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THE PROPAGATION OF LIGHT (SIGNAL) AROUND ROTATING BODIES

It is shown that the one way speed of light on the surface of a rotating body depends on the direction of its propagation, while the average speed of light on the way back and forth along the trajectories of the points on the rotating surface remains equal to constant \(C\).

The constancy of the average speed of light results from the transverse radial contraction of revolving bodies, following from the longitudinal Lorentz contraction of the elements of the revolving surface, which does not contradict the invariance of the transverse dimensions of bodies in the absence of their deformation.

The propagation of the signal along the equator

The Einstein clock synchronization is based on an arbitrary, as noted by Einstein [1], assumption about the equality of the speed of light in opposite directions.

All earlier experiments related to the measurement of the speed of light were carried out by means of a single clock by doubling the distance between a transmitter/receiver and a reflecting mirror, and by the signal propagation time on the way to the mirror and back.

Poincare, Reichenbach, Tyapkin, Brillouin [2-5] and many others noted that only the measurement of the speed of light by means of a pair of clocks pre-synchronized at points \(A\) and \(B\) in space would give the unidirectional speed of light – from point \(A\) to point \(B\). All other methods, even including, as shown by Karlov [4] the method of astronomical observations used by Roemer, give the mean value of the speed of light in opposite directions.

However, in order to synchronize two spatially separated clocks one must know that same speed of light in the direction from point \(A\) to point \(B\), which must be measured. The transport of the clocks pre-synchronized at one point in space also does not allow to unambiguously synchronize the clocks, since, without having pre-selected the arbitrary criterion of synchronization, one cannot say which of the speeds of the transport of the clocks does not lead to their desynchronization.

Considering the Einstein method of synchronization, Brillouin [5] referred to it as arbitrary and “even metaphysical”, since the substantiation of Einstein synchronization is supported by “the unexpected and unverifiable hypothesis about the equality of signal speed from east to west and from west to east, whereas Michelson's experiment makes it possible to only measure the mean value of these two speeds”.

That this hypothesis is unverifiable in the strictly inertial reference systems should, apparently, be considered true. It is likewise true that having positioned, for example, one clock in the west of London and another in the east of this city, it is not possible to measure the speed of light from the west to the east of London without pre-synchronization of the clocks.

But it does follow from this that the speed of light from west to east (or vice versa) cannot be measured in general?

That such measurement is possible and that it does not require pre-synchronization of two spatially separated clock is true for the following reasons.

Imagine that a short-wave radar sending a narrow-beam signal eastward is stationed on the very equator near the city of Quito. Also imagine that along the entire equator line a set of reflectors is stationed, which deflect the radar signal emitted in Quito in such a way that it, propagating close to the earth's surface, orbits it along the equator and returns to the radar in Quito from the west.

Knowing the length of the radar signal propagation line and the time needed for the signal to orbit the earth, the relay station operator can calculate the propagation speed of the signal going orbiting the earth from east to west or in the opposite direction. That these speeds will not be the same and different from the constant C is true for the following reasons.

Let us mentally place into a remote from the earth point of an imaginary rotational axis of the earth an outside nonrotating observer, stationary in reference to the center of the earth, and who observes the earth's northern hemisphere revolving counterclockwise below us, mentally tracking the propagation of the signal.

In the reference system of the outside observer the speed of light propagating in space is equal to the fundamental constant \(C\). If the Earth did not revolve, then the signal to go around the hypothetically non-revolving Earth would need the time equal to the length of the line covering the earth on the equator, divided by the constant \(C\).

But the earth does revolve!

When the signal returns to the starting point of the space of the outside observer, the radar in Quito city will move approximately 62 meters eastward and the signal arriving from the west will need extra time equal to two ten-millionths of a second to return to the locator.

If the operator turns the antenna 180 degrees and directs the signal westward, then the signal will need two ten-millionths of a second less to orbit the earth and return to the radar, since over the earth-orbiting time of the signal the radar will move 62 m eastward and the signal coming from the east will not have to cover these 62 meters. The signal lag is the effect of the first order with respect to the value \(v/c\), where \(v\) is the linear speed of the surface of the revolving earth, and it is high enough in comparison with the relativistic effects of the second order of smallness.

In the case of simultaneous emission of signals by the locator in opposite directions - eastward and westward, the signals that have orbited the earth and returned to the radar will spend different time on it and return to the radar at different times. The difference in the signal return times will then be equal to about four ten-millionths of a second. This effect is basically the Sagnac effect [6-7] used in optical gyroscopes.

