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C. R. Physique 18 (2017) 563–569
Contents lists available at ScienceDirect
Comptes Rendus Physique
www.sciencedirect.com
The Comptes rendus de l’Académie des sciences throughout history / Les Comptes rendus de l’Académie des sciences à travers l’histoire
The Sagnac effect and its interpretation by Paul Langevin
L’effet Sagnac et son interpretation by Paul Langevin
Gianni Pascoli
Université de Picardie Jules-Verne, Faculté des sciences, Département de physique, 33, rue Saint-Leu, 80000 Amiens, France

article info
Article history: Available online 31 October 2017
Keywords: Sagnac effect Aether Langevin Interferometer Relativity

a b s t r a c t
The French physicist Georges Sagnac is nowdays frequently cited by the engineers who work on devices such as ring-laser gyroscopes. These systems operate on the principle of the Sagnac effect. It is less known that Sagnac was a strong opponent to the theory of special relativity proposed by Albert Einstein. He set up his experiment to prove the existence of the aether discarded by the Einsteinian relativity. An accurate explanation of the phenomenon was provided by Paul Langevin in 1921.
© 2017 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
r é s u m é
Le nom de Georges Sagnac est aujourd’hui très connu des ingénieurs travaillant sur les systèmes de navigation, tant maritime qu’aérienne, qui exploitent sa découverte de 1913. Ce que l’on sait moins est que ce physicien était un farouche opposant à la relativité, théorie révolutionnaire qui avait été élaborée par Albert Einstein quelques années auparavant, en 1905. L’expérience de Sagnac a été pensée pour prouver l’existence de l’éther luminifère. Son interprétation correcte (relativiste) a été fournie par Paul Langevin en 1921.
© 2017 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Historical development
It would seem useful to recall the context of the discovery of the Sagnac effect. Around 1910, a very strong conservatism reigned concerning the paradigm of absolute space and its counterpart, the aether, the hypothetical medium of propagation of light. To this statement, we must add the presence of a very large number of skeptics, both physicists and philosophers, who opposed a non-sequitur to the new vision of space and time that was implied by the very recent Einsteinian theory of relativity. Some key points should also be highlighted.
E-mail address: pascoli@u-picardie.fr.
https://doi.org/10.1016/j.crhy.2017.10.010
1631-0705/© 2017 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

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G. Pascoli / C. R. Physique 18 (2017) 563–569

