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Open Physics 2023; 21: 20230110

Research Article
Gianfranco Spavieri and Espen Gaarder Haug*
The reciprocal linear eﬀect, a new optical eﬀect of the Sagnac type

https://doi.org/10.1515/phys-2023-0110 received June 08, 2023; accepted August 23, 2023
Abstract: The Sagnac eﬀect can be demonstrated with light propagating either along a circular contour or, as done by Wang et al., back and forth along a linear contour. In the linear Sagnac eﬀect, the emitter–receiver device is in motion relative to the contour where light propagates. In the reciprocal linear Sagnac eﬀect (RLSE), the device is stationary and the contour is in motion. When the contour changes direction of motion, some special features of the linear Sagnac eﬀect are not fully reciprocal to the RLSE, which foresees variations of the ﬁrst order in v∕c in the round-trip time taken by a light signal to cover the contour. The RLSE can be tested with present technology and, if conﬁrmed experimentally, it might have interesting technological applications. Presently, it can be important for testing light-speed invariance, simultaneity, and the relativity principle.
Keywords: light propagation, Sagnac eﬀect, one-way speed of light, relative simultaneity, Lorentz invariance
1 Introduction
The experiment related to the circular Sagnac eﬀect [1], shown in Figure 1(a), was performed in 1913, and the one corresponding to the linear eﬀect, shown in Figure 1(b), was accomplished by Wang et al. [2] in 2003. In both eﬀects, the measuring device C* (emitter–receiver clock or interferometer) is in motion relative to a stationary contour, and C* measures the diﬀerence ΔT of the round-trip times of two light signals counterpropagating around the contour. Following Post [3], ΔT is given by,
 * Corresponding author: Espen Gaarder Haug, Norwegian University of Life Sciences, Christian Magnus Falsensvei 18, 1433 As, Norway, e-mail: espenhaug@mac.com Gianfranco Spavieri: Centro de Fiiisica Fundamental, Universidad de Los Andes, Mérida, 5101, Venezuela, e-mail: gspavieri@gmail.com

ΔT = T⇐ − T⇒

2L

2L

=

−

γ(c − v) γ(c + v)

(1)

4γvL 2γvP = c2 = c2 ,

where T⇐ and T⇒ represent the round-trip time of the coand counter-moving light signals (or photons) along the contour of perimeter P = 2πr or 2L in the circular and linear eﬀects, respectively. For the circular Sagnac eﬀect, with v = ωr, result (1) is usually expressed [3] as ΔT = 4ω ⋅ A∕c2, where A is the area enclosed by the light path.
In the linear Sagnac eﬀect of Figure 1(b), the arm AB of the contour is stationary and the measuring device C* is moving clockwise with uniform speed v along the contour, going from the lower to the upper section of the contour and vice versa. While sliding around the pulley of radius R during the short time η, the device C* changes the direction of motion at the pulley A (or B). Locally, the speed v of C* relative to the contour is always constant. In the linear experiment performed by Wang et al. [2], the device C* is always in uniform rectilinear motion on the lower contour section during the round-trip time T ≃ T⇐ ≃ T⇒. Therefore, in this experiment, C* does not turn around the pulleys A or B. However, since the relative speed v between C* and the contour is constant, there is no reason to suppose that the result diﬀers from the theoretical prediction (1) if C* turns around the pulley during the round-trip time T .
In Section 2, we consider the reciprocal linear Sagnac eﬀect (RLSE), shown in Figure 2, where the device C* is stationary and the contour is in relative motion. We ﬁnd that, for counter-propagating light signals, the same observable ΔT in (1) is foreseen in the RLSE and, thus, with regard to ΔT , the RLSE is reciprocal to the linear Sagnac eﬀect.
Nevertheless, within the context of standard special relativity based on light speed invariance, there are observable features of the linear Sagnac eﬀect that are not reciprocal to those of the RLSE. In fact, when in the interval T⇒ (or T⇐) the device C* turns around the pulley in the linear Sagnac eﬀect of Figure 1(b), for the corresponding RLSE of Figure 2, the Lorentz transformations (LTs) foresee variations in the ﬁrst order in v∕c for the interval T⇒ (or T⇐),

Open Access. © 2023 the author(s), published by De Gruyter.

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2  Gianfranco Spavieri and Espen Gaarder Haug

Figure 1: (a) In the circular Sagnac eﬀect, two counter-propagating photons are emitted from the device C* and travel along the circumference of the rotating platform (only a single photon is shown). C* measures the diﬀerence ΔT of the arrival times after a round trip. (b) In the linear Sagnac eﬀect, the counter-propagating photons travel in an optical ﬁber that may slide frictionless around the two pulleys A and B. The segment AC* of length D represents the initial position of device C* relative to A, the left end point of the contour. If AC* = D > 2vL∕c = (v∕c)P, the countermoving photon performs a round trip and gets back to C* when this is still moving on the contour lower section.
which diﬀer from that given by Eq. (1). Thus, if experimentally conﬁrmed, the RLSE represents a new optical eﬀect, which diﬀers from the standard Sagnac eﬀect.
In Section 3, we describe a realistic experiment, feasible with present technology, which can measure the mentioned variations related to the velocity v.
Figure 2: In the RLSE, the emitter–receiver C* is stationary and the contour of length ≃2L = P is moving with a relative speed v. (a) The device C* emits a photon that travels on the contour lower section from the position AC* = D < 2vL∕c2. (b) When point A reaches C*, the contour changes direction of motion while sliding in the perpendicular direction at speed v on C* for a short negligible time interval η. (c) After the contour has resumed its motion in the horizontal direction, after a roundtrip, C* receives the returning photon on the contour upper section.

There are no problems in interpreting the circular and linear Sagnac eﬀect in the inertial frame of reference Sc of the stationary contour where the speed of light is assumed to be c, while diﬀerent interpretations of the Sagnac eﬀects are given [4–12] in the frame SC* comoving with the device C*.
Some of the calculations that support our results are presented in the Appendix. Moreover, in the Appendix, we discuss some of the interpretations of the Sagnac eﬀects in the wider scenario of relativistic theories [4–13], involving tests on simultaneity conﬁrming that relative and absolute simultaneity are not physically equivalent. In this scenario, we show how the RLSE can be used as a test of light speed invariance.
2 The reciprocal linear Sagnac eﬀect
In the RLSE of Figure 2, the measuring device C* (clock or interferometer) is stationary, while the whole contour is moving back and forth with uniform speed v relative to C*. Ideally, the motion of the contour should be such that, locally, the relative speed between the contour and C* would always be v. In Figure 2(a), the contour is moving with uniform motion relative to C*, which stays on the contour lower section until pulley A (not shown) reaches C*. At this moment, since C* is stationary, the contour has to move slightly in the direction perpendicular to AB during the short time η when sliding around C* at A, as shown in Figure 2(b). After changing direction of motion, the contour is moving relative to C*, now on the upper section as shown in Figure 2(c).
If the pulleys have diameter 2R, the motion in the perpendicular direction 2H ≃ 2R takes place during the ﬁnite time interval η ≃ πR∕v when the contour is accelerating and changing its direction of motion relative to C*. However, for the purpose of simplifying calculations, we omit the process taking place during the negligible short ﬁnite interval η by assuming that the round-trip time T is much greater than η (T ≫ η and L ≫ ηv ≃ 2H ≃ 2R, while (v∕c)L ≫ ηc). Thus, in the following, we consider the simple ideal case of a linear contour moving back and forth in the direction AB at the uniform speed v relative to C*. However, in the Appendix, we consider also the more realistic case of a rectangular contour where η is not negligible and the motion perpendicular to AB is taken into account.
Denoting by P = 2L the perimeter of the contour, assuming the light speed to be c and using the LT, we ﬁnd the following results valid for the RLSE:

