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P7. General Sagnac Effect and Invalid Relativistic Explanation
C P Viazminsky and P K Vizminiska

Abstract

We show that the experimentally verified formula of Wang et el [1] underlying

Sagnac interference is an immediate result of the theory of universal space and time

(TUST). The general Sagnac effect which includes rotational and translational types is

discussed. We demonstrate that a certain form of Sagnac’s effect occurs under

rotation even if the optical circuit is not closed, and under translation provided the

circuit is not closed and its enclosing segment is not perpendicular to its velocity [2].

We finally show that the Sagnac effect cannot be explained within the framework of

the special relativity theory.

1.. Introduction

We start by a review of basic results pertaining to the optical length of a rod

discussed in P5 [3], which is needed to study a more general type of Sagnac

interference. Next, we show that the Wang formula which quantifies the general

Sagnac effect is an immediate consequence of the scaling transformation of second

type (STII) in the theory of universal space and time.

Suppose that a rod ob of geometric length L is translating in the universal

frame S at a constant velocity v that makes an angle with the vector ob. We found

in P5 [3] that the optical length of the light trip

(Figure 1a) is given by either

side of the scaling transformation of the second type STII:

(1.1)

In both figures the rode ob is translating in S at a constant velocity v. The red arrow

points in the direction of a light pulse that starts from one end (an emitter) and ends

at the other (receiver). The angle between the radius vector of the emitter relative to

the receiver and the velocity v is in Fig. 1a and

in Fig.1b.

where

,

(1.2)

is the Euclidean factor, and is the duration of the trip. Note that

is

the angle between the velocity v and the radius vector of the light’s source b

relative to the receiver o.

The duration of the latter light’s trip can also be written in the form

(1.3)

where

(1.4)

is the scaling factor.

1

2

We have subscripted the duration t of the trip by (bo) to emphasize the light’s

trip’s beginning b and end o. It is easily seen that the duration of the trip

is

(1.5)

where

is the angle between the radius vector of the emitter o

relative to the receiver b.

2. Deriving Wang Formula from the STII

We show here that the experimentally verified formula of Wang [1] is an

immediate result of the scaling transformation of the second type.

Suppose that at the same instant

a light’s pulse emanates from each end

of the rod ob and propagates towards the other end (Figure 2). By equations (1.5) and

(1.3) the difference in the durations of the two trips

is

(2.1)

which is positive for

, negative for

and vanishes for

The difference

reduces to

(2.2)

for

and to

for

In both latter cases the pulse travelling

opposite to the rod’s velocity arrives at the other end before the pulse travelling along

the velocity direction.

The difference (2.1) can be expressed in a form of inner product

(2.3)

The latter relation is valid even if the rod ob is of infinitesimal geometric length .

Consider thus the situation in which light is guided to follow an arbitrary path (optical

circuit) which itself is moving in S, and think of two light beams traveling in opposite

directions along the path, and imagine that the path is divided into infinitesimal line

elements of geometric lengths dl. The difference between the durations of two

opposite light’s beams traversing one such line element is

(2.4)

The total difference between the durations of two light’s trips each starting from an end and following the given path till arriving at the other is obtained through integrating the latter expression over the whole path:

(2.5)

The Wang formula for Sagnac interference

(2.6)

approximates equation (2.5) for low velocities. Indeed, for

we have

(2.7)

2

3

which reduces (2.5) to Wang formula. However for high velocities Wang formula must be replaced by equation (2.5) which was derived from STII in the theory of universal space and time.
It is important to note however that equation (2.5) is valid as long as the line element is invariant under the path’s motion, which implies that the optical circuit moves like a rigid body keeping the distances between all the circuit’s points constant. Noting that the line element, or the metric, is invariant only under translation or rotation, or any transformation composed of them, we deduce that (2.5) is valid only when the circuit executes such motions, and thus is moving like a rigid body.
In a uniform rectilinear motion all points of the path moves at a constant vector velocity, and equation (2.5) takes the form