1.1. Newton and the absolute space (end of the 17th century)
For Isaac Newton, the concept of absolute space (or pure extension) exists, regardless of the presence or not of any material. In Newton’s worldview, absolute space was a pre-existing frame, empty and immutable, a fundamental background to any moving body. But its undeﬁned nature made it a kind of metaphysical entity, close to transcendence. In the Scholium, at the beginning of the Principia (Deﬁnitiones), Newton expatiated on time, absolute or relative space and motion [1]. He suggested that an absolute motion can be distinguished from a relative motion. To justify his argument, he introduces a thought experiment that is still famous. A bucket ﬁlled with water is subjected to a fast rotation. There are two phases. (i) The bucket rotates, but the surface of the water remains ﬂat. (ii) The movement is ﬁnally transmitted to the water. It is then observed that the latter leaks from the center and ascends towards the walls, the free surface taking the form of a paraboloid. During the ﬁrst phase, there is indeed a relative movement water/bucket, but no centrifugal force appears. It is only in the second phase, when the movement of rotation is communicated to the water, that these forces develop. For Newton, the conclusion was that these forces are not correlated with the relative movement of the water/bucket, but are generated by the movement of water in relation to absolute space. A second thought experiment was also devised by Newton. Two spheres are connected by a rope and the assembly is rotated. The rope tightens and one can measure the tension that develops there. Newton’s conclusion is the same.
1.2. Ernst Mach and the relativity of motion (end of the 19th century)
The concept of absolute space as introduced by Newton was strongly criticized by Ernst Mach in his book published in 1893 [2]. Mach assumed a positivist vision from the outset. He noted that Newton’s experiment simply demonstrates that the shape of the water surface (a paraboloid) is not induced by the relative movement of water in relation to the immediate environment (the bucket in particular). But Newton did not extend his thought suﬃciently far. It did not consider the distant masses (represented by the stars). According to Mach, it is legitimate to attribute the deformation of the surface of the water to a rotation, but not relative to an absolute space as Newton had suggested, but to a relative rotation of the water and massive distant bodies. Suppose that the bucket is ﬁxed and that the whole Universe is put in rotation. Would the surface of the water take the form of a paraboloid? For Mach, the answer is yes.
We know that Mach’s ideas had a considerable inﬂuence on the development of Albert Einstein’s reasoning, especially during the ﬁrst ﬁve years of the 20th century. But most physicists did not share Mach’s point of view, and were very opposed to getting rid of the concepts of absolute space and aether.
1.3. The luminiferous aether
The idea of a medium for the propagation of light waves goes back to Christian Huygens (1690). Unfortunately, owing to Newton’s great aura, Newton’s corpuscular theory, exposed in his treatise Opticks (1704), completely dominated for more than 100 years and took precedence over Huygens’ wave description. But Huyghens’ hypothesis enjoyed a sudden revival at the very beginning of the 19th century, thanks to Augustin Fresnel and Thomas Young. Since then, the aether was considered as the absolute frame of reference, especially in relation to which Maxwell’s equations (dating from 1862) had to be written and the celerity of the light was the constant c. This aether was automatically identiﬁed with the absolute space of Newton. But the Earth moves in this absolute space. Does it carry the aether with it as it moves? To answer this question considered as fundamental by the physicists at that time, a number of experiments were carried out in the ﬁrst half [3,4] and at the end of the 19th century [5,6]. In spite of the fact that all these experiments, taken together, showed that the aether was an illusive medium, it must be remarked that as late as 1910, Hendrick Lorentz and Henri Poincaré had still not adopted a very clear position concerning the status of the aether [7]. For a more complete analysis of the debate on the light-bearing aether, see [8–11].
2. The experiment of Sagnac
In 1899, Georges Sagnac1 had developed a theory of the existence of a motionless mechanical aether [12]. His aim was to explain within this theoretical framework all the optics phenomena, and especially the Fresnel–Fizeau experiment for the drag of light in a moving medium [4]. In 1910, he conceived a rotating interferometer for testing his ideas and to see the optical whirlwind effect, as he called it (Fig. 1).
The effect discovered by Sagnac, and published in the Comptes rendus de l’Académie des sciences in 1913 [13,14],2 has led to many applications in positioning technologies (Global Positioning System, Galileo, etc.). The sensors (gyrolasers and gyrometers) that measure angular velocities with respect to an inertial frame of reference, exploit the physics of the Sagnac effect. Generally numbering three, these sensors form a part of the inertial stations, in particular those used to determine the orientation of all types of vehicles: aircraft, boats, submarines, space probes, etc. These devices have also been applied

1 This author is also known as the ﬁrst in France to work on X-rays, following the German scientist W.C. Röntgen. 2 A more extended version has also been published by Sagnac in the Journal of Physics the following year in 1914 [15].

G. Pascoli / C. R. Physique 18 (2017) 563–569

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Fig. 1. Sagnac interferometer mounted on a turntable (reproduced from his paper of 1913).

Fig. 2. Propagation of the light beams seen from the laboratory.

to geodesy and seismology [16–18]. Since the 1960s, a large number of studies have been devoted to the narrative of the discovery of this effect, its interpretation and to its practical consequences [19–22]. On 10 October 2013, a symposium held at the Fondation Simone-et-Cino-del-Duca (Paris) was organized to celebrate the centenary of the Sagnac effect [23]. Even more and ironically enough, the Sagnac device was initially conceived to prove that the special relativity is wrong, is today used for testing the general relativity. For instance, in the GINGER (Gyroscopes IN GEneral Relativity) project a three-dimensional array of large-size ring-lasers will be conceived for measuring in a ground-based laboratory the de Sitter and Lense–Thirring effects [24].
Let us consider a rotating disk on whose rim two light beams circulate in opposite directions relative to each other.3 Both beams initially start from the same point, located on the rim of the disk. In order to simplify the reasoning but without loss of generality, it will be assumed that the source and the receiver are combined and both attached to the disk. Let
R be the radius of the disk and ω its angular velocity. This velocity is assumed to be constant and the disk rotates in the counterclockwise sense for the inertial observer, L linked to the laboratory. For this observer, the receiver travels the distance l (circular arc) at the speed v = Rω. The beam in the sense (+) must therefore travel the distance l + l before
reaching the receiver. The duration of the path of the light ray, t+, is deduced from the equality (Fig. 2):

t+

=

l

+ c

l=

l v

(1)

or

t+

=

l c−v

.

The

same

reasoning

applies

to

the

other

ray,

moving

in

the

opposite

sense

(−),

and

the

traveled

distance

is

l −

l.

The

journey

time

is

t−

=

l c+v

.

