# Noninertial Proper Motions of the Minkowski Metric, the Sagnac Effect, and the Twin Paradox

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## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem

## 3. Complete Group of Proper Motion in Space ${\mathbb{R}}_{\mathbf{1},\mathbf{2}}^{\mathbf{3}}$

## 4. An Explanation of the Sagnac Effect

## 5. The Additional Group of Motions and the Twin Paradox

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Sagnac, G. On the proof of the reality of the luminiferous aether by the experiment with a rotating interferometer. Comptes Rendus
**1913**, 157, 1410–1413. [Google Scholar] - Bershtein, I.L. Sagnac’s Experiment with Radiowaves. Dokl. Akad. Nauk SSSR
**1950**, 75, 635–638. (In Russian) [Google Scholar] - Vysotskii, V.I.; Vorontsov, V.I.; Kuz’min, R.N.; Bezirganyan, P.A.; Rostomyan, A.G. The Sagnac experiment with X-radiation. Phys.-Uspekhi.
**1994**, 37, 289–302. [Google Scholar] [CrossRef] - Atwood, D.K.; Horne, M.A.; Shull, C.G.; Arthur, J. Neutron Phase Shift in a Rotating Two-Crystal Interferometer. Phys. Rev. Lett.
**1984**, 52, 1673–1676. [Google Scholar] [CrossRef] - Hasselbach, F.; Nicklaus, M. Sagnac experiment with electrons: Observation of the rotational phase shift of electron waves in vacuum. Phys. Rev. A
**1993**, 48, 143–151. [Google Scholar] [CrossRef] [PubMed] - Riehle, F.; Kisters, T.; Witte, A.; Helmcke, J. Optical Ramsey Spectroscopy in a Rotating Frame: Sagnac Effect in a Matter-Wave Interferometer. Phys. Rev. Lett.
**1991**, 67, 177–180. [Google Scholar] [CrossRef] [PubMed] - Werner, S.A.; Staudenmann, J.-L.; Colella, R. Effect of Earth’s rotation on the quantum mechanical phase of the neutron. Phys. Rev. Lett.
**1979**, 42, 1103–1106. [Google Scholar] [CrossRef] - Anderson, R.; Bilger, H.R.; Stedman, G.E. “Sagnac” effect: A century of Earth-rotated interferometers. Am. J. Phys.
**1994**, 62, 975–985. [Google Scholar] [CrossRef] - Vali, V.; Shorthill, R.W. Fiber ring interferometer. Appl. Opt.
**1976**, 15, 1099–1100. [Google Scholar] [CrossRef] - Chow, W.W.; Gea-Banacloche, J.; Pedrotti, L.M.; Sanders, V.E.; Schleich, W.; Scully, M.O. The ring laser gyro. Rev. Mod. Phys.
**1985**, 57, 61. [Google Scholar] [CrossRef] - Post, E.J. Sagnac Effect. Rev. Mod. Phys.
**1967**, 39, 475–493. [Google Scholar] [CrossRef] - Malykin, G.B. The Sagnac effect: Correct and incorrect explanations. Phys. Uspekhi
**2000**, 43, 1229–1252. [Google Scholar] [CrossRef] - Logunov, A.A.; Chugreyev, Y.V. Special theory of relativity and the Sagnac effect. Sov. Phys. Uspekhi
**1988**, 31, 861–864. [Google Scholar] [CrossRef] - Gourgoulhon, E. Special Relativity in General Frames; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar] [CrossRef]
- Ashtekar, A.; Magnon, A. The Sagnac effect in general relativity. J. Math. Phys.