(2.8)

Example: Electromagnetic Transmission Round the Equator

In [5] the TUST was employed to calculate the time delay (advance) for an

electromagnetic wave heading eastward (westward) and making a full revolution

round the Earth’s equator. The TUST agrees fully with the common expression of

Sagnac interference equation where both predict 207 ns advance (delay)[6]. By the

TUST the time difference (2.8), in which we set

is

.
3. Translational Sagnac’s Effect. It is demonstrated here that “a certain form” of Sagnac’s effect is intimate to
non-closed optical paths translating at a constant velocity relative to a universal frame S (i.e. in a uniform rectilinear motion in S). Indeed, if the optical circuit, shown in red (Figure 3), is translating at a constant velocity v in S, then equation (2.8) takes the form

Figure 3. An open optical circuit translating at a constant velocity v. At the same instant of time a pulse of light originates from each end and travels to the other.

(3.1)

where

denote the position vector of the beginning b of line integral (the

circuit) and its end e. Note that

-The path need not be planar.

- The time difference in (3.1) refers to (time recorded at e when receiving the

light)- (time recorded at b when receiving the light). Thus if and denote the

instants at which light emitted at

from both ends is received at b and e

respectively then

(3.2)

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The quantity in (3.1), which is positive for figure 3, will of course reverse sign if

the beginning and end of the line integral were interchanged.

-The phrase “a certain form of Sgnac effect” we have used in the abstract and at

beginning of this section refers to the difference in time arrival of the light’s beams

that started simultaneously from the opposite ends. Only when the circuit’s two ends

coincide one may observe fringe’s shift.

Equation (3.1) shows that

(a)The Sagnac effect is intimate to any constant translational motion of a non-closed

optical circuit; it vanishes however when its closing vector is perpendicular to its

velocity v.

(b) When it is closed, the beginning and end of the circuit are coincident, and

on the account of

in (3.1). Therefore, there is no Sagnac effect associated with

a closed circuit executing a uniform translational motion.

(c) Two optical circuits with respective ends

and

yields the same time

difference if

. It follows that, under uniform translational motion, every

circuit is equivalent to its closure, and all circuits that have equal vector closure are

Sagnac equivalent,

4. Rotational Sagnac Effect

We mentioned in section 2 that equation (2.5) holds for any motion in which

the circuit moves as a rigid body. We retrieve here the familiar expression that

quantifies the rotational Sagnac effect [4],

(4.1)

which pertains to a polygonal circuit of area rotating about an axis with a constant

angular velocity . The relation which we shall derive using (2.5) applies however to

any shape of a planar circuit including that which is not even closed.

Consider a rigid planar optical circuit of any shape that is rotating at a

constant angular velocity about an axis . The axis of rotation, which is not

necessarily perpendicular the circuit’s plane, intersects the latter at o (Figure 4). The

velocity of a point of the circuit is

and hence

(4.2)

The quantity

represents twice the area of the triangle with base and

vertex at the point o. Substituting in (2.5) yields

(4.3)

which is the expression of rotational Sagnac interference as implied by TUST.

Although the velocity magnitude is a function in the position (and ),

in some cases the factor

can be pulled out of the integral;

these include the following:

(a). All the circuit’s points are at equal distances from the center of rotation, and

hence their velocities have the same magnitude.

(b) The dimensions of the circuit are much smaller that its distance from the axis of

rotation. In this case one can neglect the variation in

and deal with it as a

constant function. The interferometer of Michelson and Gale experiment, consisting

of 4 mirrors situated on Earth surface, and rotating with the earth about the (South-

North pole) axis is an example.

(c) The velocities of the circuits components are small .

In all cases above we may make the approximation

(4.4)

or even set

4

5

Figure 4.

where A is the area subtended by the

circuit and the lines connecting its ends e and b with the center of rotation.