If an operator sends a signal to the east and ensures the possibility for the signal arriving from the west to be reflected from the locator's auxiliary reflector in the opposite direction, and, having completed the return route, to return to the locator from the east, then the time, necessary for the dual “round-the-world journey” of the signal first from west to east, and after being reflected, from east to west, does not practically differ from the time, which the signal would have spent on a similar journey around the hypothetically non-revolving earth. In this case, the measurement of the speed of light on the way back and forth would give the value with an accuracy of at least to second-order quantity equal to the fundamental constant \(C\).

Knowing the equatorial speed of light from west to east or/and in the opposite direction, it is possible to synchronize any pair or set of the clocks on the equator.

In this case the clocks prove to be synchronized in such a way that the mentioned outside observer at any moment of time "observes" the identical readings of the clocks stationed at different points on the equator. If terrestrial experimenters attempt to synchronize any pair of equatorial clocks by the Einstein method, assuming that the speed of light from west to east is exactly equal to constant \(C\), then they will encounter serious problems.

First, the clocks synchronized in this way, which are stationed under absolutely identical symmetrical conditions, will at all times give different readings to an outside observer at the mentioned above point of the earth's axis. Second, having chosen, for example, the readings of the clock in Quito as the reference time and consecutively synchronizing each pair of adjacent clocks, the terrestrial observers passing from one pair of the clocks to another, will arrive at the starting point to the reference clock in Quito, finding that the reference clock in Quito is asynchronous to itself, with the "asynchronism" making up those two ten-millionths of a second.

Then, the synchronization of the clocks with regard to the inequality of the speeds of light from west to east and from east to west produces the same result as the synchronization of the clocks by a synchronizing signal emitted by the outside observer from the point on the imaginary rotation axis of the Earth to all points on the equator.

The motion of the signal around the rotating ring

Let us take a closer look at the propagation of a signal around the revolving body using the following example.

Imagine two adjacent identical thin rings \(k\) and \(k’\), put without any clearance on the cylindrical surface of an absolutely smooth supporting non-rotating cylinder Z of radius R.

We will accept the thickness of the rings as negligibly small compared to the radius of the cylindrical surface of the cylinder Z.

Let a certain section of ring \(k’\) be fitted with a transmitter/receiver (TR) capable of emitting electromagnetic signals and measuring the time between the moments of emission and reception of the signal by means of a single internal “clock”, with reflectors fitted around the ring to ensure propagation of the signal around the ring along the circumference \(O\) covering the surface of the cylinder.

In this case, the TR-emitted pulse that has orbited the ring and returned to TR from the other side would require the time to orbit the ring equal to \(2\pi{R/c}\), where \(R\) is the radius of the circle \(O\), equal to the radius of the supporting cylinder, and \(C\) is the speed of light in the vacuum.

Let us now assume that the ring \(k’\) is set into rotation with linear speed \(v\), and the observers stationed on the ring \(k’\), perform the measurement of the propagation time of the signal directed along the circle \(O\) around the ring \(k’\) from TR toward TR in the direction of the ring rotation and in the opposite direction. Let us also assume that the material of the ring is light and strong enough, and the diameter of the ring is big enough so that the centrifugal forces existing on the ring would not lead to its noticeable tension and formation of clearance between it and the supporting cylinder.

The observers in the inertial reference system \(K\) rigidly connected to the supporting cylinder and to the ring \(k\), while monitoring the propagation of the signal, would find that the signal emitted by TR on the ring \(k’\) in the direction of the rotation of the ring requires more time to orbit the ring and return to TR than the signal emitted by TR in the direction opposite to the direction of rotation of the ring.

