# 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 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 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

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

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

$T=T_{\rm Kep}+T_{\rm Gvm}=T_{\rm Kep}\pm{S}/{Mc^2},$ 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 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 corotation with respect to the rotation of the central body, whereas the sign is for counter-rotation. That is, if the satellite revolves in the same direction as the planet spins, it takes more time to describe a full orbital revolution, whereas if it moves oppositely with respect to the planet's rotation its orbital period gets shorter.

## 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. doi:10.1002/(SICI)1521-3889(199902)8:2.
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.
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.
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. Bibcode:2000AdSpR..25.1255L. doi:10.1016/S0273-1177(99)00997-7.
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.
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.
8. ^ Mashhoon, B.; Gronwald, F; Lichtenegger, H.I.M. (2001). "Gravitomagnetism and the Clock Effect". Lecture Notes in Physics 562: 83–108. doi:10.1007/3-540-40988-2_5.
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.
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.
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.
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.