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Gravitomagnetism

Gravimagnetism (sometimes Gravitomagnetism, Gravitoelectromagnetism, 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. The GEM equations coincide with equations which were first published in 1893, before general relativity, by Oliver Heaviside as a separate theory expanding Newton's law.

1 Background
2 Equations

2.1 In Planck units
2.2 Lorentz force

3 Higher-order effects
4 Torsion field of Earth
5 Gravimagnetic field of a pulsar
6 The interaction between electromagnetic and gravitational fields
7 See also
8 References
9 Further reading
10 External links

Background

Gravimagnetic forces and the corresponding field (gravitomagnetic field and 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 magnetic field and therefore magnetic force is possible.

Currently, verification of gravimagnetic forces are doing with the help of satellites, ^[1] and in some experiments. ^[2][3]

Indirect validations of gravimagnetic 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 gravimagnetic 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 gravimagnetism.

A group at Stanford University is currently analyzing data from the first direct test of GEM, the Gravity Probe B satellite experiment, to see if they are consistent with gravimagnetism. The Apache Point Observatory Lunar Laser-ranging Operation also plans to observe gravimagnetism effects.

Equations

According to general relativity, the 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. ^[9] Sergey Fedosin with the help of Lorentz-invariant theory of gravitation (LITG), derived the same equation in special relativity, and then compiled them for non-inertial reference systems using mathematical general relativity approach to finding the metric of spacetime. ^[10] Subsequently, Agop, Buzea and Ciobanu, ^[11] and others have confirmed the validity of GEM equations in International System of Units in the following form: ^[12] ^[13]

GEM equations	Maxwell's equations
$\nabla \cdot \mathbf{G} = -4 \pi \gamma \rho \$	$\nabla \cdot \mathbf{E} = \frac{\rho_\text{em} }{\epsilon_0} \$
$\nabla \cdot \mathbf{\Omega } = 0 \$	$\nabla \cdot \mathbf{B} = 0 \$
$\nabla \times \mathbf{G} = -\frac{\partial \mathbf{\Omega } } {\partial t} \$	$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B} } {\partial t} \$
$\nabla \times \mathbf{\Omega } = -\frac{4 \pi \gamma }{c^2_g} \mathbf{J} + \frac{1}{c^2_g} \frac{\partial \mathbf{G}} {\partial t}$	$\nabla \times \mathbf{B} = \frac{1}{\epsilon_0 c^2} \mathbf{J}_\text{em} + \frac{1}{c^2} \frac{\partial \mathbf{E}} {\partial t}$

where:

$~ \mathbf{G}$ is gravitational field strength or gravitational acceleration, also called gravielectric for the sake of analogy;
$~\mathbf{\Omega}$ is intensity of torsion field or simply torsion, also called gravitomagnetic field;
E is electric field;
B is magnetic field;
ρ is mass density of moving substance;
ρ_em is charge density of moving substance;
J is mass current density (J = ρ v_ρ, where v_ρ is the velocity of the mass flow generating the gravimagnetic field);
J_em is electric current density;
$γ$ is gravitational constant;
ε₀ is vacuum permittivity;
c_g is speed of propagation of gravity (equal to, by general relativity, the speed of light).

The equations in such form were actually published in 1893 by Oliver Heaviside as a separate theory expanding Newton's law.^[14]

The similarity of gravitational equations and Maxwell's equations for electromagnetic field highlighted in Maxwell-like gravitational equations.

In Planck units

From comparison of GEM equations and Maxwell's equations in Table it is obvious that −1/(4πγ) is the gravitational analog of vacuum permittivity ε₀. Adopting Planck units normalizes γ , c and 1/(4πε₀) to 1, thereby eliminating these constants from both sets of equations. The two sets of equations then become identical but for the minus sign preceding 4π in the GEM equations. These minus signs stem from an essential difference between gravitation and electromagnetism: electrostatic charges of identical sign repel each other, while masses attract each other. Hence the GEM equations are simply Maxwell's equations with mass (or mass density) substituting for charge (or charge density), and − γ replacing the Coulomb force constant 1/(4πε₀).

