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Heliyon 9 (2023) e19590
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Heliyon
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Research article
The One-Way Linear Eﬀect, a ﬁrst order optical eﬀect
Gianfranco Spavieri a, Espen Gaarder Haug b,∗
a Centro de Física Fundamental, Universidad de Los Andes, Mérida, 5101 Venezuela b Norwegian University of Life Sciences, Ås, Norway

ARTICLE INFO
Keywords: Light propagation Sagnac eﬀect One-way speed of light Relative simultaneity Lorentz invariance

A B S T R A C T
In the One-Way Linear Eﬀect, the round-trip time interval 𝑇0′ taken by light to propagate around a moving linear closed contour, is measured by a device. If the contour changes velocity by Δ𝑣, the round-trip interval 𝑇 ′(𝑋′, Δ𝑣) turns out to depend on the device position 𝑋′ and Δ𝑣. The variation Δ𝑇 ′ = 𝑇 ′(𝑋′, Δ𝑣) − 𝑇0′, related to Δ𝑣∕𝑐, is experimentally observable by means of standard interferometry, ring laser techniques, or high precision time-delay detectors of light pulses. Being sensitive to velocity variations, if experimentally conﬁrmed the One-Way Linear Eﬀect may have relevant applications in inertial guidance systems and related areas. Furthermore, the One-Way Linear Eﬀect can be used to conﬁrm Lorentz invariance by testing relative simultaneity versus absolute simultaneity.

1. Introduction

Among the relevant optical experiments historically related to light speed invariance, let’s mention the following. The MichelsonMorley experiment [1] of 1887 provided a surprising null result that gave support to the theory of special relativity (SR) proposed by Einstein in 1905 [2] assuming light speed invariance and the Lorentz transformations (LT). The more sensitive Michelson-Gale experiment of 1925 [3], performed with an interferometer ﬁxed on Earth, provided a non-null result. The standard Sagnac experiment [4], shown in Fig. 1-a, was carried out in 1913. The linear Sagnac experiment, shown in Fig. 1-b, was performed by Wang et al. [5], [6] in 2003. The linear Sagnac eﬀect is considered to be equivalent to the standard Sagnac eﬀect. In the Sagnac eﬀects the emitter-receiver C* (clock or interferometer) is moving relative to a ﬁxed contour where two light signals are counter-propagating and C* measures the diﬀerence Δ𝑇 = 𝑇⟸ − 𝑇⟹ of their round-trip times 𝑇⟸ and 𝑇⟹.
Recently, Spavieri and Haug [7] have proposed the Reciprocal Linear Sagnac Eﬀect, shown in Fig. 1-c, where the measuring device C* is stationary and the contour is in relative motion. In the reciprocal eﬀect, the observable Δ𝑇 is foreseen to be the same as in the linear Sagnac eﬀect. There are, however, observable features of the linear Sagnac eﬀect that diﬀer from those of the reciprocal eﬀect [7].
Following Post [8], for the Sagnac eﬀects the diﬀerence Δ𝑇 is given by,

Δ𝑇

= 𝑇⟸ − 𝑇⟹ =

2𝜋𝑟 𝛾(𝑐 − 𝑣)

−

2𝜋𝑟 𝛾(𝑐 + 𝑣)

=

4𝛾 𝑣𝜋 𝑟 𝑐2

=

2𝛾 𝑣𝑃 𝑐2

=

4𝜔⋅𝐀 𝑐2

(1)

Δ𝑇

=

𝑇⟸

−

𝑇⟹

=

2𝐿 𝛾(𝑐 −

𝑣)

−

2𝐿 𝛾(𝑐 + 𝑣)

=

4𝛾 𝑣𝐿 𝑐2

=

2𝛾 𝑣𝑃 𝑐2

,

* Corresponding author. E-mail addresses: gspavieri@gmail.com (G. Spavieri), espenhaug@mac.com (E.G. Haug).

https://doi.org/10.1016/j.heliyon.2023.e19590 Received 23 March 2023; Received in revised form 25 August 2023; Accepted 28 August 2023

Available online 1 September 2023 2405-8440/© 2023 The Author(s). Published by Elsevier Ltd. (http://creativecommons.org/licenses/by/4.0/).

