zotero-db/storage/294UP94A/.zotero-ft-cache

Physical Review & Research International
3(3): 161-175, 2013
SCIENCEDOMAIN international www.sciencedomain.org

Method for Constraining Light Speed Anisotropy by Using Fiber Optics Gyroscope
Experiments
A. Sfarti1*
1UC Berkeley, CS Dept, 387 Soda Hall, Berkeley, CA 94720, USA.
Author’s contribution
This work was carried by the author. Author AS designed the study, performed the experiment and the statistical analysis, wrote the protocol, and wrote the first draft of the
manuscript. The author read and approved the final manuscript.

Research Article

Received 6th February 2013 Accepted 23rd March 2013 Published 2nd April 2013

ABSTRACT
The Mansouri-Sexl theory is a well known test theory of relativity. Mansouri and Sexl dealt with the theory of the Michelson-Morley, Kennedy-Thorndike and Ives-Stilwell experiments but left out the very interesting Sagnac experiment. In the following paper we will present a
novel way of detecting anisotropy effects in (L)v / c2 via a reenactment of the Sagnac experiment using fiber optic gyroscopes (FOG) where L is the length of the fiber and 
is the angular speed of the FOG. We show how the fiber optics gyroscopes are used for constraining light speed anisotropy in the framework of the Mansouri-Sexl test theory. We also show an interesting amplification effect due to the use of the Mansouri-Sexl slow clock transport equations in conjunction with FOGs. Our paper is divided into four main sections: in the first one we give an overview of the Mansouri-Sexl test theory of special relativity, in the second one we give a historical perspective of the Sagnac experiment, in the third section we formulate the Mansouri-Sexl theory for the Sagnac experiment and we conclude with experimental setup and results.

Keywords: Mansouri-sexl test theory; light speed anisotropy; fiber optic gyroscopes.

____________________________________________________________________________________________ *Corresponding author: Email: egas@pacbell.net;

Physical Review & Research International, 3(3): 161-175, 2013

1. INTRODUCTION - THE MANSOURI - SEXL TEST THEORY

The test theories [1-4] of special relativity are used to examine potential alternate theories to
special relativity (SR) - such alternate theories predict particular values of the parameters of
the test theory, which may easily be compared to values determined by experiments. The existing experiments put rather strong constraints on any alternative theory. One of these
theories, the Robertson-Mansouri-Sexl theory, starts by admitting that there is one
preferential inertial frame  in which the light propagates isotropically. In such a frame, light speed in a refractive medium is c0  c / n where c is the light speed in vacuum and n is the refraction index of the optic fiber. All other frames in motion with respect to  are
considered non-preferential and the light speed is anisotropic. The light speed in the nonpreferential frames can be deduced via simple calculations described in3. We start with the
Mansouri-Sexl transforms (with c=1):

x



d

(v)X



b(v)

 v2

d

(v)

v(

vx)



b(v)

vT

(1.1)

t  a(v)T  ε(v)x

where v is the relative velocity between S and  , (x,t) are the coordinates in S while (X,T) represent their correspondents in  . Exactly like in the original Mansouri-Sexl paper [4] by transforming the light cone X 2  c02T 2  0 into S and by neglecting the terms in v2 and
higher we obtain [2]:

c ( )  1 v (1 2 ) cos

(1.2)

c0

c0

where  is the angle between the light ray direction and the x axis. Expression (1.2) is an
approximation valid if slow clock transport [2-4] synchronization has been used. In this case, the following expressions also hold [2]:

a(v)



1





(v)

v2 c2

b(v)  d(v) 1

(1.3)

  2v

According to Mansouri and Sexl, the one-way light speed is a measurable quantity in this
case and it is direction dependent for   0.5. The larger the term 1  2 in (1.2), the larger
the light speed anisotropy. We will exploit this property in the Mansouri-Sexl theory of the
FOG experiment constructed later in our paper.

On the other hand, according to Mansouri and Sexl [2], if Einstein clock synchronization is used, no first order effects exist and the second order effects are expressed as:

162

Physical Review & Research International, 3(3): 161-175, 2013

c( )



1

(



1/

2)(

v

)2

c0 sin 2





(





 1)(

v

)2

(1.4)

c0

c0

where  , are parameters originating from the Taylor expansion of b, d respectively. In
this case, the Sagnac effect cannot be used for measuring light speed anisotropy because
the second order effects are too small to measure for any reasonable value for the speed v .