The reciprocal linear eﬀect, a new optical eﬀect of the Sagnac type  3

(a) AC* = D > 2vL∕c = (v∕c)P. C* remains always on one of the sections of the contour in the interval T .
The segment AC* of length D represents the initial position of device C* relative to A, the left end point of the contour. For light counter-propagation (photon moving counter-clockwise) starting from the device C*, when D > 2vL∕c2 the description of the RLSE is reciprocal to that shown in (Figure 1(b)), where C* remains always on the lower section of the contour, while the photon performs a round trip in the interval T⇒ and gets back to C*. The Lorentz-contracted moving contour has length L∕γ and the round-trip times for counter-, T⇒, and co-propagation, T⇐, have the expected values,

T⇒ =

2L γ(c +

v)

=

2γL c

⎛⎝1

−

v⎞ c⎠

T⇐ =

2L γ(c −

v)

=

2γL c

⎛⎝1

+

v⎞ c⎠

(2)

4γvL 4γvL 2γvP ΔT = T⇐ − T⇒ = c2 = c2 = c2 ,

where in Eq. (2), T⇐, T⇒, and ΔT are the same as given in Eq. (1). In some cases, we may approximate the results to the ﬁrst order in v∕c to simplify calculations. In Appendix (A.3), we calculate the round-trip interval T⇒ for the counterpropagating photon emitted by C* from the distance AC* = D > 2vL∕c, and show in (A15) that T⇒ is independent of D when C* stays on the same track (lower contour section, in this case) during the interval T⇒.
In this case (D > 2vL∕c), the RLSE foresees results that are the same as those of the standard linear Sagnac eﬀect. Moreover, the symmetry is such that T⇐ is the same as T⇒ by changing v to −v. It follows that, for AC* = D > 2vL∕c, results (2) for the RLSE are equivalent to results (1) of the standard Sagnac eﬀect. In fact, in agreement with the principle of relativity, provided that C* and the contour are in uniform relative motion in the interval T⇒, the two eﬀects are the same when observed either from the rest frame of the contour or the device C*.
(b) AC* = D < 2vL∕c. C* and the contour change their direction of relative motion in the interval T .
In this case, for light counter-propagation, the successive positions of the contour relative to C* are shown in Figure 2, indicating that C* and the contour change their direction of relative motion during the round-trip time.
Special case D = D0 = vL∕c. Starting from D0∕γ (where the factor γ takes into account the length contraction of the moving contour) on the lower section of the contour, the photon travels at speed c toward the moving point B that, at time t, is at the distance L∕γ − D0∕γ + vt = (L∕γ)(1 − v∕c) + vt. Then, the signal reaches point B when ct = (L∕γ)(1 − v∕c) + vt, i.e., after the time interval tout = L∕(γc). Moreover, since point

A is moving toward C* at speed v from the initial distance AC* = D0, point A reaches C* when vt = D0∕γ = vL∕(γc), i.e., at tout = L∕(γc). Hence, the two events “photon at B” and “A at the position of C*” are simultaneous in the reference frame SC* where C* is at rest, as shown in Figure 2(b). The return time interval of the photon from B to C* on the contour upper section is obviously tret = L∕(γc), the return interval tret being independent of the motion of the contour. Then, in the RLSE of Figure 2(b), the return interval tret is the same regardless of whether the contour keeps moving to the right or starts moving to the left of the stationary C*.
Working out the calculations also for the co-propagating signal, for T⇐, we ﬁnd the exact result (A18) of Appendix (A.3). Then,

2L 4γvL

2L

T⇐ = γc + c2 , T⇒ = γc

4γvL 2γvP

(3)

ΔTRLSE = T⇐ − T⇒ = c2 = c2

= ΔT = invariant.

We see that, in the special case considered earlier where C* is initially on the lower and then on the upper section of the moving contour, the round-trip time T⇒ in Eq. (3) is greater than the corresponding value in Eq. (2). However, also the round-trip time T⇐ turns out to be greater than the corresponding value in (2). Then, for counter-propagating light signals, the diﬀerence ΔTRLSE in Eq. (3) is still the same as in Eq. (2) and, as far as ΔT is concerned, ΔT = ΔTRLSE = invariant and the RLSE is equivalent to the standard linear Sagnac eﬀect.
Special features of the RLSE. The important interesting feature of the RLSE is that in Eq. (3) the round-trip time T⇒ of a single photon (i.e., the counter-propagating one) diﬀers from the corresponding T⇒ in Eq. (2). The difference persists as long as 0 < D < 2vL∕c and C* moves from the lower to the upper section during the roundtrip time T⇒. Then, the function T⇒ = T⇒(D) varies from the minimum value given in Eq. (2) (for D = 0 or D > 2vL∕c) to the maximum one given in Eq. (3) (for D = D0 = vL∕c). We must stress that T⇒ is an observable measurable by clock C* and, for the standard linear Sagnac eﬀect, the theory foresees T⇒ = 2L∕γ(c + v), independent of the value of D. Analogous considerations can be made for T⇐.
Thus, results (2) and (3) indicate that there are special features of the RLSE not equivalent to those of the standard Sagnac eﬀect. In case (b), when velocity variations are involved and C* and the contour change their direction of relative motion in the interval T , the diﬀerences indicate that the relativity principle does not hold. The diﬀerences between the two eﬀects are interpreted below in the context of special relativity and the LT, showing that can be linked to relative simultaneity.

4  Gianfranco Spavieri and Espen Gaarder Haug

Since the standard RLSE, where ΔT = T⇐ − T⇒ is measured, is equivalent to the standard linear Sagnac eﬀect, the

usual approach based on measuring ΔT = ΔTRLSE = invariant does not reveal the special features of the RLSE. Then, for the

purpose of pointing out these special features foreseen by the

LT, we consider below the special reciprocal linear Sagnac

eﬀect (S-RLSE), and show how the S-RLSE can be tested and

represents an optical eﬀect with properties that diﬀer from

those of the standard linear Sagnac eﬀect.