This is an open access article under the CC BY license

G. Spavieri and E.G. Haug

Heliyon 9 (2023) e19590

Fig. 1. a) In the circular Sagnac eﬀect, two counter-propagating photons (only a single photon is shown) are emitted from the device C* and travel on the circumference of the rotating platform. C* measures the diﬀerence Δ𝑇 of the photons’ arrival times after a round trip. b) In the linear Sagnac eﬀect, device C* is moving with velocity 𝑣 relative to the stationary frame AB and emits counter-propagating photons traveling in an optical ﬁber that slides frictionless around pulley A and B. c) In the reciprocal Sagnac eﬀect, the frame AB of the optical ﬁber moves with velocity 𝑣 relative to the stationary device C* that emits photons counter-propagating along the optical ﬁber.
where 𝑇⟸ and 𝑇⟹ represent the round-trip time of the co- and counter-moving light signals (or photons) along the contour of perimeter 𝑃 = 2𝜋𝑟 in the standard (or circular) Sagnac eﬀect and 𝑃 = 2𝐿 in the linear eﬀect respectively. For the circular Sagnac eﬀect, with 𝑣 = 𝜔𝑟, the result (1) is usually expressed [8] in terms of the angular velocity 𝜔 and the area 𝐀 enclosed by the light path.
Result (1) can be applied to the Michelson-Gale experiment, usually interpreted [9], [10] as a circular Sagnac eﬀect, where the emitter-receiver device (or clock C*) is ﬁxed on the frame of the interferometer where light propagates, as in the case of the Michelson-Morley experiment. However, in the Michelson-Gale experiment, the light path has the form of a rectangle of considerable area and, being placed on the surface of the rotating Earth, it rotates relative to the Earth Centered Inertial (ECI) frame. Hence, being analogous to the circular Sagnac eﬀect, the Michelson-Gale experiment can measure the angular velocity 𝜔𝐸 of the Earth [10]. At present, experiments based on the Sagnac eﬀect using ring laser techniques are performed routinely on Earth [11], [12] to measure 𝜔𝐸 with a precision 𝛿𝜔∕𝜔𝐸 < 10−8. Therefore, the Michelson-Gale experiment can be considered to be equivalent to a circular Sagnac experiment and its non-null result is due to the fact that the system composed of C* and the rectangular light path ﬁxed on the Earth’s surface is rotating relative to the ECI frame, where the speed of light is assumed to be 𝑐.
In the Michelson-Gale and the equivalent circular Sagnac experiment, the velocity 𝑣 of C* is uniform and the two experiments can detect velocity variations due to rotation. It is thus feasible that there is an experimental set up, here denoted as One-Way Linear Eﬀect based on the LT and light speed invariance, capable of detecting linear velocity variations. The purpose of our Letter is to show how the One-Way Linear Eﬀect can be theoretically veriﬁed. A realistic test of the One-Way Linear Eﬀect, possible with present technology, is described.
The Sagnac eﬀect ﬁnds important applications in inertial guidance systems extremely sensitive to rotations, such as ring laser gyroscope and other optical systems. It is premature to indicate the applications that the One-Way Linear Eﬀect can have. Nevertheless, what is important is that the techniques involved in this type of eﬀects are very sensitive. Then, as well as in the case of the reciprocal linear Sagnac eﬀect ([7]), potential uses of the One-Way Linear Eﬀect might be the detection of movements of seismic origin (and corresponding direction) or velocity variations of aircraft due to air turbulence.
2. Round-trip time interval of a particle moving along a linear contour
The circular and linear Sagnac eﬀects are usually described in the inertial frame of reference of the stationary contour, where the speed of light is assumed to be 𝑐. Interpretations of the Sagnac eﬀects in the frame comoving with device C* are given by several authors [13–28].
In the linear Sagnac eﬀect of Fig. 1-b the arm AB of the contour is stationary and clock 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 direction of motion at the pulley A (or B). Locally, the speed 𝑣 of C* relative to the stationary contour is always constant. In the linear experiment by Wang et al. [5], device C* stays on the lower contour section in uniform rectilinear motion during the round-trip time 𝑇 ≃ 𝑇⟸ ≃ 𝑇⟹ and therefore C* does not turn around the pulleys A or B. Nevertheless, since the relative speed 𝑣 between C* and contour is constant, the theoretical prediction (1) for 𝑇⟸, 𝑇⟹, and Δ𝑇 is maintained, even when C* turns around the pulley during the round-trip time 𝑇 .
In the reciprocal linear Sagnac eﬀect of Fig. 1-c, the device C* is stationary and the contour is moving relative to C*. For this eﬀect, as long as C* is always on the lower (or upper) track of the contour during the round-trip interval 𝑇 , the predictions for Δ𝑇
2

