Abstract

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).

Index Terms

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 r₀, initial constant angular velocity ω₀, initial velocity v₀ << c, c the speed of light, due to the expansion of the universe the final radius r_f = ar₀, 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 ω₀ = ω_f and gravitational force F_g equals centripetal force

Since ω < ω₀, 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 − E₀ = T₀a² − 2T₀a⁻¹ + T₀

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 = 2T₀a + 2T₀a⁻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 expansion 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

Hodge Dual Definitions

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).

Complete General Relativity Field Equations

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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References

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2. Roger Penrose (2006) El camino a la realidad. Random House Mondadori, Barcelona 619.
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6. Frank J Tipler (1996) La Fsica de la inmortalidad. Alianza Editorial, Madrid 572.
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9. Sharpe RW (1997) Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York. Section 6.1 discusses the decomposition. Versions of the decomposition also enter into the discussion of conformal and projective geometries, in chapters 7 and 8.
10. Singer IM, Thorpe JA (1969) The curvature of 4-dimensional Einstein spaces, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press 355-365.