If the observers stationed in the inertial reference system \(K\) find that the time of signal propagation from TR to the same TR in the direction of rotation is equal to \(\Delta{t_1}\), then the signal in its reference system travels the path of length \(L+v\Delta{t_1}\), where \(L\) is the length of the circle \(O\) of the cross section of the cylinder, and \(v\) is the TR motion speed.

ince the length of the path travelled by the signal in the reference system \(K\) is equal to \(c\Delta{t_1}\), then \(L+v\Delta{t_1}=c\Delta{t_1}\), hence

$$\Delta{t_1}=L/(c-v). \quad\text{(1)}$$ Similarly, for the signal propagating in the direction opposite to the direction of rotation of the ring \(k’\), for the time \(t_2\) of the propagation of the signal we obtain:

$$\Delta{t_2}=L/(c+v). \quad\text{(2)}$$ In view of the slowdown of the motion of the internal clock of TR in the inertial reference system \(K\), which is the only clock on the ring \(k’\), the times \(\Delta{t’_1}\) and \(\Delta{t’_2}\) of the propagation of light from TR towards TR respectively in the direction of rotation of the ring \(k’\) and vice versa, which would be obtained by the observers on the ring \(k’\), turn to be \(\Gamma\) times smaller than the times \(\Delta{t_1}\) and \(\Delta{t_2}\), where \(\Gamma=1/[1-(v/c)^2]^{1/2}\), and are equal to: \(\Delta{t’_1}=\Delta{t_1}/\Gamma\); \(\Delta{t’_2}=\Delta{t_2}/\Gamma\). Hence, in view of (1) and (2):

$$\Delta{t’_1}= (L/c)([1+(v/c)]/[1-(v/c])^{1/2}\quad\text{(3)}$$ and

$$\Delta{t’_1}= (L/c)([1-(v/c)]/[1+(v/c])^{1/2}\quad\text{(4)}$$ Now assume that the signal sent from TR in the direction of rotation of the ring \(k’\), upon coming to TR is reflected by the TR reflector in the opposite direction, and after travelling the return path comes to TR again. The total time \(\Delta{t’_1}+\Delta{t’_2}\), which, according to the measurements of the observers on the ring, would be required for the signal to travel the path from TR, first in the direction of rotation of the ring to the reflector, and then, after being reflected to TR in the opposite direction, in view of (3) and (4) is equal to

$$\Delta{t’_1}+\Delta{t’_2}=2\Gamma{L/c}\quad\text{(5)}$$ Since the total time of propagation of the signal on the way back and forth differs from the time \(2L/c\) \(\Gamma\) times, then one may form an impression that for the observers on the ring the average speed of light on the way back and forth must also differ \(\Gamma\) times from the constant \(C\). This conclusion seems improbable since numerous experiments showed the equality of the average speed of light on the earth on the way back and forth to the constant \(C\). To fit the formula (5) to experimental results is only possible upon the assumption that the length \(L’\) of the ring \(k’\), according to measurement results by the observers on ring \(k’\), exceeds the length \(L\) of this ring \(\Gamma\) times, measured in the reference system \(K\).

It is often asserted in literature that due to contraction of a sufficiently short standard of length directed along the revolving ring, fixed by inertial observers, the length of the revolving ring for the observers on the ring itself must actually \(\Gamma\) times exceed the value of the length of the circle, obtained by observers in the inertial reference system, which coincides with the ring [8]. This assertion makes it possible to explain the invariance of the average speed of light on the way back and forth, though it is not fully understandable in terms of physics.

This assertion seems especially strange if we imagine that a ring consists of a certain number of \(n\) elements, the length of each being sufficiently small, though several times exceeding the length of the standard.

If we assume that the length of a sufficiently short standard directed along the ring contracts, which should supposedly lead to an extension of the ring, then one should also agree that each of the \(n\) elements making up the ring should also contract \(\Gamma\) times. However, the number of elements that make up the ring does not depend which reference system the ring is considered from, which leads to inconsistency in the conclusion that the rotating ring is extended for the observers on the ring.

To show the physical essence of what happens on a rotating ring, let us consider the spinning of the ring, during which it passes from the state of rest in the inertial reference system \(K\) to the state of rotation.

However, before considering the acceleration of the ring, we turn to a simpler case of acceleration of the rod in an inertial reference system.

The acceleration of the rod in the inertial reference system

Consider the behaviour of a rod accelerated to a certain speed \(v\) in the inertial reference system \(K\).

