Abstract
We investigate the Sagnac effect, representing the difference in travel time or phase shift observed for relativistic matter or light beams counter-propagating in a rotating interferometer. This is done for the Schwarzschild-like and Kerr-like vacuum solutions of conformal gravity which describe the field around a massive static and rotating object respectively, in the fourth order theory. To do this we employ the formal analogy between the Sagnac and Aharonov–Bohm effects that has been used earlier to study this effect in flat and curved spacetimes in general relativity. In particular we show that the linear term in the static metric of conformal gravity has a diminishing effect on the Sagnac time delay. Moreover using the expression for the Sagnac effect in the rotating Kerr-like spacetime, we show that the detection of the Weyl gravity contributions to the Sagnac time delay in the case of the Earth, lies beyond the capability of current experiments.
Similar content being viewed by others
Notes
A detailed pedagogical approach towards Cattaneo’s work on the splitting of spacetime and the development of the gravitomagnetic analogy of the Aharonov–Bohm effect can be found in Appendix A of Ref. [10].
Note that the result is independent on the choice of \(\bar{r}\) and \(s\) [which should still satisfy (29) for the chosen values of \(u\) and \(v\)] since for equatorial orbits \(y = \cos \theta = 0\).
References
Lodge, O.J.: Aberration problems. A discussion concerning the motion of the ether near the Earth, and concerning the connexion between ether and gross matter; with some new experiments. Philos. Trans. R. Soc. Lond. 184, 727–804 (1893)
Sagnac, M.G.: Luminous ether demonstrated by the effect of relative wind of ether in a uniform rotation of an interferometer. C. R. Acad. Sci. Paris 157, 708–710 (1913)
Goy, F., Selleri, F.: Time on a rotating platform. Found. Phys. Lett. 10, 17–29 (1997)
Vigier, J.P.: New non-zero photon mass interpretation of the Sagnac effect as direct experimental justification of the Langevin paradox. Phys. Lett. A 234, 75–85 (1997)
Anastasowksi, P.K., et al.: Self-inconsistencies of the U(1) theory of electrodynamics: Michelson interferometry. Found. Phys. Lett. 12, 579–584 (1999)
Anandan, J.: Sagnac effect in relativistic and non-relativistic physics. Phys. Rev. D. 24, 338–346 (1981)
Weber, T.A.: Measurements on a rotating frame in relativity, and the Wilson and Wilson experiment. Am. J. Phys. 65, 946–953 (1997)
Rizzi, G., Tartaglia, A.: Speed of light on rotating platforms. Found. Phys. 28, 1663–1683 (1998)
Rodrigues Jr, W.A., Sharif, M.: Rotating frames in SRT: the sagnac effect and related issues. Found. Phys. 31, 1767–1783 (2001)
Rizzi, G., Ruggiero, M.L.: The relativistic Sagnac effect: two derivations. arXiv:gr-qc/0305084 (2003)
Rizzi, G., Ruggiero, M.L.: A direct kinematical derivation of the relativistic Sagnac effect for light or matter beams. Gen. Relativ. Gravit. 35, 2129–2136 (2003)
Rostomyan, A.H., Rostomyan, A.M.: X-ray resonators. 3. Applications. Phys. Status Solidi A 126, 29–39 (1991)
Zimmermann, J.E., Mercereau, J.E.: Compton wavelength of superconducting electrons. Phys. Rev. Lett. 14, 887–888 (1965)
Werner, S.A., Staudenmann, J.L., Colella, R.: Effect of Earth’s rotation on the quantum-mechanical phase of the neutron. Phys. Rev. Lett. 42, 1103–1106 (1979)
