From GR, one deduces the LGR 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 [11] (with □   = 1 c 2 ∂ 2 ∂ t 2 − Δ and Δ = ∇ 2 ):

∂ 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 [12] for the vector potential H i . In our study, we are going to define:

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 (determined in [9] ) a new constant defined by:

G K = c 2 (8)

This definition is K − 1 ~ 7.4 × 10 − 28   kg ⋅ m − 1 very small compared to G.

The field Equations (1) can be then written (Poisson equations):

Δ φ = 4 π G ρ ;     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 (gravitic field), those relations can be obtained from the following equations (also called gravitomagnetism) with the differential operators “ r o t = ∇ ∧ ”, “ g r a d = ∇ ” and “ d i v = ∇ ⋅ ”:

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: All previous relations can be retrieved starting with the parameterized post-Newtonian (PPN) formalism and with the traditional gravitomagnetic field B g . From [13] one has:

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 traditional 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 GR (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 gravitic 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)

The interest of our notation ( k instead of B g ) is that the field equations are strictly equivalent to Maxwell’s idealization, in particular, the speed of the gravitational wave obtained from these equations is the light celerity c 2 = G K just like in EM c 2 = 1 / μ 0 ε 0 . Only the movement equations are different with the factor “4”. But of course, all the results of our study can be obtained in the traditional notation of gravitomagnetism with the relation k = B g 4 .

In the classical approximation ( ‖ v ‖ ≪ c ), the linearized general relativity gives the following movement equations from (13) with m i the inertial mass and m p the gravitational mass:

m i d v d t = m p [ g + 4 v ∧ k ] (22)

The traditional computation of rotation speeds of galaxies consists of obtaining the force equilibrium from the three following components: the disk, the bugle, and the halo of dark matter. More precisely, one has [14] :

Then the total speed squared can be written as the sum of squares of each of the three-speed components:

Disk and bulge components are obtained from gravity field. They are not modified in our solution. So, our goal is now to obtain only the traditional dark matter halo component from the linearized general relativity. According to this idealization, the force due to the gravitic field takes the following form and it corresponds to previous term. As explained in [9] , the natural evolution to the equilibrium state justifies that one assumes the approximation. This assumption is important because it leads to several important predictions. In particular, the motion of dwarf satellite galaxies of a host should be roughly in a plane (). It then gives the following equation:

Our idealization means that:

The equation of dark matter (gravitic field in our explanation) is then:

This equation gives us the curve of rotation speeds of the galaxies as we wanted. Because we know the curves of speeds that one wishes to have for DM component, one can then deduce the curve of the gravitic field inside the galaxy:

This solution of DM as the gravitic field has been studied in [9] for 16 galaxies (Table 1). It shows that this solution is mathematically possible but with two physical mandatory unexpected behavior for. First, the curve of the gravitic field becomes necessarily flat at the end of the galaxies. For such a field (similar mathematically to a magnetic field in EM) it is only possible if the galaxies are immersed in a uniform gravitic field. Second, the value of this field for these 16 galaxies is in the interval:

From these data (Table 1), one can deduce a mean value of and a mean value of:

The position is the position where the gravitic component of the galaxy becomes negligible compared to the external uniform gravitic term explaining DM. It roughly represents the beginning of the flat part of the rotation speed curve of the galaxies. We will use these two values at the end of the article in the demonstration of the Tully-Fisher law.

The Tully-Fisher law is written and can be rewritten . Several couples can be obtained experimentally for this law [15] depending on the masses considered and the methods of obtaining the characteristic speed of rotation as previously said.

Let’s rewrite our expression (33):

which gives:

In order to get an expression that looks like the Tully-Fisher law, we are going to use the following approximation for large r (in the flat part of the rotation speed curve):

Furthermore, in our explanation of DM, is a uniform field, i.e.. By replacing the occurrence of in the brackets, one has:

If one has:

The expression becomes:

And finally:

The couples of are in general given for a graph whose velocities are in km·s⁻¹ and the masses in solar mass (). One can then rewrite with and:

We want to verify that the values of form the Tully-Fisher law allow retrieving the values of the gravitic field explaining the DM to ensure that our theoretical solution is consistent with the Tully-Fisher law. For that, let’s rewrite:

For these calculations, one will use:

This value of r justify the previous approximation (36)

In [2] the observations give the following couple:

:

In [15] , one has the 4 following couples:

:

:

:

:

Let’s verify the previous approximation (38) with the smallest value of:

These calculations show that the expected values of to explain DM without exotic material () are consistent with the Tully-Fisher law. We will now go further by proving this law within the framework of the LGR and our explanation of DM.

As we noticed in the previous paragraph, in our relation, there is the parameter position r which appears while the Tully-Fisher relation does not explicitly depend on it. To no longer explicitly depend on this parameter, we need to define a procedure to determine a characteristic position r. It should be noted that we are in the same situation for the TULLY-FISHER law to define the characteristic speed of rotation to be considered. Several methods are used [5] and the mean velocity along the flat part of the rotation curve seems to minimize the scatter of the relation [4] . The question in our context is somehow to find which method was adopted to define the characteristic position r corresponding to the characteristic velocity on the curve of the rotational velocities of the galaxies. This characteristic speed is linked to the flat zone of the speed curve, the characteristic position will certainly be a characteristic position in this zone. We can imagine 2 simple methods.

A 1st would consist of finding the position of the beginning of the flat zone then placing the cursor at a fixed relative position, with respect to the beginning of this zone, and finally determining the value of. We will see what it will be. This method makes it possible to find punctually the good values for the law of Tully-Fisher but it doesn’t give the good slope of the law.

A 2nd method would consist of finding a threshold for the value of the gravitational forces from which the gravitational forces are sufficiently weak so that the influence of the DM dominates (a priori in the flat part of the curve). This method allows for demonstrating the Tully-Fisher law, values, and slope. But certainly, also the extent of the area to which this law is applicable.