# Physics/Essays/Fedosin/Gravitoelectromagnetism

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.

## Background

Gravitomagnetic forces and corresponding field (gravitomagnetic field in general relativity and gravitational torsion field in alternatives to general relativity) should be considered in all reference frames that move relative to a source of 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.

## Equations

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, 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:
    $~\nabla \cdot \mathbf {E_{g}} =-4\pi G\rho ,$
    $~\nabla \cdot \mathbf {B_{g}} =0,$
    $~\nabla \times \mathbf {E_{g}} =-{\frac {\partial \mathbf {B_{g}} }{\partial t}},$
    $~\nabla \times \mathbf {B_{g}} ={\frac {1}{c^{2}}}\left(-4\pi G\mathbf {J} +{\frac {\partial \mathbf {E_{g}} }{\partial t}}\right),$
where:
* $~\mathbf {E_{g}}$ is gravitoelectric field;
* $~\mathbf {B_{g}}$ is called gravitomagnetic field;
* G is gravitational constant;
* ρ is mass density of moving matter;
* J is mass current density (J = ρ v$_ρ$, where v$_ρ$ is velocity of the mass current);
* c is speed of light.

### Lorentz force

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:
    $\mathbf {F} _{m}=m\left(\mathbf {E_{g}} +k\mathbf {v} \times \mathbf {B_{g}} \right),$
where:
* m is mass of test particle;
* v is instantaneous velocity of test particle.
Acceleration of any test particle is simply:
    $\mathbf {a} =\mathbf {E_{g}} +k\mathbf {v} \times \mathbf {B_{g}} .$
The second component of the gravitational force is responsible for collimation of relativistic jets in gravitomagnetic fields of galaxies, active galactic nuclei and rapidly rotating stars (eg, jet accreting neutron stars).
In general relativity, it is assumed that either $~k=2$, or in some papers $~k=4$.

### Comparison with electromagnetism

The above equations of gravitational field (equation GEM) can be compared with Maxwell's equations:
    $~\nabla \cdot \mathbf {E} ={\frac {\rho _{q}}{\varepsilon _{0}}},$
    $~\nabla \cdot \mathbf {B} =0,$
    $~\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}},$
    $~\nabla \times \mathbf {B} ={\frac {1}{c^{2}}}\left({\frac {\mathbf {j} }{\varepsilon _{0}}}+{\frac {\partial \mathbf {E} }{\partial t}}\right),$
where:
* $~\mathbf {E}$ is the electric field,
* $~\varepsilon _{0}$ is the electric constant,
* $~\mathbf {B}$ is the magnetic field,
* $~\mathbf {j} =\rho _{q}\mathbf {v} _{q}$ is the charge current density,
* $~\rho _{q}$ is charge density of moving matter,
* $~\mathbf {v} _{q}$ is the velocity of the electric current which produces electric and magnetic fields,
* $~c$ is the speed of light.
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 $~q$, is given by:
    $~\mathbf {F} _{q}=q\mathbf {E} +q\mathbf {v} _{q}\times \mathbf {B} .$

## Comparison with LITG

Sergey Fedosin with the help of Lorentz-invariant theory of gravitation (LITG), derived gravitational equation in special relativity.
In the electromagnetic wave representation these equations have the form:
    $~\nabla \cdot \mathbf {\Gamma } =-4\pi G\rho ,$
    $~\nabla \times \mathbf {\Gamma } =-{\frac {\partial \mathbf {\Omega } }{\partial t}},$
    $~\nabla \cdot \mathbf {\Omega } =0,$
    $~\nabla \times \mathbf {\Omega } ={\frac {1}{c^{2}}}\left(-4\pi G\mathbf {J} +{\frac {\partial \mathbf {\Gamma } }{\partial t}}\right),$
where:
* $~\mathbf {\Gamma }$ is gravitational field strength,
* $~\mathbf {\Omega }$ is gravitational torsion field, which has dimension as frequency.
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:
    $~\mathbf {F} _{m}=m\mathbf {\Gamma } +m\mathbf {v} _{m}\times \mathbf {\Omega } .$
The above equations are also presented in the articles.