Riehle, F., Kisters, Th, Witte, Th, Helmcke, J., Bordé, ChJ: Optical Ramsey spectroscopy in a rotating frame—Sagnac effect in a matter-wave interferometer. Phys. Rev. Lett. 67, 177–180 (1991)
Hasselbach, F., Nicklaus, M.: Sagnac experiment with electrons - Observation of the rotational phase-shift of electron waves in vacuum. Phys. Rev. A. 48, 143–151 (1993)
Post, E.J.: Sagnac effect. Rev. Mod. Phys. 39, 475–493 (1967)
Chow, W.W., Gea-Banacloche, J., Pedrotti, L.M., Sanders, V.E., Schleich, W.P., Scully, M.O.: The ring laser gyro. Rev. Mod. Phys. 57, 61–104 (1985)
Stedman, G.E.: Ring-laser tests of fundamental physics and geophysics. Rep. Prog. Phys. 60, 615–688 (1997)
Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485–491 (1959)
Rizzi, G., Ruggiero, M.L.: The Sagnac phase shift suggested by the Aharonov-Bohm effect for relativistic matter beams. Gen. Relativ. Gravit. 35, 1745–1760 (2003)
Cattaneo, C.: General relativity—relative standard mass, momentum, energy and gravitational field in a general system of reference. Il Nuovo Cimento 10, 318–337 (1958)
Ruggiero, M.L.: The Sagnac effect in curved space-times from an analogy with the Aharonov-Bohm effect. Gen. Relativ. Gravit. 37, 1845–1855 (2005)
Tartaglia, A.: General relativistic corrections to the Sagnac effect. Phys. Rev. D. 58, 064009 (1998)
Hu, P.H., Wang, Y.J.: Sagnac effect in the Kerr-Newman and Reissner-Nordstrom fields. Chin. Phys. Lett. 23, 2341–2343 (2006)
Mannheim, P.D., Kazanas, D.: Exact vacuum solution to conformal Weyl gravity and galactic rotation curves. Astrophys. J. 342, 635–638 (1989)
Riegert, R.J.: Birkhoff’s theorem in conformal gravity. Phys. Rev. Lett. 53, 315–318 (1984)
Nandi, K.K., Alsing, P.M., Evans, J.C., Nayak, T.B.: Brans-Dicke corrections to the gravitational Sagnac effect. Phys. Rev. D 63, 084027 (2001)
Okawara, H., Yamada, K., Asada, H.: Possible daily and seasonal variations in quantum interference induced by Chern-Simons gravity. Phys. Rev. Lett. 109, 231101 (2012)
Okawara, H., Yamada, K., Asada, H.: Possible latitude effects of Chern-Simons gravity on quantum interference. Phys. Rev. D 87, 084038 (2013)
Mannheim, P.D., Kazanas, D.: Solutions to the Reissner-Nordstrom, Kerr, and Kerr-Newman problems in 4th-order conformal Weyl gravity. Phys. Rev. D 44, 417–423 (1991)
Mannheim, P.D.: Conformal cosmology with no cosmological constant. Gen. Relativ. Gravit. 22, 289–298 (1990)
Mannheim, P.D.: Conformal gravity and the flatness problem. Astrophys. J. 391, 429–432 (1992)
Mannheim, P.D., Kazanas, D.: Newtonian limit of conformal gravity and the lack of necessity of the 2nd-order Poisson equation. Gen. Relativ. Gravit. 26, 337–361 (1994)
Said, J.L., Sultana, J., Zarb Adami, K.: Exact static cylindrical solution to conformal Weyl gravity. Phys. Rev. D 85, 104054 (2012)
Said, J.L., Sultana, J., Zarb Adami, K.: Charged cylindrical black holes in conformal gravity. Phys. Rev. D 86, 104009 (2012)
Bekenstein, J.D.: Relativistic gravitation theory for the modified Newtonian dynamics paradigm. Phys. Rev. D 70, 083509 (2004)
Mannheim, P.D.: Linear potentials and galactic rotation curves. Astrophys. J. 419, 150–154 (1993)
Mannheim, P.D.: Are galactic rotation curves really flat? Astrophys. J. 479, 659–664 (1997)
Mannheim, P.D., O’Brien, J.G.: Impact of a global quadratic potential on galactic rotation curves. Phys. Rev. Lett. 106, 121101 (2011)
Mannheim, P.D., O’Brien, J.G.: Fitting galactic rotation curves with conformal gravity and a global quadratic potential. Phys. Rev. D. 85, 124020 (2012)