G. Spavieri and E.G. Haug

Heliyon 9 (2023) e19590

are the same as those of Fig. 1-b. However, the one-way predictions for 𝑇⟸ and 𝑇⟹ are diﬀerent when C* is ﬁrst on the lower and then on the upper track during the interval 𝑇 ([7]).
Previous works [18–20] discuss the role of simultaneity in special relativity in the context of the circular and linear Sagnac eﬀects and other special cases [21]. In these previous works, the device C* is moving relative to the contour where light propagates, but in the present One-Way Linear Eﬀect the device C* is always ﬁxed to and co-moving with the contour, while the contour velocity may be varying relative to an inertial frame. We show in Section 3 that, if the velocity of the contour changes by Δ𝑣, the relevant feature of the One-Way Linear Eﬀect foreseen by the LT is that the one-way round-trip time interval 𝑇 ′ measured by C* depends not only on Δ𝑣, but also on the position 𝑋′ of C* on the contour, so that: 𝑇 ′ = 𝑇 ′(𝑋′, Δ𝑣). This special feature is not present in the physical eﬀects discussed in previous works [18–22] or other works related to the interpretation of the Sagnac eﬀect.
A signiﬁcant characteristic of the One-Way Linear Eﬀect is that it represents an optical eﬀect of the Sagnac type capable of detecting velocity variations. If its existence is conﬁrmed experimentally, the One-Way Linear Eﬀect may likely ﬁnd important useful technological applications in inertial guidance systems and related areas, as has been the case for the Sagnac eﬀect.
Another important aspect of the One-Way Linear Eﬀect is that its special feature is foreseen by the LT based on relative simultaneity. Yet, as shown in the Appendix, this special feature is not predicted by theories based on absolute simultaneity. This fact suggests that, by exploiting the One-Way Linear Eﬀect, it is possible in principle to test relative simultaneity versus absolute simultaneity. Tests discriminating relative from absolute simultaneity are uncommon in literature, because in the context of relativistic theories it is generally assumed that synchronization of spatially separated clocks is arbitrary and relative and absolute simultaneity are not observable [29]. However, in our eﬀect, the measuring device is only the single clock C*, which does not require synchronization. Thus, the experiment on the One-Way Linear Eﬀect described in Section 3 appears feasible for testing Lorentz invariance. Being capable of discriminating relative from absolute simultaneity, the One-Way Linear Eﬀect represents a deﬁnite advance for the comprehension of the role of simultaneity in relativistic theories.
In the One-Way Linear Eﬀect experiment described in the next section we focus on the one-way prediction for 𝑇 ′, as foreseen by the LT.

2.1. Contour in uniform motion

Let us consider the linear contour of Fig. 2-a co-moving with frame 𝑆′ at speed 𝑣 relative to the lab frame 𝑆. The device C* is
ﬁxed to the contour at the end point A. A particle is moving on the contour upper section from A to B in the interval 𝑡𝑜𝑢𝑡 in the out trip, and back on the lower section from B to A in the interval 𝑡𝑟𝑒𝑡 in the return trip, traveling at the local speed 𝑢′ = 𝑢0 relative to the contour rest length AB = 𝐿0. If the contour consists of an optical ﬁber of refractive index 𝑛, the particle can be a photon traveling at speed 𝑢0 = 𝑐∕𝑛. The results derived are valid also for the case, to be discussed elsewhere, of a curved contour forming a circular arc
AB moving along the circumference.
In agreement with the relativistic velocity transformations, the particle speed in frame 𝑆 is 𝑢𝑜𝑢𝑡 = (𝑢0 + 𝑣)∕(1 + 𝑢0𝑣∕𝑐2) in the out trip and 𝑢𝑟𝑒𝑡 = (𝑢0 − 𝑣)∕(1 − 𝑢0𝑣∕𝑐2) in the return trip. With,

𝑢𝑜𝑢𝑡

−𝑣

=

𝑢0 𝛾2(1 + 𝑢0𝑣∕𝑐2)

𝑢𝑟𝑒𝑡

+

𝑣

=

𝛾 2 (1

𝑢0

,

− 𝑢0𝑣∕𝑐2)

the resulting time intervals are,

𝑡𝑜𝑢𝑡

=

𝐿0 𝛾(𝑢𝑜𝑢𝑡 − 𝑣)

=

𝛾𝐿0(1 + 𝑢0𝑣∕𝑐2) 𝑢0

→

𝛾𝐿0(1 + 𝑣∕𝑐) 𝑐

(2)

𝑡𝑟𝑒𝑡

=

𝐿0 𝛾(𝑢𝑟𝑒𝑡 + 𝑣)

=

𝛾𝐿0(1 − 𝑢0𝑣∕𝑐2) 𝑢0

→ 𝛾𝐿0(1 − 𝑣∕𝑐) 𝑐

𝑇

= 𝑡𝑜𝑢𝑡 + 𝑡𝑟𝑒𝑡 =

2𝛾 𝐿0 𝑢0

→

2𝛾 𝐿0 𝑐

𝑇

′

=

𝜏′

=

𝑡′𝑜𝑢𝑡

+

𝑡′𝑟𝑒𝑡

=

𝑇 𝛾

=

2𝐿0 𝑢0

→

2𝐿0 𝑐

,

where in (2) 𝑇 ′ = 𝜏′ represents the round-trip proper time interval measured by C*. The expressions to the right in (2) after the arrow

→ represent the results in free space when 𝑢0 = 𝑐. Since C* is ﬁxed to and co-moving with the contour, the round-trip interval 𝑇 calculated from frame 𝑆 is the same whether

the particle travels one way clockwise or anti-clockwise along the contour starting from C* and returning to C*. Results (2) are

independent of the position of C* relative to A, as long as the contour is in uniform motion during the interval 𝑇 .