The

difference

in

the

journey

times

is

therefore:

tL

= t+

− t−

=

2l v c2 − v2

=

4π R2ω c2 (1

−

R

2ω2
c2

)−1

4Aω
c2

(2)

3 The generalization to any closed (non-circular) loop is immediate.

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G. Pascoli / C. R. Physique 18 (2017) 563–569

Fig. 3. Propagation of the light beams from the point of view of a newtonian aether supporter linked to the disk.

where A = π R2 denotes the area of the disk. The difference in the duration of travel induces a phase shift between the two
waves, hence the appearance of interference fringes (Sagnac made a differential measurement by rotating the disk in one direction and then in the other). The phase shift (expressed in wavelength) is given by the Sagnac formula:

ϕ = 8π Aω

(3)

λc

What is the reasoning of a non-inertial observer (D) related to the disk? According to Georges Sagnac, the light has a medium of propagation which is the aether. In the reference frame linked to the laboratory (observer L) the propagation velocity (celerity) is c. On the other hand, for an observer linked to the rotating disk the light speed propagates differently,

according to whether the beam propagates counterclockwise (the direction of rotation of the disk by convention), or in reverse. It is then respectively c − v and c + v (Fig. 3). For the observer D, we have for the difference in time of course:

l

l

tD = c − v − c + v

4A ω
c2

(4)

which is in perfect agreement with the result found by the observer L. Numerically, the area of the polygonal interferometer was 0.0860 m2, the rotation rate of order 2 Hz and the resulting fringe shift δϕ = 0.07 ± 0.01. In his paper, Sagnac also predicts that a similar effect would be observed from the diurnal Earth rotation (the Earth taken as a rotating turntable) using a very large interferometer.4
For Sagnac, this result proves the existence of the aether. The argument of this author appears a priori quite convincing. The existence of a propagation medium for light waves seems to be conﬁrmed by his experiment. But in science, an hypothesis cannot be validated only if it allows one to explain all the experiments, not one in particular. Other experiments deﬁnitively invalidate the hypothesis of the aether. Perhaps most famous is the crucial experiment of Michelson and Morley [5]. The aim of this experiment was to measure the displacement of the Earth with respect to the aether. Using an interferometric device, they showed that the velocity of the aether wind, if any, was less than 5 km/s (we should remember that the velocity of the Earth in the solar system is 30 km/s, and therefore much greater). The procedure has obviously been repeated several times within a year (in case at a given time, and by an extraordinary coincidence, the Earth’s velocity at the time of measurement had been zero with respect to the aether), but no seasonal effects have been found. The Michelson–Morley experiment has been repeated with ever-increasing accuracy. The Michelson-type experiment carried out by Georg Joos in 1930 allowed him to give a bound to the velocity of the aether wind of less than 1.5 km/s [27]. More recently, experiments have been carried out using various sophisticated devices, such as lasers and masers, optical resonators and microwaves, etc. [28]. The conclusion is always the same: no measurable aether wind. We can thus conclude that the relative velocity of the Earth with respect to the aether is experimentally found to be very close to zero, with high precision. But if we only focus on the Michelson–Morley experiment, it is still perfectly legitimate to imagine that the Earth completely drags the aether in its translatory motion (Fresnel had already suggested in 1818 the hypothesis of a partial dragging of the aether with moving substances [29]). In this case, the negative result of the Michelson experiment has a trivial explanation. Unfortunately, however, such an assumption cannot be maintained. There are two immediate reasons for this. First of all, it is inconsistent with the aberration of ﬁxed stars (as we know, during a year the stars describe a small ellipse on the background of the sky. This effect cannot occur if the aether is fully dragged by the Earth). Secondly, the experiment of Sagnac was repeated by Michelson and Gale in 1925 [30], but this time taking the Earth as a rotating disk (as already suggested by Sagnac himself). These authors observed a displacement of the fringes of interferences, as had Sagnac

4 Let us note that the so-called Sagnac effect had been approached by Sir Oliver Lodge as early as 1893 [25]. This effect had also been glimpsed by Harres in 1911 [26], but this author does not seem to have paid much attention to it. When Sagnac realized his experiment, the idea was therefore in the air.

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in his own experiment. This positive result undoubtedly conﬁrms that the Earth does not drag the hypothetical aether in its rotation (it is therefore illogical to admit that it drags this medium in its translation). The only acceptable conclusion that can be drawn from these two experiments, Michelson–Morley, on the one hand, and Michelson–Gale, on the other hand, is that the hypothesis of the existence of a medium of propagation for light is not tenable (unlike sound, the light can spread in the vacuum). In the classical context, it is clear that the Sagnac effect cannot at all be explained.

