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A recent publication revealed unexpected observations about dark matter. In particular, the observed baryonic mass should probably be sufficient to explain the observed rotation curves ( i.e. without dark matter) and their observations gave an empirical relation for weak accelerations. This present work demonstrated that the equations of general relativity allow explaining the term of dark matter (without new matter) in agreement with the results of this publication and allow retrieving this empirical relation (observed values and characteristics of this correlation’s curve). These observations constrain drastically t he possible gravitational potential in the frame of general relativity to explain the term of dark matter. This theoretical solution has already been studied with several unexpected predictions that have recently been observed. For example, an article revealed that early galaxies (ten billion years ago) didn’t have dark matter and a more recent paper showed unlikely alignments of galaxies. To finish the main prediction of this solution, it is recalled: the term of dark matter should be a Lense-Thirring effect, around the earth, of around 0.3 and 0.6 milliarcsecond/year.

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 the more shared and certainly the 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.

Starting with the publication [

But before getting to the heart of our explanation, let’s recall the results of the following publication [

・ statistically the gravitational potential, g o b s , deduced from the observed rotation curves of the galaxies and the gravitational potential, g b a r , deduced from the observed distribution of the baryonic mass is strongly correlated;

・ this potential g b a r is a solution of the Poisson equation;

・ the value of the gravitational potential is around g b a r ~ 10 − 10.5 ;

・ the relation of the potential in the weak accelerations is g o b s ∝ g b a r ;

・ the correlation’s curve deviates from the line of unity for values smaller than around g b a r ~ 10 − 10 .

As mentioned in [

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 [

(With = 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 Equation (1) 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 (gravitic 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 Equation (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)

This is this relation that we are going to use to explain the observed results of [_{g}/4 (notation of [

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

v 2 ( r ) r = ( ∂ φ ( r ) ∂ r ) with φ = φ d i s k + φ b u l g e + φ h a l o (14)

Or:

v 2 ( r ) r = ( ∂ φ d i s k ( r ) ∂ r ) + ( ∂ φ b u l g e ( r ) ∂ r ) + ( ∂ φ h a l o ( r ) ∂ r ) = v d i s k 2 ( r ) r + v b u l g e 2 ( r ) r + v h a l o 2 ( r ) r (15)

According to the linearized general relativity, the gravitational force is composed of the gravity fields (represented by ∂ φ d i s k ( r ) / ∂ r and ∂ φ b u l g e ( r ) / ∂ r in the previous equation) and by the gravitic field that we assume to be able to explain the dark matter (represented by ∂ φ h a l o ( r ) / ∂ r ). Consequently, here, ∂ φ h a l o ( r ) / ∂ r gathers the gravitic force of all the components (disk, bulge, …). This force due to the gravitic field k takes the following form ‖ F k ‖ = m p 4 ‖ v ∧ k ‖ = m p ∂ φ h a l o ( r ) / ∂ r (from the Equation (13)). To simplify our computation, we idealize a situation where we have the approximation v ⊥ k . We can demonstrate that this perpendicularity is finally the more natural situation, meaning that this situation is very general, as demonstrated in [

This situation gives the following equation:

v 2 ( r ) r = ( ∂ φ d i s k ( r ) ∂ r ) + ( ∂ φ b u l g e ( r ) ∂ r ) + 4 k ( r ) v ( r ) = v d i s k 2 ( r ) r + v b u l g e 2 ( r ) r + 4 k ( r ) v ( r ) (16)

Far from the center of the galaxies, when the gravitational field becomes negligible, the contribution of ( ∂ φ d i s k ( r ) / ∂ r ) + ( ∂ φ b u l g e ( r ) / ∂ r ) is negligible. It is the area where dark matter dominates. So, far from the center of the galaxies, the equation becomes:

v 2 ( r ) r = v h a l o 2 ( r ) r = 4 k ( r ) v ( r ) (17)

If we get back to the results of [

g o b s = v 2 ( r ) r = g b a r 1 − e − g b a r / g † (18)

Let’s see what happens where the dark matter dominates, i.e. at the end of the galaxies. This area is characterized by:

・ very large value of r ≫ 15 kpc (where the dark matter dominates);

・ negligible gravity fields;

・ low accelerations.

