_{1}

In a previous paper, we demonstrated that the linearized general relativity could explain dark matter (the rotation speed of galaxies, the rotation speed of dwarf satellite galaxies, the movement in a plane of dwarf satellite galaxies, the decreasing quantity of dark matter with the distance to the center of galaxies’ cluster, the expected quantity of dark matter inside galaxies and the expected experimental values of parameters Ω
_{dm}
of dark matter measured in CMB). It leads, compared with Newtonian gravitation, to taking in account the second component (gravitational field) of the gravitation (imposed by general relativity)
without changing the gravity field (also known as gravitomagnetism). In this explanation, dark matter would be a uniform gravitational field that embeds some very large areas of the universe generated by the clusters. In this article we are going to see that this specific gravitational field, despite its weakness, could be soon detectable, allowing testing this explanation of dark matter. It should generate a slight discrepancy in the expected measure of the Lense-Thirring effect of the Earth. In this theoretical frame, the Lense-Thirring effect of the “dark matter” would be a value between around 0.3 milliarcsecond/year and 0.6 milliarcsecond/year in the best case. In the LAGEOS or Gravity Probe B experiments, there was not enough precision (around 0.3% for the expected 6606 mas·y
^{﹣1}
geodetic and around 19% for the expected 39 mas·y
^{﹣1}
frame-dragging precessions). In the GINGER experiment, there could be enough one; the expected accuracy would be around 1%. If this discrepancy was verified, it would be the first direct measure of the dark matter.

One of the most important mysteries of astrophysics is the problem of dark matter. This latter component represents at least five times the quantity of the ordinary mass. And until now, this term cannot be explained. We therefore find that our theories operate in a highly coherent and precise manner both on our scale and at large astrophysical scales. But as far as large scales are concerned, this coherence and precision is only possible on the condition of making the hypothesis of the existence of this new term of dark matter. A way to solve this problem is to propose new theories (MOND theories for example), but we can note that the term of dark matter, even if it is an ad hoc term, doesn’t generate any contradiction inside our current theories. In fact, we could even pretend that this term demonstrates the extraordinary consistency of our current theories because of the multiplicity of the ways to deduce the quantity of this term leading to its more and more coherent and precise measure. Another way is to propose that this term represents a new exotic matter. This explanation is more shared and certainly more studied. But until now, no new matter has been directly detected, despite more and more experiences. A third way is to propose an explanation in the frame of current theories. That’s the purpose of the present work. The term of dark matter will be explained by a physical phenomenon of general relativity that is generally neglected.

General relativity implies the existence of two gravitational components. In addition to the gravity field, there is a gravitational field (together giving what is called the gravitomagnetism) just like the magnetic field in electromagnetism. The new gravitational field can be measured by its precession effect, known as Lense-Thirring effect. Several experiments have validated this effect for the Earth gravitational field, NASA's LAGEOS satellites or Gravity Probe B [

In [

Before recalling the theoretical idealization used and some of the results of [

From general relativity, one deduces the linearized general relativity in the approximation of a quasi-flat Minkowski space ( g μ ν = η μ ν + h μ ν ; | h μ ν | ≪ 1 ). With the following Lorentz gauge, it gives the following field equations as in [

□ = 1 c 2 ∂ 2 ∂ t 2 − Δ ):

∂ μ h ¯ μ ν = 0 ; □ h ¯ μ ν = − 2 8 π G c 4 T μ ν (1)

With:

h ¯ μ ν = h μ ν − 1 2 η μ ν h ; h ≡ h σ σ ; h ν μ = η μ σ h σ ν ;

h ¯ = − h (2)

The general solution of these equations is:

h ¯ μ ν ( c t , x ) = − 4 G c 4 ∫ T μ ν ( c t − | x − y | , y ) | x − y | d 3 y (3)

In the approximation of a source with low speed, one has:

T 00 = ρ c 2 ; T 0 i = c ρ u i ; T i j = ρ u i u j (4)

And for a stationary solution, one has:

h ¯ μ ν ( x ) = − 4 G c 4 ∫ T μ ν ( y ) | x − y | d 3 y (5)

At this step, by proximity with electromagnetism, one traditionally defines a scalar potential φ and a vector potential H i . There are in the literature several definitions as in [

h ¯ 00 = 4 φ c 2 ; h ¯ 0 i = 4 H i c ; h ¯ i j = 0 (6)

With gravitational scalar potential φ and gravitational vector potential H i :

φ ( x ) ≡ − G ∫ ρ ( y ) | x − y | d 3 y H i ( x ) ≡ − G c 2 ∫ ρ ( y ) u i ( y ) | x − y | d 3 y = − K − 1 ∫ ρ ( y ) u i ( y ) | x − y | d 3 y (7)

With K a new constant defined by:

G K = c 2 (8)

This definition gives K − 1 ~ 7.4 × 10 − 28 very small compare to G .