It is noted that the circuit considered need not be closed and A is the area

subtended by the circuit and the lines

, where b and e are the beginning and

end points of the circuit. When the circuit is closed A reduces to the circuit’s area and

the familiar relation (4.1) approximates (4.4) on neglecting terms of order

and

higher.

5. Point Circuits

Examples of optical point circuits are the Sagnac interferometer which consists

of a number of small mirrors

, or a set of satellites revolving in one

orbit, or a set of transmission stations on a great circle on Earth’s surface.

By the STII the time duration of a light’s trip along the side

of an

n-polygonal circuit in S and in any other frame is

(5.1)

where

is the geometric length of the side

and

If the polygon is regular, and the rotation is taking place about its center the

difference between the durations of the trips

and

is given

by

(5.2)

where

is the speed of any vertex (in the unit c). The latter expression is the

familiar translational Sagnac effect as given by (2.3), and which corresponds to the

rod making an angle

with its velocity.

From (5.2) we may proceed to reach either the usual expression of the

rotational Sagnac effect (4.4), or the Wang’s formula. To reach the expression (4.4) of

a circuit rotating at an angular velocity , we proceed from (5.2) as follows

(5.3)

where we identify

by

. The latter relation yields the time difference

between the durations of two light beams propagating in opposite directions along the

ring interferometer:

(5.4)

Another interesting expression for this time difference is the following:

(5.5)

where n is the number of the sides of the regular polygon and u is the speed of any vertex of the interferometer when rotating about its center. Indeed,

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Figure 5. Geometric representation of (5.6)

where

is a vector whose length is equal to the circumference nL of the

polygon and is a vector velocity that makes an angle

with

and

whose magnitude

is equal to of the speed of any vertex when the

interferometer rotates about its center (Figure 5).

6. Special Relativistic Explanation of Sagnac Effect is Invalid We show here that the special relativistic explanation of Sagnac effect as
given in Wikipedia (prior to July 12, 2020) is incorrect, and in the next section we demonstrate that the special relativity theory cannot explain this effect.

We quote from Wikipedia: “Relativistic derivation of Sagnac formula[edit]

Fig. 4: A closed optical fiber moving arbitrarily in space without stretching.
Consider a ring interferometer where two counter-propagating light beams share a common optical path determined by a loop of an optical fiber, see Figure 4. The loop may have an arbitrary shape, and can move arbitrarily in space. The only restriction is that it is not allowed to stretch. (The case of a circular ring interferometer rotating about its center in free space is recovered by taking the index of refraction of the fiber to be 1.)
Consider a small segment of the fiber, whose length in its rest frame is . The time intervals, , it takes the left and right moving light rays to traverse the segment in the rest frame coincide and are given by

Let

be the length of this small segment in the lab frame. By the relativistic length

contraction formula,

correct to first order in the velocity v of the segment.

The time intervals for traversing the segment in the lab frame are given by Lorentz

transformation as:

correct to first order in the velocity v. In general, the two beams will visit a given segment at slightly different times, but, in the absence of stretching, the length dl is the same for both beams. It follows that the time difference for completing a cycle for the two beams is
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” - quotation finished.

The above argument which is sound in a relativistic sense contradicts

experimental findings. According to special relativity the time intervals taken by

the left and right moving light in the rest frame to traverse the given segment

coincide:

. Therefore the durations of a full trip round the ring in opposite

directions are equal:

and the two beams arrive back at the point from which they started at the same instant. This means that the Sagnac effect is not observed in the rest frame, which is not true. Indeed, the fringes shift is a physical reality and the Sagnac effect is observed in the lab frame as well as in the rest frame. The Michelson and Gale experiment is a cogent proof that the Sagnac effect is observed in the rest frame. The location of the latter experiment on the surface of the rotating Earth’s is the rest frame.

