Gravitomagnetic tensor is derived from General Relativity comparing the action for a continuous distribution of matter and the action for the electromagnetic field.
Similarly to the spin–orbit interaction there is a magnetic field acting on the star S2 induced by the supermassive black hole gravitational field. The gravitational magnetic force is derived from Special Relativity force transformations. Energy associated with this new force is stored in the dark matter energy tensor D (4, 0) related to the Riemann curvature.
Dark matter gravity is generated by D(4,0) energy related directly to the S(4,0) tensor, however this gravity is attributed to exotic particles never detected, galaxies in our universe are rotating with such speed that the gravity generated by their observable matter could not possibly hold them together.
Total Energy T(4,0) and energy tensors are defined to complete the General Relativity field equations. The Ricci decomposition is a way of breaking up the Riemann curvature tensor into three orthogonal tensors, Z(4,0), Weyl tensor C(4,0) and S(4,0).
Gravitomagnetic Tensor; General Relativity; Dark Matter; Expansion of the Universe; Riemann Curvature; Ricci Decomposition; Special Relativity; Gravitational Magnetic Field; Gravitational Magnetic Force; Star S2; Galactic Center Supermassive Black Hole , General Relativity Field Equations.
We shall consider a continuous distribution of matter whose velocity varies continuously from one point to the neighboring one. We shall set up an action principle for this matter in interaction with the gravitational field in the form [14].
(1)
Where Ig is the gravitational part of the action and Im is the matter part of the action [15].
(2)
and for Iem, the electromagnetic field action, we get for the total variation [16].
(3)
where 
is the stress-energy tensor of the electromagnetic field [16].
(4)
Comparing equations (2) and (3) is evident that gravitomagnetic tensor is defined by
(5)
Gravitational magnetic force on the star S2 orbiting the Galactic center supermassive black hole
In quantum physics, in the spin-orbit interaction there is a magnetic field B acting on the electron in the rest frame of the electron, where v is the velocity of the electron and E is the electric field, in the non-relativistic limit,
, c the speed of light, 
[11].
(6)
A particle without electric charge (q = 0) and rest mass m0 is orbiting a gravitational source with rest mass M0; m0 << M0, and velocity v at its closest approach distance r, since q = 0 the only force is the gravitational force.
The gravitational magnetic field is derived from Special Relativity force transformations [12], when velocities point to the same direction a repulsive gravitational magnetic force is induced FBg
(7)
When velocities point to the opposite direction an attractive gravitational magnetic force is induced FBg.
(8)
So, with v = 7650 km/s, r = 120 AU, M0 = 4.31 million Mo, m0 = 10 Mo, [13] we obtain the FBg value on the star S2 orbiting the Galactic center supermassive black hole at its closest approach
FBg = (0:00130399869725077405480262194772)Fg (9)
FBg = 46048607207718957314308202785:368Nw (10)
(11)
FBg is an extra attractive gravitational magnetic force pulling from the star; however this force is attributed to exotic particles never detected. You can also see discussions about dark matter and the expansion of the Universe in [10].
Conformal energy U defined as a combination of C and the Hodge dual of C, dark matter energy D defined as a combination of S and the Hodge dual of S, similar definitions for V/Z and T/R
The Ricci decomposition is a way of breaking up the Riemann curvature tensor into three orthogonal tensors, Z, Weyl tensor C and S, S tensor generates the dark matter gravity [6-8].
(12)
(13)
(14)
where Rabcd is the Riemann tensor, Rab is the Ricci tensor, R is the Ricci scalar (the scalar curvature).
The conformal energy tensor U can be defined as a combination of C and the Hodge dual of C [1-3].
(15)
The new dark matter energy tensor D can be defined as a combination of S and the Hodge dual of S
(16)
The new energy tensor V can be defined as a combination of Z and the Hodge dual of Z
(17)
The new Total Energy tensor T can be defined as a combination of the Riemann tensor R and the Hodge dual of R
(18)
The Hodge dual definition for Electromagnetic tensor and Weyl tensor [4].
(19)
(20)
The Hodge dual definition for dark matter S tensor, Z and R tensors
(21)
Weyl tensor C (4, 0) is related to the new Conformal Energy tensor U (4, 0). Dark matter tensor S (4, 0) is related to the new dark matter energy tensor D (4, 0). Z (4, 0) tensor is related to the new energy tensor V (4, 0). Riemann tensor R (4, 0) is related to the new Total Energy tensor T (4, 0).
(22)
(23)
(24)
(25)
(26)
(27)
(28)
In the general theory of relativity the Einstein field equations relate the geometry of spacetime to the distribution of matter [5].
(29)
Gravitomagnetic tensor leads to gravitomagnetic waves that will be detected in the future and gravitational magnetic forces.
The maximum acceleration on the star S2 due to the magnetic gravitational force is 0.0023144098 at its closest approach, the supermassive black hole gravitational field is inducing an extra attractive gravitational magnetic force pulling from the star, and however this force is attributed to exotic particles never detected.
When a charged particle is moving it generates a magnetic field that interacts with charged particles in motion, similarly when a particle with mass is moving it generates a gravitational magnetic field that interacts with other particles in motion.
This new force, the gravitational magnetic force has been detected by astronomers for decades but has been misinterpreted, in a galaxy there are no dark matter particles generating the dark matter force, moving stars generate gravitational magnetic fields that interact with other stars in motion.
The gravitational magnetic force is derived from Special Relativity force transformations, if Special Relativity is correct the gravitational magnetic force is irrefutable.
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