Gravitomagnetic clock effect
Encyclopedia
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
, such as a typical planet
or star
.
, 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
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
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.
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, will be suffered by a particle in orbit around a (slowly) spinning body endowed with angular momentum
, such as a typical planetPlanet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
or star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
.
Explanation
According to general relativityGeneral relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, in its weak-field and slow-motion linearized approximation, a slowly spinning material body induces an additional component of the gravitational field
Gravitational field
The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
which acts on a freely-falling test particle with a non-central, gravitomagnetic Lorentz
Lorentz
Lorentz is a name derived from the Roman surname, Laurentius, which mean "from Laurentum". It is the German form of Laurence.Lorentz may refer to:- Literature :* Friedrich Lorentz, author of works on the Pomeranian language...
-like force.
Among its consequences on the particle's orbital motion there is a small correction to Kepler's third law, namely
where TKep is the particle's period, M is the mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
of the central body, and a is the semimajor axis of the particle's ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
. If the orbit of the particle is circular and lies in the equatorial plane of the central body, the correction is
where S is the central body's angular momentumAngular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
and c is the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
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.