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PHYSICS ESSAYS 31, 2 (2018)

Circular and rectilinear Sagnac effects are dynamically equivalent and 1 contradictory to special relativity theory

2

Ramzi Suleimana)

3

Triangle Center for Research and Development (TCRD), PO-Box 2167, Kfar Qari 30075, Israel, Department

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of Psychology, University of Haifa, Abba Khoushy Avenue 199, Haifa 3498838, Israel, and Department of

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Philosophy, Al Quds University, East Jerusalem and Abu Dies, P.O.B. 51000, Palestine

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(Received 19 June 2016; accepted 14 April 2018; published online xx xx xxxx)

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Abstract: The Sagnac effect, named after its discoverer, is the phase shift occurring between two

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beams of light, traveling in opposite directions along a closed path around a moving object. A

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special case is the circular Sagnac effect, known for its crucial role in the global positioning system

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(GPS) and ﬁber-optic gyroscopes. It is often claimed that the circular Sagnac effect does not contra-

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dict special relativity theory (SRT) because it is considered an accelerated motion, while SRT

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applies only to uniform, nonaccelerated motion. It is further claimed that the Sagnac effect, mani-

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fest in circular motion, should be treated in the framework of general relativity theory (GRT). We

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counter these arguments by underscoring the fact that the dynamics of rectilinear and circular types

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of motion are completely equivalent, and that this equivalence holds true for both nonaccelerated

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and accelerated motion. With respect to the Sagnac effect, this equivalence means that a uniform

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circular motion (with constant w) is completely equivalent to a uniform rectilinear motion (with

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constant v). We support this conclusion by convincing experimental ﬁndings, indicating that an

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identical Sagnac effect to the one found in circular motion, exists in rectilinear uniform motion.

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We conclude that the circular Sagnac effect is fully explainable in the framework of inertial sys-

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tems, and that the circular Sagnac effect contradicts SRT and calls for its refutation. VC 2018

Physics Essays Publication. [http://dx.doi.org/10.4006/0836-1398-31.2.215]

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Re´sume´: L’effet Sagnac, nomme´ d’apre`s son de´couvreur, est le de´phasage qui se produit entre

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deux faisceaux de lumie`re voyageant dans des sens oppose´s le long d’un chemin ferme´ autour d’un

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objet en mouvement. Un cas particulier est l’effet circulaire de Sagnac, connu pour son roˆle crucial

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dans le syste`me Global Positioning System (GPS) et les gyroscopes a` ﬁbre optique. On dit souvent

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que l’effet circulaire de Sagnac ne viole pas la the´orie de la relativite´ restreinte, parce qu’il

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s’agirait d’un mouvement acce´le´re´, alors que cette the´orie ne s’applique qu’aux mouvements

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uniformes non acce´le´re´s. On dit aussi que l’effet Sagnac, qui se manifeste dans le mouvement

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circulaire, doit eˆtre traite´ dans le cadre de la the´orie de la relativite´ ge´ne´rale. Nous allons a`

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l’encontre de ces afﬁrmations en soulignant le fait que les dynamiques des mouvements rectilignes

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et circulaires sont absolument e´quivalentes, et que cette e´quivalence vaut pour les mouvements

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aussi bien non acce´le´re´s qu’acce´le´re´s. En ce qui concerne l’effet Sagnac, cette e´quivalence signiﬁe

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qu’un mouvement circulaire uniforme (a` constante w) est totalement e´quivalent a` un mouvement

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rectiligne uniforme (a` constante v). Nous soutenons cette conclusion par des re´sultats

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expe´rimentaux convaincants qui indiquent qu’un effet de Sagnac identique a` celui trouve´ dans le

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mouvement circulaire existe en mouvement rectiligne uniforme. Nous concluons que l’effet

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circulaire de Sagnac est pleinement explicable dans le cadre des syste`mes inertiels, qu’il contredit

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la the´orie de la relativite´ restreinte et qu’il appelle a` la re´futation de cette the´orie.

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Key words: Sagnac Effect; Special Relativity Theory; Lorentz Invariance; Systems Equivalence; GPS.