1.1. THE SPECIAL RELATIVITY THEORY OF THE SAGNAC EXPERIMENT USING FOG

A fiber optic gyroscope (FOG) senses changes in orientation, thus performing the function of a mechanical gyroscope. However its principle of operation is instead based on the interference of light which has passed through a coil of optical fiber. Two beams from a laser are injected into the same fiber but in opposite directions. Due to the Sagnac effect, the beam travelling against the rotation experiences a slightly shorter path delay than the other beam. The resulting differential phase shift is measured through interferometry, thus translating one component of the angular velocity into a shift of the interference pattern which is measured.

Fig. 1. Explanation of the sagnac experiment
The right hand side of Fig. 1 illustrates what happens if the loop itself is rotating. The symbol
 denotes the angular displacement of the loop during the time required for the pulses to travel once around the loop. For any positive value of  , the pulse traveling in the same
direction as the rotation of the loop must travel a slightly greater distance than the pulse traveling in the opposite direction. As a result, the counter-rotating pulse arrives at the "end"
point slightly earlier than the co-rotating pulse. Quantitatively, if we let  denote the angular
speed of the loop, then the circumferential tangent speed of the end point is  R . The
respective angles traveled by the two light fronts are:
163

Physical Review & Research International, 3(3): 161-175, 2013





2





ct R

(2.1)

for the co-rotating front





2





ct R

(2.2)

for the counter-rotating front,
where c  c  c in vacuum and:

  t

(2.3)

for the co-rotating front
  t

(2.4)

for the counter-rotating front. Substituting (2.3) in (2.1) and (2.4) in (2.2) we get:

t



2 R c R

(2.5)

for the co-rotating front

t



2 R c R

(2.6)

for the counter-rotating front. From (2.5) and (2.6) it follows that:

Ttotal

 t

 t



4 R2 c2  R2 2



c2

4 A  R2 2

(2.7)

where A is the area of the interferometer loop. The above is the exact formula. For R  c
we recover the formula used in practice for detecting angular speed via the Sagnac experiment [5,6]:

Ttotal



4 A c2

(2.8)

The formula shows that the phase difference between the two counter-propagating light signals is, at low angular speeds, proportional to the angular speed and to the area enclosed by the interferometer loop. The first to perform a ring interferometer experiment aimed at observing the correlation of angular velocity and phase-shift was G. Sagnac [6] in 1913 with

164

Physical Review & Research International, 3(3): 161-175, 2013

the purpose of detecting "the effect of the relative motion of the ether". In 1926 a very ambitious ring interferometry experiment was set up by A. Michelson and H.Gale [7]. The aim was to find out whether the rotation of the Earth has an effect on the propagation of light in the vicinity of the Earth. The Michelson-Gale experiment used a very large ring interferometer, with a perimeter of 1.9 kilometer, so it was large enough to detect the angular velocity of the Earth. The outcome of the experiment was that the angular velocity of the Earth as measured by astronomical methods was confirmed to within measuring accuracy. The situation is a little more complicated in the case of using a fiber optic of refraction index
n:

c R

c



n 1

R

nc

(2.9)

c R

c



n 1

R

nc

Substituting (2.9) into (2.5)-(2.6):

t



2 R c   R



2

R

1 R nc
c  (R)2

n nc

for the co-rotating front

(2.10)

t



2 R c   R



2

R

1 R nc
c  (R)2

n nc

(2.11)

for the counter-rotating front, resulting into a total time:

Ttotal _ SR

 t

 t



4 R2 c2  R22

(2.12)

Interestingly enough, the outcome of the experiment does not depend on the refraction index of the fiber optic. The SR prediction from expression (2.12) fully coincides with the experimental results [9]. One of the important advantages of FOGs, besides the absence of any moving parts is the fact that the optic cables can be wrapped around k times resulting into an “amplification” of the net effect:

165

Physical Review & Research International, 3(3): 161-175, 2013

Ttotal _ SR



k

4 R2 c2  R2 2

(2.13)

The resulting phase difference is:

Stotal _ SR

 cTtotal _ SR



kc

c

4 2

R2 R2

2



4 R2k c

(2.14)

that is, the effect is the first order in


, “amplified” by the length of the fiber,

L  2 Rk

and

c

by the radius of the gyroscope, R .

2. THE MANSOURI-SEXL THEORY OF THE FOG EXPERIMENT

Light speed is propagating with the isotropic speed c0 in the preferential frame. In the non preferential frame S associated with the center of the rotating FOG device light speed
propagates at the speeds c in the direction of rotation and c in the direction against the
rotation of the device (Fig. 2).