If a longitudinal force is applied to the end \(B\) of the rod \(AB\), which is small enough so that irreversible deformation of the rod does not occur, then picking up the velocity \(v\), according to the Lorentz transformation, the rod, which has not changed its own length \(L_0\), will shorten \(\Gamma\) times and acquire in the reference system \(K\) the length equal to \(L_v=L_0/\Gamma\). Here the point \(A\) of the rod during acceleration will in the reference system \(K\) travel a slightly greater distance than point \(B\). It is clear that the rod can be accelerated by applying force to the ends \(B\) and \(A\) (and to its midpoints), however, in order not to deform the rod, the ends \(A\) and \(B\) (and midpoints) of the rod must be accelerated in such a way that they moved in accordance with the laws of the Lorentz contraction of the rod of constant proper length, accelerating in the given reference system, i.e. so that at every moment of time the rod, having reached a certain velocity \(v\), the length \(l_v\) of any section of the moving rod is \(\Gamma\) times smaller than the proper length \(l_0\) of this section.

Let us call the acceleration of the rod, meeting this condition, non-synchronous (in a third-party inertial reference system \(K\)) non-deforming acceleration.

The rod \(AB\) can be forced to accelerate so that the ends \(A\) and \(B\) are accelerated in coordination, remaining at any time \(t\) of the reference system \(K\) at a constant distance from each other in the reference system \(K\). In this case, the length \(L_v\) of the accelerating rod is retained during acceleration; however, the proper length \(L_o=\Gamma{L_v}\) increases \(\Gamma\) times, and the rod, after it stops accelerating and becomes an inertial body, becomes force stretched in the inertial reference system rigidly connected to the rod [9].

If the rod is resilient, then efforts are required to retain its length after acceleration is completed. If the rod is ductile, then it elongates without residual deformation stresses.

We will call the acceleration of the rod, in which in the external inertial reference system \(K\) at each moment of time, the rod reaching a certain speed \(v\), the length \(l_v\) of any section of the moving rod remains invariable, the synchronous (in the external inertial reference system \(K\)) deformative acceleration.

The deformative acceleration can also be realized by applying to it a distributed load that does not violate this condition.

If the rod has its own standard of length \(St’\) and if both ends of the own standard \(St’\) are rigidly fixed to the ring, then after the synchronous deformative acceleration the numerical value of the length of the rod, measured by the observers on the rod, will be retained; however, this will be due to stretching (i.e., in fact change) of the \(St’\) standard. If the observers on the rod release the ends of the standard \(St’\), then the standard will become \(\Gamma\) times shorter and the numerical value of the length will increase. In this case, the measured value will correspond to reality expressed by the lengthening of the rod.

The synchronous deformative spinning of the ring

Coming back to the consideration of the ring, we note that in order to ensure uniform spinning of the ring without creating local deformation jumps in some sections of the ring, it is only possible to set all its sections in motion simultaneously and consistently in the reference system \(K\). This is owing to the fact that in the process of clock synchronization on a rotating ring by the Einstein method, a time “jump” point [9] appears on the ring, where a given moment of time becomes indefinite.

If the observers in the reference system \(K\) set the ring into spinning, forcing each of its points to accelerate in exactly the same way in terms of time and leaving each point on the surface of the reference cylinder, then the observers on the ring \(k’\) will find that the ring \(k’\) is stretched \(\Gamma\) times under coordinated acceleration in a way similar to that of the rod considered above.

In this case, the ring does not move away from the cylindrical surface, since its length in the reference system \(K\) is retained. The extension of the ring is explained by the invariance of the Lorentz length of any sufficiently small element of the ring \(k’\) in the reference system \(K\) and, as a consequence, by an increase in the proper length of this element.

If both ends of the observers' own standard \(St’\) on the ring \(k’\) are rigidly fixed to the ring, then the numerical value of the ring length measured by the observers on the ring will be retained, however, as in the case with the rod, this will be due to the elongation of the standard \(St’\)'.

If the observers on the ring \(k’\) release the ends of the standard \(St’\), then the standard will become shorter and the length of the ring obtained from its measurement using this standard will be \(\Gamma\) times greater than the length it had prior to its spinning. The lengthening of the ring is due to its actual physical stretching.

Since the value of the length \(L’\) of the extended ring \(k’\) for observers on the ring is \(\Gamma\) times greater than the value of the length \(L\) of the circumference of the supporting cylinder, measured in the reference system \(K\), i.e. as

$$L’=\Gamma L\quad\text{(6)}$$ then the average speed of light obtained by dividing the length \(2L’\) by the time \(\Delta{t’_1}+\Delta{t’_2}=2\Gamma{L/c}\) becomes equal to \(C\).

