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Annales de la Fondation Louis de Broglie, Volume 40, 2015 149

Classical and Relativistic Derivation of the

Sagnac Eﬀect

Wolfgang Engelhardt1

retired from:

Max-Planck-Institut f¨ur Plasmaphysik, D-85741 Garching, Germany

R´

ESUM´

E. Aussi bien la loi classique que la loi relativiste de composi-

tion des vitesses sont appliqu´ees pour la recalculation de l’eﬀet Sagnac.

Les formules qui r´esultent du d´ecalage des franges d’interf´erences

diﬀ`erent d´ej`a au premier ordre de v/c. L’´equation classique est con-

ﬁrm´ee par les mesures interf´erom´etriques et ´egalement v´eriﬁ´ee par le

syst`eme GPS, alors que ce n’est pas le cas pour le r´esultat relativiste.

ABSTRACT. Both the classical and the relativistic composition law

for velocities are applied to re-calculate the Sagnac Eﬀect. The ensuing

formulae for the fringe shift are found to diﬀer already in ﬁrst order of

v/c. Whilst the classical formula is validated by interferometric mea-

surements and veriﬁed by the GPS-system, this is not the case for the

relativistic result.

P.A.C.S.: 07.60.Ly, 42.81.Pa, 04.20.-q, 03.30.+p

1 Introduction

In 1925 Michelson and Gale built a huge earth-ﬁxed Sagnac Interfer-

ometer in Illinois [1] demonstrating that the light velocity is anisotropic

on the rotating earth. For Sagnac this result did not come as a surprise

having explained the underlying eﬀect on the basis of the ether theory in

1913 [2]. The Special Relativity Theory (SRT), however, had predicted

on the basis of the Lorentz Transformation (LT) that the velocity of

light is isotropic in all inertial systems [3]. Of course, the earth is not an

inertial system, but due to the earth’s rotation the motion of Michelson’s

1Home address: Fasaneriestrasse 8, D-80636 M¨unchen, Germany

E-mail address: wolfgangw.engelhardt@t-online.de

150 W. Engelhardt

device was only about 2 mm during the passage of a light beam round

the interferometer. The deviation of the mirrors from a straight orbit

was less than one hundredth of an atomic diameter so that non-inertial

eﬀects should be entirely negligible. Nevertheless, the linear term xv/c2

in the LT – which is responsible for c= const – was simply not there.

The experimental result was consistent with the Galilei Transformation

(GT), i.e. light seemed to travel at the velocity c−vfrom west to east

and c+von the return path from east to west, where vis the local

rotational velocity of the earth’s surface.

Strictly speaking, the rotating earth must be described by the Gen-

eral Relativity Theory, but it is hard to see how a very slow rotation

should invalidate the LT and re-establish the GT as found by Michelson

and Gale. Indeed, Post admits in his great review article [4]: “The search

for a physically meaningful transformation for rotation is not aided in

any way whatever by the principle of general space-time covariance, nor

is it true that the space-time theory of gravitation plays any role in es-

tablishing physically correct transformations.” Consequently, he sticks

to the classical formula valid in the ether theory making only a slight

modiﬁcation which is too small to be veriﬁed by experiment.

In this paper we re-derive the Sagnac Eﬀect on the basis both of

the classical and of the relativistic composition law of velocities. This

method allows to calculate the phase velocity of light along the circum-

ference of a Sagnac interferometer rotating at constant velocity in an

analogous way both in the classical framework of the linear GT and in

the relativistic framework of the linear LT [5-8]. Determining the ensu-

ing travel times of light round the interferometer we ﬁnd that the LT –

due to its linear term xv/c2– does not predict any Sagnac Eﬀect, but

results in c= const also in a rotating system as it does in an inertial

system. This explains then why Ashby [9], e.g., uses the Newtonian or

Galilean time transformation t0=trather than t0=γ(t−xv/c2) when he

calculates the Sagnac Eﬀect in the GPS-System. A similar observation

was made by Carroll Alley in a comment at the end of an engineering

presentation on GPS and Relativity [10].