166

Physical Review & Research International, 3(3): 161-175, 2013

Fig. 2. Detail of the experiment with anisotropic light speed

where, for an infinitesimal angle of rotation d :

ct  Rd  Rt

(3.1)

for the co-rotating front

ct  Rt  Rd

(3.2)

for the counter-rotating front.

t



Rd c  R

(3.3)

for the co-rotating front

t



Rd c  R

(3.4)

for the counter-rotating front. From (3.3) and (3.4) it follows that the phase difference element is:

167

Physical Review & Research International, 3(3): 161-175, 2013

s



c t



ct



R2d

(c

(c  c )  R)(c  R)

(3.5)

Formula (3.5) is a generalization of formula (21) in reference [8]. On the other hand,
according to Fig. 2, light speed appears to be anisotropic in frame S, associated with the
center of the rotating FOG, such that for slow clock transport synchronization and for
 [0, ] the following holds by (1.2):

c  1 v (1 2 ) cos(   )  1 v (1 2 ) sin

(3.6)

c0

c0

2

c0

for the co-rotating front

c  1 v (1 2 ) cos(   )  1 v (1 2 )sin

(3.7)

c0

c0

2

c0

for the counter-rotating front.
For  [ , 2 ] c and c exchange roles. Substituting (3.6) and (3.7) into (3.5) we obtain:

s



c02

2R2c0d [v(1 2 ) sin



 R]2

(3.8)

The total phase differential between the two light paths obtained through the integration of the phase difference element is:


Stotal_MS (v,,)  4R2c0k
0

d c02 [v(12)sin R]2

(3.9)

The doubling of the integral (3.9) is caused by c and c exchanging roles in the interval
 [ , 2 ]. Using the notation A(v,)  v(1 2) ( A is a function of v and  ) and
B   R we obtain:

 S total _ MS  2 R 2 k (

1



(B  c0 )2  A2

1

)

(B  c0 )2  A2

(3.10)

A quick sanity check shows that in SR   0.5 (i.e. A  0 in (3.10)) so we recover the well known SR expression (2.7):

Stotal _ SR



4 R2kc0 c02  (R)2

(3.11)

168

Physical Review & Research International, 3(3): 161-175, 2013

That is, in SR the phase difference is independent of the speed between the lab and the
“preferential” frame  . Given that v c0 , the effect is very close to null for non-rotating
devices. The difference:

Sviolation  Stotal_MS Stotal_SR

(3.12)

is the actual Mansouri-Sexl violation expressed in terms of fraction of a fringe (in m) and it
is a function of the Mansouri-Sexl parameter , the angular speed  of the FOG with respect to the lab frame S and the relative speed of the lab v with respect to the preferential frame  . As it can be seen from (3.9), any deviation from -0.5 for the parameter  attracts a dependency of the result in terms of the speed v between S and  . As opposed to the
case of the SR formula (2.12), the Mansouri-Sexl formula (3.10) depends on the refraction
index via c0  c / n restricting the constraining of the parameter  to experiments that must
use fiber optics with refraction indexes close to unity. The difference is due to the fact that formulas (3.6) and (3.7) are just approximations in the Mansouri-Sexl theory whereas formula (2.9) is exact. In our experiment, we made use of the above prediction in order to set constrains on light speed anisotropy.

The laboratory velocity v(t) has contributions [10-19] from the motion of the Sun with respect to frame  with a constant velocity vs  377km / s , while Earth’s orbital motion around the Sun ve  30km / s . For example, in the case of the references [10-19] the Earth’s daily rotation speed is vd  0.33km / s while for Berkeley, where the experiment was executed (latitude 37o52’18” N) vd  0.355km / s . Finally, vr   R is the active
rotation speed of the FOG so:

v(t)  vs  ve sin[y (t  t0 )]cos E  vd sin[d (t  td )]cos  A  vr sin(t) cos B

(3.13)

Here

 A

 8o

is the angle between the equatorial plane and the velocity of the sun.

 E  6o is the declination between the plane of Earth’s orbit and the velocity of the Sun,

B  33o is the declination between the plane of FOG plane and the velocity of the Sun,

2 /  y  1yr , 2 / d  1 sidereal day, t0 and td are determined by the phase and

start date of the measurement, respectively. If   0.5 then the sinusoidal time variation of

v will be reflected in the phase difference (3.10). In other words, the phase difference (3.10)

will exhibit a characteristic time signature when measured over a sufficiently long time. In
order to constrain the parameter  we will take a series of measurements at different angular speeds  over periods of time long enough such that we could integrate the

sinusoidal effects shown in (3.13). From expression (3.13) we can see that v(t)  c0 . This

enables us to further simplify expression (3.10) and, subsequently, (3.12) by using Taylor
expansion such that we can express the translational effects in v in a simpler form:

