DARK MATTER, a DIRECT DETECTION Stéphane Le Corre

DARK MATTER, A DIRECT DETECTION Stéphane Le Corre

To cite this version:

Stéphane Le Corre. DARK MATTER, A DIRECT DETECTION. 2016. hal-01276745

HAL Id: hal-01276745 https://hal.archives-ouvertes.fr/hal-01276745 Preprint submitted on 19 Feb 2016

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. DARK MATTER, A DIRECT DETECTION

Stéphane Le Corre (E-mail : [email protected]) No affiliation

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 of dark matter measured in CMB). It leads, compared with Newtonian gravitation, to add a new component (gravitic field) to gravitation without changing the gravity field (also dm known as gravitomagnetism). In this explanation, dark matterΩ would be a uniform gravitic field that embeds some very large areas of the universe. In this article we are going to see that this specific gravitic 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 / and 0.6 / . In the LAGEOS or Gravity Probe B experiments, there was not enough precision (around 10% for the expected 6606 . geodetic and 39 . frame-dragging precessions𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚). In the GINGER𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 experiment,𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 there could be𝑦𝑦 𝑦𝑦𝑦𝑦𝑦𝑦enough, the expected−1 accuracy would be around−1 1%. If this discrepancy were verified, it would be the first direct measure of the dark matter.𝑚𝑚𝑚𝑚𝑚𝑚 𝑦𝑦 𝑚𝑚𝑎𝑎𝑠𝑠 𝑦𝑦 Keywords: gravitation, gravitic field, dark matter

