Gravitoelectromagnetism (sometimes Gravitomagnetism, Gravimagnetism, abbreviated GEM), refers to a set of formal analogies between Maxwell's field equations and an approximation, valid under certain conditions, to the Einstein field equations for general relativity. The most common version of GEM is valid only far from isolated sources, and for slowly moving test particles.
Gravitomagnetic forces and the corresponding field (gravitomagnetic field and gravitational torsion field in alternatives to general relativity) should be considered in all reference frames that move relative to a source of the static gravitational field. Similarly, the relative motion of an observer with respect to an electrical charge creates a magnetic field and therefore magnetic force is possible.
Currently, verification of gravitoelectromagnetic forces are doing with the help of satellites,  and in some experiments.  
Indirect validations of gravitomagnetic effects have been derived from analyses of relativistic jets. Roger Penrose had proposed a frame dragging mechanism for extracting energy and momentum from rotating black holes. This model was used to explain the high energies and luminosities in quasars and active galactic nuclei; the collimated jets about their polar axis; and the asymmetrical jets (relative to the orbital plane).  All of those observed properties could be explained in terms of gravitomagnetic effects. Application of Penrose's mechanism can be applied to black holes of any size. Relativistic jets can serve as the largest and brightest form of validations for gravitomagnetism.
According to general relativity, the weak gravitational field produced by a moving or rotating object (or any moving or rotating mass-energy) can, in a particular limiting case, be described by equations that have the same form as the equations in classical electromagnetism. Starting from the basic equation of general relativity, the Einstein field equation, and assuming a weak gravitational field or reasonably flat spacetime, the gravitational analogs to Maxwell's equations for electromagnetism, called the "GEM equations", were derived by Lano. 
Subsequently, Agop, Buzea and Ciobanu,  and others have confirmed the validity of GEM equations in International System of Units in the following form:
For a test particle whose mass m is "small," in a stationary system, the net (Lorentz) force acting on it due to a GEM field is described by the following GEM analog to the Lorentz force equation:
Acceleration of any test particle is simply:
The second component of the gravitational force responsible for the collimation of relativistic jets in the gravitomagnetic fields of galaxies, active galactic nuclei and rapidly rotating stars (e.g., jet accreting neutron stars).
In general relativity, due to the alleged tensor nature of gravitation considered that the effective mass for the gravitomagnetic field is twice the usual body mass. Because of this, it is assumed that either ,  or in some papers .  
The above equations of the gravitational field (equation GEM) can be compared with Maxwell's equations:
It can be seen that the form of the gravitational and electromagnetic fields equations is almost the same, except for some factors and minus signs in GEM equations arising from the fact that the masses are attracted, and the electric charges of the same sign repel each other.
Lorentz force, acting on a charge , is given by:
Sergey Fedosin with the help of Lorentz-invariant theory of gravitation (LITG), derived gravitational equation in special relativity. 
The equations coincide with equations which were first published in 1893 by Oliver Heaviside as a separate theory expanding Newton's law. 
These equations, called the Heaviside equations, are Lorentz covariant, unlike equations of gravitoelectromagnetism. The similarity of Heaviside gravitational equations and Maxwell's equations for electromagnetic field highlighted in Maxwell-like gravitational equations.
The gravitational force in LITG is as follows:
In contrast to general relativity, where spin of gravitons is equal to 2, Lorentz-invariant theory of gravitation (LITG) relies on vectorial gravitons with spin equal to 1. Accordingly, in LITG body mass for gravitational and torsion fields is the same.
The above equations are also presented in the articles.    
Some higher-order effects can reproduce effects reminiscent of the interactions of more conventional polarized charges. In torsion field appears momentum of force acting on a rotating particle with the spin :
This leads to precession of the particle spin with angular velocity around direction of .
The mechanical energy of the particle with spin in torsion field will be:
If two disks are spun on a common axis, the mutual gravitational attraction between the two disks arguably ought to be greater if they spin in opposite directions than in the same direction. This can be expressed as an attractive or repulsive force component. When disks rotate in opposite directions, the energy will be negative, and additional force of gravitation is equal to
where torsion field of one disk acts on the angular momentum of another disk.