4π appears in both the GEM and Maxwell equations, because Planck units normalize γ and 1/(4πε₀) to 1, and not 4π γ and ε₀.

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:

$\mathbf{F}_{m} = m \left( \mathbf{G} + k \mathbf{v} \times \mathbf{\Omega } \right) ,$

where:

m is the mass of the test particle;
v is the instantaneous velocity of the test particle.

Acceleration of any test particle is simply:

$\mathbf{a} = \mathbf{G} +k \mathbf{v} \times \mathbf{\Omega } .$

The second component of the gravitational force responsible for the collimation of relativistic jets in the torsion (gravitomagnetic) fields of galaxies, active galactic nuclei and rapidly rotating stars (eg, jet accreting neutron stars).

In general relativity, due to the alleged tensor nature of gravitation considered that the effective mass for the torsion field is twice the usual body mass, so that was adopted . 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 and body mass for gravitational and torsion fields is the same.

Higher-order effects

Some higher-order gravimagnetic effects can reproduce effects reminiscent of the interactions of more conventional polarized charges. In 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 gravimagnetic component. When disks rotate in opposite directions, the energy is 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 other disk.

Due to the torsion field becomes possible effect of gravitational induction.

Torsion field of Earth

Formula for torsion field $~ \mathbf {\Omega}$ near a rotating body can be derived from the GEM equations and is: ^[10]

$\mathbf{\Omega } = \frac{\gamma }{2 c^2_g} \frac{\mathbf{L} - 3(\mathbf{L} \cdot \mathbf{r}/r) \mathbf{r}/r}{r^3},$

where $~ \mathbf {L}$ is angular momentum of the body, $~\mathbf {r}$ is radius-vector from the center of the body to the point, where defined the torsion field.

A detailed derivation of this formula is contained in the book. ^[15] At the equatorial plane, r and L are perpendicular, so their dot product vanishes, and this formula reduces to:

$\mathbf{\Omega } = \frac{\gamma }{2 c^2_g } \frac{\mathbf{L}}{r^3}.$

Magnitude of angular momentum of a homogeneous ball-shaped body is:

$L=\omega I_\text{ball} = \frac{2 m r^2}{5} \frac{2 \pi}{T},$

where:

$I_\text{ball} = \frac{2 m r^2}{5}$ is the moment of inertia of a ball-shaped body;
$\omega \$ is angular velocity;
m is mass;
r is radius;
T is rotational period.

Therefore, magnitude of Earth's torsion field at its equator is:

$\Omega_\text{Earth} = \frac{\gamma }{5 c^2_g} \frac{m}{r} \frac{2 \pi}{T} = \frac{2 \pi r g}{5c^2_g T},$

where $g = \frac{\gamma m}{r^2}$ 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 ball, from this calculation it follows that Earth's equatorial gravimagnetic field is about $\Omega_\text{Earth} = 8.5 \cdot 10^{-15}$ rad/s 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 $~ r_\text{Earth}=6.378 \cdot 10^{6}$ 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.

Gravimagnetic field of a pulsar

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 16 km, and its mass as two solar masses, then we have

$\Omega = \frac{2 \pi \gamma m}{5rc^2_g T}$

equals about $1.7 \cdot 10^{2}$ rad/s. 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 gravimagnetic and gravielectric forces can be applied only as a very rough approximation.

The interaction between electromagnetic and gravitational fields

It is clear that charged and massive bodies that interact with each other two similar forces (Lorentz force for charges and gravimagnetic 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 sharing of both fields, based on the similarity of the field equations. For example, Fedosin combines both fields into a single electrogravitational field. ^[10] Naumenko offered his version of combination of the fields. ^[16] Alekseeva builds model of electro-gravimagnetic field with the help of biquaternions. ^[17] The interaction of gravitation and electromagnetism is described in some papers of Evans. ^[18]

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. ^[1⁹^] The rotation of the disc increases the effect. 2) Using a disk of toroidal form. ^[²⁰^] The impact of rotation of superconducting disk on accelerometer is found in some experiments. ^[2¹^]

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 in 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 substance integrity under action of photons of electromagnetic radiation, associated with the charge of the object. ^[22]

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 interact with photons, we should expect the influence of electric currents in substance 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.