In this area, the results of [

g o b s ∝ g b a r (19)

By definition, g b a r = v b a r 2 ( r ) / r , it then gives for the low accelerations:

g o b s ∝ v b a r ( r ) r (20)

Furthermore, for very large value of r ≫ 15 kpc , the curve r is extremely flat. It evolves very slowly and can be considered as constant with a very good approximation. It means that in this area the empirical relation can be written:

g o b s ∝ v b a r ( r ) (21)

If we make the assumption that dark matter doesn’t exist, at the ends of the galaxies the speed v b a r ( r ) is in fact the speed of the ordinary mass v ( r ) . One has then:

g o b s ∝ v ( r ) (22)

The previous Equation (17) from the general relativity gives at the ends of the galaxies (where the effects of the bulge and disk fields are negligible):

g o b s = v 2 ( r ) r = 4 k ( r ) v ( r ) (23)

If we compare this relation with the empirical relation (22), the only way to conciliate them is to suppose that the gravitic field k ( r ) at the ends of the galaxies is approximately constant. In other words, without dark matter and without modifying dynamic laws, to be in agreement with the observations, the general relativity implies the existence of an approximatively uniform gravitic field k ( r ) ~ k 0 .

By this way, the general relativity verifies and explains the observations of [

In fact, this solution has already been studied in [

We are now going to see the accuracy of this solution. For the galaxy NGC 7331 (

is in agreement with the results of [

More precisely, all along the galaxy, we can write k = k 1 + k 0 as in [

In term of g b a r = 4 k 0 v , it leads to the interval 10 − 11.32 < g b a r < 10 − 10.2 . The solution in [

And one can focus on the remarkable agreement between the theoretical expectation and the experimental observations. The observed interval in [

One can also note that, as demonstrated in [

The fact that this gravitic field is uniform also means that it cannot come from the galaxy (otherwise it should decrease with the distance to the center). This result is in agreement with the simulations in [

A recent publication [

A more recent article [

Since the publication of [

1) The satellite dwarf galaxies are distributed according to plans;

2) For nearby galaxies, the plans of satellite dwarf galaxies must have the same orientation;

3) The plans must be aligned on the equatorial axis of the cluster they belong;

4) The clusters of neighboring galaxies must have a strong tendency to align;

5) A calculation of the order of magnitude provides that these alignments can extend over distances of tens of Mpc at least;

6) Statistically, the spin vectors of galaxies must be oriented in the same half-space (that of the cluster rotation vector).

The article [

With the constraints of the observations of [

・ The correlation between the baryonic mass distribution and the rotation’s speed of the galaxies;

・ the value of the gravitational potential;

・ the fact that this potential is a solution of the Poisson equation;

・ the relation of the potential in the weak accelerations g o b s ∝ g b a r (demonstrated with the general relativity);

・ the characteristics of the correlation’s curve (deviation from the line of unity for values smaller than around g b a r ~ 10 − 10 ).

Furthermore, we have seen that this solution also implied several unexpected predictions that have been recently verified:

・ The satellite dwarf galaxies are distributed according to plans;

・ For nearby galaxies, the plans of satellite dwarf galaxies must have the same orientation;

・ The plans must be aligned on the equatorial axis of the cluster they belong to;

・ The clusters of neighboring galaxies must have a strong tendency to align;

・ A calculation of the order of magnitude provides that these alignments can extend over distances of tens of Mpc at least;

・ Statistically, the spin vectors of galaxies must be oriented in the same half-space (that of the cluster rotation vector);

・ The galaxies of early clusters must be characterized by the absence of dark matter.

But this solution also implies others predictions not yet verified, in particular, a discrepancy in the measurement of the expected Earth’s Lense-Thirring effect, as demonstrated in [

One can also note that this solution could be adapted to explain the dark energy, another mystery of astrophysics more important than dark matter, as proposed in [

All these first results, in agreement with the observations and general relativity, are to compare with the failure of the detection of the dark matter (as an exotic matter) until now. One can also focus on a remarkable consistency between this solution and the characteristic of dark matter. Indeed, the value of the gravitic field is so weak that it explains why dark matter has not been yet detected. And by the same time, because this term is so weak, its effect become measurable for large distance, as it can be seen with the Expression (23), explaining why dark matter appears only at large scale. Irony of history, dark matter and dark energy began to be seen as a potential failure of the general relativity, but with this solution, they would become two new great victories of general relativity. If it was the case, dark matter and dark energy will be certainly part of the top five of the most remarkable verifications of general relativity (just like space’s curvature, gravitational waves and Lense-Thirring effect).

Le Corre, S. (2017) Dark Matter without New Matter Is Compliant with General Relativity. Open Access Library Journal, 4: e3877. https://doi.org/10.4236/oalib.1103877