In Sec. 2 we derive the classical Sagnac Eﬀect for a circular light

beam, and in Sec. 3 we do the same making use of the LT. Sec. 4

discusses the implications of our results.

Sagnac Eﬀect 151

2 Sagnac Eﬀect explained in the framework of the

ether theory

The classical transformation law between inertial systems is the GT

which reads:

~x 0=~x −~v t , t 0=t(1)

Although it is formulated only for a linear constant velocity between

the systems, it can also be adapted for a constant rotation by deriving

a composition law for circular velocities. Considering inﬁnitesimal line

elements on a circular orbit one can write:

ds 0=ds −v0dt , dt 0=dt (2)

where v0= Ω Ris now the rotational velocity with angular frequency Ω

on the circular orbit sat radius R. Dividing by the time increment on

both sides of equation (2) one obtains with ds/dt =vϕthe composition

law for constant circular velocities:

v0

ϕ=vϕ−v0(3)

In fact, facilitating the analysis most treatments of the Sagnac Eﬀect

consider a circular interferometer where the light beam is guided on a

circle either by a large amount of tangential mirrors or by optical ﬁbres

[4-8]. The travel time of light to complete the orbit L= 2π R around

the circumference of the interferometer is then

t±=L

vϕ±v0

=L

c±v0

(4)

depending on whether the light propagates parallel or anti-parallel to

the rotational velocity. Two coherent light beams starting at the beam

splitter in opposite directions return at diﬀerent times leading to a fringe

shift due to the time diﬀerence as calculated with (4):

∆t=L

c−v0

−L

c+v0

=2L v0

c2−v2

0

=2L v0

c2

1

1−v2

0c2(5)

The linear dependence on the rotational velocity has been conﬁrmed

by many experiments, whereas the quadratic term is too small to be

measured in practice. In most textbooks result (5) is expressed by the

fringe shift in units of the wavelength λ0

∆Z=4~

A·~

Ω

c λ0

(6)

152 W. Engelhardt

where, in general, the scalar product of the oriented area ~

Aenclosed by

the light path with the vector angular velocity ~

Ω enters. One can show

that the fringe shift is independent of the shape of ~

Aand of the position

of the rotational axis, but depends on the cosine of the angle between

~

Aand ~

Ω.

Whereas Sagnac could rotate his device in the laboratory at variable

speed in order to verify formula (6), Michelson and Gale had to vary

the area enclosed by the light path in order to prove the validity of (6).

This experiment was important as it demonstrated that the ether is not

co-rotating with the earth like air. Apparently, an “ether wind” must

exist due to the rotation of the earth which leads to the observed time

diﬀerence (5). Any translational motion, which surely exists, cannot be

detected by the circular interferometer, as the inﬂuence of a constant

translational velocity cancels on the roundtrip of the light. In this re-

spect the Sagnac Interferometer is as insensitive as the Michelson-Morley

Interferometer [11] where the enclosed area Avanishes.

Although the experimental results obtained by Sagnac Interferome-

ters are in full agreement with the ether theory, as shown above, they

are generally not considered as disproving the SRT which postulates c=

const and is thus at variance with the ether composition law (3). In the

next Section we will show why it is commonly held that there is no dis-

crepancy between the classical composition law (3) and the relativistic

philosophy concerning the behaviour of light. The reason is a defective

application of SRT.

