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<title>Relativistic analysis of the Michelson-Gale experimental result | Scientific Reports</title>
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Relativistic analysis of the Michelson-Gale experimental result
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<li class=c-article-identifiers__item>Published: <time datetime=2024-04-30>30 April 2024</time></li>
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<h1 class=c-article-title data-test=article-title data-article-title>Relativistic analysis of the Michelson-Gale experimental result</h1>
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<ul class="c-article-author-list c-article-author-list--short" data-test=authors-list data-component-authors-activator=authors-list><li class=c-article-author-list__item><a data-test=author-name data-track=click data-track-action="open author" data-track-label=link href=#auth-Yang_Ho-Choi-Aff1 data-author-popup=auth-Yang_Ho-Choi-Aff1 data-author-search="Choi, Yang-Ho" data-corresp-id=c1>Yang-Ho Choi<svg width=16 height=16 focusable=false role=img aria-hidden=true class=u-icon><use xmlns:xlink=http://www.w3.org/1999/xlink xlink:href=#icon-eds-i-mail-medium></use></svg></a><sup class=u-js-hide><a href=#Aff1>1</a></sup> </ul>
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<b data-test=journal-volume><span class=u-visually-hidden>volume</span> 14</b>, Article number: <span data-test=article-number>9956</span> (<span data-test=article-publication-year>2024</span>)
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<section aria-labelledby=Abs1 data-title=Abstract lang=en><div class=c-article-section id=Abs1-section><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id=Abs1>Abstract</h2><div class=c-article-section__content id=Abs1-content><p>The result of the Michelson-Gale experiment, which shows fringe shifts by the interference between two light beams traversing a rectangular loop in opposite directions, has been nonrelativistically analyzed based on the Galilean transformation. We relativistically analyze it via the transformation under the constant light speed (TCL) and via the framework of Mansouri and Sexl (MS). The TCL provides a coordinate transformation between the isotropic frame and a rotating frame, in which the two-way speed of light is a constant <i>c</i> irrespective of direction on the surface that has the same radius of rotation. When using TCL, we assume that the Solar System is isotropic so that the one-way speed of light is <i>c</i> in it. On the contrary, considering its movement, the analysis is carried out without the assumption of isotropy based on the MS framework. The analysis results via the TCL and via the MS framework correspond to each other and are in agreement with the result of the experiment. It is shown that the difference between the travel times of the counter-propagating light beams, which results in the fringe shift, takes place due to the two factors, the anisotropy of the one-way speed of light in inertial frames and the different rotation radii at different latitudes on the Earth surface.</p></div></div></section>
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<div class=c-article-recommendations-card__img><img src="data:image/png;base64,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
|
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|
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|
|
<div class=c-article-recommendations-card__main>
|
|||
|
|
<h3 class=c-article-recommendations-card__heading itemprop="name headline">
|
|||
|
|
<a class=c-article-recommendations-card__link itemprop=url href="https://www.nature.com/articles/s41586-024-07315-1?fromPaywallRec=false" data-track=click data-track-action="click recommendations inline - 1" data-track-label=10.1038/s41586-024-07315-1>The solar dynamo begins near the surface
|
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</a>
|
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</h3>
|
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|
|
<div class=c-article-meta-recommendations>
|
|||
|
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<span class=c-article-meta-recommendations__item-type>Article</span>
|
|||
|
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<span class=c-article-meta-recommendations__access-type>Open access</span>
|
|||
|
|
<span class=c-article-meta-recommendations__date>22 May 2024</span>
|
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|
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</div>
|
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|
|
</div>
|
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</article>
|
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</div>
|
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|
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<div class=c-article-recommendations-list__item>
|
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|
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<article class=c-article-recommendations-card itemscope itemtype=http://schema.org/ScholarlyArticle>
|
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|
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<div class=c-article-recommendations-card__img><img src="data:image/png;base64,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
|
|||
|
|
|
|||
|
|
<div class=c-article-recommendations-card__main>
|
|||
|
|
<h3 class=c-article-recommendations-card__heading itemprop="name headline">
|
|||
|
|
<a class=c-article-recommendations-card__link itemprop=url href="https://www.nature.com/articles/s41550-024-02277-w?fromPaywallRec=false" data-track=click data-track-action="click recommendations inline - 2" data-track-label=10.1038/s41550-024-02277-w>An emission-state-switching radio transient with a 54-minute period
|
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|
|
</a>
|
|||
|
|
</h3>
|
|||
|
|
<div class=c-article-meta-recommendations>
|
|||
|
|
<span class=c-article-meta-recommendations__item-type>Article</span>
|
|||
|
|
<span class=c-article-meta-recommendations__access-type>Open access</span>
|
|||
|
|
<span class=c-article-meta-recommendations__date>05 June 2024</span>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</article>
|
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|
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</div>
|
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|
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|
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<div class=c-article-recommendations-list__item>
|
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|
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<article class=c-article-recommendations-card itemscope itemtype=http://schema.org/ScholarlyArticle>
|
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<section data-title=Introduction><div class=c-article-section id=Sec1-section><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id=Sec1>Introduction</h2><div class=c-article-section__content id=Sec1-content><p>Michelson had shown a great passion to search for the luminiferous ether. He continued his efforts in the Michelson-Gale (MG) experiment<sup><a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 1" title="Michelson, A. A. & Gale, H. G. The effect of the Earth’s rotation on the velocity of light: Part I, part II. Astrophys. J. 61, 137 (1925)." href=#ref-CR1 id=ref-link-section-d263354249e314>1</a></sup>, more than 35 years after the null result in the famous Michelson-Morley (MM) experiment<sup><a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 2" title="Michelson, A. A. & Morley, E. W. On the relative motion of the Earth and the luminiferous ether. Am. J. Sci. 34, 333 (1887)." href=#ref-CR2 id=ref-link-section-d263354249e318>2</a></sup>, and at last had observed fringe shifts. The MG experiment employed a large rectangular loop with a perimeter of about 1.9 km that two light beams traverse in opposite directions. The fringe shift is due to the difference between the travel times of the counter-propagating light beams that travel the same distance. Though nearly 100 years have passed since then, very few relativistic analyses on the experiment result are found, which may indicate the difficulty that the special and general relativity suffers in consistently handling circular motions<sup><a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref title="Klauber, R. D. Relativistic rotation: A comparison of theories. Found. Phys. 37, 198 (2007)." href=#ref-CR3 id=ref-link-section-d263354249e322>3</a>,<a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref title="Rizzi, G. & Ruggiero, M. L. (eds) Relativity in Rotating Frames (Kluwer Academic, 2004)." href=#ref-CR4 id=ref-link-section-d263354249e322_1>4</a>,<a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 5" title="Choi, Y.