### Higher-order effects

Some higher-order effects in LITG can reproduce effects reminiscent of interactions of more conventional polarized charges. In gravitational torsion field $~\mathbf {\Omega }$ appears momentum of force acting on a rotating particle with the spin $~\mathbf {L}$:
    $~\mathbf {K} ={\frac {1}{2}}\mathbf {L} \times \mathbf {\Omega } .$
This leads to precession of the particle spin with angular velocity $~\mathbf {w} =-{\frac {\mathbf {\Omega } }{2}}$ around direction of $~\mathbf {\Omega }$.
The mechanical energy of the particle with spin in torsion field will be:
    $~U=-{\frac {1}{2}}\mathbf {L} \cdot \mathbf {\Omega } .$
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
    $~\mathbf {F} ={\frac {1}{2}}\nabla \left(\mathbf {L} \cdot \mathbf {\Omega } \right),$
where torsion field $~\mathbf {\Omega }$ of one disk acts on the angular momentum $~\mathbf {L}$ of another disk.
Due to the torsion field becomes possible effect of gravitational induction.

### Torsion field of the Earth and pulsar

The formula for torsion field $~\Omega$ near a rotating body can be derived from the Heaviside equations and is:
    $~\mathbf {\Omega } ={\frac {G}{2c^{2}}}{\frac {\mathbf {L} -3(\mathbf {L} \cdot \mathbf {r} /r)\mathbf {r} /r}{r^{3}}},$
where $~\mathbf {L}$ is angular momentum of the body, $~\mathbf {r}$ 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:
    $\mathbf {\Omega } ={\frac {G}{2c^{2}}}{\frac {\mathbf {L} }{r^{3}}}.$
Magnitude of angular momentum of a homogeneous ball-shaped body is:
    $L=\omega I_{\text{ball}}={\frac {2ma^{2}}{5}}{\frac {2\pi }{T}},$
where:
* $I_{\text{ball}}={\frac {2ma^{2}}{5}}$ is the moment of inertia of a ball-shaped body;
* $\omega \$ is angular velocity;
* m is mass;
* a is the body radius;
* T is rotational period.
Therefore, magnitude of Earth's torsion field at its equator is:
    $\Omega _{\text{Earth}}={\frac {G}{5c^{2}}}{\frac {m}{R}}{\frac {2\pi }{T}}={\frac {2\pi Rg}{5c^{2}T}},$
where $g={\frac {Gm}{R^{2}}}$ 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 $\Omega _{\text{Earth}}=8.5\cdot 10^{-15}$ s$^{−1}$ for the observer, fixed relative to the stars. Here were used the following data: the angular momentum of the Earth $~L_{\text{Earth}}=5.879\cdot 10^{33}$ J • s, radius of the Earth $~R=6.378\cdot 10^{6}$ m. Such a field is extremely weak and requires very sensitive measurements to be detected. One experiment to measure the 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 $R_{p}=16$ km, and its mass as two solar masses, then we have
    $\Omega ={\frac {2\pi Gm}{5R_{p}c^{2}T}}$
equals about $1.7\cdot 10^{2}$ 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.

## Interaction between electromagnetic and gravitational fields

It is clear that charged and massive bodies which interact with each other by two similar forces (Lorentz force for charges and gravitoelectromagnetic force for masses), and create around themselves in the space similar in shape 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 similarity of 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 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 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 a 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. In addition, it becomes possible to imagine the operating principle of an engine that draws energy from electrogravitational vacuum.
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 gravitational 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. For vector fields, it was possible not only to present all possible results of Lagrangian formalism for curved space-time in continuously distributed matter, but also to derive covariant formulas for generalized four-momentum and for total four-momentum of a physical system with particles and fields.