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, ^{[1]} and in some experiments. ^{[2]}^{ }^{[3]}
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.^{[4]} 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).^{[5]} ^{[6]} All of those observed
properties could be explained in terms of gravitomagnetic effects.^{[7]} Application of Penrose's mechanism can be applied to
black holes of any size.^{[8]} 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
massenergy) 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. ^{[9]}
Subsequently, Agop, Buzea and Ciobanu, ^{[10]} and
others have confirmed the validity of GEM equations in International System of
Units in the following form:
where:
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:
where:
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 , ^{[11]} or in
some papers . ^{[12]} ^{[13]}
The above equations of the gravitational field (equation
GEM) can be compared with Maxwell's equations:
where:
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 Lorentzinvariant theory of gravitation
(LITG), derived gravitational equation in special relativity. ^{[14]}
where:
The equations coincide with equations which were first
published in 1893 by Oliver Heaviside as a separate theory expanding Newton's
law. ^{[15]}
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 Maxwelllike
gravitational equations.
The gravitational force in LITG is as follows:
In contrast to general relativity, where spin of
gravitons is equal to 2, Lorentzinvariant 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. ^{[16]} ^{[17]}
^{[18]} ^{[19]}
Some higherorder 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: ^{[14]}
where is angular momentum of the body, is radiusvector 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. ^{[20]}
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
ballshaped body is:
where:
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
fastestspinning pulsar known, PSR J17482446ad (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. ^{[14]} Naumenko offered his version of combination of the fields. ^{[21]} Alekseeva builds the model of
electrogravitomagnetic field with the help of biquaternions. ^{[22]} The interaction of gravitation and electromagnetism
is described in some papers of Evans. ^{[23]}
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. ^{[24]} The rotation
of the disc increases the effect. 2) Using a disk of toroidal form. ^{[25]} The impact of rotation of superconducting disk on
accelerometer is found in some experiments. ^{[26]}
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. ^{[27]} 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. ^{[28]} ^{[29]}
1.
Everitt, C.W.F., et al., Gravity Probe B: Countdown to Launch. In: Laemmerzahl, C., Everitt, C.W.F., Hehl,
F.W. (Eds.), Gyros, Clocks, Interferometers...: Testing Relativistic Gravity in
Space. – Berlin, Springer, 2001, pp. 52–82.
2.
Fomalont E.B., Kopeikin S.M. The Measurement of the Light
Deflection from Jupiter: Experimental Results (2003), Astrophys.
J., 598, 704. (astroph/0302294)
3.
Graham, R.D., Hurst, R.B., Thirkettle, R.J.,
Rowe, C.H., and Butler, B.H., "Experiment to Detect FrameDragging in a
Lead Superconductor," (2007). [1]
4.
Roger Penrose (1969). "Gravitational collapse: The role
of general relativity". Rivista de Nuovo Cimento, Numero Speciale 1: 252–276.
5.
R.K. Williams (1995). "Extracting x rays, Ύ rays, and
relativistic e^{−}e^{+} pairs from supermassive Kerr black
holes using the Penrose mechanism". Physical Review 51 (10): 5387–5427.
6.
R.K. Williams (2004). "Collimated escaping vortical
polar e^{−}e^{+} jets intrinsically produced by rotating black
holes and Penrose processes". The Astrophysical Journal 611: 952–963. doi:10.1086/422304.
7.
R.K. Williams (2005). "Gravitomagnetic field and
Penrose scattering processes". Annals of the New York Academy of Sciences 1045: 232–245.
8.
R.K. Williams (2001). "Collimated energymomentum
extraction from rotating black holes in quasars and microquasars
using the Penrose mechanism". AIP Conference Proceedings 586: 448–453 (arXiv:
astroph/0111161).
9.
R.P. Lano (1996).
"Gravitational Meissner Effect". arXiv: hepth/9603077.
10.
M. Agop, C. Gh.
Buzea, B. Ciobanu (1999). "On Gravitational
Shielding in Electromagnetic Fields". arXiv: physics/9911011.
11.
M. L. Ruggiero, A. Tartaglia. Gravitomagnetic effects. Nuovo Cim. 117B
(2002) 743—768 ( grqc/0207065
), формулы (24) и (26).
12.
Mashhoon, Gronwald, Lichtenegger (19991208). "Gravitomagnetism and the Clock Effect".
arXiv: General Relativity and Quantum Cosmology 9912027.
13.
Clark, S J; R W Tucker (2000). "Gauge symmetry and gravitoelectromagnetism". Class. Quantum Grav. 17: 41254157.
14.
^{14.0} ^{14.1} ^{14.2} Fedosin, S.G. (1999), written at Perm, pages 544, Fizika i filosofiia podobiia ot preonov
do metagalaktik, ISBN 5813100121.
15.
Heaviside, Oliver, A gravitational
and electromagnetic analogy. The Electrician, 1893.
16.
Nyambuya G.G. Fundamental Physical Basis for
MaxwellHeaviside Gravitomagnetism. Journal of Modern
Physics, Vol. 6, pp. 12071219 (2015). http://dx.doi.org/10.4236/jmp.2015.69125.
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W.D., Japaridze G.S. Photon deflection and precession
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(2004). https://doi.org/10.1088/02649381/21/7/007.
18. Behera H.
Comments on gravitoelectromagnetism of Ummarino and Gallerati in “Superconductor in a weak static gravitational
field” vs other versions. Eur. Phys. J. C. Vol. 77, Article number 822 (2017). https://doi.org/10.1140/epjc/s1005201753864.
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20.
Fedosin S.G. Fizicheskie
teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, Tabl.
21, Pic. 41, Ref. 289. ISBN 9785990195110. (in Russian).
21.
Naumenko Y.V. Unified Theory of Vector Fields
(from Maxwell's electrodynamics to a unified field theory). Armavir,
Armavir polygraph company, 2006. (In Russian)
22.
Alekseeva L.A. One biquaternionic
model of electrogravimagnetic field. Field analogs of Newton's laws. 11
Mar. 2007. (In Russian).
23.
Myron W. Evans. Gravitational
Poynting theorem: interaction of gravitation and electromagnetism. Paper
168. Alpha
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24.
Eugene Podkletnov and R. Nieminen. A Possibility
of Gravitational Force Shielding by Bulk YBa2Cu3O7x Superconductor, Physica C, 1992, pp. 441443.
25.
E. Podkletnov and A.D. Levit. Gravitational
shielding properties of composite bulk Y Ba2Cu3O7x superconductor below 70 K
under electromagnetic field, Tampere University of Technology report MSUchem,
January 1995.
26.
M. Tajmar, et. al. Measurement of Gravitomagnetic and
Acceleration Fields Around Rotating Superconductors. 17 October 2006.
27.
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2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. ISBN
9785990195110. (in Russian).
28.
Fedosin S.G. The procedure of
finding the stressenergy tensor and vector field equations of any form.
Advanced Studies in Theoretical Physics, Vol. 8, no. 18, pp. 771779 (2014). http://dx.doi.org/10.12988/astp.2014.47101.
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Source: http://sergf.ru/gmen.htm