-H. Uniqueness of the isotropic frame and usefulness of the Lorentz transformation. J. Korean Phys. Soc. 72(10), 1110 (2018)." href=#ref-CR5 id=ref-link-section-d263354249e325>5</a></sup>. It is stated in ref.<sup><a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 6" title="Moon, P., Spencer, D. E. & Moon, E. E. The Michelson-Gale experiment and its effect on the postulates on the velocity of light. Phys. Essays 3(3), 421 (1990)." href=#ref-CR6 id=ref-link-section-d263354249e329>6</a></sup> that “an imposing list of more than a thousand books and papers on the subject of the velocity of light makes no mention of this experiment.” In ref.<sup><a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 6" title="Moon, P., Spencer, D. E. & Moon, E. E. The Michelson-Gale experiment and its effect on the postulates on the velocity of light. Phys. Essays 3(3), 421 (1990)." href=#ref-CR6 id=ref-link-section-d263354249e333>6</a></sup>, the travel times of the light beams were nonrelativistically analyzed, under the assumption that the speed of light is constant regardless of direction in the Solar System. In ref.<sup><a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 7" title="Pascoli, G. The Sagnac effect and its interpretation by Paul Langevin. Comptes Rendus Phys. 18(9–10), 563–569 (2017)." href=#ref-CR7 id=ref-link-section-d263354249e338>7</a></sup>, mentioning that the hypothesis of a dragging of the ether is not valid as an explanation about the null result, t
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(1)
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</div></div><p>where <span class=mathjax-tex>\({\mathbf{T}}_{L} ({\varvec{\beta}}_{k} )\)</span> is the Lorentz transformation matrix,<div id=Equ2 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$${\mathbf{T}}_{L} ({\varvec{\beta}}_{k} ) = \left[ {\begin{array}{*{20}c} {\gamma_{k} } & { - i\gamma_{k} {\varvec{\beta}}_{k}^{T} } \\ {i\gamma_{k} {\varvec{\beta}}_{k} } & {(\gamma_{k} - 1)\hat{\user2{\beta }}_{k} \hat{\user2{\beta }}_{k}^{T} + {\mathbf{I}}} \\ \end{array} } \right],$$</span></div><div class=c-article-equation__number>
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(2)
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</div></div><p>with,<div id=Equ3 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$\gamma_{k} = (1 - |{\varvec{\beta}}_{k} |^{2} )^{ - 1/2} ,$$</span></div><div class=c-article-equation__number>
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(3)
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</div></div><p>and <span class=mathjax-tex>\({\mathbf{I}}\)</span> an identity matrix. Since <span class=mathjax-tex>\(d{\mathbf{p}} = {\mathbf{T}}_{L}^{ - 1} ({\varvec{\beta}}_{i} )d{\mathbf{p}}_{i}\)</span>, the transformation from one inertial frame <span class=mathjax-tex>\(S_{i \cdot }\)</span> to another <span class=mathjax-tex>\(S_{j \cdot }\)</span> is expressed as<sup><a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 5" title="Choi, Y.-H. Uniqueness of the isotropic frame and usefulness of the Lorentz transformation. J. Korean Phys. Soc. 72(10), 1110 (2018)." href=#ref-CR5 id=ref-link-section-d263354249e1782>5</a>,<a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 15" title="Choi, Y.-H. Multiple velocity composition in the standard synchronization. Open Phys. 20(1), 155 (2022)." href=#ref-CR15 id=ref-link-section-d263354249e1785>15</a></sup>.<div id=Equ4 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$d{\mathbf{p}}_{j} = {\mathbf{T}}_{L} ({\varvec{\beta}}_{j} ,\;{\varvec{\beta}}_{i} )d{\mathbf{p}}_{i} ,$$</span></div><div class=c-article-equation__number>
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(4)
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</div></div><p>where,<div id=Equ5 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$${\mathbf{T}}_{L} ({\varvec{\beta}}_{j} ,\;{\varvec{\beta}}_{i} ) = {\mathbf{T}}_{L} ({\varvec{\beta}}_{j} ){\mathbf{T}}_{L}^{ - 1} ({\varvec{\beta}}_{i} ).$$</span></div><div class=c-article-equation__number>
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(5)
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</div></div><p>It is obvious that <span class=mathjax-tex>\({\mathbf{T}}_{L}^{ - 1} ({\varvec{\beta}}_{k} ) = {\mathbf{T}}_{L}^{T} ({\varvec{\beta}}_{k} )\)</span>, which leads to <span class=mathjax-tex>\({\mathbf{T}}_{L}^{ - 1} ({\varvec{\beta}}_{j} ,\;{\varvec{\beta}}_{i} )\, = {\mathbf{T}}_{L}^{T} ({\varvec{\beta}}_{j} ,\;{\varvec{\beta}}_{i} )\)</span>.<p>Proper time (PT) is independent of synchronization schemes and can be obtained in any inertial frame if relative velocity is known. We use a subscript ‘<span class=mathjax-tex>\(\circ\)</span>’ in PT to distinguish it from the adjusted time (AT) through the synchronization of clocks. The PT interval is measured at the same place. From (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ1>1</a>) and (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ4>4</a>), the PT interval of an observer <span class=mathjax-tex>\(O_{j}\)</span> who is at rest in <span class=mathjax-tex>\(S_{j}\)</span> is expressed as <span class=mathjax-tex>\(d\tau_{{j^{^\circ } }} = d\tau_{i} /\gamma_{ji} = d\tau /\gamma_{j}\)</span>, which is valid even if <span class=mathjax-tex>\(i\)</span> and <span class=mathjax-tex>\(j\)</span> are interchanged.<h3 class=c-article__sub-heading id=Sec4>In TCL</h3><p>An observer <span class=mathjax-tex>\(\tilde{O^{\prime}}\)</span> is located at a radius <span class=mathjax-tex>\(r^{\prime}\)</span> in a primed rotating frame <span class=mathjax-tex>\(\tilde{S}^{\prime}\)</span>, the coordinate vector of which is represented as <span class=mathjax-tex>\({\mathbf{\tilde{p}^{\prime}}}\, = [t^{\prime},\;r^{\prime},\;\tilde{\varphi^{\prime}},z^{\prime}]^{{T}}\)</span> in the cylindrical coordinate system where <span class=mathjax-tex>\(\tilde{\varphi^{\prime}}\)</span> indicates an azimuth angle. The observer is rotating at an angular velocity <span class=mathjax-tex>\(\omega\)</span> in the isotropic frame <i>S</i>, the coordinate vector of which is denoted by <span class=mathjax-tex>\({\mathbf{p}}\, = [t,\;r,\;\varphi ,z]^{T}\)</span>. In TCL, the coordinate transformation between <span class=mathjax-tex>\(\tilde{S}^{\prime}\)</span> and <i>S</i> is given by,<div id=Equ6 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$t^{\prime}\, = \frac{t}{\gamma },\,r^{\prime} = \gamma {\kern 1pt} r\,,\,\tilde{\varphi^{\prime}} = \varphi - \omega \,t,\,z^{\prime} = z.$$</span></div><div class=c-article-equation__number>
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(6)