40 I. INTRODUCTION
41 The Sagnac effect is a phase shift observed between two 42 beams of light traveling in opposite directions along the 43 same closed path around a moving object. Called after its 44 discoverer in 1913,1 the Sagnac effect has been replicated in 45 many experiments.2–5
a)suleiman@psy.haifa.ac.il

The circular Sagnac effect is a special case of the general 46
Sagnac effect, which has crucial applications in ﬁber-optic 47 gyroscopes (FOGs)6–10 and in navigation systems such as the 48 global positioning system (GPS).2,11 The amount of the cir- 49
cular Sagnac effect is calculated using a Galilean summation 50
of the velocity of light and the velocity of the rotating frame 51 (c 6 xr). The difference in time intervals of two light beams 52
sent clockwise and counterclockwise around a closed path 53 on a rotating circular disk is Dt ¼ 2cv2l, where v ¼ xR is the 54

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Physics Essays 31, 2 (2018)

55 speed of the circular motion, and l ¼ 2pR is the circumfer56 ence of the circle. In fact, the Galilean summation of c and 57 6wr contradict special relativity theory’s (SRT’s) second 58 axiom and the Lorentz transformations. Nonetheless, it is 59 consensual that the Sagnac effect does not falsify SRT,12 60 because it is manifested in circular motion, which is consid61 ered an accelerated motion,13–15 while SRT applies only to 62 inertial (nonaccelerated) systems. Based on this consensus, 63 in the GPS, concurrent corrections for the Sagnac effect and 64 SRT’s time dilation are made. Moreover, some theoreticians 65 claimed that the Sagnac effect manifest in circular motion, 66 should be treated in the framework of general relativity the67 ory (GRT) and not SRT.16,17 68 The view that the Sagnac effect is a property of rota69 tional systems is strongly disproved by Wang and his 70 colleagues18–20 who conducted experiments demonstrating 71 that an identical Sagnac effect, to the one found in circular 72 motion, exists in rectilinear uniform motion.21 Using an opti73 cal ﬁber conveyor, the authors measured the travel-time dif74 ference between two counter propagating light beams in a 75 uniformly moving ﬁber. Their ﬁnding revealed that the 76 travel-time difference in a ﬁber segment of length Dl moving 77 at a speed v was equal to Dt ¼ 2vDl/c2, whether the segment 78 was moving uniformly in rectilinear or circular motion. The 79 existence of a Sagnac effect in rectilinear uniform motion is 80 at odds with the prediction of SRT, and with the Lorentz 81 invariance principle and, thus, should qualify as a strong ref82 utation of both theories. However, despite the fact that Wang 83 and his colleagues published their ﬁndings in well-respected 84 mainstream journals, their falsiﬁcation of SRT’s second 85 axiom, and the Lorentz transformations, has been completely 86 ignored. To the best of my knowledge, no effort was done by 87 SRT experimentalists to replicate Wang et al.’s falsifying 88 test of SRT. 89 In this short note, we provide strong theoretical support 90 to the aforementioned ﬁndings regarding the identity 91 between the rectilinear and circular Sagnac effects, by under92 scoring the fact that, in disagreement with the acceptable 93 Newton’s deﬁnition of inertial motion, the dynamics of 94 rectilinear and circular types of motion are completely equiv95 alent, and that this equivalence holds true for both nonaccel96 erated and accelerated motion. We elucidate this fact in 97 Section II and in Section III we draw conclusions regarding 98 the contradiction between the rectilinear and circular Sagnac 99 effects, and the predictions of SRT.
100 II. ON THE EQUIVALENCE BETWEEN CIRCULAR AND 101 RECTILINEAR KINEMATICS
102 The common view in physics is that the above103 mentioned two types of motion are, in general, qualitatively 104 different. Linear motion with constant velocity is considered 105 inertial, while circular motion, even with constant radial 106 velocity, is considered an accelerated (noninertial) motion. 107 The above view is not restricted to the Sagnac effect, or to 108 relativistic motion, but it is believed to be a general distinc109 tion in classical mechanics as well, and is repeated in all 110 books on physics. This common view maintains that the cen111 tripetal force acting on a rigid rotating mass causes continual

change in its velocity vector, reﬂected in change in its direc- 112

tion (keeping it in a tangential direction to the circular path). 113

Here, we challenge this convention by claiming that 114

there is a one-to-one correspondence between the linear and 115

circular types of motion. In the language of systems analysis, 116

the two types of motion are completely equivalent sys- 117 tems.22,23 The proof for our claim is trivial. To verify that, 118

consider a dynamical system of any type (physical, biologi- 119

cal, social, etc.), which could be completely deﬁned by a set 120

of dynamical parameters pi (i ¼ 1, 2, …, 6), and a set of 121

equations R deﬁned as

122


R ¼ p2 ¼ p_1; p3 ¼ p€1; p5 ¼ p3p4;