3. Contour in variable motion

If the velocity of the contour changes by Δ𝑣, the interval 𝑇 ′ = 𝑇 ′(𝑋′, Δ𝑣) depends on the initial distance 𝑋′ of C* from point A and on Δ𝑣.
The contour of rest length 2𝐿0 shown in Figs. 2-b is in motion relative to the inertial frame 𝑆. A particle, emitted from C* located at the distance 𝑋′ from A, travels (Fig. 2-b1) along the contour at the local speed 𝑢0 and reaches point B where is reﬂected. When the particle has covered the distance 𝐿0 from C*, it reaches point Z (Fig. 2-b2), being the distance of Z from B: ZB = 𝑋′. At this moment
3

G. Spavieri and E.G. Haug

Heliyon 9 (2023) e19590

Fig. 2. a) The contour AB is moving at speed 𝑣 relative to frame 𝑆. After being emitted by C* at A, the particle moves always at the local speed 𝑢0 relative to the contour. When reaching point B, the particle changes direction and returns to C*, while the contour keeps moving at speed 𝑣. b-1) C*, placed on the contour upper
section at the distance 𝑋′ from A, emits a particle that, traveling at the local speed 𝑢0, reaches B. b-2) The particle changes direction at B and moves on the lower section toward point Z. b-3) When the particle has covered the distance 𝐿0 from C* and passes by Z, the contour changes speed from 𝑣 to 𝑣𝑓 . When reaching A, the particle changes direction and returns to C*.

the contour velocity is slowed by Δ𝑣 to 𝑣𝑓 = 𝑣 − Δ𝑣 in the negligible interval 𝜂 (Fig. 2-b3) and the particle travels from Z to A, where it is reﬂected, and completes its round trip by reaching C* in the time interval 𝑇 .
For simplicity’s sake, let’s consider the ideal case where 𝜂 is negligible in comparison to 𝑇 and, relative to 𝑆′, the particle travels
the short ﬁnite distance 𝜂𝑢0 ≪ 2𝑣𝐿0∕𝑐 during the interval 𝜂. With this assumption, we may omit taking into account the details of the process when the contour changes velocity.
The particle is moving along the contour performing what we deﬁne as a one-way clockwise round trip starting from C*. A
two-way trip consists of a clockwise trip from C* and back to C*, plus a counter-clockwise trip from C* and back to C*.
In the trip from C* to Z, the contour has uniform speed 𝑣. According to the LT, relative to frame 𝑆 the speed of the particle is 𝑢+ = (𝑢0 + 𝑣)∕(1 + 𝑢0𝑣∕𝑐2) while traveling from C* to B. The time interval 𝑡1 from C* to B is obtained from the equation 𝑢+𝑡 = (𝐿0 − 𝑋′)∕𝛾 + 𝑣𝑡. While traveling from B to Z the particle speed is 𝑢− = (𝑢0 − 𝑣)∕(1 − 𝑢0𝑣∕𝑐2) and the time interval 𝑡2 from B to Z is obtained from the equation 𝑋′∕𝛾 − 𝑢−𝑡 = 𝑣𝑡. With

𝑢+ − 𝑣 =

𝑢0

𝛾2(1 + 𝑢0𝑣∕𝑐2)

we ﬁnd,

𝑢− + 𝑣 =

𝑢0

,

𝛾2(1 − 𝑢0𝑣∕𝑐2)

𝑡1

=

𝐿0 − 𝑋′ 𝛾(𝑢+ − 𝑣)

=

𝛾 (𝐿0

−

𝑋′)(1 𝑢0

+

𝑢0𝑣∕𝑐2)

→

𝛾(𝐿0 − 𝑐

𝑋′)

+

𝛾𝑣(𝐿0 − 𝑐2

𝑋′)

(3)

𝑡2

=

𝑋′ 𝛾(𝑢− + 𝑣)

=

𝛾𝑋′(1 − 𝑢0𝑣∕𝑐2) 𝑢0

→

𝛾𝑋′ 𝑐

−

𝛾 𝑣𝑋 ′ 𝑐2

𝑡1

+ 𝑡2

=

𝛾 𝐿0 𝑢0

+

𝛾 𝑣𝐿0 𝑐2

−

2𝛾 𝑣𝑋 ′ 𝑐2

→

𝛾 𝐿0 𝑐

+

𝛾 𝑣𝐿0 𝑐2

−

2𝛾 𝑣𝑋 ′ 𝑐2

Since in the short interval 𝜂 the contour has decelerated to speed 𝑣𝑓 = 𝑣 − Δ𝑣, from Z to A the return speed of the particle is 𝑢−𝑓 = (𝑢0 − 𝑣𝑓 )∕(1 − 𝑢0𝑣𝑓 ∕𝑐2) and the corresponding time interval 𝑡3 is obtained from the equation (𝐿0 − 𝑋′)∕𝛾𝑓 − 𝑢−𝑓 𝑡 = 𝑣𝑓 𝑡. For the trip
from A to C* the speed is, 𝑢+𝑓 = (𝑢0 + 𝑣𝑓 )∕(1 + 𝑢0𝑣𝑓 ∕𝑐2) and for the corresponding interval 𝑡4 the equation is 𝑢+𝑓 𝑡 = 𝑋′∕𝛾𝑓 + 𝑣𝑓 𝑡. With,