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</div></div><p> where <span class=mathjax-tex>\(\gamma = (1 - \beta^{2} )^{ - 1/2}\)</span> with <span class=mathjax-tex>\(\beta = r\omega /c\)</span>. The elapsed time and the radius in the primed are different from those in the unprimed. As a result, the angular velocity <span class=mathjax-tex>\(\omega^{\prime}\)</span> as seen in the primed becomes different from <span class=mathjax-tex>\(\omega\)</span>. It is convenient to introduce the primed inertial frame <span class=mathjax-tex>\(S^{\prime}\)</span> corresponding to <i>S</i> and the unprimed rotating frame <span class=mathjax-tex>\(\tilde{S}\)</span> corresponding to <span class=mathjax-tex>\(\tilde{S}^{\prime}\)</span>. The coordinate transformations between <i>S</i> and <span class=mathjax-tex>\(\tilde{S}\)</span> in the unprimed and between <span class=mathjax-tex>\(S^{\prime}\)</span> and <span class=mathjax-tex>\(\tilde{S}^{\prime}\)</span> in the primed are nonrelativistic Galilean. The azimuth angle <span class=mathjax-tex>\(\varphi^{\prime}\)</span> in <i>S′</i> is the same as <span class=mathjax-tex>\(\varphi\)</span> in <i>S</i><sup><a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 8" title="Choi, Y.-H. Consistent coordinate transformation for relativistic circular motion and speeds of light. J. Korean Phys. Soc. 75(3), 176 (2019)." href=#ref-CR8 id=ref-link-section-d263354249e3173>8</a></sup>. The <span class=mathjax-tex>\(\tilde{S}\)</span> rotates at the angular velocity <span class=mathjax-tex>\(\omega\)</span> in <i>S</i> while the <span class=mathjax-tex>\(\tilde{S}^{\prime}\)</span> does at the angular velocity <span class=mathjax-tex>\(\omega^{\prime}\)</span> in <span class=mathjax-tex>\(S^{\prime}\)</span>, where <span class=mathjax-tex>\(\omega^{\prime}\)</span> and <span class=mathjax-tex>\(\omega\)</span> are related by <span class=mathjax-tex>\(\omega^{\prime} = \gamma {\kern 1pt} \omega\)</span><sup><a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 8" title="Choi, Y.-H. Consistent coordinate transformation for relativistic circular motion and speeds of light. J. Korean Phys. Soc. 75(3), 176 (2019)." href=#ref-CR8 id=ref-link-section-d263354249e3365>8</a></sup>. It should be noted that <span class=mathjax-tex>\(r^{\prime}\)</span> is the radius seen by the observer moving with the tangential speed of <span class=mathjax-tex>\(r^{\prime}\omega^{\prime}\)</span> in <span class=mathjax-tex>\(S^{\prime}\)</span>.<p>The two-way speed of light is the constant <span class=mathjax-tex>\(c\)</span> in TCL<sup><a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 8" title="Choi, Y.-H. Consistent coordinate transformation for relativistic circular motion and speeds of light. J. Korean Phys. Soc. 75(3), 176 (2019)." href=#ref-CR8 id=ref-link-section-d263354249e3465>8</a></sup>. In other words, when <span class=mathjax-tex>\(r^{\prime}\)</span> is fixed, the two-way speed is constant regardless of direction in TCL, which is consistent with the result of the MM experiment.</p></div></div></section><section data-title="Analysis of the MG experiment result"><div class=c-article-section id=Sec5-section><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id=Sec5>Analysis of the MG experiment result</h2><div class=c-article-section__content id=Sec5-content><p>We investigate the result of the MG experiment with the transformation (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ6>6</a>) in Subsection "<a data-track=click data-track-label=link data-track-action="section anchor" href=#Sec6>With TCL</a>", assuming that the Solar System is isotropic, and based on the MS framework without the assumption in Subsection "<a data-track=click data-track-label=link data-track-action="section anchor" href=#Sec7>Based on the MS framework</a>". Michelson had speculated about a
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(7)
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</div></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test=figure data-container-section=figure id=figure-1 data-title="Figure 1"><figure><figcaption><b id=Fig1 class=c-article-section__figure-caption data-test=figure-caption-text>Figure 1</b></figcaption><div class=c-article-section__figure-content><div class=c-article-section__figure-item><a class=c-article-section__figure-link data-test=img-link data-track=click data-track-label=image data-track-action="view figure" href=https://www.nature.com/articles/s41598-024-60515-7/figures/1 rel=nofollow><picture><img aria-describedby=Fig1 src="data:image/png;base64,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(8)
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</div></div><p>When the light beams <span class=mathjax-tex>\(b_{ + }\)</span> and <span class=mathjax-tex>\(b_{ - }\)</span> return to the detector, their travel times are different, which brings about a fringe shift. The times that are taken for <span class=mathjax-tex>\(b_{ + }\)</span> and <span class=mathjax-tex>\(b_{ - }\)</span> to transverse the segments <span class=mathjax-tex>\(P_{0} P_{3}\)</span> and <span class=mathjax-tex>\(P_{1} P_{2}\)</span>, by symmetry between them, are equal and thus the travel time difference results from the others. For convenience, we use <span class=mathjax-tex>\(L_{1}\)</span> and <span class=mathjax-tex>\(L_{2}\)</span> to represent the segments <span class=mathjax-tex>\(P_{0} P_{1}\)</span> and <span class=mathjax-tex>\(P_{3} P_{2}\)</span>, respectively.<h3 class=c-article__sub-heading id=Sec6>With TCL</h3><p>The Solar System is assumed to be isotropic. Our Earth and Solar System correspond to <span class=mathjax-tex>\(\tilde{S}^{\prime}\)</span> and <i>S</i>, respectively. The angular velocity of <span class=mathjax-tex>\(\tilde{S}\)</span> is <span class=mathjax-tex>\(\omega\)</span> in <i>S</i> while that of <span class=mathjax-tex>\(\tilde{S}^{\prime}\)</span> is <span class=mathjax-tex>\(\omega^{\prime}( = \gamma {\kern 1pt} \omega )\)</span> in <span class=mathjax-tex>\(S^{\prime}\)</span>. If we know the speeds in <span class=mathjax-tex>\(\tilde{S}^{\prime}\)</span> of <span class=mathjax-tex>\(b_{ \pm }\)</span> their travel times can be calculated. The speed of light is known in <i>S</i>. Using the speed of light in <i>S</i>, we can obtain the speeds of <span class=mathjax-tex>\(b_{ \pm }\)</span> in <span class=mathjax-tex>\(\tilde{S}^{\prime}\)</span>. If <span class=mathjax-tex>\(r^{\prime}\)</span> is fixed, so is <span class=mathjax-tex>\(r\)</span> and vice versa. Then the squared line element on the surface of a cylinder of radius <span class=mathjax-tex>\(r\)</span> is written in <i>S</i> as,<div id=Equ9 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$ds^{2} = - c^{2} dt^{2} + \;\;r^{2} d\varphi^{2} + dz^{2} .$$</span></div><div class=c-article-equation__number>
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(9)
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</div></div><p>Substituting (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ6>6</a>) into (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ9>9</a>) gives,<div id=Equ10 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$ds^{2} = - c^{2} (\gamma {\kern 1pt} dt^{\prime})^{2} + \;r^{{\prime}{2}} (d\tilde{\varphi^{\prime}} + \omega^{\prime}dt^{\prime})^{2} /\gamma^{2} + dz^{{\prime}{2}} .$$</span></div><div class=c-article-equation__number>
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(10)
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</div></div><p>The differential interval is independent of <span class=mathjax-tex>\(dr^{\prime}\)</span> since <span class=mathjax-tex>\(r^{\prime}\)</span> is fixed. Noting <span class=mathjax-tex>\(r^{\prime}\omega^{\prime} = c\gamma^{2} \beta\)</span>, (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ10>10</a>) is rewritten as,<div id=Equ11 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$ds^{2} = - (cdt^{\prime})^{2} + \;2\beta \,r^{\prime}d\tilde{\varphi^{\prime}}(cdt^{\prime}) + r^{{\prime}{2}} d\tilde{\varphi^{\prime}}^{2} /\gamma^{2} + dz^{{\prime}{2}} .$$</span></div><div class=c-article-equation__number>
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(11)
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</div></div><p>For light signals, <span class=mathjax-tex>\(ds\)</span> reduces to zero, which leads to,<div id=Equ12 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$cdt^{\prime} = \beta \,r^{\prime}d\tilde{\varphi^{\prime}} + dl^{\prime},$$</span></div><div class=c-article-equation__number>
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(12)
