into account propagation velocity of the signals. It is synchronization " naturelle".

The Sagnac effect is universal: the temporal shift does not depend on the speed of the signals used.

• the local anisotropy of the signals is artificial in Selleri (parameter of Selleri), in particular for signals speed lower than $c$. It is function number of revolutions and the anisotropy is thus not related to the geometry (with the ray of the disc). Its only physical interpretation is that it is proportional to the centripetal force (centrifugal in R'). However this one is perpendicular to the anisotropy and cannot justify its direction. The anisotropy is thus arbitrary because one cannot find relation with some things of physics and room to him. It is badly seen how the local speed of a signal can be influenced by the fact that other signals can carry out one traverses total (the turn of the disc) particular (as in the Paradoxe of Selleri).
• In the gauge of Einstein, it is not need to introduce an arbitrary local anisotropy. Total anisotropy having a clear geometrical significance. And the only condition, in all the cases (universal effect), is the isotropy in any local reference mark.
• Rappelons that the Principe of relativity is a natural condition, an ideal, that one must respect. However the gauge of Selleri violates this principle explicitly. Even if one considers reference marks R 1, R 2,… inertial, the synchronization of Selleri violates the principle of relativity. That does not have anything astonishing since one chooses an absolute reference mark arbitrarily what is only one decision of mathematical and nonphysical nature. Since this violation also exists in inertial reference marks, that also shows that this choice cannot be related to the existence of a physical effect such as the centrifugal force.

It is always possible, by violating the principle of relativity, to choose an arbitrary absolute reference mark and to issue that these are the clocks which dictate time and not the clocks moving. It is enough in any point to any reference mark to reading time on the clock of R located at this place at the time when one sees it passing (it is even more radical than the synchronization of Selleri and illustrates the artificial character well). But the time thus defined in R1 is completely abstract. It does not correspond to natural synchronization (the speed of the signals, for example, would depend on their direction, completely arbitrarily, without apparent reason, whereas their return ticket duration would be identical) and worse time thus defined does not correspond to than an observer of R1 would note on a clock than it transports with him.

The choice of Selleri is a little more subtle than that (it takes account of the dilation of time) but remains completely arbitrary.

While in the gauge of Einstein, the principle of relativity is respected. On the other hand there is well the presence of a reference mark R privileged due to the existence of rotation. This one has physical effects which justify a modification of the geometry and a total violation of the principle of relativity (when the signals make a turn of the disc, which is obviously connected to rotation). While locally (isotropy speed of light) the principle is respected. There is something of physics which must be responsible for this rotation (if not O' straight would continue), from the centripetal forces, directed towards the center of the circle, which maintain it in rotation. Whatever the nature of these forces, there exists something of physics, related to R, which is responsible for rotation east justifies that in R' the situation is different. And this time, contrary to the gauge of Selleri, while passing to the inertial reference marks, the violation disappears at the same time as the physical cause.

It thus seems that the synchronization of Einstein is most suitable at the same time to describe the physical aspects but also in any general information (not only for Sagnac), without artificially having to choose an absolute reference mark, by adopting the same procedure in any local inertial reference mark. I.e. the gauge of Einstein is covariante.

But let us recall that remains a choice (like choosing the gauge of Coulomb, not covariante, in electromagnetism), a convention, and the only important thing when one affirms a certain number of thing in relativity is always well to specify synchronization used and the fact that the unusual results presented are possibly only one artefact due to the choice of the gauge. I.e. it is necessary to have the physical results while being freed from synchronization. For example, which sees (with the clean direction) an observer (for example the apparent rate/rhythm of a distant clock, as in the Doppler effect) or what it measures locally (for example moments of such events or the time put by a signal to make the turn of the disc) should not depend on synchronization.

You can check which is the speed of the signals in R1 in the gauge of Selleri. You will see that it is possible for certain signals to go up time! Whereas they do not do it in R. That illustrates well the completely artificial character of this gauge. It is a mathematical curiosity on which we will not insist more.

See too

• the Effect Sagnac

• the Paradox of Ehrenfest
• the Paradox of Selleri
• the Calculation of the Sagnac effect in restricted relativity
• the Geometry of the space time in the revolving reference marks
• the Paradox of the twins and the effect Sagnac
• the compact spaces
• the Paradox of the twins in compact spaces
• the restricted Relativity
• the General relativity
• the Principle of relativity
• the Paradox of the twins