The synchronous non-deformative spinning of the ring

If now the observers in the reference system \(K\) remove the supporting cylinder and, having ensured that the linear velocity of the ring remained \(v\), allow the ring to contract, then the rotating ring will shrink according to the results of their measurement, reducing the diameter and the length \(\Gamma\) times. According to the results of the measurements by the observers on the ring \(k’\), the ring restores its original length, which it had before its acceleration, but the ring \(k\) after shrinkage of the ring \(k’\) becomes longer. That means that the observers on the ring \(k’\) and the observers in the reference system \(K\) will find that the non-rotating ring \(k\) is located outside the ring \(k’\).

The same result would be obtained if the observers in the reference system \(K\), synchronously spinning the ring \(k’\), made it possible for it to contract during the synchronous acceleration. The average speed of the signal on its route from TR to TR and back to TR along the surface of the ring \(k’\) would be equal to \(C\) also in case the observers on the ring \(k’\) measured it after the contraction of the ring \(k’\). But in this case the observers in the reference system \(K\) would find that the signal does not propagate along the circumference \(O\) of the length \(L\) as dealt with above in the present paper, but along the circumference \(O_1\) of a smaller length \(L_1\), equal to \(L/\Gamma\). In this case, the time \(\Delta{t’_1}+\Delta{t’_2}\) would become, according to their calculations, not equal to \(2\Gamma{L/c}\), but somewhat less and equal to \(2\Gamma{L_1/c}\). But \(\Gamma{L_1}=L\), thus \(\Delta{t’_1}+\Delta{t’_2}=2L/c\). Since for the observers on the ring \(k’\)' the value of the length \(L_1\) of the circumference \(O_1\) matches the value of the length \(L\) of the circumference \(O\) obtained by the observers in the reference system \(K\), then, calculating the average speed and dividing the value \(2L_1\), equal to \(2L\), by the time \(\Delta{t’_1}+\Delta{t’_2}\) equal to \(2L/c\), they would obtain the speed equal to the constant \(C\).

The rotation of large rings

It is widely believed that a large-sized body cannot be set into spinning to arbitrarily large angular velocities since in this case the points of the body that are far from the axis of rotation would move at speeds exceeding the speed of light, which is impossible [10].

Imagine a ring of radius \(R_0\), which is gradually set into spinning to an arbitrarily large angular velocity \(\omega\). At a certain angular velocity \(\omega\), the linear velocity \(v\) of the ring approaches the speed of light. However, in this case the radius of the ring in the inertial reference system \(K\) shrinks \(\Gamma\) times, as shown above, and becomes equal to \(R=R_0/\Gamma\).

We write \(\Gamma\) in the form \(1/[1-(\omega{R/c})^2]^{1/2}\), where \(\omega{R}\) is the linear velocity \(v\) of the ring.

Then $$R=R_0[1-(\omega{R/c})^2]^{1/2}\quad\text{(7)}$$ From (7) after an elementary transformation we find

$$R=R_0[1+(\omega{R/c})^2]^{1/2}\quad\text{(8)}$$ It is clear from the formula (8) that as the angular velocity \(\omega\) increases, the radius \(R\) of the rotating body decreases, while the linear velocity \(v=\omega{R}\), which, taking into account (8), is equal to

$$v=R_0[1/\omega^2+(R_0/c)^2]^{1/2}\quad\text{(9)},$$ cannot in the inertial reference system \(K\) exceed the speed of light \(C\).

Bibliography

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3. A.A. Tyapkin. Uspekhi Fiziceskih Nauk (Advances in physical sciences), Moscow, 1972, 106, p. 617-659

4. L. Karlov. Australian journal of physics. 23, 1970, p. 243-253.

5. Brillouin L. Relativity Reexamined. 1972. Academic. London Press. 1970.

6. G.B. Malykin. Uspekhi Fiziceskih Nauk (Advances in physical sciences). 2000, volume 170, no. 12, p. 1325-1349.

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8. M.Born. Einstein's theory of relativity. Moscow, Mir publishing house, 1972, 308-310.

9. D.V. Skobeltsyn. The twins paradox in the theory of relativity. Moscow, Nauka (Science) Publishing House, 1966.

10. V.N. Matveev. In the third millennium without physical relativity? Moscow, CheRo Publishing House, 2000.

Links

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Vadim N. Matveev, Oleg V. Matvejev, Øyvind Grøn, "A relativistic trolley paradox"

Wikipedia, "Lorentz transformations"

Wikipedia, "Einstein synchronization"

Wikipedia, "One way speed of light"

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