Scarpa, R., Marconi, G., Gilmozzi, R.: Using globular clusters to test gravity in the weak acceleration regime. Astron. Astrophys. 405, L15–L18 (2003)
Kazanas, D.: Cosmological inflation: a personal perspective. In: Contopoulos, G., Patsis, P.A. (eds.) Chaos in Astronomy, Conference 2007. Astrophysics and Space Science Proceedings, pp. 485–496. Springer, Berlin (2009)
Edery, A., Paranjape, M.B.: Classical tests for Weyl gravity: deflection of light and time delay. Phys. Rev. D 58, 024011 (1998)
Amore, P., Arceo, S., Fernandez, F.M.: Analytical formulas for gravitational lensing: higher order calculation. Phys. Rev. D 74, 083004 (2006)
Sultana, J., Kazanas, D.: Bending of light in conformal Weyl gravity. Phys. Rev. D 81, 127502 (2010)
Sultana, J., Kazanas, D., Said, J.L.: Conformal Weyl gravity and perihelion precession. Phys. Rev. D 86, 084008 (2012)
Aharonov, Y., Carmi, G.: Quantum aspects of the equivalence principle. Found. Phys. 3, 493–498 (1973)
Ashtekar, A., Magnon, A.: The Sagnac effect in general relativity. J. Math. Phys. 16, 341–344 (1975)
Sakurai, J.J.: Comments on quantum-mechanical interference due to the Earth’s rotation. Phys. Rev. D 21, 2993–2994 (1980)
Semon, M.D.: Experimental-verification of an Aharonov-Bohm effect in rotating reference frames. Found. Phys. 12, 49–57 (1982)
Papini, G.: Particle wave functions in weak gravitational fields. Nuovo Cimento B 52, 136–137 (1967)
Wisnisvesky, D., Aharonov, Y.: Nonlocal effects in classical and quantum theories. Ann. Phys. 45, 479–492 (1967)
Stodolsky, L.: Matter and light-wave interferometry in gravitational-fields. Gen. Relativ. Gravit. 11, 391–405 (1979)
Anandan, J.: Gravitational and rotational effects in quantum interference. Phys. Rev. D. 15, 1448–1457 (1977)
Kajari, E., Buser, M., Feiler, C., Schleich, W.P.: Rotation in relativity and the propagation of light. Riv. Nuovo Cimento 32, 339–438 (2009)
Cohen, J.M., Mashhoon, B.: Standard clocks, interferometry, and gravitomagnetism. Phys. Lett. A. 181, 353–358 (1993)
Kajari, E., et al.: Sagnac effect of Godel’s Universe. Gen. Relativ. Gravit 36, 2289–2316 (2004)
Carter, B.: Black hole equilibrium states. In: DeWitt, C., DeWitt, B.S. (eds.) Black Holes, Proceedings of the Les Houches School of Physics, Les Houches, France, 1972, pp. 57–214. Gordon and Breach, New York (1973)
Straumann, N.: General Relativity and Relativistic Astrophysics. Springer, Berlin (1984)
Lense, J., Thirring, H.: Über den Einfluss der Eigenrotation der Zentralkrper auf die Bewegung der Planeten und Monde nach der Einsteinschen Gravitationstheorie. Phys. Z. 19, 156–163 (1918)
de Sitter, W.: On Einstein’s theory of gravitation, and its astronomical consequences. Mon. Not. R. Astron. Soc. 77, 155–184 (1916)
Schreiber, K.U., et al.: How to detect the Chandler and the annual wobble of the Earth with a large ring laser gyroscope. Phys. Rev. Lett. 107, 173904 (2011)
Bosi, F., et al.: Measuring gravitomagnetic effects by a multi-ring-laser gyroscope. Phys. Rev. D 84, 122002 (2011)
Tartaglia, A.: Experimental determination of gravitomagnetic effects by means of ring lasers. arXiv:1212.2880 (gr-qc)(2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sultana, J. The Sagnac effect in conformal Weyl gravity. Gen Relativ Gravit 46, 1710 (2014). https://doi.org/10.1007/s10714-014-1710-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10714-014-1710-6