ð


p6 ¼

p5dp1;

p7

¼

1 2

p4p22

(1)

If we think of p1, p2, p3, as representing rectilinear posi- 123 tion x, velocity v, and acceleration a, respectively, and of p4, 124 p5, p6, p7, as mass m, rectilinear force F, work W, and kinetic 125 energy E, respectively, then the dynamical system deﬁned by 126

R gives a full description of a classical rectilinear motion 127

(see Table I). Alternatively, if we think of p1, p2, p3, as rep- 128 resenting angular position h, velocity w, and acceleration a, 129

respectively, and of p4, p5, p6, p7, as radial inertia I, torque s, 130 work W, and kinetic energy E, respectively, then the dynami- 131

cal system deﬁned by R gives a full description of a classical 132

circular motion (Q.E.D.).

133

It is worth noting that the equivalence between rectilin- 134

ear and circular dynamical systems is not restricted to the 135

special case of rotation with constant angular velocity or 136

even with constant acceleration.

137

We note here that the equivalence demonstrated above 138

between the dynamics of uniform rectilinear and uniform cir- 139

cular types of motion is inconsistent with Newton’s ﬁrst law, 140

which states that, unless acted upon by a net unbalanced 141

force, an object will remain at rest, or move uniformly 142 forward in a straight line.24 According to this deﬁnition of 143

inertial motion, which was adopted by Einstein, a circular 144

motion with uniform radial velocity is considered an acceler- 145

ated motion. However, the above demonstrated equivalence 146

is at odds with Newton and Einstein’s views of inertial sys- 147

tems. In fact, based on Newton’s mechanics, the ﬁrst law for 148

TABLE I. Dynamical equations of rectilinear and circular systems.

Variable Position Velocity
Acceleration
Mass/Inertia Newton’s second law Work Kinetic energy

Rectilinear
x v ¼ dx
dt a ¼ dv
dt M F ¼ ma Ð W ¼ Fdx E ¼ 1 mv2 2 …….

Circular
h x ¼ dh
dt a ¼ dx
dt
I s¼Ia
Ð W ¼ sdh E ¼ 1 I x2
2 …….

General

p1

p2

¼

dp1 dt

p3

¼

dp2 dt

p4

p5 ¼ p4 p3 Ð
p6 ¼ p5dp1

p7

¼

1 2

p4

p22

…….

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Physics Essays 31, 2 (2018)

217

149 circular motion could be derived simply by replacing, in the 150 original statement of the law, the words “straight line” by the 151 word “circle,” thus yielding the following law:

152

“A body in circular motion will continue its

153

rotation in the same direction at a constant

154

angular velocity unless disturbed.”

155 Quite interestingly, our view of what deﬁnes an inertial

156 system is in complete agreement with Galileo’s interpreta-

157 tion of inertia. In Galileo’s words: “All external impediments

158 removed, a heavy body on a spherical surface concentric

159 with the earth will maintain itself in that state in which it has

160 been; if placed in movement toward the west (for example), 161 it will maintain itself in that movement.”25 This notion,

162 which is termed “circular inertia” or “horizontal circular

163 inertia” by historians of science, is a precursor to Newton’s 164 notion of rectilinear inertia.26,27

165 A deeper inquiry of the different opinions of the notion

166 of “inertia” throughout the history of physics is beyond the

167 scope and aims of the present paper. Nonetheless, we dare to

168 put forward the following deﬁnition of an inertial motion,

169 which agrees well with Galileo’s conception. According to

170 the proposed deﬁnition, a rigid body is said to be in a state

171 of inertial motion if and only if the scalar product between

172 the sum of all the forces acting on the body, and its velocity

173 vector is always equal to zero, or

 X !FiðtÞ :~vðtÞ ¼ 0 for all t:

(2)

174 Note that the condition in Eq. (2) is satisﬁed (under ideal 175 conditions) only by two types of motion: The rectilinear and 176 the circular types of motion.