3 Sagnac Eﬀect interpreted in the framework of the

Special Relativity Theory

Most textbooks assure the reader that the Sagnac Eﬀect is a “relativis-

tic” eﬀect which incidentally may also be derived by the ether theory

leading practically to the same result. Looking closer one ﬁnds, how-

ever, that a truly relativistic derivation is not pursued. A good example

is Post’s procedure in Sec. III [4]. He also uses the circular geometry (see

Fig. 8) which our analysis in the previous Section was referring to. For

the stationary observer he comes up with the time diﬀerence (7 P) which

is identical with our formula (5) derived from the classical composition

Sagnac Eﬀect 153

law for velocities:

∆t=2L v0

c2−v2

0

This must also hold for the co-moving observer, if Ashby’s [9] Galilean

time transformation t0=tinto the rotating system is valid.

Post insists, however, that the time increment between the rotating

and the stationary system must transform according to his equation (11

P):

dt =γ dt 0=dt 0

q1−v2c2

which can also be found in Malykin’s paper [5]. Consequently, both

authors claim that the correct relativistic time diﬀerence as measured on

the rotating beam splitter must be Post’s equation (23 P) or Malykin’s

(5 M) which is our classical formula (5) divided by the γ-factor. One

notices that the LT was only applied in a mutilated form when (23 P)

was derived, since (11 P) is at variance with the complete LT formula.

A diﬀerent conclusion is reached by employing the correct expression,

i.e. by applying the truly relativistic composition law for velocities that

cannot be obtained from (11 P). For the case where the light beam

propagates parallel to the x-axis the LT reads:

x0=γ(x−v t), t0=γt−x vc2(7)

In order to adapt these formulae to a curved light path, we consider

inﬁnitesimal time increments in analogy to equation (2) assuming a con-

stant rotational velocity v0:

dt0=γdt −v0dsc2(8)

where sdenotes the curved light path in the rotating interferometer.

This formula can also be found in Post’s article (29 P) where he quotes

papers by Langevin [12] and Trocheris [13]. Of course, equation (8)

is only justiﬁed as long as the light beam propagates parallel to the

rotational velocity, but this is ensured by the use of an optical ﬁbre, for

example. The spatial increment

ds0=γ(ds −v0dt) (9)

154 W. Engelhardt

is not given by Post, nor does he calculate the composition rule resulting

from division of (9) by (8):

v0

ϕ=vϕ−v0

1−vϕv0c2(10)

Although not explicitly derived, this formula can also be found in papers

[5-8]. These authors hesitate, however, to draw the obvious conclusion:

If the phase velocity of the light in the laboratory is

vϕ=c(11)

it follows from (10) that the phase velocity in the rotating system is also

v0

ϕ=c(12)

This result does not really come as a surprise, since it only reﬂects Voigt’s

postulate c= const from which the LT (7) was derived [3]. If this

transformation holds between inertial systems, it must also hold between

an inertial system and a system rotating with constant velocity as just

demonstrated.

The obvious consequence of formula (10) above is that coherent

beams leaving the beam splitter at the same time in opposite direc-

tions will return at the same time as they both travel at the same speed

caccording to (12). The relativistically correct result is, therefore, not

Post’s formula (23 P), but simply

∆t0=t0+−t0 − =L/c−L/c= 0 (13)

In other words, the SRT correctly applied to a rotating Sagnac Interfer-

ometer does not predict the Sagnac Eﬀect.

4 Implications of the demonstrated results

The Michelson-Gale experiment of 1925 had already shown that Voigt’s

postulate c= const is untenable. When the GPS-system delivered more

and more precise measurements based on the propagation of microwaves,

it became clear that the Sagnac Eﬀect had to be taken into account.

This implied the use of the classical formulae derived in Sec. 2 on the

Sagnac Eﬀect 155

basis of the ether theory. Ashby drew the correct conclusion and re-

introduced the Galilean time transformation t0=tbetween the inertial

ECI and the rotating ECEF system [9]. From Alley’s remark at the

end of [10] we conclude that the GPS-software – developed by engineers

for providing precise position measurements – ignores the Lorentz time

transformation, following Ashby’s line instead. There are, of course,

other indications voiced by Hatch [14] which suggest that the SRT should

not be applied to GPS-measurements.

