User:Christillin/gravitational clock effect

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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 $S$ , 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

T_{\rm {Kep}}=2\pi {\sqrt {a^{3}/GM}}

where T_Kep 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

T=T_{\rm {Kep}}+T_{\rm {Gvm}}=T_{\rm {Kep}}\pm {S}/{Mc^{2}},

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 T_Gvm 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

mv^{2}/r=GMm/r^{2}+mvh+2mv\;\omega _{\rm {rot}},

where h stands for the gravitomagnetic field produced by the rotation ω_rot of the earth which also contributes to the Coriolis force (last term). ^[13] ^[14]

^ 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.{{cite journal}}: CS1 maint: date and year (link)
^ 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.{{cite journal}}: CS1 maint: date and year (link)
^ 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.{{cite journal}}: CS1 maint: date and year (link)
^ 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.{{cite journal}}: 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.
^ 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.{{cite journal}}: CS1 maint: date and year (link)
^ 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.{{cite journal}}: 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.
^ 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.{{cite journal}}: CS1 maint: date and year (link)
^ 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.{{cite journal}}: CS1 maint: date and year (link)
^ 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.{{cite journal}}: CS1 maint: date and year (link)
^ 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.{{cite journal}}: CS1 maint: date and year (link)
^ Christillin, P. (2012). "The Machian contribution of the Universe to geodetic precession, frame dragging and gravitational clock effect". arXiv:1206.4593 [physics.gen-ph]. {{cite arXiv}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
^ Christillin, P. (2012). "Gravitomagnetic forces and quadrupole gravitational radiation from special relativity". arXiv:1205.3514 [physics.gen-ph]. {{cite arXiv}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

Category:Clocks

[1] 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.{{cite journal}}: CS1 maint: date and year (link)

[2] 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.{{cite journal}}: CS1 maint: date and year (link)

[3] 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.{{cite journal}}: CS1 maint: date and year (link)

[4] 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.{{cite journal}}: CS1 maint: date and year (link)

[5] 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.

[6] 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.{{cite journal}}: CS1 maint: date and year (link)

[7] 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.{{cite journal}}: CS1 maint: date and year (link)

[8] 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.

[9] 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.{{cite journal}}: CS1 maint: date and year (link)

[10] 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.{{cite journal}}: CS1 maint: date and year (link)

[11] 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.{{cite journal}}: CS1 maint: date and year (link)

[12] 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.{{cite journal}}: CS1 maint: date and year (link)

[13] Christillin, P. (2012). "The Machian contribution of the Universe to geodetic precession, frame dragging and gravitational clock effect". arXiv:1206.4593 [physics.gen-ph]. {{cite arXiv}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

[14] Christillin, P. (2012). "Gravitomagnetic forces and quadrupole gravitational radiation from special relativity". arXiv:1205.3514 [physics.gen-ph]. {{cite arXiv}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

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