177 III. CONCLUSIONS AND GENERAL REMARKS

178 Although it is not the subject of the present paper, our dem-

179 onstration of the complete equivalence between the circular

180 and the rectilinear dynamics, based on Newtonian dynamics,

181 calls for a reformulation of Newton’s ﬁrst law, which is in line

182 with Galileo’s view of inertial motion. Such reformulation is

183 far from being semantic. By accepting the fact that the circular

184 and rectilinear dynamics are completely equivalent, it becomes

185 inevitable but to conclude that the Sagnac effect in uniform cir-

186 cular motion is completely equivalent to the Sagnac effect in

187 uniform rectilinear motion, and that both effects contradict

188 SRT.

189 Moreover, the claim that the circular Sagnac effect

190 should be treated in the framework of GRT simply does not

191 make sense. In most Sagnac experiments, the experimental

192 apparatus is of small physical dimensions, allowing us to

193 assume that the gravitational ﬁeld in the apparatus is uni-

194 form, thus excluding any GRT effects.

195 Another erroneous justiﬁcation for the coexistence

196 between SRT and the Sagnac effect is that the observed 197 effect could be derived from SRT,28,29 e.g., by using Lorentz

198 transformations expressed in coordinates of a rotating frame.

199 This claim is based on fact that the difference between the

200 detected effect, and the one predicted by SRT, amounts to

201

1 2

ðvcÞ2,

which

is

claimed

to

be

negligible

for

all

practical

cases

and applications. We argue that this line of reasoning is erro- 202

neous in more than one aspect: (1) The directionality of the 203

Sagnac effect is dependent on the direction of light travel 204

with respect to the rotating object, whereas the time dilation 205

effect is independent of the direction of motion; (2) Special 206

relativity is founded on the axiom postulating that the motion 207

of the source of light, relative to the detector, has no effect 208

on the measured velocity of light, whereas in the Sagnac 209

effect, the Galilean kinematic composition of velocities 210

(c þ v, c-v) is the reason behind its appearance; (3) At rela- 211

tivistic velocities, for which SRT predictions become practi- 212

cally relevant, the second order of v/c can amount to values 213

approaching one; and (4) The aforementioned difference, 214

even if inﬁnitesimally small, as in the case of GPS, could not 215

be overlooked because it is a systematic deviation between 216

the model’s prediction and reality, and not some kind of sta- 217

tistical or system’s error.

218

Finally, we note that the abundance of experimental 219

ﬁndings in support of SRT, mainly its prediction of time 220 dilation,30–33 is no more than what Carl Popper calls 221

“conﬁrmation tests” of the theory. What is needed is to sub- 222

ject SRT to stringent tests, i.e., to what Carl Popper has 223 termed a “risky” or “severe” falsiﬁcation test.34,35 Evidently, 224

the Sagnac effect, arising in rectilinear and in circular 225

motion, qualiﬁes as a severe test of SRT. But such experi- 226

ments have already been performed in linear and circular 227 motion by Wang and his colleagues,18–20 and we have shown 228

here that the two types of motion are completely equivalent. 229

We have no other way but to conclude that the physics 230

community is acting irrationally and unscientiﬁcally. The logic 231

behind the second axiom of SRT is shaky, and Herbert Din- 232 gle’s argument36–38 that it leads to contradiction has never 233

been answered without violating the principle of relativity 234

itself. On the experimental side, the Sagnac effect detected in 235

linear motion is a clear falsiﬁcation of the theory, and we have 236

closed the loophole by showing here that what applies to recti- 237

linear motion applies to circular motion.

238

In science, it takes one well-designed and replicated 239

experiment to falsify a theory. As put most succinctly by 240

Einstein himself: “If an experiment agrees with a theory it 241

means ‘perhaps’ for the latter… but If it does not agree, it 242 means ‘no’.”39 (p. 203). Meanwhile, an experiment falsify- 243

ing SRT is ﬂying above our heads in the GPS and similar 244

systems, but there are no good and brave experimentalists to 245

observe them and register their results.

246

We are not aware of a similar case in the history of 247

modern science, where a theory, which deﬁes reason, and 248

contradicts with the ﬁndings of crucial tests, holds ﬁrm. We 249

believe that it is due time for a serious reconsideration of 250

SRT and the Lorentz transformations.

251

ACKNOWLEDGMENTS

252

I thank Dr. Ibrahim Yehia, and two anonymous 253

reviewers for their helpful remarks on earlier drafts of the 254

paper.

255

256

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