Testing the S-RLSE with an experiment

We pointed out that the round-trip T⇒ of the counterpropagating signal is an observable that can be measured

by the single clock C*. Therefore, the special features of the

RLSE can be revealed with this type of measurements. In

general, for precise measurements, we need to compare

the observable T⇒ with the corresponding observable T * of a signal on a diﬀerent light path. In the case of the S-

RLSE, for the optical light path of the co-propagating

photon, we choose a contour with the same perimeter

P = 2L, but stationary in the device frame SC*. Hence, the main diﬀerence between the RLSE and the

S-RLSE is that in the RLSE, we have T * = T⇐, while in the SRLSE, the round-trip time interval of the co-propagating

signal is now T * = 2L∕c. With our choice, by means of (1),

for the standard Sagnac eﬀect, we have ΔTS*agnac = T * − T⇒ = 2L∕c − 2γ(L∕c)(1 + v∕c) ≃ Δ∗Sagn = 2vL∕c2 constant and inde-

pendent of D. Instead, for the S-RLSE, we have ΔT *=

ΔT *(D) = T * − T⇒ = T * − T⇒(D) dependent on D. With the help of (2) and (3), we ﬁnd,

2vL D≳ ⇒
c

ΔT *(D) = T *

−

T⇒

=

2L c

−

2γL(1 − c

v∕c)

≃

2vL c2

=

ΔTS*agnac

=

Δ∗Sagn

(4)

vL 2vL D = D0 ≃ c ≲ c ⇒

2L 2L ΔT *(D0) = T * − T⇒(D0) ≃ c − c = 0.

According to results (4), the function ΔT *(D) is constant for D ≳ 2vL∕c and given by 2vL∕c2 = ΔS∗agn = ΔTS*agnac, which is the same for the RLSE and the linear Sagnac eﬀect. However, for D ≲ 2vL∕c, the function ΔT *(D) varies from the maximum values 2vL∕c2 (D = 0 and D = 2vL∕c) to zero (D = D0 = vL∕c), as shown in Figure 4. Analogous results, given in the following section and in the Appendix, are obtained when light propagates in a medium of refractive index n.
Thus, by assuming light speed invariance and the LT, the S-RLSE foresees the variant ΔT *(D) of (4) that can be observed. The variant special feature of (4) is discussed in

the Appendix within the wider scenario of relativistic theories. To fully understand the diﬀerence between the D-dependent S-RLSE results (4) and the D-independent results of the linear Sagnac eﬀect in Eq. (2) and in Eq. (1), we consider the following corresponding interpretations.
2.1 Interpreting the RLSE and the linear Sagnac eﬀect using the LTs
In the linear Sagnac eﬀect and in the RLSE, the measurements are made by device C*. Why are the round-trip time intervals of the propagating signals diﬀerent in the two eﬀects? Let us, then, consider an observer comoving with clock C* (let us call it “observer C*” or simply “C*”) and check ﬁrst what happens in the linear Sagnac eﬀect of Figure 1(b) when C* is near point A and, during the roundtrip interval T⇒, changes the direction of motion in the negligible time interval η moving from the lower to the upper section of the contour. While on the lower section, C* is comoving with the inertial frame S″ and with the inertial frame S′ when on the upper section.
If the counter-propagating photon is emitted by C* at AC* = D0 = (v∕c)L when in the lower section frame S″, C* reaches A when, simultaneously, the photon reaches point B, as shown in Figure 3. Then, for the relative position of photon and C*, the situation is the same as that in the RLSE of Figure 2(b).
However, when (after reaching A and turning around the pulley) C* starts comoving with frame S′ on the upper section, we can see in Figure 3 that now the position of the photon is not at B, but at K, being AK ≃ L(1 − 2v∕c) < L and, thus, the situation diﬀers from that of Figure 2(b) of the RLSE.
The diﬀerence is due to the fact that, in the linear Sagnac eﬀect, the mechanism of relative simultaneity takes place when C* changes velocity by moving from frame S″ on the lower section to the frame S′ on the upper section [7]. The two events, “C* at A” and “photon at B,” are simultaneous for observer C* on frame S″ in the lower section, but no longer simultaneous for C* on frame S′ in the upper section because S′ is in motion with speed ≃2v relative to S″.
As shown in Figure 3, in the linear Sagnac eﬀect, clock C* is reached by the photon returning on the upper section after the proper time interval τret ≃ (L∕c)(1 − 2v∕c) and the round-trip time interval is, as expected, (T⇒)Sagnac = τout + τret ≃ L∕c + (L∕c)(1 − 2v∕c) = 2L∕(c + v). In this case, relative to C*, in the return trip, the photon covers at speed c the shorter distance L(1 − 2v∕c) only.

The reciprocal linear eﬀect, a new optical eﬀect of the Sagnac type  5

However, in the case of the RLSE (Figure 2(b)), C* is stationary and, since the mechanism of relative simultaneity does not apply, in the return trip, the photon covers at speed c the full distance ≃L in the time interval tret = L∕(γc) ≃ L∕c. Thus, for the special case D = D0 = vL∕c2, in the RLSE, the round-trip time is (T⇒)RLSE ≃ 2L∕c, which is greater than the corresponding one (T⇒)Sagnac ≃ 2L∕(c + v) of the linear Sagnac eﬀect.
As shown in ref. [7], in the case of the linear Sagnac eﬀect, the “time gap” δt ≃ 2vL∕c2, due to relative simultaneity between S′ and S″, corresponds to the length diﬀerence cδt ≃ 2vL∕c not covered by the photon when C* is on the upper section. Therefore, we ﬁnd that (T⇒)RLSE ≃ (T⇒)Sagnac + δt, showing that the diﬀerence can be related to relative simultaneity.
In general, in the linear Sagnac eﬀect, the round-trip time T of light signals is independent of D and the stationary contour shape, while in the RLSE, the round-trip time T depends on D and on the shape of the moving contour. Then, the RLSE represents an optical eﬀect not fully equivalent to the linear Sagnac eﬀect.
3 Experiment testing the S-RLSE

Appendix) of sizes L and H . With H ≃ 2R, where R is the radius of the pulley in the linear Sagnac eﬀect, the shape of this contour represents more realistically the RLSE.
For the rectangular contour of Figure A1, we assume that the contour slides continuously at the local speed v relative to C*. This is a mechanical diﬃculty that, in principle, can be surmounted, although it increases the cost of the experiment in comparison to the standard Sagnac experiments. Actually, for the experiment, the rectangular contour can move relative to C* for a short distance, starting with C* on the lower section L and ending on the left side H , as shown in Figure A1. For the size of the contour to be manageable, it is convenient to have light propagating in an optical ﬁber with a high refractive index n. In this case, the round-trip time is increased to T⇒ ≃ nP∕c.
In the case of light propagation in a medium, we have to distinguish the case when the medium is locally at rest along the contour from the case when the medium is locally at rest with C* and sliding along the contour. The speed of light in the moving medium is obtained from the standard relativistic velocity transformations. The results using the LT, derived in the Appendix (neglecting dispersion), are shown schematically below.
S-RLSE with medium locally at rest with the contour P = 2L + 2H

Here, we discuss the results obtained for the rectangular contour of Figure 4 (shown also in Figure A1 of the

nP T* = ,

ΔT * ≃ n2 v P ,

independent of D

c

cc

Figure 3: Eﬀect of relative simultaneity observed by C* in the linear Sagnac eﬀect. The two events “C* at A” and “photon at B” are simultaneous on frame S″ comoving with C* when on the contour lower section. After turning around the pulley in the negligible time interval η and changing velocity, C* starts comoving with frame S′ on the upper section at t′ = t″ = 0. However, due to relative simultaneity between S′ and S″, for observer C* on S′, the photon is no longer at B but is at K already.