The field equations can be then written (Poisson equations):

Δ φ = 4 π G ρ ; Δ H i = 4 π G c 2 ρ u i = 4 π K − 1 ρ u i (9)

With the following definitions of g (gravity field) and k (gravitational field), those relations can be obtained from the following equations (also called gravitomagnetism):

g = − g r a d φ ; k = r o t H r o t g = 0 ; d i v k = 0 ; d i v g = − 4 π G ρ ; r o t k = − 4 π K − 1 j p (10)

With the Equations (2), one has:

h 00 = h 11 = h 22 = h 33 = 2 φ c 2 ; h 0 i = 4 H i c ; h i j = 0 (11)

The equations of geodesics in the linear approximation give:

d 2 x i d t 2 ~ − 1 2 c 2 δ i j ∂ j h 00 − c δ i k ( ∂ k h 0 j − ∂ j h 0 k ) v j (12)

It then leads to the movement equations:

d 2 x d t 2 ~ − g r a d φ + 4 v ∧ ( r o t H ) = g + 4 v ∧ k (13)

Remark: Afterwards, we will use relations defined with the parameterized post-Newtonian formalism (PPN). All previous relations can be retrieved starting with the PPN formalism. From [

g 0 i = − 1 2 ( 4 γ + 4 + α 1 ) V i ; V i ( x ) = G c 2 ∫ ρ ( y ) u i ( y ) | x − y | d 3 y (14)

The gravitomagnetic field and its acceleration contribution are:

B g = ∇ ∧ ( g 0 i e i ) ; a g = v ∧ B g (15)

And in the case of general relativity (that is our case):

γ = 1 ; α 1 = 0 (16)

It then gives:

g 0 i = − 4 V i ; B g = ∇ ∧ ( − 4 V i e i ) (17)

And with our definition:

H i = − δ i j H j = G c 2 ∫ ρ ( y ) δ i j u j ( y ) | x − y | d 3 y = V i ( x ) (18)

One then has:

g 0 i = − 4 H i ; B g = ∇ ∧ ( − 4 H i e i ) = ∇ ∧ ( 4 δ i j H j e i ) = 4 ∇ ∧ H (19)

B g = 4 r o t H

With the following definition of gravitational field:

k = B g 4 (20)

One then retrieves our previous relations:

k = r o t H ; a g = v ∧ B g = 4 v ∧ k (21)

A last remark: The interest of our notation is that the field equations are strictly equivalent to Maxwell idealization (in particular the speed of the gravitational wave obtained from these equations is the light celerity). Only the movement equations are different with the factor “4”. But of course, all the results of our study could be obtained in the traditional notation of gravitomagnetism

with the relation k = B g 4 .

The article [

10 − 16.62 < ‖ k 0 ‖ < 10 − 16.3 (22)

As demonstrated in [

Just like for the electromagnetism with the magnetic field, this gravitational field implies a phenomenon of precession. It is known as the Lense-Thirring effect. As this gravitational field of clusters should embed large area of the Universe, it should in particular embed the earth. So, instead of taking into account only the own gravitational field of the earth, we are also going to take into account the hypothetical embedding gravitational field that explains the dark matter. We are first going to recall what the equations in the general relativity are for the Lense-Thirring effect. And secondly, we will use it to test this solution by calculating the contribution to the precession effect generated by this gravitational field that explains the dark matter.