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</div></div><p>where <span class=mathjax-tex>\(dl^{\prime} = (r^{{\prime}{2}} d\tilde{\varphi }^{{\prime}{2}} + \;dz^{{\prime}{2}} )^{1/2}\)</span>. It can be easily shown from (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ12>12</a>) that the two-way speed is the constant <span class=mathjax-tex>\(c\)</span> irrespective of direction. Suppose that a light beam takes a round trip along a differential <span class=mathjax-tex>\(dl^{\prime}\)</span>. The sign of <span class=mathjax-tex>\(d\tilde{\varphi^{\prime}}\)</span> at one path in the round trip is reversed at the other. The round trip time is thus <span class=mathjax-tex>\(dt^{\prime}_{ \updownarrow } = 2dl^{\prime}/c\)</span>, and the two-way speed of light becomes <span class=mathjax-tex>\(c^{\prime}_{ \updownarrow } = c\)</span>. When a light beam traverses <span class=mathjax-tex>\(L_{1}\)</span> or <span class=mathjax-tex>\(L_{2}\)</span>, <span class=mathjax-tex>\(dz^{\prime}\)</span> is zero and <span class=mathjax-tex>\(dl^{\prime} = r^{\prime}|d\tilde{\varphi^{\prime}}|\)</span>. The speeds of the co-rotating and counter-rotating light beams, which are denoted by <span class=mathjax-tex>\(c^{\prime}_{ + }\)</span> and <span class=mathjax-tex>\(c^{\prime}_{ - }\)</span> respectively, are given by,<div id=Equ13 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$c^{\prime}_{ \pm } = \frac{{dl^{\prime}}}{{dt^{\prime}}} = \frac{c}{1 \pm \beta }.$$</span></div><div class=c-article-equation__number>
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(13)
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</div></div><p>According to the second equation of (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ6>6</a>), <span class=mathjax-tex>\(r^{\prime}_{m}\)</span> and <span class=mathjax-tex>\(r_{m}\)</span>, <span class=mathjax-tex>\(m = 1,\;2\)</span>, are related by <span class=mathjax-tex>\(r^{\prime}_{m} = \gamma_{m} r_{m}\)</span> with <span class=mathjax-tex>\(\gamma_{m} = (1 - \beta_{m}^{2} )^{ - 1/2}\)</span> where <span class=mathjax-tex>\(\beta_{m} = r_{m} \omega /c\)</span>. From (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ13>13</a>), the elapsed times of <span class=mathjax-tex>\(b_{ \pm }\)</span> during the travel to the segment <span class=mathjax-tex>\(L_{1}\)</span> are calculated, respectively as,<div id=Equ14 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$${\kern 1pt} t^{\prime}_{1 \pm } = \frac{{(1 \pm \;\beta_{1} \,)\,l^{\prime}_{w1} }}{c},$$</span></div><div class=c-article-equation__number>
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(14)
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</div></div><p>and in the case of the travel to <span class=mathjax-tex>\(L_{2}\)</span>,<div id=Equ15 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$t^{\prime\prime}_{2 \pm } = \frac{{(1 \mp \beta_{2} \,)\,l^{\prime}_{w2} }}{c}.$$</span></div><div class=c-article-equation__number>
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(15)
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</div></div><p>One can also confirm from (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ14>14</a>) and (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ15>15</a>) that the two-way speed of light is <span class=mathjax-tex>\(c\)</span>. For example, <span class=mathjax-tex>\(c^{\prime}_{1 \updownarrow } = 2\,l^{\prime}_{w1} /({\kern 1pt} t^{\prime}_{1 + } + t^{\prime}_{1 - } ) = c\)</span> where <span class=mathjax-tex>\(c^{\prime}_{1 \updownarrow }\)</span> is the two-way speed at <span class=mathjax-tex>\(L_{1}\)</span>. The elapsed times <span class=mathjax-tex>\(t^{\prime\prime}_{2 \pm }\)</span> are measured at <span class=mathjax-tex>\(L_{2}\)</span>. What we try to attain is the time difference at the detector, which is located at <span class=mathjax-tex>\(L_{1}\)</span>. Therefore the <span class=mathjax-tex>\(t^{\prime\prime}_{2 \pm }\)</span> should be converted into the times by the clock of the detector, which are written as<div id=Equ16 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$${\kern 1pt} t^{\prime}_{2 \pm } = \xi_{21} t^{\prime\prime}_{2 \pm } ,$$</span></div><div class=c-article-equation__number>
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(16)
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</div></div><p>where <span class=mathjax-tex>\(\xi_{21} = \gamma_{2} /\gamma_{1}\)</span>. The time intervals <span class=mathjax-tex>\(t^{\prime\prime}_{2 \pm }\)</span> at <span class=mathjax-tex>\(L_{2}\)</span> are observed as <span class=mathjax-tex>\({\kern 1pt} \gamma_{2} t^{\prime\prime}_{2 \pm }\)</span> in <i>S</i>, which correspond to <span class=mathjax-tex>\({\kern 1pt} t^{\prime}_{2 \pm }\)</span> when seen by the clock of the detector. The difference between the travel times of <span class=mathjax-tex>\(b_{ \pm }\)</span> is expressed as,<div id=Equ17 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$\Delta {\kern 1pt} t^{\prime}_{d} = {\kern 1pt} \;t^{\prime}_{ + } - t^{\prime}_{ - } = \frac{{2(\beta_{1} {\kern 1pt} l^{\prime}_{w1} - \xi_{21} \beta_{2} {\kern 1pt} l^{\prime}_{w2} )}}{c},$$</span></div><div class=c-article-equation__number>
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(17)
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</div></div><p>where <span class=mathjax-tex>\(t^{\prime}_{ \pm } = t^{\prime}_{1 \pm } + \,{\kern 1pt} t^{\prime}_{2 \pm }\)</span>.<p>The tangential speed at the equator is less than 500 m/s, and <span class=mathjax-tex>\(\beta_{1}^{{}} ,\;\beta_{2}^{{}} < < 1\)</span>. The fringe shift <span class=mathjax-tex>\(N\)</span> is given, to a first-order approximation by,<div id=Equ18 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$N = \frac{{4l^{\prime}_{w1} l^{\prime}_{h} \omega^{\prime}\cos \alpha_{1} }}{\lambda c},$$</span></div><div class=c-article-equation__number>
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(18)
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</div></div><p>where <span class=mathjax-tex>\(\lambda\)</span> is the wavelength of light. For derivation of (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ18>18</a>), see the <a data-track=click data-track-label=link data-track-action="supplementary material anchor" href=#MOESM1>Supplementary Information</a>. The quantity <span class=mathjax-tex>\(l^{\prime}_{w1} l^{\prime}_{h}\)</span> corresponds to the area of the rectangular loop. Equation (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ14>18</a>) agrees with the result of the MG experiment<sup><a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 1" title="Michelson, A. A. & Gale, H. G. The effect of the Earth’s rotation on the velocity of light: Part I, part II. Astrophys. J. 61, 137 (1925)." href=#ref-CR1 id=ref-link-section-d263354249e7457>1</a></sup>.<h3 class=c-article__sub-heading id=Sec7>Based on the MS framework</h3><p>In reality, our Solar System moves in the Milky Way and it would be different from the isotropic frame <span class=mathjax-tex>\(S\)</span>. Though it moves, we can consider that it belongs to an inertial frame during a very short time that the light beams traverse the closed loop. We denote the Solar System by <span class=mathjax-tex>\(S_{i \cdot }\)</span>, which includes the orbital motion of the Earth. The speed of light is <span class=mathjax-tex>\(c\)</span> with respect to AT, <span class=mathjax-tex>\(t_{i}\)</span>, in <span class=mathjax-tex>\(S_{i \cdot }\)</span>. The closed loop in Fig. <a data-track=click data-track-label=link data-track-action="figure anchor" href=#Fig1>1</a> is divided into an infinite number of differential elements. A differential segment <span class=mathjax-tex>\(d{\mathbf{l}}_{j}\)</span>, which can be located on the segment <span class=mathjax-tex>\(L_{1}\)</span> or <span class=mathjax-tex>\(L_{2}\)</span>, belongs to an inertial frame <span class=mathjax-tex>\(S_{j \cdot }\)</span>. The direction of <span class=mathjax-tex>\(d{\mathbf{l}}_{j}\)</span> is defined such that it is the same as the direction of travel of the light beam <span class=mathjax-tex>\(b_{ + }\)</span>.<p>From (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ4>4</a>), <span class=mathjax-tex>\(|d{\mathbf{p}}_{i} |\; = \;|d{\mathbf{p}}_{j} |\)</span>. Since <span class=mathjax-tex>\(S_{i \cdot }\)</span> and <span class=mathjax-tex>\(S_{j \cdot }\)</span> are standard-synchronized, the time that is taken for a light beam to travel a distance <span class=mathjax-tex>\(dl_{j}\)</span> is <span class=mathjax-tex>\(dl_{j} /c\)</span> and so <span class=mathjax-tex>\(d\tau_{j} = idl_{j}\)</span>. When <span class=mathjax-tex>\(d{\mathbf{p}}_{j} = [idl_{j} ,\;d{\mathbf{l}}_{j}^{T} ]^{T}\)</span>, the corresponding differential vector in <span class=mathjax-tex>\(S_{i \cdot }\)</span> is <span class=mathjax-tex>\(d{\mathbf{p}}_{i} = [d\tau_{i} ,\;d{\mathbf{l}}_{i}^{T} ]^{T}\)</span>. For the light travel, <span class=mathjax-tex>\(|d{\mathbf{p}}_{j} |\; = 0\)</span> and thus <span class=mathjax-tex>\(d\tau_{i}^{2} + \;|d{\mathbf{l}}_{i} |^{2} = 0\)</span>. Equivalently,<div id=Equ19 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$dl_{i}^{{}} = \;|d\tau_{i} |.$$</span></div><div class=c-article-equation__number>
