_{1}

The star SO-2 at the galactic center was recently at its closest distance to the supermassive black hole (SMBH). It allowed measuring relativistic effects. The observations confirm that the general relativity is the best gravitational theory compared to other alternative theories [1]. But although an excellent agreement with the observations is obtained all along the orbit, a discrepancy in the redshift of SO-2 is measured at its periastron. An excess of around 20% (240 km/s instead of 200 km/s) has been observed. This discrepancy has been predicted in the paper [2]. It should come from the second component (gravitic field) of the general relativity generated by SgrA*. In general, the expected value of this component of general relativity is negligible. But in the frame of the explanation of dark matter ( as the gravitic field of the clusters [3]), the gravitic field of the galaxies should be larger than expected. It is in this theoretical frame (without exotic matter and in agreement with general relativity) that this discrepancy was predicted. Furthermore, its value is in agreement with the expected gravitic field computed in [3]. These results would mean that the gravitic field of large astrophysical structures (galaxies, clusters, ...) would be greater than expected and that the explanation of dark matter by the gravitic field could be a pertinent solution (in agreement with general relativity and without exotic matter). Furthermore, this explanation implies necessarily movements of dwarf satellite galaxies along planes, movements that seem to be more and more likely and allows retrieving experimental relations. This dark matter explanation would then solve several kinds of problems with this specific component of general relativity currently neglected.

General relativity implies the existence of two gravitational components. In addition to the gravity field, there is a gravitic field just like the magnetic field in electromagnetism. These both components give what is called the gravitomagnetism, obtained from the linearization of the general relativity. This new gravitic field can be measured by its effect, known as Lense-Thirring effect. Several experiments have validated this effect for the Earth gravitic field, NASA’s LAGEOS satellites or Gravity Probe B [

Let’s first demonstrate what general relativity gives for the expressions of the gravitic field; it will allow giving us an expression for the gravitational redshift.

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, because these approximations lead to an idealization equivalent to 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)

From relation (11), one deduces the metric in a quasi flat space:

d s 2 = ( 1 + 2 φ c 2 ) c 2 d t 2 + 8 H i c c d t d x i − ( 1 − 2 φ c 2 ) ∑ ( d x i ) 2 (14)

In the approximation of a quasi-flat Minkowski space, one has:

H i d x i = − δ i j H j d x i = − H ⋅ d x (15)

We retrieve the known expression [

d s 2 = ( 1 + 2 φ c 2 ) c 2 d t 2 − 8 H ⋅ d x c c d t − ( 1 − 2 φ c 2 ) ∑ ( d x i ) 2 (16)

Remark: The interest of our notation (compare to the traditional notation of gravitomagnetism) 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 .

In our approximation, we are going to seek for a relation of the gravitational potential that contain the gravitic component in addition to the gravity’s term. It could be seen as a correction of the traditional gravitational potential that take in account only the gravity field. By this way (in the linearized approximation), others relations in which the gravitational potential intervenes could be adapted to take in account the gravitic field.

From the previous relation (16), one can write:

d s 2 = ( 1 + 2 φ c 2 − 8 H ⋅ d x c 2 d t ) c 2 d t 2 − ( 1 − 2 φ c 2 ) ∑ ( d x i ) 2 (17)

d s 2 = ( 1 + 2 ( φ − 4 H ⋅ ( d x / d t ) ) c 2 ) c 2 d t 2 − ( 1 − 2 φ c 2 ) ∑ ( d x i ) 2 (18)

One can then define, in this approximation, a gravitational potential that could be qualified as “corrected” in the sense that the gravitic term is added to the traditional gravitational potential as a corrected term (with the velocity of the test particle, v = d x / d t ):

φ C O R R = φ − 4 H ⋅ v = φ + φ K 1 (19)

We can then apply these relations for the gravitational redshift:

1 + z = ( 1 − 2 c 2 φ C O R R , E 1 − 2 c 2 φ C O R R , R ) 1 / 2 ~ ( 1 − 2 c 2 ( φ E + φ K 1 , E ) 1 − 2 c 2 ( φ R + φ K 1 , R ) ) 1 / 2 (20)

