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Classical and Relativistic Derivation of the Sagnac Effect

Authors:
  • JET, Max-Planck-Institut für Plasmaphysik (retired)

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 measurements and verified by the GPS-system, this is not the case for the relativistic result.
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 cvfrom 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(txv/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= 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=vϕv0(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
cv0
L
c+v0
=2L v0
c2v2
0
=2L v0
c2
1
1v2
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·~
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
c2v2
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
q1v2c2
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(xv t), t0=γtx 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=vϕv0
1vϕv0c2(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/cL/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),
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[8] Guido Rizzi and Matteo Luca Ruggiero, The relativistic Sagnac effect:
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156 W. Engelhardt
[9] Neil Ashby, Relativity in the Global Positioning System, Living Rev. Rel-
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(http://www.livingreviews.org/lrr-2003-1).
[10] Henry F. Fliegel, Raymond S. DiEsposti, GPS and Relativity: An Engi-
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(http://tycho.usno.navy.mil/ptti/1996/Vol%2028 16.pdf).
[11] Wolfgang Engelhardt, Phase and frequency shift in a Michelson Interfer-
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[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
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(https://www.ion.org/publications/abstract.cfm?articleID=937).
[15] Woldemar Voigt: Ueber das Doppler’sche Princip , Nachrichten von der
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)
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