Dark matter gravity force is generated by the expansion of the universe, gravitational force is not strong enough and an extra centripetal force is needed due to this expansion, this model also explains the high angular momentum of the Earth–Moon system, energy related to this new force is stored in the dark matter energy tensor D (4, 0) related to the Riemann curvature.

The cosmological redshift energy loss due to the expansion of the universe is stored in the dark matter energy tensor D (4, 0). The Cosmic Microwave Background energy loss has been stored in the dark matter energy tensor D (4, 0).

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).

Dark Matter; Expansion of the Universe; Riemann Curvature; Earth Moon System; Cosmic Microwave Background Redshift; General Relativity Field Equations

**Dark matter gravity force is generated by the expansion of the universe**

In an expanding universe a satellite is in a uniform circular orbit around a planet, satellite mass m *<< *planet mass M, initial radius *r*_{0}, initial constant angular velocity *ω*_{0}, initial velocity *v*_{0} *<< c*, c the speed of light, due to the expansion of the universe the final radius *r*_{f} = *ar*_{0}*, a > *1, final constant angular velocity *ω*_{f}, initial gravitational force *F*_{0g} equals centripetal force.

In the expansion of the universe angles remain invariant, so *ω*_{0} = *ω*_{f} and gravitational force *F*_{g} equals centripetal force

Since *ω < ω*_{0}, gravitational force *F*_{g} is not strong enough and an extra centripetal force is needed, this extra force is attributed to exotic particles never detected.

*E*_{dm} = *E*_{f} − *E*_{0} = *T*_{0}*a*^{2} − 2*T*_{0}*a*^{−}^{1} + *T*_{0}

Dark Matter energy *E*_{dm} has been stored in the dark matter energy tensor D (4, 0) defined below, differentiating we get the dark matter force *F*_{dm}

F_{dm} = 2T_{0}a + 2T_{0}a^{−2}

_{This model also explains the high angular momentum of the Earth Moon system [7]. }

_{}

_{The definition of the dark matter energy leads to think that dark matter effect is zero in a newborn galaxy (a = 1) and is increasing when it is becoming older while the universe is expanding (increasing a). I suggest comparing observational data of dark matter effect for similar galaxies nowadays and in the past in order to confirm this equation.}

**Cosmological redshift energy loss due to the ex****pansion of the universe**

In an expanding closed universe, regardless our universe is closed or open, the final wave length of the photon when the closed universe reaches its maximum radius is [6].

where h is the Planck constant and *E*_{l} is the energy loss, in a closed universe after reaching its maximum expansion the universe comes backwards and will reach *r*_{now} again restoring the loss of energy to the photon, so this *E*_{l} has been stored and was not definitely lost.

The cosmological redshift energy loss due to the expansion of the universe is stored in the dark matter energy tensor D (4, 0) related directly to the S (4, 0) tensor; however this gravity is attributed to exotic particles never detected.

The Cosmic Microwave Background energy loss has been stored in the dark matter energy tensor D (4, 0); gravitational waves and other fields are subject to the same redshift phenomena.

**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 [8] [9] [10].

where Rabcd is the Riemann tensor, Rab is the Ricci tensor, R is the Ricciscalar (the scalar curvature)

The conformal energy tensor U can be defined as a combination of C and the Hodge dual of C [1] [2] [3].

The new dark matter energy tensor **D **can be defined as a combination of **S **and the Hodge dual of **S**

The new energy tensor V can be defined as a combination of Z and the Hodge dual of **Z**

The new Total Energy tensor T can be defined as a combination of the Riemann tensor **R** and the Hodge dual of **R**

The Hodge dual definition for Electromagnetic tensor and Weyl tensor [4]

The Hodge dual definition for dark matter S tensor, Z and R tensors

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).

The complete field equations are described by a new T(4,0) tensor for Total Energy, the new conformal energy tensor U(4,0), the new energy tensor V(4,0) and the new dark matter energy tensor D(4,0).

In the general theory of relativity the Einstein field equations relate the geometry of spacetime to the distribution of matter [5].

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