169

Physical Review & Research International, 3(3): 161-175, 2013

Stotal _ MS

 4 R2k  4 R2k(1 2) v

c0

c02

 (4 R2kn)  2Rn2(1 2) (2 Rk)v

c

c2

(3.14)

The amplification of the effect due to the presence of large values of the coil number k is a game changer since we can achieve 2 Rk  v for suitable optical cable lengths even
with moderate angular speeds of rotating the FOG. In his analysis, made 14 years ago,
Stedman [5] expressed pessimism that FOGs can be used in the detection of light speed
anisotropy but FOGs have made huge advancements in the past decade, not only in terms
of precision but also in terms of the fiber optic length. For example, in our experimental
setup, 2 Rk  1200m . In order to constrain the parameter  we will take a series of measurements at different angular speeds  over periods of time long enough such that we
could integrate the sinusoidal effects shown in (3.13). Substituting (3.13) into (3.14) we
obtain:

Stotal _ MS  C0  C11 sin(yt)  C12 cos(yt)  C21 sin(dt)  C22 cos(dt)  C3 sin(t)

(3.15)

where

C0

4 R2kn  4 R2kn2 (1 2 ) vs

c

c

c

C11

4

R2kn2 c

(1 

2 )

ve c

cos(yt0 )

cos

E

C12



4

R2kn2 c

(1 

2

)

ve c

sin( yt0

)

cos

 E

C21

4 R2kn2 c

(1 

2 )

vd c

cos(d td

) cos  A

C22

4 R2kn2 c

(1 

2 )

vd c

sin(dtd ) cos  A

C3

4

R2kn2 c

(1 

2 )

R c

cos B

(3.16)

For

  0.5

we recover the SR prediction

Stotal _MS

 Stotal _SR



4R2kn c

.

The second

observation is that the coefficient C0 is much larger than the other coefficients in the Fourier

expansion.

3. THE EXPERIMENTAL SETUP AND THE RESULTS

We used two experimental setups, both based on commercially available FOGs. One uses
EMP-1.2k, (1.2km coil, n  1.1) and the other one uses EMP-1 ( 200m coil, n  1.1), both
from Emcore Inc mounted on a Yaskawa SGMJV Sigma-5 high precision turntable with variable angular speed (Fig. 3). We made four sets of measurements, alternating between the two FOGs, in a 24 hr interval, at 6 hours intervals in order to best capture the effects of the variation of the Earth speed expressed in (3.14) as well as the diurnal changes of temperature affecting the FOGs. Each set of measurements is composed of 10 runs, labeled

170

Physical Review & Research International, 3(3): 161-175, 2013
0-9. We repeated the measurements sets four times, at different angular speeds, varying
from   30 to   150 . The Sviolation measurements, expressed in m, including the
calculation of the error bars, are presented in Tables 1 through 4.
Table 1. Sviolation measurements at 6 am
Table 2. Sviolation measurements at 12 pm
171

Physical Review & Research International, 3(3): 161-175, 2013
Table 3. Sviolation measurements at 6 pm
Table 4. Sviolation measurements at 12 am
172

Physical Review & Research International, 3(3): 161-175, 2013

Based on the measurements we developed a best fit approximation in the form of a Fourier expansion:

^

^

^

^

^

^

^

S  C0  C11 sin(yt)  C12 cos(yt)  C21 sin(dt)  C22 cos(dt)  C3 sin(t) (4.1)

The standard error in the determination of

^
C0

is equal to1.331014. Comparing with (3.16)

and taking into considerations the characteristics of the FOGs employed, this results into a

constraint of | 0.5|(1.20.83)106 for the parameter  .

Fig. 3. The experimental setup
The measurement errors can be attributed in totality to the systematic errors introduced by the FOG devices and the turntable, better results will be obtained in the next generation of the experiments when more precise FOGs become available and we can get a better control over maintaining constant angular speed of the underlying turntable.
3.1. FUTURE WORK AND COMPARISON WITH OTHER METHODS
Presently, the method using FOGs results into lesser constraints than the methods using resonating cavities [10-19]. On the other hand, our results are better by an order of magnitude than the ones of Champeney et al. [20] while being one order of magnitude less restrictive than the experiment executed by Isaak [21]. We plan to repeat the measurements as higher precision FOGs become available. The nice aspect about using FOGs is that they have no moving parts and that their technology is advancing very quickly, both reasons for increasing precision over time. Thus, we can put ever tightening constraints over the
Mansouri-Sexl parameter  using commercially available equipment that costs a fraction of
the price of the specially designed equipment for such kind of experiments.
4. CONCLUSION
We have developed the Mansouri-Sexl theory for the FOG experiment. We have shown that
the Mansouri-Sexl violation is a function of the Mansouri-Sexl parameter , the angular speed  and of the relative speed of the lab v with respect to the preferential frame  . We
have shown how the FOG experiment can be used in order to detect light speed anisotropy
173