Figure 4: Time diﬀerence ΔT *(D) for the counter-moving photon in the S-RLSE. For the Sagnac eﬀect, ΔT * =vP∕c2 is constant and represented by
the dotted line. For the S-RLSE, starting from right to left with C* on the contour lower section L from the position D > 2vL∕c2, we have ΔT * = vP∕c2 until D ≃ 2vL∕c. When D < 2vL∕c and until D = D0 = vL∕c, ΔT *(D) decreases up to vL∕c2. Then, ΔT * increases and reaches again the value vP∕c2 at D = 0 and afterwards when C* is on the left side H of
the moving contour. The broken line refers to the S-RLSE of Figure 2, where P = 2L and the dip is 2vL∕c2 with ΔT * = 0 at D = D0.

6  Gianfranco Spavieri and Espen Gaarder Haug

vnL D = D0 ≃ c2 ,

T⇒ ≃

nP c

−

n2 v P cc

+

vL c2

vnL D = D0 ≃ c2 ,

ΔT *(D0)

=

T*

−

T⇒ ≃

Δ∗Sagn

−

vL c2

(5)

2vnL D > c2 ,

T⇒ ≃

nP c

−

n2 v P cc

2vnL D > c2 ,

ΔT *(D) = T * −T⇒ ≃ Δ∗Sagn.

S-RLSE with medium locally

at rest with C*, P = 2L + 2H

nP T* = ,
c

Δ∗Sagn ≃

v c

P ,
c

independent of D

vnL

nP v P vL

D = D0 ≃ c2 , T⇒ ≃ c − c c + c2

vnL D = D0 ≃ c2 ,

ΔT *(D0)

=

T*

−

T⇒ ≃

Δ∗Sagn

−

vL c2

(6)

2vnL

nP v P

D > c2 ,

T⇒ ≃

c

− cc

2vnL D > c2 ,

ΔT *(D) = T * − T⇒ ≃ Δ∗Sagn.

For the S-RLSE with medium locally at rest with device C*, Figure 4 shows ΔT * = ΔT *(D) as a function of D. For the Sagnac eﬀect, ΔT * = ΔTS*agnac = Δ∗Sagn = vP∕c2 is constant. For the S-RLSE, ΔT *(D) is represented by the solid line that varies as a function of the initial position D of clock C*. For the S-RLSE, the LT foresee, the observable dip by vL∕c2 in the function ΔT *(D). The broken line refers to the S-RLSE of Figure 2, where P = 2L and the dip is 2vL∕c2 with ΔT * = 0 at D = D0.
Expected precision for the S-RLSE The variation ΔT *(D) may be measured with the techniques of ring interferometry or, after some adaptations, ring lasers [3]. Ring lasers Sagnac experiments, which can measure the angular velocity ωE of the Earth with a precision δω∕ωE < 10−8, are performed routinely on the Earth [14,15]. For the eﬀect of refractive index of an optical medium on the rotation frequency of ring lasers and resonators, there are various experimental expressions as functions of n, considered by Malykin in ref [16] (2014). In relation to the sensitivity of detectors measuring δτA and the smallest measurable time interval, there are techniques capable of resolving femtosecond (10−15 s) [17] or even attosecond (10−18 s) [18] pulses of laser light, although better limits may be obtained by means of advanced interferometry. The dip by vL∕c2, to be measured to conﬁrm the prediction of standard relativity based on the LT, is comparable to that measured in the usual experiments testing the Sagnac eﬀect and, thus, well within the range of the sensitivity of available detectors and not too diﬃcult to observe.

In the experiment by Wang et al. [2], the device C* is in motion relative to the contour. In the S-RLSE, the device C* is stationary and the contour in relative motion. Assuming the same relative velocity, we expect that the sensitivity achievable for our S-RLSE experiment is approximately the same as that achieved in the experiment by Wang et al. [2]. The main diﬀerence with respect to the experiment by Wang et al. is related to the mechanical diﬃculties involved with the motion of the contour, which implies more an increase in the complexity and cost of the experiment, rather than a loss in sensitivity.
Possible applications of the S-RLSE if conﬁrmed experimentally.
After more than a century, the Sagnac eﬀect is employed in current technology and is used in inertial guidance systems, ring laser gyroscope (extremely sensitive to rotations), and other optical systems. Being not yet experimentally conﬁrmed, it might be premature to indicate what kind of applications the RLSE might have. However, considering that the circular Sagnac eﬀect (sensitive to rotations) can detect very accurately the angular velocity of objects, such as the Earth, the RLSE (sensitive to velocity changes) might be suitable to detect velocity variations and also point out the corresponding direction. The realization of these applications requires, however, to take into account other features related to the RLSE that will be considered in a future contribution.
4 Conclusions
We have described an optical eﬀect, denoted as RLSE, where the emitter–receiver device is stationary and the contour along which light propagates is in motion. The RLSE provides the same results of the standard Sagnac eﬀect for the round-trip time diﬀerence ΔT of light signals counter-propagating along the closed contour. However, the RLSE possesses features that are not equivalent to the corresponding ones of the Sagnac eﬀect. When the contour changes velocity, standard special relativity foresees that the round-trip time interval T⇒ of a counter-moving light signal diﬀers from the corresponding one of the Sagnac eﬀect. The diﬀerence is observable with our S-RLSE experiment and given by the dip by vL∕c2 in the measured roundtrip time diﬀerence ΔT *(D) = T * − T⇒ of counter-propagating photons, plotted in Figure 4. The dip may be related to the diﬀerent role played by relative simultaneity in the linear Sagnac eﬀect and the RLSE.
If conﬁrmed experimentally, the S-RLSE stands to be a new relativistic optical eﬀect. In this case, being sensitive to velocity changes, the S-RLSE might have important

The reciprocal linear eﬀect, a new optical eﬀect of the Sagnac type  7

technological applications in inertial guidance systems by detecting velocity variations and corresponding direction.
In any event, the diﬀerent features between the standard linear Sagnac eﬀect and the S-RLSE can be exploited for testing Lorentz and light speed invariance.
Funding information: Our research has been supported by the CDCHTA of the Universidad de Los Andes, Mérida, Venezuela, and the ‘Braingain’ grant of the International Center for Theoretical Physics (ICTP), Trieste, Italy, for promoting teaching and researching in Venezuela.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conﬂict of interest: The authors state no conﬂict of interest.
References
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8  Gianfranco Spavieri and Espen Gaarder Haug