𝑢−𝑓

+

𝑣𝑓

=

𝑢0 𝛾2(1 − 𝑢0𝑣𝑓 ∕𝑐2)

𝑢+𝑓

− 𝑣𝑓

=

𝛾 2 (1

𝑢0

,

+ 𝑢0𝑣𝑓 ∕𝑐2)

we ﬁnd,

𝑡3

=

𝐿0 − 𝑋′ 𝛾𝑓 (𝑢−𝑓 + 𝑣𝑓 )

=

𝛾𝑓 (𝐿0

−

𝑋′)(1 − 𝑢0

𝑢0𝑣𝑓 ∕𝑐2)

→

𝛾𝑓 (𝐿0 − 𝑋′) 𝑐

−

𝛾𝑓 𝑣(𝐿0 − 𝑐2

𝑋′)

(4)

𝑡4

=

𝑋′ 𝛾𝑓 (𝑢+𝑓 − 𝑣𝑓 )

=

𝛾𝑓 𝑋′(1 + 𝑢0𝑣𝑓 ∕𝑐2) 𝑢0

→

𝛾𝑓 𝑋′ 𝑐

+

𝛾𝑓 𝑣𝑋′ 𝑐2

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G. Spavieri and E.G. Haug

Heliyon 9 (2023) e19590

𝑡3

+ 𝑡4

=

𝛾𝑓 𝐿0 𝑢0

−

𝛾𝑣𝑓 𝐿0 𝑐2

+

2𝛾𝑣𝑓 𝑋′ 𝑐2

The round-trip proper time interval 𝑇 ′ = 𝜏′ measured by C* can be derived from the time transformation of the LT, which is

𝑡′ = 𝛾(𝑡 − 𝑣𝑥∕𝑐2) when the contour speed is 𝑣 relative to 𝑆. In the interval Δ𝑡 clock C* moves by Δ𝑥 = 𝑣Δ𝑡 and the proper time

interval is Δ𝜏′ = Δ𝑡′ = 𝛾Δ𝑡(1 − 𝑣2∕𝑐2) = Δ𝑡∕𝛾. Alternatively, we may use the inverse transformation, 𝑡 = 𝛾(𝑡′ + 𝑣𝑥′∕𝑐2) and consider the

variation, Δ𝑡 = 𝛾(Δ𝑡′ + 𝑣Δ𝑥′∕𝑐2). Since Δ𝑡′ = Δ𝜏′ is measured by C* at the same position (Δ𝑥′ = 0), we ﬁnd again, Δ𝜏′ = Δ𝑡′ = Δ𝑡∕𝛾

(i.e., 𝜏′ = 𝑇 ∕𝛾). After the time interval Δ𝑡 = 𝑡1 + 𝑡2, when the particle has reached Z, the contour and the device C* change velocity. Then, for the successive time interval Δ𝑡𝑓 = 𝑡3 + 𝑡4, C* is moving at speed 𝑣𝑓 and now, Δ𝑡′𝑓 = Δ𝑡𝑓 ∕𝛾𝑓 . Hence, with Δ𝑣 = 𝑣 − 𝑣𝑓 relations (3) and (4) give,

𝑇 ′(𝑋′, Δ𝑣) = Δ𝑡′

+ Δ𝑡′𝑓

=

𝐿0 𝑢0

+

𝑣𝐿0 𝑐2

−

2𝑣𝑋′ 𝑐2

+

𝐿0 𝑢0

−

𝑣𝑓 𝐿0 𝑐2

+

2𝑣𝑓 𝑋′ 𝑐2

(5)

=

2𝐿0 𝑢0

+

Δ𝑣 𝑐

𝐿0 𝑐

−

2Δ𝑣 𝑐

𝑋′ 𝑐

→

2𝐿0 𝑐

+

Δ𝑣 𝑐

𝐿0 𝑐

−

2Δ𝑣 𝑐

𝑋′ 𝑐

.

We can see from (5) that, besides depending on 𝑋′, the interval 𝑇 ′(𝑋′, Δ𝑣) depends on Δ𝑣 and is sensitive to velocity variations of

the ﬁrst order in Δ𝑣∕𝑐.