As demonstrated in [

z G R ~ − 1 c 2 φ SgrA* − 1 c 2 φ K 1 − SgrA* = z G + z H φ SgrA* = − G M SgrA* r SO-2 ↔ SgrA* φ K 1 − SgrA* = − 4 K 1 v r SO-2 ↔ SgrA* (21)

with K 1 the factor of the gravitic field in the punctual definition (from Poisson Equation (9) the gravitic field can be written k ∝ K 1 / r 2 and in this punctual

approximation ‖ H ‖ ~ K 1 r ) [

The two components of redshift of SO-2 due to general relativity is then:

z G ~ G M SgrA* r SO-2 ↔ SgrA* c 2 z H ~ 4 K 1 v r SO-2 ↔ SgrA* c 2 (22)

Because the term z H depends on the velocity of SO-2 (v), the effect of this term can be detectable only at the periastron at which the velocity is large enough.

Let’s remind that there is also another component for the computation of the redshift due to special relativity z S R ~ z G .

In the paper explaining the dark matter [

10 24 ≤ K 1 _GAL ≤ 10 24.9 (23)

If we adapt these values of whole galaxy, K 1 _GAL (for a typical mass of 2 × 10 42 kg ) to SMBH of the center of galaxy, K 1 ( M SgrA* ~ 4 × 10 6 × 2 × 10 30 kg ) [

10 18.6 ≤ K 1 ≤ 10 19.5 (24)

with the distance of SO-2 to SgrA* ( r SO-2 ↔ SgrA* ~ 2 × 10 13 m ) and the velocity at the periastron of SO-2 v ~ 8 × 10 6 m ⋅ s − 1 [

z H ~ 4 K 1 8 × 10 6 2 × 10 13 × 9 × 10 16 ~ 2 K 1 × 10 − 23 (25)

And with the previous values of K 1 (24), one obtains:

10 − 4.1 ≤ z H ≤ 10 − 3.2 8 × 10 − 5 ≤ z H ≤ 6 × 10 − 4 (26)

If we compare to z G , (22) gives:

z G ~ 6 × 10 − 11 × 8 × 10 36 2 × 10 13 × 9 × 10 16 ~ 3 × 10 − 4 (27)

Let’s remind that the expected redshift (expected for the main part of the researchers) is only z = z G + z S R with z S R ~ z G , it would then give for the redshift of SO-2:

z = z G + z S R ~ 6 × 10 − 4 (28)

The added term that I predict is then at least close to the same order of magnitude of each term ( z S R or z G ).

Translated in term of velocity, these redshifts ( v z H for the sample of the studied galaxies) are:

30 km ⋅ s − 1 ≤ v z H ≤ 180 km ⋅ s − 1 v z S R ~ 100 km ⋅ s − 1 v z G ~ 100 km ⋅ s − 1 (29)

Instead of the expected value v z ~ 200 km ⋅ s − 1 , one then should have:

v z ≥ 230 km ⋅ s − 1 (30)

If we look at the recent observations (

But we also detect a discrepancy, only at the periastron. Instead of v z ~ 200 km ⋅ s − 1 , the measure gives v z ~ 240 km ⋅ s − 1 . The value matches well with the gravitic field (30) expected in the frame of the explanation of the dark matter [

We can specify that the effect of the gravitic field of SgrA* on SO-2 was expected to be negligible all around the orbit of SO-2. Only at the periastron, a discrepancy was expected. The value of this discrepancy was expected to be at least v z H ≥ 30 km ⋅ s − 1 . The observation of [

With the passage of SO-2 star at its closest position to SgrA*, the gravitational theories can be tested. Once again, General Relativity is the best gravitational theory to explain these observations. But these measures on the redshift of SO-2

reveal a discrepancy at its periastron. It doesn’t mean that general relativity failed. Indeed, in a previous paper [

We can remind that others success of this solution has been obtained. This solution predicted necessarily the movement of dwarf satellites galaxies along planes [

The author declares no conflicts of interest regarding the publication of this paper.

Le Corre, S. (2018) Dark Matter: SO-2’s Gravitational Redshift Could Give an Important Clue to Solve the Dark Matter Mystery. Open Access Library Journal, 5: e5028. https://doi.org/10.4236/oalib.1105028