Appendix
A.1 Special reciprocal linear Sagnac eﬀect for light propagation in a medium of refracting index n on a rectangular contour of sides L and H
For the S-RLSE, the round-trip interval T⇒ diﬀers from the corresponding one of the linear Sagnac eﬀect if, on the contour lower section, to the ﬁrst order in v∕c, D ≃ nvL∕c and the photon reaches the right corner when the left corner reaches C*.
S-RLSE with medium locally at rest with the contour. P = 2L + 2H .
For the contour of Figure A1(a), we calculate with the LT to the ﬁrst order in v∕c the time intervals taken by the counter-propagating photon to cover the sides of the rectangular contour in motion with velocity v relative to the stationary C*.
Position 1: Starting from D = vt ≃ nvL∕c, and using the equation Cnt = L − D + vt with Cn the light speed relative to C*. Then, we have,

S-RLSE medium locally at rest with the device C*. P = 2L + 2H .
For the contour of Figure A1(b), we calculate with the LT to the ﬁrst order in v∕c the time intervals taken by the counter-propagating photon to cover the optical ﬁber sliding on the moving rectangular contour.
Position 1: Starting from D = vt, by means of the equation Cnt = L − D + vt with Cn the light speed relative to C* along the ﬁber at rest, we have,

c Cn = n

L nL

t1 = Cn =

. c

Position 2: From the equation Cnt = H − vt and with the optical ﬁber at speed 2v relative to C*,

c∕n − 2v Cn = 1 − 2v∕nc

H nH(1 − 2v∕nc) t2 = Cn + v ≃ c(1 − nv∕c)

nH n2vH 2vH

≃+ c

c2

− c2 .

c∕n + v Cn = 1 + v∕nc
L nL(1 + v∕nc) nL n2vL vL t1 = Cn = c(1 + nv∕c) ≃ c − c2 + c2 .

Position 2: By means of the equation Cnt = H − vt,

c∕n − v Cn = 1 − v∕nc

H nH(1 − v∕nc) nH vH

t2 = Cn + v ≃

c

≃ c − c2 .

Position 3: With the equation Cnt = L,

c Cn ≃ n

L nL

t3 = Cn ≃

. c

Position 4: By means of the equation, Cnt = H − v(t2 + t3),

c∕n + v Cn = 1 + v∕nc

nH (1 + v∕nc) n2vH n2vL t4 = c(1 + nv∕c) − c2 − c2

nH n2vH vH n2vH n2vL

=− c

c2

+ c2 −

c2

−

c2

T⇒ = t1 +

t2 + t3 +

t4 =

nP c

− n2 v P cc

+

vL c2 .

Figure A1: (a) Optical ﬁber of index n locally at rest on the rectangular contour moving with velocity v relative to the stationary clock C*. The photon starts from C* at the distance D from the contour left side H and returns to C* in the interval T⇒ after traveling along the contour sides at diﬀerent speeds Cn relative to C*. (b) Optical ﬁber of index n locally at rest with C* and sliding on the rectangular contour moving with velocity v relative to the stationary clock C*. In position 1, the ﬁber in the lower contour section is at rest relative to C*. In 2, the sliding ﬁber has speed 2v relative to C*. In 3, the relative speed is v and in 4, the ﬁber is at rest relative to C*.

The reciprocal linear eﬀect, a new optical eﬀect of the Sagnac type  9

Position 3: By means of the equation Cnt = L and with the optical ﬁber at speed v,

c∕n − v Cn ≃ 1 − v∕nc

L nL(1 − v∕nc) t3 = Cn = c(1 − nv∕c)

nL n2vL vL

≃+ c

c2

− c2 .

Position 4: From the equation Cnt = H − v(t2 + t3) and with the optical ﬁber at rest,

c nH n2vH n2vL Cn = n t4 = c − c2 − c2

T⇒ =

t1

+

t2

+

t3

+

t4

=

P c

⎛⎝n

−

v⎞ c⎠

+

vL c2 .

A.2 Interpreting optical eﬀects and the RLSE in the scenario of relativistic theories
Several authors [4,13,19–23] argue that the LT are physically equivalent to alternative coordinate transformations that diﬀer from the LT by an arbitrary clock synchronization parameter only. Thus, these authors claim that the one-way light speed is undetermined, or conventional, when measured by two spatially separated clocks that can be arbitrarily synchronized. In the framework of relativistic theories [13], the coordinate transformations alternative to the LT, more often used to describe physical phenomena, are denoted as LTs based on absolute simultaneity (LTA). The LT and the LTA are expressed as,
LT t′ = γ(t − vx∕c2) x′ = γ(x − vt) (A1)
LTA t′ = t∕γ x′ = γ(x − vt),
where in (A1)v indicates the velocity of frame S′ relative to frame S and the transformations y′ = y; z′ = z are understood. With the factor γ = (1 − v2∕c2)−1∕2 depending on v, LT stands for the LTs, based on standard synchrony and relative simultaneity. LTA stands for the LTs based on absolute synchrony and simultaneity. To the ﬁrst order in v∕c, the LTA coincides with the Galileo transformations. The LTA (or ALT in Ref. [9]) are known in literature also as the Tangherlini–Selleri transformations [6,19,20], used by several authors [4–9].
Because of the arbitrariness of synchronization and one-way light speed, many physicists consider the LT and LTA to be physically equivalent and, according to them, the LTA can be used to interpret all the experiments supporting standard special relativity [13,21–23]. Certainly, the equivalence should hold in the cases when the one-

way light speed is conventional because measured by two spatially separated clocks that are arbitrarily synchronized. However, in the Sagnac eﬀects, the one-way speed of light around the contour can be determined with the single clock C* and no arbitrary clock synchronization is involved. Thus, in relation to the diﬃculties pointed out in the interpretations of the Sagnac eﬀect [6–9,11], several authors [4–9] have shown that the diﬃculties are surmounted if, in lieu of the LT based on relative simultaneity, coordinate transformations based on conservation of simultaneity (LTA) are adopted.
Nevertheless, epistemologists [24] claim that the basic postulates of a meaningful physical theory must be testable (i.e., falsiﬁable). Then, if one of its basic postulates is not falsiﬁable, it may be argued by physicists and epistemologists [24] that the theory is not physically meaningful. If the LT (with relative simultaneity) are equivalent to the LTA (with absolute simultaneity) and the speed of light is conventional, the standard theory of special relativity has a drawback because its fundamental postulate of one-way light speed invariance cannot be tested [13,19–23]. For this reason, in the more recent formulation of special relativity found in mainstream physics journals where light speed is conventional [4], the constant c in the second postulate is no longer the one-way light speed, but “the round-trip speed of light (i.e., the average speed of light during the round-trip from A to B and then back to A),” which is observable.
Nevertheless, standard special relativity is based on the LT and, for determining the good standing of this theory and the LT, it is essential to establish whether the LTA and the LT are, or are not, physically equivalent.
It is worth mentioning that in practically all the experiments supporting special relativity what is being tested are eﬀects related to relativistic time dilation and length contraction, equally foreseen by the LT and the LTA [4–9,13]. Special relativity is a very well tested theory conﬁrmed by high precision experiments [13,25–28]. The experiments performed so far are likely suitable to test [13] the so-called relativistic eﬀects only, which are of second (or higher) order in v∕c and are not dedicated experiments capable of testing the special diﬀerent features of the LT and LTA, relative versus absolute simultaneity, because these diﬀerences vanish on average. In literature, there are proposals of dedicated experiments of ﬁrst order in v∕c that might test the diﬀerent features of simultaneity, relative or absolute (see, for example, [7,20]). However, up to now and as far as we know, no experiments of this type have been performed. Yet, as shown below, these diﬀerent features can be tested by means of the RLSE also.