There is a remarkable diﬀerence between 𝑇 ′ in (2) and 𝑇 ′(𝑋′, Δ𝑣) in (5). As seen from an observer co-moving with the contour

when in uniform motion, result (2) indicates that the particle, traveling at the local speed 𝑢0, in the interval 𝑇 ′ covers the ground distance 2𝐿0 as expected. However, if the contour changes velocity by Δ𝑣, result 𝑇 ′(𝑋′, Δ𝑣) in (5) indicates that the particle covers the diﬀerent ground distance 2𝐿0 + 2𝑢0Δ𝑣(𝐿0 − 2𝑋′)∕𝑐2 → 2𝐿0 + 2Δ𝑣(𝐿0 − 2𝑋′)∕𝑐.
Results (5) have been derived in the inertial frame 𝑆 where the speed of light is assumed to be 𝑐 and using the velocity transfor-

mations provided by the LT. The same results (5) can be derived from the inertial frame 𝑆′, 𝑆′′, or any other inertial frame taking

into account the eﬀect due to relative simultaneity, as considered in the Appendix.

Testing the one-way eﬀect

In the case considered above, the round-trip interval 𝑇 ′ = 𝑇⟸ ′ for a given velocity variation Δ𝑣, refers to a one-way (clockwise) co-moving particle traveling along the contour. In the Sagnac eﬀect the interval 𝑇⟹ ′ for a counter-moving particle diﬀers from 𝑇⟸ ′ and Δ𝑇 ′ = 𝑇⟸ ′ − 𝑇⟹ ′ ≠ 0. In our case, however, since C* is ﬁxed on the contour, the interval 𝑇⟹ ′ (𝑋′, Δ𝑣) for a counter-moving particle is the same as in (5) for the co-moving particle. Then, for the usual case of counter-propagating particles we have the null

result, Δ𝑇 ′ = 𝑇⟸ ′ (𝑋′, Δ𝑣) − 𝑇⟹ ′ (𝑋′, Δ𝑣) = 0 and, in our case, the dependence on 𝑋′ and Δ𝑣 of the one-way interval 𝑇⟸ ′ (𝑋′, Δ𝑣) cannot be revealed.

However, the one-way round-trip interval 𝑇 ′(𝑋′, Δ𝑣) can be compared with the round-trip time interval 𝑇0′ = 2𝐿0∕𝑢0 taken by a particle to propagate along an optical ﬁber of length 2𝐿0 forming a coil that can be placed at the position of C* and co-moving
with it. Let 𝑑0 represent the dimension of the coil where the spiraling particle moves back and forth. According to (2), for uniform motion the time interval corresponding to each turn in the coil is 𝜏𝑝′𝑎𝑟𝑡𝑖𝑎𝑙 = 2𝑑0∕𝑢0, independent of velocity. Therefore, during the time interval when the coil is moving at the uniform speed 𝑣 relative to 𝑆, by adding up the partial intervals 𝜏𝑝′𝑎𝑟𝑡𝑖𝑎𝑙 until the length 𝐿0 is covered, the expected proper time interval is 𝑇0′∕2 = 𝐿0∕𝑢0. Again, during the time interval when the coil is moving at the uniform speed 𝑣𝑓 relative to 𝑆, we have 𝑇0′∕2 = 𝐿0∕𝑢0. Then, the total length 2𝐿0 of the coil is covered after the round-trip proper time interval 𝑇0′ = 2𝐿0∕𝑢0, which is the same as in the case of a stationary coil. Hence, the device C* can measure the non-null diﬀerence,

Δ𝑇 ′(𝑋′, Δ𝑣) = 𝑇 ′(𝑋′, Δ𝑣) − 𝑇0′

=

Δ𝑣𝐿0 𝑐2

−

2Δ𝑣𝑋′ . 𝑐2

(6)

The plot of Δ𝑇 ′(𝑋′, Δ𝑣) as a function of the position 𝑋′ of C* is given in Fig. 3, where we assume that the contour changes

direction of motion with velocity variation Δ𝑣 = 2𝑣.

For the one-way eﬀect described above, we have made some assumptions in order to simplify calculations, such as that the interval

𝜂 is negligible and the velocity change occurs when the particle covers the distance 𝐿0. However, in principle the calculations can be performed in general and, thus, we expect to have 𝑇 ′ = 𝑇 ′(𝑋′, Δ𝑣, 𝜂, 𝐿, ...) depending on the various parameters that can aﬀect the

result.

Expected precision of the measuring device.

