J_ID: PEP DOI: 10.4006/0836-1398-31.2.215 Date: 28-April-18 Stage: Page: 215 Total Pages: 4 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 4 of Psychology, University of Haifa, Abba Khoushy Avenue 199, Haifa 3498838, Israel, and Department of 5 Philosophy, Al Quds University, East Jerusalem and Abu Dies, P.O.B. 51000, Palestine 6 (Received 19 June 2016; accepted 14 April 2018; published online xx xx xxxx) 7 Abstract: The Sagnac effect, named after its discoverer, is the phase shift occurring between two 8 beams of light, traveling in opposite directions along a closed path around a moving object. A 9 special case is the circular Sagnac effect, known for its crucial role in the global positioning system 10 (GPS) and fiber-optic gyroscopes. It is often claimed that the circular Sagnac effect does not contra- 11 dict special relativity theory (SRT) because it is considered an accelerated motion, while SRT 12 applies only to uniform, nonaccelerated motion. It is further claimed that the Sagnac effect, mani- 13 fest in circular motion, should be treated in the framework of general relativity theory (GRT). We 14 counter these arguments by underscoring the fact that the dynamics of rectilinear and circular types 15 of motion are completely equivalent, and that this equivalence holds true for both nonaccelerated 16 and accelerated motion. With respect to the Sagnac effect, this equivalence means that a uniform 17 circular motion (with constant w) is completely equivalent to a uniform rectilinear motion (with 18 constant v). We support this conclusion by convincing experimental findings, indicating that an 19 identical Sagnac effect to the one found in circular motion, exists in rectilinear uniform motion. 20 We conclude that the circular Sagnac effect is fully explainable in the framework of inertial sys- 21 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] 22 Re´sume´: L’effet Sagnac, nomme´ d’apre`s son de´couvreur, est le de´phasage qui se produit entre 23 deux faisceaux de lumie`re voyageant dans des sens oppose´s le long d’un chemin ferme´ autour d’un 24 objet en mouvement. Un cas particulier est l’effet circulaire de Sagnac, connu pour son roˆle crucial 25 dans le syste`me Global Positioning System (GPS) et les gyroscopes a` fibre optique. On dit souvent 26 que l’effet circulaire de Sagnac ne viole pas la the´orie de la relativite´ restreinte, parce qu’il 27 s’agirait d’un mouvement acce´le´re´, alors que cette the´orie ne s’applique qu’aux mouvements 28 uniformes non acce´le´re´s. On dit aussi que l’effet Sagnac, qui se manifeste dans le mouvement 29 circulaire, doit eˆtre traite´ dans le cadre de la the´orie de la relativite´ ge´ne´rale. Nous allons a` 30 l’encontre de ces affirmations en soulignant le fait que les dynamiques des mouvements rectilignes 31 et circulaires sont absolument e´quivalentes, et que cette e´quivalence vaut pour les mouvements 32 aussi bien non acce´le´re´s qu’acce´le´re´s. En ce qui concerne l’effet Sagnac, cette e´quivalence signifie 33 qu’un mouvement circulaire uniforme (a` constante w) est totalement e´quivalent a` un mouvement 34 rectiligne uniforme (a` constante v). Nous soutenons cette conclusion par des re´sultats 35 expe´rimentaux convaincants qui indiquent qu’un effet de Sagnac identique a` celui trouve´ dans le 36 mouvement circulaire existe en mouvement rectiligne uniforme. Nous concluons que l’effet 37 circulaire de Sagnac est pleinement explicable dans le cadre des syste`mes inertiels, qu’il contredit 38 la the´orie de la relativite´ restreinte et qu’il appelle a` la re´futation de cette the´orie. 39 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 ISSN 0836-1398 (Print); 2371-2236 (Online)/2018/31(2)/215/4/$25.00 215 VC 2018 Physics Essays Publication ID: satheeshkumaro Time: 12:37 I Path: //chenas03.cadmus.com/Home$/satheeshkumaro$/ES-PEP#180025 J_ID: PEP DOI: 10.4006/0836-1398-31.2.215 Date: 28-April-18 Stage: Page: 216 Total Pages: 4 216 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 ……. ID: satheeshkumaro Time: 12:37 I Path: //chenas03.cadmus.com/Home$/satheeshkumaro$/ES-PEP#180025 J_ID: PEP DOI: 10.4006/0836-1398-31.2.215 Date: 28-April-18 Stage: Page: 217 Total Pages: 4 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. 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