10  Gianfranco Spavieri and Espen Gaarder Haug

A.2.1 The LTA and LT foresee diﬀerent results for the RLSE
In the interpretation of the optical eﬀects of the Sagnac type with the LTA, we are looking for the preferred frame where the one-way light speed can be assumed to be c. The natural choice is given by the contour inertial rest frame Sc where Maxwell’s equations are valid, and the electromagnetic waves, locked on the contour, propagate at speed c. With this choice of the preferred frame, from the contour inertial frame Sc, the LTA and LT provide an equivalent interpretation with the same results.
If, for the RLSE of Figure 2, we choose again the contour to be the preferred frame, then, we may assume that while moving with uniform velocity relative to C* the contour is carrying along the electromagnetic waves, always propagating at the local speed c relative to the contour. With the aforementioned assumptions, we ﬁnd that in the framework of the LTA, the RLSE is fully reciprocal to the linear Sagnac eﬀect. In fact, save for the negligible time interval η when the contour changes velocity, the observer Sc comoving with the contour of Figure 2 is on an inertial frame before and after the interval η. Then, as far as the RLSE is concerned, the result is independent of whether the relative change of motion is performed by C* or the contour. Since simultaneity is conserved with the LTA, when the contour changes velocity, there are no variations in the relative positions of photon and C*, unlike what happens with relative simultaneity in Figure 3. Therefore, if calculated from the contour frame Sc, the invariant proper time interval T⇒ is the same as in the linear Sagnac eﬀect. However, when calculated from clock frame SC* we have to take into account that with the LTA light speed is no longer invariant.
To verify that, with the LTA, the invariant interval T⇒ in the RLSE is independent of D and the same as in the linear Sagnac eﬀect, we calculate it from the clock frame SC* for the case when the contour changes velocity in the interval T⇒, as in Figure 2. Let the photon start from C* at D and travel toward point B moving to the right (Figure 2(a)). Relative to C*, by addition of velocities, the light speed along the moving contour is ≃c + v, where for simplicity we use the ﬁrst-order approximation in v∕c. In the out trip and at time t, point B is at the position L − D + vt to the right of C* and the photon reaches B when (c + v)t = L − D + vt. Considering ﬁrst the special case when simultaneity takes place, we set D = vt and, thus, t = tout = L∕(c + v) and D = vtout ≃ vL∕c. Then, the two events “photon at B” and “A at C*” are simultaneous in every frame. In the return trip on the upper section (Figure 2(b)), the contour is now moving to the left, while the photon travels on the upper section toward the stationary

C* at the speed ≃c + v. With the photon coming from B at L, the return time interval is tret = L∕(c + v) and the round-trip is T⇒ = tout + tret ≃ 2L∕(c + v), which is the same as in the linear Sagnac eﬀect shown in (2).
It can be shown that the same result, independent of D, is obtained in general, e.g., when the photon reaches B before A reaching C* (or when A reaches C* before the photon reaching B) and when C* is always on the lower (or upper) section of the contour in the round-trip interval T⇒.
In conclusion, with the LTA based on absolute simultaneity, the RLSE is fully equivalent to the linear Sagnac eﬀect and the relativity principle is holding. For the S-RLSE experiments described earlier, contrary to the LT, the LTA foresee that the variant (for the LT) function ΔT *(D) is (for the LTA) always ΔT * = ΔTS*agnac, independent of D. It follows that, by means of the S-RLSE, the diﬀerent predictions of the LT and LTA, and light speed invariance, can be tested.
Another example indicating that the LT and LTA are not equivalent, and cannot be arbitrarily interchanged in the description of physical phenomena, is given by the Thomas precession, discussed below.
A.2.2 Using the Thomas precession to discriminate relative (LT) from absolute (LTA) simultaneity
In the following, we consider the relativistic phenomenon of the Thomas precession [29] and ﬁnd that it is predicted by the LT but not by the LTA. Since the spacetime symmetry of the LT diﬀers from that of the LTA, it is not surprising that the two transformations predict diﬀerent results. Standard special relativity is routinely used in several areas of modern physics employing the symmetry of the LTs based on relative simultaneity and the Thomas precession derived from the LT is well known for predicting the correct spin–orbit interaction energy splitting [29,30] and is thus considered to be observable. Moreover, Thomas’ precession is included in the equation of spin motion of the BMT equation of elementary particle physics [31]. However, so far there are no direct tests of the Thomas precession.
Deriving the Thomas precession with the LT Here, we consider the textbook derivation made by Jackson [30] and indicate schematically the steps that lead to the Thomas precession. In deriving the spin precession for the case of spin–orbit interaction, Thomas considers an electron orbiting on a plane at the peripheral velocity v around the nucleus of an atom. Thomas shows that, because of the motion of the electron in its circular orbit with the acceleration a due to the Coulomb ﬁeld, the electron spin acquires an extra angular velocity ωT of

The reciprocal linear eﬀect, a new optical eﬀect of the Sagnac type  11

purely relativistic kinematical origin. The Thomas angular velocity ωT , corresponding to the rotation per unit of time of the electron frame, can be calculated from the inﬁnitesimal LT along the electron path (shown in Figure A2), indicating that successive transformations consist of a pure boost and a rotation.
To point out the origin of the rotation of the particle rest frame S″, in his approach to the Thomas precession, Jackson starts from the LT between the two frames S″ and S given by (Ref. [30], Section 11.19),

x′ = Aboost(β)x x″ = Aboost(β + δβ)x,

where β = v∕c and β + δβ = u∕c = v∕c + δβ, with the factor γu = γ + γ3βδβ1 to the ﬁrst order in δβ. The resulting inﬁnitesimal transformation between S′ and S″ is,

x″ = ALTx′,

(A3)

where the transformation matrix,

ALT = Aboost(β + δβ)Ab−o1ost(β),

(A4)

x0″ = γu(x0 − β ⋅ x) = γu(x0 − β1x1 − β2x2)

x1″ = x1

+

γu − β2

1 (β1x1

+

β2x2)β1

−

γβ1x0

(A2)

x2″ = x2 +

γu − β2

1

(β1x1

+

β2x2)β2 −

γβ2x0

x3″ = x3,

where, as shown in Figure A2, S″ is in motion with velocity
components ux = u1 and uy = u2 relative to the laboratory frame S and in (A2)γu = (1 − u2∕c2)−1∕2, u2 = u12 + u22, β1 = u1∕c, and β2 = u2∕c.
The connection between the two sets of rest frame coordinates, x′ at time t and x″ at time t + δt, corresponding to
the frames instantaneously co-moving with the particle in
its accelerated motion, are given by,

is,

⎛1

− γ2δβ1 − γδβ2 0⎞

⎜

⎜− γ2δβ1

1

ALT = ⎜ ⎜ −γδβ2 ⎜

γ−1 − β δβ2

γ−1

⎟

β

δβ2

0⎟ ⎟.