For testing the existence of the optical eﬀect it is suﬃcient to perform, for example, measurements at 𝑋′ = 0 and 𝑋′ = 𝐿0∕2. The measurable diﬀerence Δ𝑇 ′ = Δ𝑇 ′(𝑋′, Δ𝑣) in (6) represents the prediction of standard special relativity based on the LT. Δ𝑇 ′ may

be measured with standard interferometry or by means of ring laser techniques [8], [30], [31]. With standard interferometry there

could be some problems of stability because of the change of speed of the contour. Alternatively, detectors that act as clocks may

be used to measure the time of ﬂight of light pulses. For detectors capable of measuring Δ𝑇 ′, there are techniques that may resolve

femtosecond (10−15(𝑠)) [32] or even attosecond (10−18(𝑠)) [33], [34] pulses of laser light. The value of Δ𝑇 ′ is comparable to that of

the standard quantities measured in the Sagnac eﬀects, which are within the range of the sensitivity of existing detectors. The main

challenge for this type of experiment is probably the one related to the mechanical diﬃculties involving the motion of the contour,

which requires speed variations for performing the test.

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Heliyon 9 (2023) e19590

Fig. 3. Special case Δ𝑣 = 2𝑣. The diﬀerence Δ𝑇 ′(𝑋′) = 𝑇 ′(𝑋′) − 𝑇0′ as a function of the position of device C* along the contour. The maximum variation occurs at 𝑋 = 0 and 𝑋 = 𝐿0. When C* is at 𝑥′ = 𝐿0∕2, the round-trip interval is 𝑇 ′(𝑋′) = 𝑇0′ = 2𝐿0∕𝑐 and Δ𝑇 ′(𝐿0∕2) = 0.

4. Conclusions

We have shown that, for our One-Way Linear Eﬀect, when the contour changes velocity, the LT foresee the 𝑋′, Δ𝑣-dependent round-trip time interval 𝑇 ′(𝑋′, Δ𝑣) for the particle, or light, propagation. When 𝑇 ′(𝑋′, Δ𝑣) is compared with the round-trip interval 𝑇0′ = 2𝐿0∕𝑐 corresponding to a stationary contour, we ﬁnd that the LT foresee the observable non-null variation Δ𝑇 ′(𝑋′, Δ𝑣) = 𝑇 ′(𝑋′, Δ𝑣) − 𝑇0′ in (6). The diﬀerence Δ𝑇 ′(𝑋′) is plotted in Fig. 4 for the special case Δ𝑣 = 2𝑣 and its maximum is 2𝑣𝐿0∕𝑐2, experimentally observable with present technology.
As in the case of the Sagnac eﬀect, which after more than a century has found important uses in the development of technology,
it is feasible that, if experimentally conﬁrmed, the One-Way Linear Eﬀect can ﬁnd similar important applications in inertial guidance
systems or other related areas.

CRediT authorship contribution statement

Gianfranco Spavieri and Espen Gaarder Haug: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Declaration of competing interest

The authors declare that they have no known competing ﬁnancial interests or personal relationships that could have appeared to inﬂuence the work reported in this paper.

Data availability statement

No data was used for the research described in the article.

Acknowledgements

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 research in Venezuela.

Appendix A

If we wish to derive results (5) from an inertial frame other than frame 𝑆, we have to take into account that, since simultaneity is not conserved with the LT, the contour ends A and B change direction of motion simultaneously in frame 𝑆, but not in the contour
or other frames. Within the assumption 𝜂𝑢0 ≪ 2𝑣𝐿∕𝑐 and, thus, neglecting the interval 𝜂 ≪ 𝑇 ′ when the velocity change takes place, the contour
frame is co-moving with the inertial frame 𝑆′ before changing direction of motion and with the inertial frame 𝑆′′ immediately after.
In short, we may see how the mechanism of relative simultaneity works between frames in relative motion, as follows. Let us consider the case 𝑋′ = 0 when the particle is emitted from C* at A and travels at speed 𝑐. Furthermore, let the contour change direction of motion with 𝑣𝑓 = −𝑣 and Δ𝑣 = 2𝑣. In this case, from (5) the round trip interval is,

𝑇′

= 𝑇𝑜′𝑢𝑡

+ 𝑇𝑟′𝑒𝑡

=

2𝐿0 𝑐

+

2𝑣𝐿0 , 𝑐2

(7)

which we interpret in terms of the out trip from A to B (𝑇𝑜′𝑢𝑡 = 𝐿0∕𝑐 + 𝑣𝐿0∕𝑐2) and the return trip from B to A (𝑇𝑟′𝑒𝑡 = 𝐿0∕𝑐 + 𝑣𝐿0∕𝑐2).