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(19)
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</div></div><p>The differential vector <span class=mathjax-tex>\(d{\mathbf{p}}_{i}\)</span> is related to <span class=mathjax-tex>\(d{\mathbf{p}}_{j}\)</span> by <span class=mathjax-tex>\(d{\mathbf{p}}_{i} = {\mathbf{T}}_{L} ({\varvec{\beta}}_{i} ,\;{\varvec{\beta}}_{j} )d{\mathbf{p}}_{j}\)</span>. Equation (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ19>19</a>) indicates that <span class=mathjax-tex>\(dl_{i}^{{}}\)</span> can be obtained if the first row of <span class=mathjax-tex>\({\mathbf{T}}_{L} ({\varvec{\beta}}_{i} ,\;{\varvec{\beta}}_{j} )\)</span> is known so that <span class=mathjax-tex>\(d\tau_{i}\)</span> is found, even though the rest is unknown. The first row of <span class=mathjax-tex>\({\mathbf{T}}_{L} ({\varvec{\beta}}_{i} ,\;{\varvec{\beta}}_{j} )\)</span> is given by <span class=mathjax-tex>\({\mathbf{T}}_{L} ({\varvec{\beta}}_{i} ,\;{\varvec{\beta}}_{j} )_{1r} = \gamma_{ij} [1,\; - i{\varvec{\beta}}_{ij}^{T} ]\)</span> where <span class=mathjax-tex>\({\varvec{\beta}}_{kl}\)</span> is the normalized velocity of <span class=mathjax-tex>\(S_{k \cdot }\)</span> relative to <span class=mathjax-tex>\(S_{l \cdot }\)</span><sup><a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 5" title="Choi, Y.-H. Uniqueness of the isotropic frame and usefulness of the Lorentz transformation. J. Korean Phys. Soc. 72(10), 1110 (2018)." href=#ref-CR5 id=ref-link-section-d263354249e8843>5</a>,<a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 15" title="Choi, Y.-H. Multiple velocity composition in the standard synchronization. Open Phys. 20(1), 155 (2022)." href=#ref-CR15 id=ref-link-section-d263354249e8846>15</a></sup>. Then <span class=mathjax-tex>\(d\tau_{i}^{{}}\)</span> is calculated as,<div id=Equ20 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$d\tau_{i}^{{}} = i\gamma_{ij} (dl_{j}^{{}} - {\varvec{\beta}}_{ij}^{T} d{\mathbf{l}}_{j}^{{}} ).$$</span></div><div class=c-article-equation__number>
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(20)
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</div></div><p>In the travel of <span class=mathjax-tex>\(b_{ + }\)</span>(<span class=mathjax-tex>\(b_{ - }\)</span>), <span class=mathjax-tex>\(d{\mathbf{l}}_{j}^{{}}\)</span> and <span class=mathjax-tex>\({\varvec{\beta}}_{ij}^{{}}\)</span> are in opposite directions at <span class=mathjax-tex>\(L_{1}\)</span>(<span class=mathjax-tex>\(L_{2}\)</span>) and in the same direction at <span class=mathjax-tex>\(L_{2}\)</span>(<span class=mathjax-tex>\(L_{1}\)</span>). Recall <span class=mathjax-tex>\(\beta_{m} = r_{m} \omega /c\)</span>, <span class=mathjax-tex>\(m = 1,\;2\)</span>. At <span class=mathjax-tex>\(L_{1}\)</span>, <span class=mathjax-tex>\(\gamma_{ij} = \gamma_{1}\)</span>, where <span class=mathjax-tex>\(\beta_{ij} = \beta_{1}\)</span>, and <span class=mathjax-tex>\({\varvec{\beta}}_{ij}^{T} d{\mathbf{l}}_{j}^{{}} = \mp dl_{j}\)</span> for <span class=mathjax-tex>\(b_{ \pm }\)</span> respectively. At <span class=mathjax-tex>\(L_{2}\)</span>, <span class=mathjax-tex>\(\beta_{ij} = \beta_{2}\)</span>, <span class=mathjax-tex>\(\gamma_{ij} = \gamma_{2}\)</span>, and <span class=mathjax-tex>\({\varvec{\beta}}_{ij}^{T} d{\mathbf{l}}_{j}^{{}} = \pm dl_{j}\)</span> for <span class=mathjax-tex>\(b_{ \pm }\)</span>. The travel distances in <span class=mathjax-tex>\(S_{i \cdot }\)</span> of <span class=mathjax-tex>\(b_{ \pm }\)</span> at <span class=mathjax-tex>\(L_{m}\)</span>, <span class=mathjax-tex>\(m = 1,\;2\)</span>, each are given from (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ19>19</a>) and (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ20>20</a>) by,<div id=Equ21 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$l_{1 \pm } = (1 \pm \beta_{1} )\,\gamma_{1} l^{\prime}_{w1} ,$$</span></div><div class=c-article-equation__number>
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(21a)
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</div></div><div id=Equ22 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$l_{2 \pm } = (1 \mp \beta_{2} )\gamma_{2} l^{\prime}_{w2} ,$$</span></div><div class=c-article-equation__number>
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(21b)
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</div></div><p>and the travel times are <span class=mathjax-tex>\(t_{m \pm } = l_{m \pm } /c\)</span>. Then the time difference in <i>S</i><sub>i</sub> is calculated as,<div id=Equ23 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$\Delta t_{d} = \sum\limits_{m = 1}^{2} {(t_{m + } - \;t_{m - } )} = \frac{{2(\beta_{1} \gamma_{1} l^{\prime}_{w1} - \beta_{2} \gamma_{2} l^{\prime}_{w2} )}}{c}.$$</span></div><div class=c-article-equation__number>
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(22)
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</div></div><p>The time difference at <span class=mathjax-tex>\(P_{0}\)</span> is,<div id=Equ24 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$\Delta t^{\prime}_{d} = \frac{{\Delta t_{d} }}{{\gamma_{1} }}.$$</span></div><div class=c-article-equation__number>
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(23)
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</div></div><p>Equation (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ19>23</a>) is the same as (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ17>17</a>) and is valid regardless of whether the spacetime of <i>S</i><sub>i∙</sub> is actually isotropic or not.<p>The time intervals <span class=mathjax-tex>\(t_{m \pm }\)</span> in the unprimed correspond to <span class=mathjax-tex>\(t^{\prime}_{m \pm } = t_{m \pm } /\gamma_{m}\)</span> in the primed. The speeds of <span class=mathjax-tex>\(b_{ \pm }\)</span> at <span class=mathjax-tex>\(L_{m}\)</span> are written from <span class=mathjax-tex>\(t_{m \pm } = l_{m \pm } /c\)</span> and (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ21>21a</a>, <a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ22>21b</a>) as,<div id=Equ25 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$c^{\prime}_{m \pm } = \frac{{l^{\prime}_{wm} }}{{t^{\prime}_{m \pm } }} = \frac{c}{{1 \mp ( - 1)^{m} \beta_{m} }},\,m = 1,\;2.$$</span></div><div class=c-article-equation__number>
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(24)