3. The relativistic interpretation

Special relativity has successfully passed tests of thousands of experiments of all types (for example, the experiments of

Kennedy–Thorndike [31] and Ives–Stilwell [32]). If it is clearly no longer questionable today, the situation was quite different at the beginning of the twentieth century.5

An experiment such as Sagnac’s was still used as a strong argument by the opponents of the theory of relativity, who

were still quite numerous at the time, including Sagnac himself. The scientiﬁc community as a whole only accepted relativity

after some time. So a lot of pseudoparadoxes issued from relativity, which seemed to arise here and there, were solved

rather slowly. The ﬁrst convincing explanation of the Sagnac effect, in the framework of relativity, was provided by Paul Langevin in 1921 [36].6

It appears rather amazing that the correct relativistic interpretation of the Sagnac effect took eight years. A seemingly

obvious reason is that Sagnac’s experiment was not very much discussed in the scientiﬁc literature, even in France after the discovery of 1913.7 Conscious of this situation, in 1919, Sagnac published ﬁve papers on his work in the Comptes rendus

[37]. The paradox is that his ideas were nevertheless borne by a French group of strong antirelativists. In 1919, Sagnac was

even rewarded with the Pierson–Perrin Prize for his achievements on this topic (ﬁrst for the experiment, seen as a rebuttal

of the relativity principle, the constancy of light, and also for having proven the reality of absolute space and time) [38].

Eventually, in 1921, the mathematician (and relativist skeptic) Émile Picard asked Paul Langevin to demonstrate the effect in

the framework of the relativity, perhaps guessing it was impossible [39]. Again any such expectations and within a couple

of months, Langevin derived the effect from the General relativity [36]. Let us specify however that the Sagnac effect was

already derived from special relativity by Von Laue one year before Langevin [40]. Many derivations were in fact proposed

all along the 20th century, issued from both special relativity and general relativity [41,42], even though the Sagnac effect

is usually deemed to be a special relativistic effect [43].

The reasoning, both simple and elegant, of Langevin can be shown in a few lines. This illustrious physicist begins by

noting that this is an experiment of the ﬁrst order in

v c

(more precisely in

Rω
c

,

if

R

denotes the radius of the disk and ω the

angular velocity, assumed to be constant), contrarily to the experiment of Michelson–Morley, which involves a second-order

effect. We start from the Minkowskian metric associated with the central observer, denoted by O , located at the center of

the rotating disk. In his article, Langevin used the cartesian coordinates, we modernize the presentation a little by choosing

the cylindrical coordinates, which are better suited to the analysis of the problem (by eliminating the third coordinate z which is superﬂuous). We write the Minkowskian ds2, expressed in these coordinates

ds2 = c2dt2 − dr2 − r2dθ 2

(5)

The observer O then performs the global transformation of the coordinates (θ ∈ [0, 2π ])

(t, r, θ ) −→ (t, r, θ + ωt)

(6)

imposing

the

constraint

r

<

R

<

c
ω

.

This transformation means that the observer O (inertial) now uses a coordinate system that accompanies the disk in its

rotation. From the inertial observer O point of view, the source, the detector and the non-inertial observer D (assumed to

be at the same place) are located on a radius with a ﬁxed orientation, for instance θ = 0. Neglecting a small second-order

term

in

Rω
c

,

the

ds2

(relativistic

invariant)

takes

the

form

ds2 = c2dt2 − dr2 − r2dθ 2 − 2r2ω dθ dt

(7)

5 Remember that Einstein was not awarded the Nobel Prize for the relativity, but for the photoelectric effect in 1921! The Nobel Committee has never given this prestigious award to anybody in recognition of the discovery of the relativity. The name of the eminent mathematician Henri Poincaré, the other important contributor to this theory, had been suggested (by a committee led by Georges Darboux in 1905 [33]); but Poincaré did not receive this prestigious award either. If he had, would Albert Einstein have jointly received the award? It is a remarkable fact that Poincaré presented the great trails of his work on special relativity in the Comptes Rendus in 1905 [34]. In the same year, a lesser known physicist, Albert Einstein, independently presented in Annalen der Physik similar results, but more complete [35]. It should be noted that Poincaré, too conservative, never totally got rid of the aether, whereas Einstein considered this concept superﬂuous.
6 Also let us note that Paul Langevin is recognized as one of the ﬁrst physicists to have defended the relativistic conceptions in France in numerous seminars; in particular the contributions of Einstein in the ﬁeld, better known as they had been eclipsed in this country by the work of Poincaré on the same topic.
7 The quotations were very scarce between this date and the 1960s, where the effect became well known for its applications in gyrolasers (ﬁrst built in 1963). The ﬁrst signiﬁcant paper in English related to the Sagnac effect is the study of Post published in 1967, which contained also some historical information [19].