References

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.
Fomalont E.B., Kopeikin S.M. The Measurement of the Light Deflection from Jupiter: Experimental Results (2003), Astrophys. J., 598, 704. (astro-ph/0302294)
Graham, R.D., Hurst, R.B., Thirkettle, R.J., Rowe, C.H., and Butler, B.H., "Experiment to Detect Frame-Dragging in a Lead Superconductor," (2007). [1]
Roger Penrose (1969). "Gravitational collapse: The role of general relativity". Rivista de Nuovo Cimento, Numero Speciale 1: 252–276.
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.
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.
R.K. Williams (2005). "Gravitomagnetic field and Penrose scattering processes". Annals of the New York Academy of Sciences 1045: 232–245.
R.K. Williams (2001). "Collimated energy-momentum extraction from rotating black holes in quasars and microquasars using the Penrose mechanism". AIP Conference Proceedings 586: 448–453 (arXiv: astro-ph/0111161).
R.P. Lano (1996). "Gravitational Meissner Effect". arXiv: hep-th/9603077.
^10.0 ^10.1 ^10.2 Fedosin, S.G. (1999), written at Perm, pages 544, Fizika i filosofiia podobiia ot preonov do metagalaktik, ISBN 5-8131-0012-1.
M. Agop, C. Gh. Buzea, B. Ciobanu (1999). "On Gravitational Shielding in Electromagnetic Fields". arXiv: physics/9911011.
B. Mashhoon, F. Gronwald, H.I.M. Lichtenegger (1999). "Gravitomagnetism and the Clock Effect". arXiv: gr-qc/9912027.
S.J. Clark, R.W. Tucker (2000). "Gauge symmetry and gravito-electromagnetism". Classical and Quantum Gravity 17: 4125–4157. doi:10.1088/0264-9381/17/19/311.
Oliver Heaviside (1893). "A gravitational and electromagnetic analogy". The Electrician 31: 81–82.
Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. ISBN 978-5-9901951-1-0. (in Russian).
Naumenko Y.V. Unified Theory of Vector Fields (from Maxwell's electrodynamics to a unified field theory). Armavir, Armavir polygraph company, 2006. (In Russian)
Alekseeva L.A. One biquaternionic model of electro-gravimagnetic field. Field analogs of Newton's laws. 11 Mar. 2007. (In Russian).
Myron W. Evans. Gravitational Poynting theorem: interaction of gravitation and electromagnetism. Paper 168. Alpha Institute for Advanced Studies (AIAS).
Eugene Podkletnov and R. Nieminen. A Possibility of Gravitational Force Shielding by Bulk YBa2Cu3O7-x Superconductor, Physica C, 1992, pp. 441-443.
E. Podkletnov and A.D. Levit. Gravitational shielding properties of composite bulk Y Ba2Cu3O7-x superconductor below 70 K under electro-magnetic field, Tampere University of Technology report MSU-chem, January 1995.
M. Tajmar, et. al. Measurement of Gravitomagnetic and Acceleration Fields Around Rotating Superconductors. 17 October 2006.
Fedosin S.G. Comments to the book: Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. ISBN 978-5-9901951-1-0. (in Russian).

External links

Gyroscopic Superconducting Gravitomagnetic Effects news on tentative result of European Space Agency (esa) research
In Search of gravitomagnetism, NASA, 20 April 2004.
Gravitomagnetic London Moment-New test of General Relativity?
Test of the Lense-Thirring effect with the MGS Mars probe, New Scientist, January 2007.
Gravimagnetism in Russian

v • d • e

Theories of gravitation

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History of gravitational theory
Newtonian gravity (NG)

Classical mechanics

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Gravimagnetism

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Yilmaz

Source: http://serg.fedosin.ru/gmen.htm

On the list of pages

На русском языке

Gravitomagnetism

Contents

Background

Equations

In Planck units

Lorentz force

Higher-order effects

Torsion field of Earth

Gravimagnetic field of a pulsar

The interaction between electromagnetic and gravitational fields

See also

References

Further reading

External links