7. Sagnac Effect is not Explicable by Special Relativity

We abide throughout this section with special relativity theory and show that it

is incapable of explaining Sagnac effect. Consider again the circuit drawn in the

previous section and let s be the rest frame of the given segment (line element) drawn

there. We name the segment’s left and right ends by 1 and 2 respectively. The rest

frame s is moving in the lab frame S at velocity v and can be considered inertial

during the extremely short duration of the experiment.

Consider the following events in the lab frame S:

- The right moving light,

passes by the left end (1) of the segment at

:

, and

arrives at the right end (2) of the segment at :

- The left moving light

passes by end (2) of the segment at :

, and

arrives at end (1) at

:

Note that the left moving light may pass from the end 2 at an instant that differs

slightly (by ) from the instant at which the right moving light passes through end 1.

However, this difference is so small and can be neglected (i.e.

with no bearing

on subsequent results.

The 4-interval between the first two events is represented by the 4-vector

in Minkowski space . Similarly, the 4-interval between the last two

events is the 4-vector

The difference between the latter vectors is the

4-vector

(7.1)

where we set

(7.2)

Let’s take the inner product of the latter vector by the velocity 4-vector of the

given circuit element,

, and denote the result by I:

(7.3)

We remind the reader that the inner product of two 4-vectors

(7.4)

in is defined by

(7.5)

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An important property of the inner product is that: It is invariant under orthogonal

transformations in and thus in particular under Lorentz transformation. In the rest

frame s of the given segment, the 4-velocity of the segment is

, and the vector

corresponding to (7.1) is

(7.6)

However, in the rest frame s,

because light takes the same duration to

traverse the segment from either end to the other, and

Thus the vector

(7.1) is transformed in s to

. Equation (7.3) and the invariance of the inner

product yield

(7.7)

In the frame S, and are the vectors from one end of the segment to the other

but at different instants of time. Each vector joins the location of the end when light

enters the segment and the location of the other end when it leaves. Neglecting

quantities of high order in smallness yields

, and hence

(7.8)

The total difference in time arrival of the two beams to their common starting point in

the circuit is obtained through integrating over the circuit:

(7.9)

However, in the rest frame of each segment of the circuit there is no difference in time arrival of the two beams at the opposite ends. As a result, the two beams return to their starting point at the same time, and no displacement in interference fringes will be observed in a frame attached rigidly to the circuit. The latter result is at contradiction with experiment.
Conclusion The Wang formula for Sagnac interference follows directly from the scaling transformation of the second type in the theory of universal space and time. In contrast, the special theory of relativity is utterly incapable of explaining Sagnac effect. It was also demonstrated that that a certain form of Sagnac’s effect occurs under rotation even if the optical circuit is not closed, and under translation provided
the circuit is not closed and its enclosing segment is not perpendicular to its velocity.
References [1] Wang, R.; Zheng, Y.; Yao, A.; Langley, D (2006). "Modified Sagnac experiment for measuring travel-time difference between counter-propagating light beams in a uniformly moving fiber". Physics Letters A. 312 (1–2): 7– [2] Viazminsky C P., November 1, 2009: The Scaling Theory - X: The General Sagnac Effect https://www.gsjournal.net/Science-Journals/Research%20Papers/View/1959 [3] Viazminsky C P and Vizminiska P, May 15, 2020: Optical Length of a Moving Rod https://www.gsjournal.net/Science-Journals/Research%20Papers/View/8249 [4] Viazminsky C P and Vizminiska P, May 16, 2020: The Sagnac's Interference, and Michelson and Gale Experiment. https://www.gsjournal.net/Science-Journals/Research%20Papers/View/8252 [5] Viazminsky C P, April 5, 2007: On the Sagnac Effect. https://www.gsjournal.net/Science-Journals/Research%20PapersMechanics%20/%20Electrodynamics/Download/1942 [6] Russo Daniele, (2007), Stellar aberration: the contradiction between Einstein and Bradley", Apeiron, 14, no 2, 95-112.

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