Physical Review & Research International, 3(3): 161-175, 2013
within the framework of the Mansouri-Sexl theory and we constrained the parameter  to
less than 0.50.83106 .
AKNOWLEDGEMENTS
The author is grateful for the suggestions of the anonymous referees. The research in this paper was self-funded.
COMPETING INTERESTS
Author has declared that no competing interests exist.
REFERENCES
1. Robertson HP. Postulate versus Observation in the Special Theory of. Relativity, Rev. of Mod. Phys. 1949;21:378-382.
2. Mansouri R, Sexl SU. A test theory of special relativity: I. Simultaneity and Clock Synchronization, Gen. Rel. Grav. 1977;8:497-513.
3. Mansouri R, Sexl SU. A test theory of special relativity: II. First Order Tests, Gen. Rel. Grav. 1977;8;515-524.
4. Mansouri R, Sexl SU. A test theory of special relativity: III. Second Order Tests,Gen. Rel. Grav. 1977;8:809-814.
5. Stedman GE. Ring-laser tests of fundamental physics and geophysics, Rep. Prog. Phys. 1997;60:615-688.
6. Sagnac G. L’Ether lumineux demontre par l’effet du vent relatif d’ether dans un interferometre en rotation uniforme, Comptes Rendus de l'Academie des Sciences (Paris). 1913;157:708-710,1410-1413.
7. Michelson A, Gale H, Pearson F. The Effect of the Earth's Rotation on the Velocity of Light (Parts I and II), Astrophysical Journal. 1925;61:137-145.
8. Rizzi G, Ruggiero ML. Relativistic Physics in Rotating Reference Frames, Kluwer Academic Publishers-Netherlands; 2004.
9. Wang R, Zheng Y, Yao A, Langley D. Modified Sagnac experiment for measuring travel-time difference between counter-propagating light beams in a uniformly moving fiber, Phys.Lett. A. 2003;312:7-10.
10. Saathoff G, Karpuk S, Eisenbarth U, Huber G, Krohn S, Horta RM, et al. Improved test of time dilation in special relativity, Phys. Rev. Lett. 2003;91(19):190403.
11. Müller H, Herrmann S, Braxmaier C, Schiller S, Peters A. Modern Michelson-Morley experiment using cryogenic optical resonators, Phys. Rev. Lett. 2003;91:020401.
12. Müller H, Herrmann S, Braxmaier C, Schiller S, Peters A. Theory and technology for a modern Michelson-Morley Test of Special Relativity, Appl. Phys. B; 2003.
13. Müller H, Braxmaier C, Herrmann S, Pradl O, Lämmerzahl C, Mlynek J, et al. Testing the foundations of relativity using cryogenic optical resonators, Int. J. Mod. Phys. D. 2002;11.
14. Müller H, Braxmaier C, Herrmann S, Peters A, Lämmerzahl C. Electromagnetic cavities and Lorentz invariance violation, Phys. Rev. D 67, 056006; 2003.
15. Lipa JA. New Limit on Signals of Lorentz Violation in Electrodynamics, Phys. Rev. Lett. 90 060403; 2003.
16. Braxmaier C, Müller H, Pradl O, Mlynek J, Peters A, Schiller S. Test of Relativity using a cryogenic optical resonator, Phys. Rev. Lett. 2002;88.
174

Physical Review & Research International, 3(3): 161-175, 2013
17. Schiller S, Antonini P, Okhapkin M. A precision test of the isotropy of the speed of light using rotating cryogenic optical cavities, Phys. Rev. Lett. 2005;95.
18. Wolf P, Bize S, Clairon A, Santarelli G, Tobar ME, Luiten AN. Improved Test of Lorentz Invariance in Electrodynamics, Phys. Rev. 2004;D(70).
19. Ahmed F, Quine BM, Sargoytchev S, Stauffer AD. A Review of Speed of Light, IJP, accepted; 2011.
20. Champeney DC, Isaak GR, Khan AM. An 'aether drift' experiment based on the Mössbauer effect, Phys. Lett. 1963;7(4):241-243.
21. Isaak GR. The Mossbauer effect: Application to relativity, Phys. Bull. 1970;21:255. _________________________________________________________________________
© 2013 Sfarti; This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Peer-review history: The peer review history for this paper can be accessed here: http://www.sciencedomain.org/review-history.php?iid=216&id=4&aid=1193
175