1 0⎟

⎟

(A5)

⎝0

0

0 1⎠

As shown by Jackson [30], the resulting inﬁnitesimal Lorentz transformation (A3) consists of a boost with velocity cΔβ and a rotation ΔΩ given by,

γ−1

ΔΩ = β2 β × δβ.

(A6)

The rotation is highlighted by the two antisymmetric

matrix elements (ALT)23 and (ALT)32 of ALT,

⎛⋯

⋯

⋯

⋯⎞

⎜⋯

ALT

=

⎜ ⎜⎜⋯

⋯ γ−1 − β δβ2

γ−1 β δβ2

⋯⎟ ⎟,

⋯

⋯⎟⎟

(A7)

⎝⋯

⋯

⋯ ⋯⎠

Figure A2: An object (electron) is in accelerated motion along a curvilinear path under the action of external forces. The moving frames S′ and S″ represent successive rest frames of the object. Starting from the origin of the laboratory frame S, the velocity of the rest frame S′ at time t is v(t) = cβ and at time t + δt the velocity of the rest frame S″ is v(t + δt) = cβ + cδβ. The resulting inﬁnitesimal LT from S′ to S″ consists of a boost and a rotation, indicating that the object acquires a precession at the Thomas angular velocity ωT of purely relativistic kinematical origin independent of other eﬀects.

indicating the existence of a rotation of the electron rest frame S″ by ΔΩ about the x3 axis. Because of the spacetime symmetry of the LT in (A2), the parameter time depends on space, t″ = t″(t, x) and t′ = t′(t, x). Therefore, through the matrix multiplication (A4), we can see that the time dependence on space, related to relative simultaneity, is essential for providing the resulting antisymmetric matrix elements (ALT)23 and (ALT)32 of ALT shown in (A7).
A direct consequence of expression (A6) is the resulting Thomas precession acquired by the object at the angular velocity,

ΔΩ γ2 a × v

ωT

=

− lim
δt→0

δt

= γ+1

c2

.

(A8)

Then, if the rotating object has a component of a perpendicular to v, the object acquires a Thomas precession of purely kinematical origin, independent of other eﬀects.

12  Gianfranco Spavieri and Espen Gaarder Haug

The LTA based on absolute simultaneity predict no Thomas precession
The LTs based on absolute simultaneity (LTA) between frame S and S″ are given by,

x0″ = γu−1x0

x1″

=

x1

+

γu − β2

1

(β1x1

+

β2x2)β1

−

γβ1x0

(A9)

x2″

=

x2

+

γu − β2

1

(β1x1

+

β2x2)β2

−

γβ2x0

x3″ = x3,

which are the same as the LT in (A2), save for the fact that the time parameter does not depend on space (absolute simultaneity) and is given by cτ″ = x0″ = γu−1x0, being τ″ the proper time on the particle rest frame S″.
Obviously, the spacetime symmetry of the LT is diﬀerent from that of the LTA and, therefore, for observables that depend on symmetry, diﬀerent predictions may be expected from the two diﬀerent transformations.
Following the procedure adopted earlier, we proceed by calculating the connection,

x″ = ALTA x′,

(A10)

using the LTA (instead of the LT) and derive the corresponding matrix ALTA (A14).
The transformations between frame S′ and S are,

x0′ = γ−1x0 ⇒ x0 = γx0′ x1′ = γ(x1 − βx0) ⇒ x1 = γ−1x1′ + γβx0′ x2′ = x2 ⇒ x2 = x2′ x3′ = x3 ⇒ x3 = x3′.

(A11)

After substituting x0 and x1 given by (A11) in expression (A9), the connection x″ = ALTA x′ of (A10) to the ﬁrst order in δβ is found to be,

x0″ = γu−1x0 = γu−1γx0′

x1″ = −γuβ1γx0′

+

⎛ ⎜1 ⎝

+

γu − β2

1

β1β1⎟⎞⎠(γ−1x1′

+

βγx0′ )

+

γu − β2

1

δβ2β1x2′

(A12)

x2″ = −γδβ2γx0′

+

γ

− β2

1

δβ2β1(γ−1x1′

+

βγx0′ )

+

x2′

x3″ = x3.

For the matrix elements (ALTA )23 and (ALTA )32, we may take into account that the velocity of S″ relative to S is β1 = β + δβ1 and β2 = δβ2 and γu ≃ γ to the ﬁrst order in δβ. Then, (A12) becomes,

x0″ = γu−1γx0′

x1″

= γ(β

+

γu − β2

1

ββ1β1

−

γuβ1)x0′

+

(1

+

γu − β2

1 β1β1 )γ−1x1′

+

γ− β

1 δβ2x2′

(A13)

γ−1

γ−1

x2″ = γ( β δβ2β1 − γδβ2)x0′ + γβ δβ2x1′ + x2′

x3″ = x3.

From the connection x″ = ALTA x′ given by Eq. (A13), we ﬁnd that the elements of the corresponding matrix ALTA relevant to our case are,

⎛⋯

⎜⋯

ALTA

=

⎜ ⎜⎜⋯

⎝⋯

⋯

⋯

⋯⎞

⋯ γ−1
γβ δβ2

γ−1 β δβ2 ⋯

⋯⎟ ⎟,
⋯⎟⎟

⋯

⋯ ⋯⎠

(A14)

indicating that both the matrix elements (ALTA )23 and (ALTA )32 are positive, not antisymmetric, and do not reﬂect the existence of a rotation about the x3 axis, as it is the case for the corresponding elements of the matrix ALT Eq. (A5) derived with the LT. In fact, the matrix elements (ALTA )23 and (ALTA )32 indicate a relative distortion of axes due to length contraction in the direction of motion, but no relative rotation about the x3 axis.
In conclusion, standard special relativity predicts the existence of the Thomas precession, which is related to the spacetime symmetry properties of the LT based on relative simultaneity. Instead, the LTA, based on conservation of simultaneity, predicts that the Thomas precession does not exist. With the examples of the Thomas precession, the RLSE, and other phenomena [6–9], we may infer that the LTA are not in general physically equivalent to the LT. Hence, the two transformations – and absolute and relative simultaneity – cannot be interchanged arbitrarily.