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</div></div><p>Equation (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ24>24</a>) is consistent with (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ13>13</a>). Virtually <span class=mathjax-tex>\(L_{1}\)</span> and <span class=mathjax-tex>\(L_{2}\)</span> can be considered to belong to certain inertial frames during the very short time of the light travel. As shown in (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ25>24</a>), the inertial frames are anisotropic, the speed of light depending on the propagation direction, which has also been observed in the experiments of the generalized Sagnac effect<sup><a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref title="Choi, Y.-H. Theoretical analysis of generalized Sagnac effect in the standard synchronization. Can. J. Phys. 95(8), 761 (2017)." href=#ref-CR11 id=ref-link-section-d263354249e10585>11</a>,<a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref title="Wang, R., Zheng, Y. & Yao, A. Generalized Sagnac effect. Phys. Rev. Lett. 93, 143901 (2004)." href=#ref-CR12 id=ref-link-section-d263354249e10585_1>12</a>,<a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 13" title="Wang, R., Zheng, Y., Yao, A. & Langley, D. Modified Sagnac experiment for measuring travel-time difference between counter-propagating light beams in a uniformly moving fiber. Phys. Lett. A 312, 7 (2003)." href=#ref-CR13 id=ref-link-section-d263354249e10588>13</a></sup>. The time difference is caused due to two factors. One is the anisotropy of the light speed at <span class=mathjax-tex>\(L_{1}\)</span> and <span class=mathjax-tex>\(L_{2}\)</span> each. The other is the difference between the rotation radii of <span class=mathjax-tex>\(L_{1}\)</span> and <span class=mathjax-tex>\(L_{2}\)</span>, which results in different tangential speeds. Although the speed of light is anisotropic, there would be no time difference, as can be seen from (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ23>22</a>), if there were no difference in radius, i.e. <span class=mathjax-tex>\(r^{\prime}_{1} = r^{\prime}_{2}\)</span>. Although the radii are different, no fringe shifts would occur if the speed of light were isotropic in inertial frames.</p></div></div></section><section data-title=Discussion><div class=c-article-section id=Sec8-section><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id=Sec8>Discussion</h2><div class=c-article-section__content id=Sec8-content><p>To find exact physical quantities, we have to use <span class=mathjax-tex>\({\mathbf{T}}_{L} ({\varvec{\beta}}_{j} ,\;{\varvec{\beta}}_{i} )\)</span>. However, the absolute velocities <span class=mathjax-tex>\({\varvec{\beta}}_{i}\)</span> and <span class=mathjax-tex>\({\varvec{\beta}}_{j}\)</span> are unknown and we cannot. Disguising the inertial frame <span class=mathjax-tex>\(S_{i}\)</span> as isotropic via the standard synchronization and then using <span class=mathjax-tex>\({\mathbf{T}}_{L} ({\varvec{\beta}}_{ji} )\)</span> instead of <span class=mathjax-tex>\({\mathbf{T}}_{L} ({\varvec{\beta}}_{j} ,\;{\varvec{\beta}}_{i} )\)</span>, nonetheless, we can exactly obtain some physical quantities such as PTs, Doppler shifts, spatial lengths, and speeds with respect to PT<sup><a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 5" title="Choi, Y.-H. Uniqueness of the isotropic frame and usefulness of the Lorentz transformation. J. Korean Phys. Soc. 72(10), 1110 (2018)." href=#ref-CR5 id=ref-link-section-d263354249e10980>5</a></sup>. It is because the first rows of <span class=mathjax-tex>\({\mathbf{T}}_{L} ({\varvec{\beta}}_{j} ,\;{\varvec{\beta}}_{i} )\)</span> and <span class=mathjax-tex>\({\mathbf{T}}_{L}
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(25)
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</div></div><p>where the symbol <span class=mathjax-tex>\(t_{ \cdot }\)</span> is used to explicitly indicate the standard-synchronized time, AT, in <span class=mathjax-tex>\(S_{i \cdot }\)</span>. The events that occur at the same <span class=mathjax-tex>\(t^{\prime}\)</span> in (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ6>6</a>) is actually simultaneous whereas the events at the same <span class=mathjax-tex>\(t_{ \cdot }\)</span> in (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ26>25</a>) is not since <span class=mathjax-tex>\(t_{ \cdot }\)</span> is AT. However <span class=mathjax-tex>\(t^{\prime}\)</span> is exact since it represents the PT interval.<p>The analysis of the MG experimental result with (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ26>25</a>) is the same as in Subsection "<a data-track=click data-track-label=link data-track-action="section anchor" href=#Sec6>With TCL</a>" except that <span class=mathjax-tex>\(t\)</span> is replaced by <span class=mathjax-tex>\(t_{ \cdot }\)</span>. Here, using (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ26>25</a>), we analyze the Sagnac effect. In the experiment of the Sagnac effect, the Earth can be considered to be in linear motion during the traverse of light beams, though it rotates. The inertial frame <span class=mathjax-tex>\(S_{i \cdot }\)</span> represents the one for a laboratory. The light detector <span class=mathjax-tex>\(\tilde{O^{\prime}}\)</span> is located on a circumference of radius <span class=mathjax-tex>\(r\)</span> in <span class=mathjax-tex>\(S_{i \cdot }\)</span>. At <span class=mathjax-tex>\(t_{ \cdot } = t^{\prime} = 0\)</span>, two light beams <span class=mathjax-tex>\(b_{ + }\)</span> and <span class=mathjax-tex>\(b_{ - }\)</span> leave a light source, which is located at the same place as the detector, and traverse the circular paths in the co- and counter-rotating directions respectively.<p>Since the transformation (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ26>25</a>) has the same form as (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ6>6</a>), the same equation as (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ12>12</a>) is obtained for the former. The angle <span class=mathjax-tex>\(\tilde{\varphi^{\prime}}\)</span> is positive in the same direction as the rotation direction of <span class=mathjax-tex>\(\tilde{O^{\prime}}\)</span>. Integrating (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ12>12</a>) with respect to <span class=mathjax-tex>\(\tilde{\varphi^{\prime}}\)</span> after the replacement of <span class=mathjax-tex>\(dl^{\prime}\)</span> by <span class=mathjax-tex>\(r^{\prime}\,|d\tilde{\varphi^{\prime}}|\)</span>, we have,<div id=Equ27 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$ct^{\prime}_{ \pm } = \int_{0}^{ \pm 2\pi } {\beta \,r^{\prime}d\tilde{\varphi^{\prime}}} + \int_{0}^{ \pm 2\pi } {r^{\prime}|d\tilde{\varphi^{\prime}}} |\; = (1 \pm \beta )l^{\prime},$$</span></div><div class=c-article-equation__number>
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(26)
|
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</div></div><p>where <span class=mathjax-tex>\(l^{\prime} = 2\pi {\kern 1pt} r^{\prime}\)</span>. The travel times of <span class=mathjax-tex>\(b_{ \pm }\)</span> are,<div id=Equ28 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$t^{\prime}_{ \pm } = \frac{{(1 \pm \beta ){\kern 1pt} l^{\prime}}}{c}.$$</span></div><div class=c-article-equation__number>
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(27)
|
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</div></div><p>The time difference is given by,<div id=Equ29 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$\Delta {\kern 1pt} t^{\prime} = \frac{{2\beta {\kern 1pt} l^{\prime}}}{c},$$</span></div><div class=c-article-equation__number>
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(28)
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</div></div><p>which corresponds to the experimental result. The travel distances of <span class=mathjax-tex>\(b_{ \pm }\)</span> are <span class=mathjax-tex>\(l^{\prime}\)</span> and the speeds of <span class=mathjax-tex>\(b_{ \pm }\)</span> with respect to PT are equal to (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ13>13</a>).<p>Using (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ26>25</a>), let us make analysis on the travel of the light beams in <span class=mathjax-tex>\(S_{i \cdot }\)</span>. From (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ26>25</a>), <span class=mathjax-tex>\(d\varphi = d\tilde{\varphi^{\prime}} + \omega \,dt_{ \cdot }\)</span> and <span class=mathjax-tex>\(dt_{ \cdot } = \gamma {\kern 1pt} dt^{\prime}\)</span>. While the light beams <span class=mathjax-tex>\(b_{ \pm }\)</span> traverse the circular loop, <span class=mathjax-tex>\(\tilde{\varphi^{\prime}}\)</span> and <span class=mathjax-tex>\(t^{\prime}\)</span> vary from 0 to <span class=mathjax-tex>\(\pm 2\pi\)</span> and from 0 to <span class=mathjax-tex>\(t^{\prime}_{ \pm }\)</span>. Integrating <span class=mathjax-tex>\(d\varphi\)</span> yields,<div id=Equ30 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$\varphi_{ \pm } = \pm 2\pi + \gamma {\kern 1pt} \omega \,t^{\prime}_{ \pm } .$$</span></div><div class=c-article-equation__number>