**1975**, 16, 341–344. [Google Scholar] [CrossRef] - Bazanski, S.L. Some properties of light propagation in relativity. AIP Conf. Proc.
**1998**, 453, 421–430. [Google Scholar] [CrossRef] - Caponio, E.; Masiello, A. A Note on the Sagnac Effect in General Relativity as a Finslerian Effect. Found. Phys.
**2022**, 52, 5. [Google Scholar] [CrossRef] - Anandan, J. Sagnac effect in relativistic and nonrelativistic physics. Phys. Rev. D
**1981**, 24, 338–346. [Google Scholar] [CrossRef] - Born, M. Die Theorie des starren Elektrons in der Kinematik des Relativitatsprinzips. Ann. Phys.
**1909**, 30, 1–56. [Google Scholar] [CrossRef] - Dubrovin, B.A.; Novikov, S.P.; Fomenko, A.T. Modern Geometry. Methods and Applications; Nauka: Moscow, Russia, 1979. (In Russian) [Google Scholar]
- Barut, A.; Raczka, R. Theory of Group Representations and Applications; World Scientific: Singapore, 1986. [Google Scholar] [CrossRef]
- Selleri, F. Noninvariant one-way velocity of light. Found. Phys.
**1996**, 26, 641–664. [Google Scholar] [CrossRef] - Klauber, R.D. Toward a Consistent Theory of Relativistic Rotation. In Relativity in Rotating Frames; Fundamental Theories of Physics; Rizzi, G., Ruggiero, M.L., Eds.; Springer: Dordrecht, The Netherlands, 2004; Volume 135, pp. 103–137. [Google Scholar] [CrossRef]
- Vigier, J.P. New non-zero photon mass interpretation of the Sagnac effect as direct experimental justification of the Langevin paradox. Phys. Lett. A
**1997**, 234, 75–85. [Google Scholar] [CrossRef] - Bergia, S.; Guidone, M. Time on a Rotating Platform and the One-Way Speed of Light. Found. Phys. Lett.
**1998**, 11, 549–559. [Google Scholar] [CrossRef] - Rizzi, G.; Tartaglia, A. Speed of Light on Rotating Platforms. Found. Phys.
**1998**, 28, 1663–1683. [Google Scholar] [CrossRef] - Rizzi, G.; Ruggiero, M.L. (Eds.) The Relativistic Sagnac Effect: Two Derivations. In Relativity in Rotating Frames; Fundamental Theories of Physics; Springer: Dordrecht, The Netherlands, 2004; Volume 135, pp. 179–220. [Google Scholar] [CrossRef]
- Rodrigues, W.A., Jr.; Sharif, M. Rotating Frames in SRT: Sagnac’s Effect and Related Issues. Found. Phys.
**2001**, 31, 1767–1783. [Google Scholar] [CrossRef] - Tartaglia, A.; Ruggiero, M.L. The Sagnac effect and pure geometry. Am. J. Phys.
**2015**, 83, 427–432. [Google Scholar] [CrossRef] - Langevin, P. L’evolution de l’espace et du temps. Scientia
**1911**, 10, 31–54. [Google Scholar] - Langevin, P. Sur la theorie de relativite et l’experience de M. Sagnac. Comptes Rendus Seances l’Academie Sci.
**1921**, 173, 831–834. [Google Scholar] - Tessarotto, M.; Cremaschini, C. Theory of Nonlocal Point Transformations in General Relativity. Adv. Math. Phys.
**2016**, 2016, 9619326. [Google Scholar] [CrossRef] - Landau, L.D.; Lifshiz, E.M. The Classical Theory of Fields; Pergamon Press: Oxford, UK, 1971. [Google Scholar]
- Wang, J.; Sun, H.; Fiori, S. Empirical Means on Pseudo-Orthogonal Groups. Mathematics
**2019**, 7, 940. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Popov, N.; Matveev, I.
Noninertial Proper Motions of the Minkowski Metric, the Sagnac Effect, and the Twin Paradox. *Axioms* **2023**, *12*, 537.
https://doi.org/10.3390/axioms12060537

**AMA Style**

Popov N, Matveev I.
Noninertial Proper Motions of the Minkowski Metric, the Sagnac Effect, and the Twin Paradox. *Axioms*. 2023; 12(6):537.
https://doi.org/10.3390/axioms12060537

**Chicago/Turabian Style**

Popov, Nikolay, and Ivan Matveev.
2023. "Noninertial Proper Motions of the Minkowski Metric, the Sagnac Effect, and the Twin Paradox" *Axioms* 12, no. 6: 537.
https://doi.org/10.3390/axioms12060537