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 [5] (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, 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 [6] 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 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 [5] with our definition of H i :

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 [2] , in the case of SO-2, the redshift due to general relativity, z G R , can be obtained with the following term (due on one hand to the gravity field of SgrA*, φ SgrA* and on the other hand to the gravitic field of SgrA*, φ K 1 − SgrA* ):

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 ) [3].

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 [3] , the gravitic field of the studied galaxies (more specifically the factor K 1 for the whole galaxy, K 1 _GAL and not for the only SMBH of the center of the galaxy) is expected to be in the following interval:

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 ) [7] [8] by applying a factor of M SgrA* / M GAL ~ 8 × 10 36 / ( 2 × 10 42 ) = 4 × 10 − 6 , (23) becomes:

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 [7] [8] , (22) gives:

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 (Figure 1) on the redshift of SO-2 [1] , we first remark that general relativity is once again confirmed as the more accurate theory of gravitation.

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 [3].

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 [1] confirms these three characteristics.