Due to the torsion field becomes possible effect of gravitational induction.
The formula for torsion field near a rotating body can be derived from the Heaviside equations and is: 
where is angular momentum of the body, is radius-vector from the center of the body to the point, where the torsion field defined.
A detailed derivation of this formula is contained in the book. 
At the equatorial plane, r and L are perpendicular, so their dot product vanishes, and this formula reduces to:
Magnitude of angular momentum of a homogeneous ball-shaped body is:
Therefore, magnitude of Earth's torsion field at its equator is:
where is the gravity of Earth. The torsion field direction coincides with the angular moment direction, i.e. north.
As the Earth is only approximately a homogeneous ball, from this calculation it follows that Earth's equatorial torsion field is about s−1 for the observer, fixed relative to the stars. Here were used the following data: the angular momentum of the Earth J • s, radius of the Earth m, the speed of gravity is assumed equal to the speed of light. Such a field is extremely weak and requires extremely sensitive measurements to be detected. One experiment to measure such field was the Gravity Probe B mission.
If we use the preceding formula for the second fastest-spinning pulsar known, PSR J1748-2446ad (which rotates 716 times per second), assuming its radius of km, and its mass as two solar masses, then we have
equals about s−1. This is simple estimation of the field. But the pulsar is spinning at a quarter of the speed of light at the equator, and its radius is only three times more than its Schwarzschild radius. When such fast motion and such strong gravitational field exist in a system, the simplified approach of separating gravitomagnetic and gravitoelectric forces can be applied only as a very rough approximation.
It is clear those charged and massive bodies that interact with each other two similar forces (Lorentz force for charges and gravitoelectromagnetic force for masses), and create around themselves in the space similar in shape and dependence on the movement electromagnetic and gravitational fields, may have even something more common. In particular, we can not exclude the fact that one field, one way or another does not affect the other field or strength of its interaction. There are some attempts to describe the connection of both fields, based on the similarity of the field equations. For example, Fedosin combines both fields into a single electrogravitational field.  Naumenko offered his version of combination of the fields.  Alekseeva builds the model of electro-gravitomagnetic field with the help of biquaternions.  The interaction of gravitation and electromagnetism is described in some papers of Evans. 
There are published articles that described a weak shielding of gravity of a test body: 1) With a superconducting disk, suspended with the help of Meissner effect.  The rotation of the disc increases the effect. 2) Using a disk of toroidal form.  The impact of rotation of superconducting disk on accelerometer is found in some experiments. 
Connection between the field of strong gravitation and the electromagnetic field of proton is given by the ratio of mass to charge of the particle. On the base of similarity of matter levels one can make the transformation of physical quantities and move from a proton to neutron star (magnetar as analogue of proton), with the replacement of strong gravitation by normal gravitation. It is assumed that magnetars not only have a strong magnetic field, but also a positive electric charge. Consideration of joint evolution of the neutron star and its constituent nucleons leads to the following conclusion: the maximum charge of object (neutron star or a proton) is restricted by condition of matter integrity under action of photons of electromagnetic radiation, associated with the charge of the object.  Then from the condition of equality of density of vacuum electromagnetic energy and the energy density of gravitation (derived from Le Sage's theory of gravitation), the assumption is that gravitons are particles like photons. In this case, since electrons are actively interacting with photons, we should expect the influence of electric currents in matter on distribution of gravitons and magnitude of gravitational forces. This approach allows explaining the above experiments with superconductors.
Another finding is interaction of strong gravitational field and electromagnetic field in a hydrogen atom, arising from the law of redistribution of energy flows. On the one hand, the equality of gravitational and electrical forces acting on atomic electron, can set the value of strong gravitation constant. On the other hand, there is a limit relation of equality of interaction energies of proton in magnetic field and gravitational torsion (gravitomagnetic) field of electron.
The concept of general field has brought together not only the electromagnetic and gravitational fields, but also other vector fields, including acceleration field, the pressure field, the dissipation field, the fields of strong and weak interactions in matter.  
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