A.3 Calculating with the Lorentz transformations the intervals T⇐, T⇒ and ΔT = T⇐ ‒ T⇒ for the RLSE
A.3.1 Showing that, for D > 2vL∕c and C* on the same (upper or lower) contour section, in the RLSE the roundtrip time interval T⇒ for the counter-moving photon is independent of D and the same as in the standard linear Sagnac eﬀect
With reference to Figures 1(b) and 2(a), we assume that the counter-moving photon is emitted by C* when D > 2vL∕c,

The reciprocal linear eﬀect, a new optical eﬀect of the Sagnac type  13

so that C* is always on the lower track during the roundtrip interval T⇒. In this case, the contour is always moving at speed +v relative to C*. Then, the photon moving at speed c reaches point B when ct = (L − D)∕γ + vt, i.e., at,

L − D γ(L − D)(1 + v∕c)

t1 = γ(c − v) =

, c

where we have used the relation 1∕γ2 = (1 − v∕c)(1 + v∕c) = 1 − v2∕c2.
Moving now on the upper track, the photon travels from B toward A and reaches A when L∕γ − ct = vt after the interval,

L

γL(1 − v∕c)

t2 = γ(c + v) =

. c

After reaching A, the photon starts moving on the lower track to return to C*. Since point A has been moving at speed v toward C* during the interval t1 + t2, the position of A relative to C* is now D∕γ − v(t1 + t2) and the photon reaches C* when D∕γ − v(t1 + t2) = ct, after the interval,

Dv t3 = γc − c (t1 + t2)

=

D γc

−

vγ ⎛ (L c⎝

−

D)(1 c

+ v∕c)

+

L(1 − v∕c) ⎞ c⎠

γD(1 − v2∕c2) vγ 2L vγ D(1 + v∕c)

=

−

+

c

cc c c

γD(1 + v∕c) vγ 2L

=

−

.

c

cc

The resulting round-trip interval T⇒ is,

T⇒ = t1 + t2 + t3

γ(L − D)(1 + v∕c) γL(1 − v∕c)

=

+

c

c

γD(1 + v∕c) vγ 2L

+

−

c

cc

γL(1 + v∕c) γL(1 − v∕c) vγ 2L

=

+

−

c

c

cc

γD(1 + v∕c) γD(1 + v∕c)

−

+

c

c

2γL(1 − v∕c)

2L

=

=

,

c

γ(c + v)

(A15)

showing that, for D > 2vL∕c, in the RLSE T⇒, is D-independent and the same as that of the standard linear Sagnac eﬀect in Eq. (1). The same conclusion holds for the roundtrip T⇐ of the co-propagating photon.

A.3.2 Showing that in general, for the RLSE the round trips intervals T⇐ and T⇒ depend on D. For two counter-propagating photons, the standard RLSE provides the same D-independent result ΔT of the standard linear Sagnac eﬀect.

When during the round-trip interval T the device C* keeps

on the same track (lower or upper contour section), there is

no problem in deriving (as shown earlier) for ΔT = T⇐ − T⇒ the D-independent results (1)–(3), which are the same as

those that can be found in literature. The calculations

become quite complicated when the device C* passes,

e.g., from the lower to the upper track and we wish to

ﬁnd results in general. Since in the variant S-RLSE, the

round-trip T⇒ has its maximum when the photon covers the maximum length 2L∕γ (occurring when D0 = vL∕(γc)),

we consider here this special case and perform the calcula-

tions for deriving the round-trip T⇐ of the co-moving

photon and determine ΔT of (3). For the co-moving photon,

we have ﬁrst to ﬁnd its position at tout = L∕(γc) when point A reaches C* and the rod changes the direction of motion.

With reference to Figure A3(a), starting from C* and moving

to the left at speed c toward point A, the photon reaches A

when −D0 + vt = ct, i.e.,

tA =

D0 c+v

=

vL γc(c +

. v)

Figure A3: Relative positions of photon and clock C* during light copropagation in the round-trip interval T⇐.

After reaching A (Figure A3(b)), the photon moves to the upper track and, in the remaining interval, Δt = tout − tA propagates on the upper track toward point B while the rod is still moving to the right at speed v. Then, since A is moving at speed v, after the interval Δt, the position of the photon relative to A is,

14  Gianfranco Spavieri and Espen Gaarder Haug

(c − v)L

h = (c − v)Δt

=

(c

−

v)(tout −

tA) =

γ(c

+

, v)

as shown in Figure A3(c). With the photon at distance h from A and the device C*
coinciding with A and displaying the reading tout, the contour changes direction of motion. Then, from h, the photon reaches B when, h + ct = L∕γ − vt, i.e., after the interval,

L∕γ − h 1 ⎡ L (c − v)L ⎤ 2vL t2 = c + v = c + v ⎣⎢ γ − γ(c + v) ⎦⎥ = γ(c + v)2

2vγL(1 − v∕c)

=

,

c(c + v)

as indicated in Figure A3(d).

After reaching B on the upper track, the photon passes

to the lower track and moves toward A, reaching it when L∕γ − ct = −vt at,

L

t3

=

γ(c

−

, v)

as in shown Figure A3(e).

At this point, the photon at A goes to the upper track

and starts moving toward C*. However, in the interval

t2 + t3, point A has moved to the distance v(t2 + t3) to the left of C*. Then, from A, the photon reaches C* when

ct = v(t2 + t3) after the interval,

t4 =

v(t2 + c

t3)

=

vL(2v(c − v) + (c + v)2) γc(c + v)2(c − v)

vγL(c2 + 4vc − v2) = c(c + v)c2 .

After some algebra, the resulting round-trip time interval for the co-propagating photon is found to be,

T⇐ = tout + t2 + t3 + t4

L 2vγL(1 − v∕c)

L

=+

+

γc

c(c + v)

γ(c − v)

vγL(c2 + 4vc − v2)

+

c(c + v)c2 .

(A16)

The sum tout + t3 can be expressed as follows:

L

L

2L L

L

tout + t3 = γc + γ(c − v) = γc − γc + γ(c − v)

(A17)

2L

vL

2L γvL(1 + v∕c)

= γc + γc2(1 − v∕c) = γc +

c2

.

By substituting Eq. (A17) in Eq. (A16), we ﬁnd,

2L γvL(1 + v∕c)

T⇐ = γc +

c2

2vγL(1 − v∕c) vγL(c2 + 4vc − v2)

+

+

c(c + v)

c(c + v)c2

2L γvL(4c2 + 4cv) 2L 4γvL

=+ γc

c(c + v)c2

=+ γc

c2 ,

(A18)

and with T⇒ = 2L∕(γc), 4γvL
ΔT = T⇐ − T⇒ = c2
in agreement with the result of the standard linear Sagnac eﬀect.