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(29)
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</div></div><p>The travel distances are calculated using (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ30>29</a>) and (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ28>27</a>) as,<div id=Equ31 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$l_{ \pm } = r|\varphi_{ \pm } |\; = \frac{l}{1 \mp \beta },$$</span></div><div class=c-article-equation__number>
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(30)
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</div></div><p>where <span class=mathjax-tex>\(l = 2\pi {\kern 1pt} r\)</span>. The speed of light is <span class=mathjax-tex>\(c\)</span> with respect to AT and the travel times of <span class=mathjax-tex>\(b_{ \pm }\)</span> measured by AT in <span class=mathjax-tex>\(S_{i \cdot }\)</span> are given by <span class=mathjax-tex>\(t_{ \cdot \pm } = l_{ \pm } /c\)</span>, which agrees with (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ28>27</a>), i.e. <span class=mathjax-tex>\(t_{ \cdot \pm } = \gamma {\kern 1pt} t^{\prime}_{ \pm }\)</span>. These analysis results substantiate the transformation (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ26>25</a>).<p>Traditionally the Sagnac effect has been analyzed usually using the Langevin metric [e.g. Refs.<sup><a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 3" title="Klauber, R. D. Relativistic rotation: A comparison of theories. Found. Phys. 37, 198 (2007)." href=#ref-CR3 id=ref-link-section-d263354249e14162>3</a>,<a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 4" title="Rizzi, G. & Ruggiero, M. L. (eds) Relativity in Rotating Frames (Kluwer Academic, 2004)." href=#ref-CR4 id=ref-link-section-d263354249e14165>4</a>,<a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 7" title="Pascoli, G. The Sagnac effect and its interpretation by Paul Langevin. Comptes Rendus Phys. 18(9–10), 563–569 (2017)." href=#ref-CR7 id=ref-link-section-d263354249e14168>7</a>,<a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 18" title="Post, E. J. Sagnac effect. Rev. Mod. Phys. 39, 475 (1967)." href=#ref-CR18 id=ref-link-section-d263354249e14171>18</a></sup>]. Since it is the first-order effect of <span class=mathjax-tex>\(\beta\)</span> as shown in (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ29>28</a>), we can approximately calculate the time difference with the Langevin metric. Neglecting the terms with higher degrees than <span class=mathjax-tex>\(\beta\)</span> in (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ26>25</a>) yields,<div id=Equ32 class=c-article-equation><div class=c-article-equation__content><span class=mathjax-tex>$$\tilde{t^{\prime}} = t_{ \cdot } ,\,\tilde{r^{\prime}} = \;{\kern 1pt} r,\,\tilde{\varphi^{\prime}} = \varphi - \omega \,t_{ \cdot } ,\,\tilde{z^{\prime}} = z.$$</span></div><div class=c-article-equation__number>
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(31)
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</div></div><p>where the symbol “tilde” is used to explicitly represent the coordinates of <span class=mathjax-tex>\(\tilde{S}^{\prime}\)</span>. The Langevin metric is found in accordance with (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ32>31</a>). Clearly the transformation (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ32>31</a>) is Galilean, which does not recognize the difference between <span class=mathjax-tex>\(\tilde{t^{\prime}}\)</span> and <span class=mathjax-tex>\(t_{ \cdot }\)</span> and between <span class=mathjax-tex>\({\kern 1pt} \tilde{r^{\prime}}\)</span> and <span class=mathjax-tex>\(r\)</span>. If the symbol “prime” in the coordinates of <span class=mathjax-tex>\(\tilde{S}^{\prime}\)</span> is removed so that for example, <span class=mathjax-tex>\(\tilde{r} = \;{\kern 1pt} r\)</span> and if <span class=mathjax-tex>\(S_{i \cdot } = S\)</span>, (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ32>31</a>) becomes the transformation between <i>S</i> and <span class=mathjax-tex>\(\tilde{S}\)</span>, from which the same time difference as (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ29>28</a>) is exactly derived with the recognition of the difference between <span class=mathjax-tex>\(\tilde{S}\)</span> and <span class=mathjax-tex>\(\tilde{S}^{\prime}\)</span> [<sup><a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 8" title="Choi, Y.-H. Consistent coordinate transformation for relativistic circular motion and speeds of light. J. Korean Phys. Soc. 75(3), 176 (2019)." href=#ref-CR8 id=ref-link-section-d263354249e14659>8</a></sup>, p. 184]. Without the recognition of the difference, the computation results using (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ32>31</a>) are only valid within the first-order approximation. In the MG paper<sup><a data-track=click data-track-action="reference anchor" data-track-label=link data-test=citation-ref aria-label="Reference 1" title="Michelson, A. A. & Gale, H. G. The effect of the Earth’s rotation on the velocity of light: Part I, part II. Astrophys. J. 61, 137 (1925)." href=#ref-CR1 id=ref-link-section-d263354249e14667>1</a></sup>, the fringe shift, which also results from the first-order effect as in (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ23>22</a>), has been calculated based on (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ32>31</a>). The frame <span class=mathjax-tex>\(\tilde{S}\)</span> is different from <span class=mathjax-tex>\(\tilde{S}^{\prime}\)</span>. Under <span class=mathjax-tex>\(\tilde{S} = \tilde{S}^{\prime}\)</span>, the analyses by the Langevin metric or (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ32>31</a>) are approximate and nonrelativistic.<p>Meanwhile, the MM experiment was devised to test the effect of <span class=mathjax-tex>\(\beta^{2}\)</span> on the round trip velocity. The Langevin metric, in which the round trip speed of light is anisotropic, fails to explain the MM experiment whereas the TCL of (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ26>25</a>) can do. The experiment had been carried out to measure the effect due to the motion of the Earth relative to the Solar System and the two arms of the interferometer, which are very small compared with the radius of the Earth, can be considered to be laid at the same rotation radius. The round trip speed of light is constant in (<a data-track=click data-track-label=link data-track-action="equation anchor" href=#Equ26>25</a>) with the radius fixed irrespective of direction. The TCL is consistent with both MM and MG experiments. It is stated in Ref.<sup><a data-track=click data-track-action="reference anchor" data-track-label=link da
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<section data-title="Data availability"><div class=c-article-section id=data-availability-section><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id=data-availability>Data availability</h2><div class=c-article-section__content id=data-availability-content>
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<p>All data generated or analyzed during this study are included in this published article and its supplementary information file.</p>
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</div></div></section><div id=MagazineFulltextArticleBodySuffix><section aria-labelledby=Bib1 data-title=References><div class=c-article-section id=Bib1-section><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id=Bib1>References</h2><div class=c-article-section__content id=Bib1-content><div data-container-section=references><ol class=c-article-references data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter=1.><p class=c-article-references__text id=ref-CR1>Michelson, A. A. & Gale, H. G. The effect of the Earth’s rotation on the velocity of light: Part I, part II. <i>Astrophys. J.</i> <b>61</b>, 137 (1925).<p class="c-article-references__links u-hide-print"><a data-track=click||click_references rel="nofollow noopener" data-track-label=10.1086/142878 data-track-item_id=10.1086/142878 data-track-value="article reference" data-track-action="article reference" href=https://doi.org/10.1086%2F142878 aria-label="Article reference 1" data-doi=10.1086/142878>Article</a>