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Heliyon 9 (2023) e19590

Fig. 4. Eﬀect of relative simultaneity between frames in relative motion. At 𝑡 = 𝑡′ = 𝑡′′ = 0 the origins of frames 𝑆, 𝑆′, and 𝑆′′ coincide at A. 𝑎) As seen from frame 𝑆, at the moment (𝑡 = 0) the contour is changing direction of motion, the particle (a photon) is at point B. 𝑏) As seen from frame 𝑆′ moving at the relative speed +𝑣, at 𝑡′ = 0 the photon is already on the lower section at the distance 𝑐𝛿𝑡′ from B. 𝑐) As seen from frame 𝑆′′ moving at the relative speed −𝑣, at 𝑡′′ = 0 the photon is still on the upper section at the distance 𝑐𝛿𝑡′′ from B.
For the out trip as seen from frame 𝑆, we consider the instant when the particle reaches point B (Fig. 4-a) and the contour has traveled the distance 𝑣𝑇𝑜𝑢𝑡 from the original position 𝑆0, shown in Fig. 4-a. Then, to keep it simple, we assume that the origins of 𝑆, 𝑆′, and 𝑆′′ coincide at point A at 𝑡′′ = 𝑡′ = 𝑡 = 0, as in Fig. 4.
According to the time transform of the LT, 𝑡′ = 𝛾(𝑡 − 𝑣𝑥∕𝑐2), the event “particle at B with 𝑥𝐵 = 𝐿0∕𝛾 at 𝑡 = 0” occurs in the past at 𝑡′𝐵 = 𝛾(0 − 𝑣𝑥𝐵∕𝑐2) = −𝑣𝐿0∕𝑐2 = −𝛿𝑡′. Therefore, at the present time 𝑡′ = 0 on 𝑆′, after reﬂection at B at 𝑡′𝐵 = −𝛿𝑡′ the particle has moved by 𝑐𝛿𝑡′ toward the contour end point A (Fig. 4-b). Hence, as seen from 𝑆′, the path length covered in the out trip is 𝐿0 from A to B on the upper contour section, and 𝑐𝛿𝑡′ from B toward A on the lower section in the interval 𝛿𝑡′, i.e., 𝐿0 + 𝑐𝛿𝑡′. In agreement with the interpretation based on relative simultaneity, the corresponding measured time interval is, 𝑇𝑜′𝑢𝑡 = 𝐿0∕𝑐 + 𝛿𝑡′ = 𝐿0∕𝑐 + 𝑣𝐿0∕𝑐2.
On the return trip, the contour changes direction of motion and we have to consider the frame 𝑆′′ co-moving with the contour just after changing direction of motion (Fig. 4-c). According to the time transform of the LT, 𝑡′′ = 𝛾(𝑡 + 𝑣𝑥∕𝑐2), for 𝑆′′ the event “particle at B with 𝑥𝐵 = 𝐿0∕𝛾 at 𝑡 = 0” will occur in the future at 𝑡′𝐵′ = 𝛾(0 + 𝑣𝑥𝐵∕𝑐2) = 𝑣𝐿0∕𝑐2 = 𝛿𝑡′′. Therefore, at the present time 𝑡′′ = 0 on 𝑆′′, the particle has not yet reached B and is on the upper contour section (Fig. 4-c), and will reach B after traveling the distance 𝑐𝛿𝑡′′. Afterwards, traveling at speed 𝑐 from B toward A, the particle covers the path length 𝐿0 and reaches C* after traveling the distance 𝐿0 + 𝑐𝛿𝑡′′, corresponding to the measured time interval 𝑇𝑟′𝑒𝑡 = 𝐿0∕𝑐 + 𝛿𝑡′′ = 𝐿0∕𝑐 + 𝑣𝐿0∕𝑐2.
In conclusion, when the contour changes direction of motion, the round-trip interval measured by C* is not 2𝐿0∕𝑐, but is given by (7) where, according to standard special relativity based on the LT, the additional contribution 2𝑣𝐿0∕𝑐2 = 𝛿𝑡′ + 𝛿𝑡′′ comes from the eﬀect of relative simultaneity between the frames in relative motion (𝑆′ and 𝑆 in the out trip, and 𝑆′′ and 𝑆 in the return trip).
Interpreting the optical eﬀect in the context of theories assuming absolute simultaneity. In the interpretation of optical phenomena in the context of theories assuming absolute simultaneity, we are looking for the preferred frame where the light speed can be assumed to be 𝑐. For the Michelson-Morley experiment, the Sagnac eﬀect, the reciprocal linear Sagnac eﬀect, and other optical phenomena, the natural choice corresponds to the inertial frame where the contour is stationary, Maxwell’s equations are valid, and the electromagnetic waves, constrained to move along the contour, propagate at speed 𝑐. Then, we may assume that the contour is carrying along in its motion the electromagnetic waves propagating at the local speed 𝑐, regardless of the contour state of motion. If simultaneity is conserved in this scenario, even with the contour changing velocity, 𝛿𝑡′ = 𝛿𝑡′′ = 0 and there are no variations in the relative positions of the particle on the contour, as it occurs with relative simultaneity in Fig. 4-a, 4-b, and 4-c. Hence, by assuming absolute simultaneity we have 𝑇 ′ = 2𝐿0∕𝑐 independent of 𝑋′ and Δ𝑇 ′(𝑋′) = 0 in (6), representing a null result, unlike the predictions of the LT based on relative simultaneity.
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