gives the following field equations (HOBSON et al., 2009) = (with 2 ): 1. Overview 1 𝜕𝜕 2 2 8 𝑐𝑐 𝜕𝜕𝜕𝜕 −=∆0 ; = 2 ( ) General relativity implies the existence of two gravitational 𝜇𝜇𝜇𝜇 𝜇𝜇𝜇𝜇 𝜋𝜋𝜋𝜋 𝜇𝜇𝜇𝜇 With: 𝜕𝜕𝜇𝜇ℎ� ℎ� − 4 𝑇𝑇 𝐼𝐼 components. In addition of the gravity field, there is a gravitic field 1 𝑐𝑐 = ; ; = ; = ( ) (together giving what is called the gravitomagnetism) just like the 2 𝜇𝜇𝜇𝜇 𝜇𝜇𝜇𝜇 𝜇𝜇𝜇𝜇 𝜎𝜎 𝜇𝜇 𝜇𝜇𝜇𝜇 magnetic field in electromagnetism. The new gravitic field can be Theℎ� generalℎ −solution𝜂𝜂 ℎ of theseℎ ≡ ℎ equations𝜎𝜎 ℎ𝜈𝜈 is:𝜂𝜂 ℎ𝜎𝜎𝜎𝜎 ℎ� −ℎ 𝐼𝐼𝐼𝐼 measured by its precession effect, known as Lense-Thirring effect. 4 ( | |, ) ( , ) = Several experiments have validated this effect for the Earth 𝜇𝜇𝜇𝜇 | | 𝜇𝜇𝜇𝜇 𝐺𝐺 𝑇𝑇 𝑐𝑐𝑡𝑡 − 𝑥𝑥⃗ − 𝑦𝑦⃗ 𝑦𝑦⃗ 3 gravitic field, NASA's LAGEOS satellites or Gravity Probe B (ADLER, In the approximationℎ� 𝑐𝑐𝑐𝑐 𝑥𝑥⃗ of− a source4 � with low speed, one𝑑𝑑 has:𝑦𝑦⃗ 𝑐𝑐 𝑥𝑥⃗ − 𝑦𝑦⃗ 2015) with an accuracy of around 10%. Some new experiments = ; = ; = will try to obtain a higher accuracy, for example GINGER And for a stationary00 solution,2 0 𝑖𝑖one has:𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖 𝑗𝑗 𝑇𝑇 𝜌𝜌𝑐𝑐 𝑇𝑇 𝑐𝑐𝜌𝜌𝑢𝑢 𝑇𝑇 𝜌𝜌𝑢𝑢 𝑢𝑢 (RUGGIERO, 2015) with an expected accuracy of around 1%. 4 ( ) ( ) = | 𝜇𝜇𝜈𝜈 | 𝜇𝜇𝜇𝜇 𝐺𝐺 𝑇𝑇 𝑦𝑦⃗ 3 In (LE CORRE, 2015), a solution is proposed to explain the dark At this step, by proximityℎ� 𝑥𝑥⃗ with− electromagnetism,4 � 𝑑𝑑 𝑦𝑦⃗ one traditionally 𝑐𝑐 𝑥𝑥⃗ − 𝑦𝑦⃗ matter. This explanation leads to the assumption that we are defines a scalar potential and a vector potential . There are in embedded in a relatively uniform gravitic field generated by larger the literature several definitions (MASHHOON, 2008𝑖𝑖 ) for the 𝜑𝜑 𝐻𝐻 structures than galaxies (likely the clusters). Just like the Earth vector potential . In our study, we are going to define: gravitic field can be measured, this hypothetical embedded 𝑖𝑖 4 4 𝐻𝐻 = ; = ; = 0 gravitic field could be measured by its precession effect. Such a 𝑖𝑖 00 𝜑𝜑 0𝑖𝑖 𝐻𝐻 𝑖𝑖𝑖𝑖 measure will be a direct measure of the “dark matter”. We are With gravitationalℎ� scalar2 potentialℎ� andℎ� gravitational vector going to see that the magnitude of this effect is at the limit of our potential : 𝑐𝑐 𝑐𝑐 𝜑𝜑 detection. And even, in the most advantageous case, the accuracy 𝑖𝑖 ( ) 𝐻𝐻 ( ) of 1% (as expected in GINGER experiment) could be enough to | | 𝜌𝜌 𝑦𝑦⃗ 3 detect it. 𝜑𝜑(𝑥𝑥⃗) ≡(−)𝐺𝐺 � 𝑑𝑑 𝑦𝑦⃗ ( ) ( ) ( ) 𝑥𝑥⃗=− 𝑦𝑦⃗ | 𝑖𝑖 | | 𝑖𝑖 | First, I recall the theoretical idealization used in this article and in 𝑖𝑖 𝐺𝐺 𝜌𝜌 𝑦𝑦⃗ 𝑢𝑢 𝑦𝑦⃗ 3 −1 𝜌𝜌 𝑦𝑦⃗ 𝑢𝑢 𝑦𝑦⃗ 3 𝐻𝐻 𝑥𝑥⃗ ≡ − 2 � 𝑑𝑑 𝑦𝑦⃗ −𝐾𝐾 � 𝑑𝑑 𝑦𝑦⃗ (LE CORRE, 2015). With a new constant𝑐𝑐 𝑥𝑥 ⃗defined− 𝑦𝑦⃗ by: 𝑥𝑥⃗ − 𝑦𝑦⃗ = 𝐾𝐾 2. Gravitation in linearized general relativity This definition gives ~7.4 × 102 very small compare to . −1 𝐺𝐺𝐺𝐺 𝑐𝑐 −28 From general relativity, one deduces the linearized general The field equations (𝐾𝐾) can be then written (Poisson equations)𝐺𝐺: relativity in the approximation of a quasi-flat Minkowski space 4 = 4 ;𝐼𝐼 = = 4 ( ) ( = + ; | | 1). With following Lorentz gauge, it 𝑖𝑖 𝜋𝜋𝜋𝜋 𝑖𝑖 −1 𝑖𝑖 𝜇𝜇𝜇𝜇 𝜇𝜇𝜇𝜇 𝜇𝜇𝜇𝜇 𝜇𝜇𝜇𝜇 ∆𝜑𝜑 𝜋𝜋𝜋𝜋𝜌𝜌 ∆𝐻𝐻 2 𝜌𝜌𝑢𝑢 𝜋𝜋𝐾𝐾 𝜌𝜌𝑢𝑢 𝐼𝐼𝐼𝐼𝐼𝐼 𝑐𝑐 𝑔𝑔 𝜂𝜂 ℎ ℎ ≪ 1

