Gravitomagnetic clock effect

General relativity
${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}$
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In physics, the gravitomagnetic clock effect is a deviation from Kepler's third law that, according to the weak-field and low-velocity approximation of general relativity, will be suffered by a particle in orbit around a (slowly) spinning body, such as a typical planet or star.

According to general relativity, in its weak-field and low-velocity 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][excessive citations]}

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.

• Introduction to general relativity
• Gravitomagnetic time delay