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 $$S$$, 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 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. 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
 * $$m v^2/r = GMm/r^2 + mv h + 2m v \; \omega_{\rm rot},$$

where h stands for the gravitomagnetic field produced by the rotation ωrot of the earth which also contributes to the Coriolis force (last term).

Category:Clocks