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G. Pascoli / C. R. Physique 18 (2017) 563–569

Fig. 4. Effect of no-synchronization of the clocks in a rotating frame.

Following Langevin, this expression is called the metric of the rotating disk. It will be noted the presence of a cross term

in dθ dt, source of the impossibility of synchronizing clocks, uniformly distributed around the periphery of the disk and

connected thereto (a relativistic phenomenon which is the essence of the Sagnac effect). The path in space–time of the light rays is determined by the cancellation of the ds2 and by putting dr = 0 and r = R

(the spatial trajectory of the light rays is circular and follows the periphery of the disk). We obtain an equation of the

second degree in dt. The solutions (Rω c) are written

⎡

⎤

dt±

=

R 2 dθ c2

⎣ω ±

c2 R2

+

ω2⎦

R 2 dθ

c

c2

ω±
R

(8)

It is then integrated on the periphery of the disk (noting that dt± > 0, that is dθ > 0 for dt+ and dθ < 0 for dt−). It yields

4π R2ω

t = t+ − t− = c2

(9)

We ﬁnd again the relationship given by eq. (4).
Fig. 4 illustrates this situation. When the disk is in rotation, the light beams are represented in thick gray lines, labeled respectively + for the beams traveling in the trigonometric direction, − for the ray moving in the opposite direction (in dashed lines are the beams for the disk at rest). For the ﬁxed observer L, the difference in velocities between the light beam + (resp. −) and the observer D is c − v (resp. c + v). If the speed of the light beams with respect to this observer is c, the same value is deﬁnitely measured by the observer D linked to the disk, whatever the direction of travel of the considered beam. The points D+ and D− shown in Fig. 4 are identiﬁed.8

4. The universality of the Sagnac effect

The Sagnac effect appears universal. More speciﬁcally speaking, one can cite the experiment carried out by Hafele and

Keating in 1971 [44]. It consisted of the following. Two planes each equipped with an atomic clock ﬂew around the Earth,

one in the direction of the Earth’s rotation and the other in the opposite direction (the two clocks are synchronized at

the start). On arrival, it is observed that the two clocks indicate different times (in fact it is simply a revisited Sagnac

experiment). A practical conclusion is that one cannot synchronize a collection of clocks distributed all along the Earth

equator. The effect in question is taken into account in the GPS calculations.

The phase shift of the Sagnac experiment was also measured in the case of coherent beams of relativistic neutrons

traveling in opposite directions along the circumference (closed path) of a rotating disk [45,46].

More generally, since the ﬁrst atom interferometry experiment in 1991, measurements of rotation using the Sagnac effect

have been currently performed with atoms [47].

Finally, the quantum mechanical phenomenon predicted by Y. Aharonov and D. Bohm [48] can also be perceived as

some kind of Sagnac effect (by replacing the light beams by charged particle beams (charge q) in the experiment, formally

substituting the rotation ω

by a magnetic ﬁeld

B

and doing

8π λc

−→

q

in the relationship (3)) [49].

8 But then we may wonder why, since the waves have the same speed in a direction and in the other for the observer D, is there a difference in the travel times measured by this observer. The subtlety is that the points D+ and D− reported in Fig. 4 are actually distinct and are identiﬁed only by abuse. A close analogy can be made with the screw dislocations seen in material sciences. A more mathematical approach can be envisioned with help of the
complex function Lnz. This function is continuous and differentiable throughout C, but a cut must be created along any line joining 0 and inﬁnity. The cut is made necessary because when the z phase jumps to ±2π , the Lnz function does not return to its initial value. From a topological (and not merely geometrical) point of view, we must therefore see the two regions represented in Fig. 4, ]−2π R, 0[ × t and [0, 2π R[ × t as a two-leaf foliation of the space–time of the rotating disk.

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5. Conclusion
In Sagnac’s view, his experiment had been initially designed to prove the existence of the aether, the assumed medium of propagation of light waves. It is now known that this hypothetical medium, with its clearly very contradictory physical properties, does not exist. In 1921, Langevin gives the appropriate (relativistic) explanation in a very short note to the Comptes rendus. The Sagnac effect is today at the basis of virtually all of our terrestrial and satellite inertial guidance systems. These technological developments were far from being imaginable when this famous experiment was conducted.
Acknowledgements
We are indebted to Professor Jacques Villain for carefully reading the manuscript and providing us with detailed comments and suggestions.
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