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<a data-track=click||click_references rel="nofollow noopener" data-track-label=link data-track-item_id=link data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=1925ApJ....61..137M" aria-label="ADS reference 1">ADS</a>
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<a data-track=click||click_references data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label=link data-track-item_id=link rel="nofollow noopener" aria-label="Google Scholar reference 1" href="http://scholar.google.com/scholar_lookup?&title=The%20effect%20of%20the%20Earth%E2%80%99s%20rotation%20on%20the%20velocity%20of%20light%3A%20Part%20I%2C%20part%20II&journal=Astrophys.%20J.&doi=10.1086%2F142878&volume=61&publication_year=1925&author=Michelson%2CAA&author=Gale%2CHG">
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</p></ol><p class="c-article-references__download u-hide-print"><a data-track=click data-track-action="download citation references" data-track-label=link rel=nofollow href="https://citation-needed.springer.com/v2/references/10.1038/s41598-024-60515-7?format=refman&flavour=references">Download references<svg width=16 height=16 focusable=false role=img aria-hidden=true class=u-icon><use xmlns:xlink=http://www.w3.org/1999/xlink xlink:href=#icon-eds-i-download-medium></use></svg></a></p></div></div></div></section></div><section aria-labelledby=author-information data-title="Author information"><div class=c-article-section id=author-information-section><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id=author-information>Author information</h2><div class=c-article-section__content id=author-information-content><h3 class=c-article__sub-heading id=affiliations>Authors and Affiliations</h3><ol class=c-article-author-affiliation__list><li id=Aff1><p class=c-article-author-affiliation__address>Department of Electrical and Electronic Engineering, Kangwon National University, Chunchon, Kangwon-do, 200-701, South Korea<p class=c-article-author-affiliation__authors-list>Yang-Ho Choi</p></ol><div class="u-js-hide u-hide-print" data-test=author-info><span class=c-article__sub-heading>Authors</span><ol class="c-article-authors-search u-list-reset"><li id=auth-Yang_Ho-Choi-Aff1><span class="c-article-authors-search__title u-h3 js-search-name">Yang-Ho Choi</span><div class=c-article-authors-search__list><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="https://www.nature.com/search?author=Yang-Ho%20Choi" class=c-article-button data-track=click data-track-action="author link - publication" data-track-label=link rel=nofollow>View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in
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<span class=c-article-identifiers><a class=c-article-identifiers__item href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Yang-Ho%20Choi" data-track=click data-track-action="author link - pubmed" data-track-label=link rel=nofollow>PubMed</a><span class=u-hide> </span><a class=c-article-identifiers__item href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Yang-Ho%20Choi%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track=click data-track-action="author link - scholar" data-track-label=link rel=nofollow>Google Scholar</a></span></p></div></div></ol></div><h3 class=c-article__sub-heading id=contributions>Contributions</h3><p>The submitted paper has been prepared and written solely by Yang-Ho Choi.<h3 class=c-article__sub-heading id=corresponding-author>Corresponding author</h3><p id=corresponding-author-list>Correspondence to
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<a id=corresp-c1 href=mailto:yhochoi@kangwon.ac.kr>Yang-Ho Choi</a>.</p></div></div></section><section data-title="Ethics declarations"><div class=c-article-section id=ethics-section><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id=ethics>Ethics declarations</h2><div class=c-article-section__content id=ethics-content>
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<h3 class=c-article__sub-heading id=FPar1>Competing interests</h3>
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<p>The authors declare no competing interests.</p>
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<p><b>Open Access</b> This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit <a href=http://creativecommons.org/licenses/by/4.0/ rel=license>http://creativecommons.org/licenses/by/4.0/</a>.</p>
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<i>Sci Rep</i> <b>14</b>, 9956 (2024). https://doi.org/10.1038/s41598-024-60515-7<p class="c-bibliographic-information__download-citation u-hide-print"><a data-test=citation-link data-track=click data-track-action="download article citation" data-track-label=link data-track-external rel=nofollow href="https://citation-needed.springer.com/v2/references/10.1038/s41598-024-60515-7?format=refman&flavour=citation">Download citation<svg width=16 height=16 focusable=false role=img aria-hidden=true class=u-icon><use xmlns:xlink=http://www.w3.org/1999/xlink xlink:href=#icon-eds-i-download-medium></use></svg></a><ul class=c-bibliographic-information__list data-test=publication-history><li class=c-bibliographic-information__list-item><p>Received<span class=u-hide>: </span><span class=c-bibliographic-information__value><time datetime=2023-09-10>10 September 2023</time></span></p><li class=c-bibliographic-information__list-item><p>Accepted<span class=u-hide>: </span><span class=c-bibliographic-information__value><time datetime=2024-04-24>24 April 2024</time></span></p><li class=c-bibliographic-information__list-item><p>Published<span class=u-hide>: </span><span class=c-bibliographic-information__value><time datetime=2024-04-30>30 April 2024</time></span></p><li class="c-bibliographic-information__list-item c-bibliographic-information__list-item--full-width"><p><abbr title="Digital Object Identifier">DOI</abbr><span class=u-hide>: </span><span class=c-bibliographic-information__value>https://doi.org/10.1038/s41598-024-60515-7</span></p></ul><div data-component=share-box><div class="c-article-share-box u-display-none sf-hidden" hidden></div></div><h3 class=c-article__sub-heading>Keywords</h3><ul class=c-article-subject-list><li class=c-article-subject-list__subject><span><a href="https://www.nature.com/search?query=Michelson-Gale%20experiment&facet-discipline=%22Science%2C%20Humanities%20and%20Social%20Sciences%2C%20multidisciplinary%22" data-track=click data-track-action="view keyword" data-track-label=link>Michelson-Gale experiment</a></span><li class=c-article-subject-list__subject><span><a href="https://www.nature.com/search?query=Coordinate%20transformation&facet-discipline=%22Science%2C%20Humanities%20and%20Social%20Sciences%2C%20multidisciplinary%22" data-track=click data-track-action="view keyword" data-track-label=link>Coordinate transformation</a></span><li class=c-article-subject-list__subject><span><a href="https://www.nature.com/search?query=Standard%20synchronization&facet-discipline=%22Science%2C%20Humanities%20and%20Social%20Sciences%2C%20multidisciplinary%22" data-track=click data-track-action="view keyword" data-track-label=link>Standard synchronization</a></span><li class=c-article-subject-list__subject><span><a href="https://www.nature.com/search?query=Speed%20of%20light&facet-discipline=%22Science%2C%20Humanities%20and%20Social%20Sciences%2C%20multidisciplinary%22" data-track=click data-track-action="view keyword" data-track-label=link>Speed of light</a></span><li class=c-article-subject-list__subject><span><a href="https://www.nature.com/search?query=Sagnac%20effect&facet-discipline=%22Science%2C%20Humanities%20and%20Social%20Sciences%2C%20multidisciplinary%22" data-track=click data-track-action="view keyword" data-track-label=link>Sagnac effect</a></span></ul><div data-component=article-info-list></div></div></div></div></div></section>
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