User:Christillin/gravitational clock effect
The gravitomagnetic clock effect is a deviation from Kepler's third law that, according to the weak-field and slow-motion approximation of general relativity, will be suffered by a particle in orbit around a (slowly) spinning body endowed with angular momentum , such as a typical planet or star.
Explanation
According to general relativity, in its weak-field and slow-motion linearized approximation, a slowly spinning material body induces an additional component of the gravitational field which acts on a freely-falling test particle with a non-central, gravitomagnetic Lorentz-like force.
Among its consequences on the particle's orbital motion there is a small correction to Kepler's third law, namely
where TKep is the particle's period, M is the mass of the central body, and a is the semimajor axis of the particle's ellipse. If the orbit of the particle is circular and lies in the equatorial plane of the central body, the correction is
- where S is the central body's angular momentum and c is the speed of light in vacuum.
Interestingly, particles orbiting in opposite directions experience gravitomagnetic corrections TGvm with opposite signs, so that the difference of their orbital periods would cancel the standard Keplerian terms and would add the gravitomagnetic ones.[1][2][3][4][5][6][7][8][9][10][11][12] Note that the sign occurs for particle's co-rotation with respect to the rotation of the central body, while the sign is for counter-rotation. That is, if the satellite revolves in the same direction as the planet spins, it takes longer time to describe a full orbital revolution, while if it moves oppositely with respect to the planet's rotation its orbital period gets shorter.
The same result has been shown NOT to be a distinctive feature of General Relativity. Under the same conditions (low energy weak field) "effective" (to O(v^2/c^2) included) vector equations have been derived just from special relativity and shown to account in a parameter free way for the gravitational quadrupole radiation as well as for geodetic precession and frame dragging
It trivially follows from the equation
where h stands for the gravitomagnetic field produced by the rotation ωrot of the earth which also contributes to the Coriolis force (last term). [13] [14]
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Cohen, Jeffrey M.; Mashhoon, Bahram (October 1993). ""Standard Clocks, Interferometry, and Gravitomagnetism". Physics Letters A. 181 (5): 353–358. Bibcode:1993PhLA..181..353C. doi:10.1016/0375-9601(93)90387-F.
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Mashhoon, B.; Gronwald, F.; Theiss, D.S.; Theiss, D. S. (February 1999). "On measuring gravitomagnetism via spaceborne clocks: a gravitomagnetic clock effect". Annalen der Physik. 8 (2): 135–152. arXiv:gr-qc/9804008. Bibcode:1999AnP...511..135M. doi:10.1002/(SICI)1521-3889(199902)8:2<135::AID-ANDP135>3.0.CO;2-N. S2CID 17353038.
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Tartaglia, A. (February 2000). "Detection of the gravitomagnetic clock effect". Classical and Quantum Gravity. 17 (4): 783–792. arXiv:gr-qc/9909006. Bibcode:2000CQGra..17..783T. doi:10.1088/0264-9381/17/4/304. S2CID 250883898.
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Tartaglia, A. (September 2000). "Geometric Treatment of the Gravitomagnetic Clock Effect". General Relativity and Gravitation. 32 (9): 1745–1756. arXiv:gr-qc/0001080. Bibcode:2000GReGr..32.1745T. doi:10.1023/A:1001998505329. S2CID 119383886.
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: CS1 maint: date and year (link) - ^ Lichtenegger, H.I.M.; Gronwald, F.; Mashhoon, B. (2000). "On detecting the gravitomagnetic field of the Earth by means of orbiting clocks". Advances in Space Research. 25 (6): 1255–1258. arXiv:gr-qc/9808017. Bibcode:2000AdSpR..25.1255L. doi:10.1016/S0273-1177(99)00997-7. S2CID 16542540.
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Iorio, L. (August 2001). "Satellite Gravitational Orbital Perturbations and the Gravitomagnetic Clock Effect". International Journal of Modern Physics D. 10 (4): 465–476. arXiv:gr-qc/0007014. Bibcode:2001IJMPD..10..465I. doi:10.1142/S0218271801000925. S2CID 119426253.
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Iorio, L. (October 2001). "Satellite non-gravitational orbital perturbations and the detection of the gravitomagnetic clock effect". Classical and Quantum Gravity. 18 (20): 4303–4310. arXiv:gr-qc/0007057. Bibcode:2001CQGra..18.4303I. doi:10.1088/0264-9381/18/20/309. S2CID 250921271.
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: CS1 maint: date and year (link) - ^ Mashhoon, Bahram; Gronwald, Frank; Lichtenegger, Herbert I.M. (2001). "Gravitomagnetism and the Clock Effect". Lecture Notes in Physics. 562: 83–108. arXiv:gr-qc/9912027. doi:10.1007/3-540-40988-2_5. ISBN 978-3-540-41236-6. S2CID 32411999.
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Mashhoon, Bahram; Iorio, Lorenzo; Lichtenegger, Herbert (December 2001). "On the gravitomagnetic clock effect". Physics Letters A. 292 (1–2): 49–57. arXiv:gr-qc/0110055. Bibcode:2001PhLA..292...49M. doi:10.1016/S0375-9601(01)00776-9. S2CID 14981533.
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Iorio, Lorenzo; Lichtenegger, Herbert; Mashhoon, Bahram (January 2002). "An alternative derivation of the gravitomagnetic clock effect". Classical and Quantum Gravity. 19 (1): 39–49. arXiv:gr-qc/0107002. Bibcode:2002CQGra..19...39I. doi:10.1088/0264-9381/19/1/303. S2CID 250919715.
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Iorio, Lorenzo; Lichtenegger, Herbert I M. (February 2005). "On the possibility of measuring the gravitomagnetic clock effect in an Earth space-based experiment". Classical and Quantum Gravity. 22 (1): 119–132. arXiv:gr-qc/0210030. Bibcode:2005CQGra..22..119I. doi:10.1088/0264-9381/22/1/008. S2CID 118903460.
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Lichtenegger, H.; Iorio, L.; Mashhoon, B. (December 2006). "The gravitomagnetic clock effect and its possible observation". Annalen der Physik. 15 (12): 868–876. arXiv:gr-qc/0211108. Bibcode:2006AnP...518..868L. doi:10.1002/andp.200610214. S2CID 9087843.
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Christillin, P. (2012). "The Machian contribution of the Universe to geodetic precession, frame dragging and gravitational clock effect". arXiv:1206.4593 [physics.gen-ph].
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Christillin, P. (2012). "Gravitomagnetic forces and quadrupole gravitational radiation from special relativity". arXiv:1205.3514 [physics.gen-ph].
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