The equations of the motion for the spin four-vector S μ have been studied in several papers. In general relativity, it leads to a precession of S μ . It can be

K 1 | k 0 | r 0 [ K 1 r 2 ~ k 0 ] | r 0 [ kpc ] | |
---|---|---|---|---|

NGC 5055 | 10^{24.6} | 10^{−}^{16.62} | 10^{2}^{0.61} | 13 |

NGC 4258 | 10^{24.85} | 10^{−}^{16.54} | 10^{2}^{0.695} | 16 |

NGC 5033 | 10^{24.76} | 10^{−}^{16.54} | 10^{2}^{0.65} | 15 |

NGC 2841 | 10^{24.85} | 10^{−}^{16.33} | 10^{2}^{0.59} | 13 |

NGC 3198 | 10^{24.9} | 10^{−}^{16.55} | 10^{2}^{0.725} | 18 |

NGC 7331 | 10^{24.18} | 10^{−}^{16.3} | 10^{2}^{0.24} | 6 |

NGC 2903 | 10^{24.71} | 10^{−}^{16.3} | 10^{2}^{0.505} | 11 |

NGC 3031 | 10^{24.15} | 10^{−}^{16.57} | 10^{2}^{0.36} | 8 |

NGC 2403 | 10^{24.59} | 10^{−}^{16.39} | 10^{2}^{0.49} | 10 |

NGC 247 | 10^{24.3} | 10^{−}^{16.3} | 10^{2}^{0.3} | 7 |

NGC 4236 | 10^{24} | 10^{−}^{16.34} | 10^{2}^{0.17} | 5 |

NGC 4736 | 10^{24.54} | 10^{−}^{16.3} | 10^{2}^{0.42} | 9 |

NGC 300 | 10^{24.27} | 10^{−}^{16.31} | 10^{2}^{0.29} | 6 |

NGC 2259 | 10^{24.2} | 10^{−}^{16.3} | 10^{2}^{0.25} | 6 |

NGC 3109 | 10^{24} | 10^{−}^{16.58} | 10^{2}^{0.29} | 6 |

NGC 224 | 10^{24} | 10^{−}^{16.5} | 10^{2}^{0.25} | 6 |

deduced from the equations seen in paragraph 2.1. From [

S ˙ = [ ( γ + α 2 ) 1 c 2 ( g r a d φ ∧ v ) + 1 4 ( γ + α ) r o t h ] ∧ S (23)

Which lead to define a geodetic vector field Ω G and a “gravito-magnetic” (frame-dragging) vector field Ω L T :

Ω G = ( γ + α 2 ) 1 c 2 ( g r a d φ ∧ v ) ; Ω L T = 1 4 ( γ + α ) r o t h (24)

These expressions use the PPN formalism. For general relativity, α = 1 and as seen before (16), γ = 1 , it leads to:

Ω G = 3 2 c 2 g r a d φ ∧ v ; Ω L T = 1 2 r o t h (25)

In our notation (20):

H = h 4 ; k = r o t H (26)

One then has

Ω L T = 2 k (27)

In our solution, around the Earth, k represents the addition of two terms, the own gravitational field of the earth k E and the external uniform gravitational field of the clusters (the dark matter) k 0 :

k = k E + k 0 (28)

In the same way, the Lense-Thirring effect Ω L T is then composed of the own Earth gravitational field term Ω L T _ E and of a new supplementary term of “dark matter” Ω L T _ D M

Ω L T = 2 k E + 2 k 0 = Ω L T _ E + Ω L T _ D M (29)

The term Ω L T _ E is the traditional frame-dragging precession:

H E = h E 4 = ( G 2 c 2 r 3 ) ( r ∧ J ) ; Ω L T _ E = G c 2 ( J r 3 − 3 r r 5 ( r ⋅ J ) ) (30)

In the Gravity Probe B experiment, the expected value for the frame-dragging precession was:

‖ Ω L T _ G P B ‖ = 39 milliarcsecond / year (31)

Let’s evaluate the order of magnitude of the external gravitational field (the dark matter) around the Earth. From [

‖ Ω L T _ D M ‖ = 2 ‖ k 0 ‖ ~ 2 × 10 − 16.5 s − 1

‖ Ω L T _ D M ‖ = 0.4 milliarcsecond / year (32)

In fact, from the sample of galaxies studied in [

0.3 < ‖ Ω L T _ D M ‖ < 0.6 ( milliarcsecond / year ) (33)

‖ Ω L T _ D M ‖ represents around 1% of ‖ Ω L T _ G P B ‖ , (indeed 0.4 mas ⋅ y − 1 ~ 0.01 × 39 mas ⋅ y − 1 ). But until now, ‖ Ω L T _ G P B ‖ is only known with a precision 19% [

But there are several aspects of the experiment that can play a role in decreasing or increasing this discrepancy. The Sun is at around 8kpc from the galactic center. In [

In [

Le Corre, S. (2017) Dark Matter, a Direct Detection. Open Access Library Journal, 4: e4219. https://doi.org/10.4236/oalib.1104219