In the light of the analysis in Sec. 3 and the irrefutable experimental

results it is obvious that Voigt’s postulate of 1887 [15] does not reﬂect

an observed physical law. It was always clear that it was invalid in the

case of acoustics, and we have no reason to adopt it in optics either. As

the SRT is a consequence of this postulate, its basic tenet is at stake.

References

[1] Albert Abraham Michelson, Henry G. Gale, The Eﬀect of the Earth’s

Rotation on the Velocity of Light II, The Astrophysical Journal 61 (1925)

140-145.

[2] Georges Sagnac, Sur la preuve de la r´ealit´e de l’´ether lumineux

par l’exp´erience de l’interf´erographe tournant, Comptes Rendus 157

(1913) 1410-1413.

[3] Wolfgang Engelhardt, On the Origin of the Lorentz Transformation,

http://arxiv.org/abs/1303.5309.

[4] Evert Jan Post, Sagnac Eﬀect, Review of Modern Physics 39 (1967) 475-

493. (http://www.orgonelab.org/EtherDrift/Post1967.pdf)

[5] G. B. Malykin, The Sagnac eﬀect: correct and incorrect explanations,

Physics-Uspekhi 43 (2000) 1229-1252, equation (2).

[6] G. B. Malykin, Sagnac eﬀect in a rotating frame of reference. Relativistic

Zeno paradox, Physics-Uspekhi 45 (2002) 907-909, equation (1).

[7] Annemarie Schied, Helium-Neon-Ringlaser-Gyroskop, Staatsexamensar-

beit, University of Ulm (2006), equation (1.1),

http://www.uni-ulm.de/fileadmin/website uni ulm/nawi.inst.220

/publikationen/LehramtAnnemarieSchied2006.pdf.

[8] Guido Rizzi and Matteo Luca Ruggiero, The relativistic Sagnac eﬀect:

two derivations, equation (22), Relativity in Rotating Frames, eds. G.

Rizzi and M. L. Ruggiero , in the series ”Fundamental Theories of

Physics”, ed. A. Van der Merwe, Kluwer Academic Publishers, Dordrecht,

(2003) (http://arxiv.org/abs/gr-qc/0305084v4).

156 W. Engelhardt

[9] Neil Ashby, Relativity in the Global Positioning System, Living Rev. Rel-

ativity 6(2003) 1. Sec. 2, equation (3)

(http://www.livingreviews.org/lrr-2003-1).

[10] Henry F. Fliegel, Raymond S. DiEsposti, GPS and Relativity: An Engi-

neering Overview. , 28th Annual Precise Time and Time Interval (PTTI)

Applications and Planning Meeting, VA, 3-5 Dec 1996, pp. 189-199

(http://tycho.usno.navy.mil/ptti/1996/Vol%2028 16.pdf).

[11] Wolfgang Engelhardt, Phase and frequency shift in a Michelson Interfer-

ometer , Physics Essays 27 (2014) 586-590.

[12] Paul Langevin, On the Experiment of Sagnac, Comptes Rendus 205

(1937) 51.

[13] M. G. Trocheris, Electrodynamics in a Rotating Frame of Reference, Phil.

Mag. 40 (1949) 1143.

[14] Ronald R. Hatch, Clock Behavior and the Search for an Underly-

ing Mechanism for Relativistic Phenomena, Proceedings of the 58th

Annual Meeting of The Institute of Navigation and CIGTF 21st

Guidance Test Symposium, Albuquerque, NM, June 2002, pp. 70-81

(https://www.ion.org/publications/abstract.cfm?articleID=937).

[15] Woldemar Voigt: Ueber das Doppler’sche Princip , Nachrichten von der

K¨oniglichen Gesellschaft der Wissenschaften und der Georg–Augusts–

Universit¨at zu G¨ottingen, No. 2, 10. M¨arz 1887.

(Manuscrit re¸cu le 12 f´evrier 2015)