Gravitomagnetic clock effect

In physics, 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, such as a typical planet or star.

Explanation

According to general relativity, in its weak-field and slow-motion linearized approximation, a slowly spinning body induces an additional component of the gravitational field that 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

${\displaystyle T_{\rm {Kep}}=2\pi {\sqrt {\frac {a^{3}}{GM}}}}$

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

${\displaystyle T=T_{\rm {Kep}}+T_{\rm {Gvm}}=T_{\rm {Kep}}\pm {\frac {S}{Mc^{2}}},}$ where S is the central body's angular momentum and c is the speed of light in vacuum.

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 corotation with respect to the rotation of the central body, whereas the − sign is for counter-rotation. That is, if the satellite orbits in the same direction as the planet spins, it takes more time to make a full orbit, whereas if it moves oppositely with respect to the planet's rotation its orbital period gets shorter.

See also

• Introduction to general relativity
• Gravitomagnetic time delay

References

1. Cohen, J.M.; Mashhoon, B. (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.
2. Mashhoon, B.; Gronwald, F.; 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. doi:10.1002/(SICI)1521-3889(199902)8:2<135::AID-ANDP135>3.0.CO;2-N.
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 9356721.
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.
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.
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 6342400.
8. Mashhoon, B.; Gronwald, F; Lichtenegger, H.I.M. (2001). "Gravitomagnetism and the Clock Effect". Lecture Notes in Physics. 562: 83–108. arXiv:gr-qc/9912027. Bibcode:2001LNP...562...83M. doi:10.1007/3-540-40988-2_5. ISBN 978-3-540-41236-6. S2CID 32411999.
9. Mashhoon, B.; Iorio, L.; Lichtenegger, H.I.M. (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.
10. Iorio, L.; Lichtenegger, H.I.M.; Mashhoon, B. (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 5941537.
11. Iorio, L.; Lichtenegger, H.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.
12. Lichtenegger, H.I.M.; 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.