With the following definitions of (gravity field) and (gravitic notation of gravitomagnetism with the relation field), those relations can be obtained from following equations: = 𝑔𝑔⃗ 𝑘𝑘�⃗ . 𝐵𝐵��𝑔𝑔⃗

= ; = 𝑘𝑘�⃗ 4 3. Gravitic field: a way to measure it = 0 ; = 0 ; 𝑔𝑔⃗ −𝑔𝑔��𝑔𝑔𝑔𝑔𝑔𝑔��⃗𝜑𝜑 𝑘𝑘�⃗ �𝑟𝑟𝑟𝑟𝑟𝑟��⃗ 𝐻𝐻��⃗ = 4 ; = 4 �𝑟𝑟𝑟𝑟𝑟𝑟��⃗𝑔𝑔⃗ 𝑑𝑑𝑑𝑑𝑑𝑑 𝑘𝑘�⃗ Just like for the electromagnetism, this gravitic field implies a −1 ��⃗�⃗ p phenomenon of precession. It is known as the Lense-Thirring With relations𝑑𝑑𝑑𝑑 ( 𝑑𝑑)𝑔𝑔⃗, one− has:𝜋𝜋𝐺𝐺 𝜌𝜌 𝑟𝑟𝑟𝑟𝑟𝑟𝑘𝑘 − 𝜋𝜋𝐾𝐾 ȷ��⃗ effect. Instead of taking into account only the own gravitic field of 2 4 = = 𝐼𝐼𝐼𝐼 = = ; = ; = 0 ( ) the earth, we are also going to take into account the hypothetical 𝑖𝑖 00 11 22 33 𝜑𝜑 0𝑖𝑖 𝐻𝐻 𝑖𝑖𝑖𝑖 external gravitic field that explains the dark matter. We are first The equations of geodesics in the2 linear approximation give: ℎ ℎ ℎ ℎ ℎ ℎ 𝐼𝐼𝐼𝐼 going to recall what the equations in the general relativity are for 1 𝑐𝑐 𝑐𝑐 ~ the Lense-Thirring effect. And secondly, we will use it to test our 2 𝑖𝑖 2 𝑑𝑑 𝑥𝑥 2 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑗𝑗 solution by calculating the contribution to the precession effect It then leads2 to the− movement𝑐𝑐 𝛿𝛿 𝜕𝜕𝑗𝑗ℎ00 equations:− 𝑐𝑐𝛿𝛿 �𝜕𝜕𝑘𝑘 ℎ0𝑗𝑗 − 𝜕𝜕𝑗𝑗ℎ0𝑘𝑘�𝑣𝑣 𝑑𝑑𝑑𝑑 generated by our gravitic field that explain the dark matter. ~ + 4 = + 4 2 𝑑𝑑 𝑥𝑥⃗ 3.1. Gravitic field and precession effect 2 − 𝑔𝑔��𝑔𝑔𝑔𝑔𝑔𝑔��⃗𝜑𝜑 𝑣𝑣⃗ ∧ ��𝑟𝑟𝑟𝑟𝑟𝑟��⃗𝐻𝐻��⃗� 𝑔𝑔⃗ 𝑣𝑣⃗ ∧ 𝑘𝑘�⃗ From relation𝑑𝑑𝑑𝑑 ( ), one deduces the metric in a quasi flat space: 2 8 2 The equations of the motion for the spin four-vector has been = 1 + + 1 𝐼𝐼𝐼𝐼 studied in several papers. In general relativity, it leads to a 2 𝜑𝜑 2 2 𝐻𝐻𝑖𝑖 𝑖𝑖 𝜑𝜑 𝑖𝑖 2 𝑆𝑆𝜇𝜇 In a𝑑𝑑𝑑𝑑 quasi-�Minkowski2 � 𝑐𝑐 space𝑑𝑑𝑑𝑑 , one has:𝑐𝑐𝑑𝑑𝑑𝑑𝑑𝑑 𝑥𝑥 − � − 2 � ��𝑑𝑑𝑥𝑥 � precession of . For example, with (ADLER, 2015), one can write 𝑐𝑐 𝑐𝑐 𝑐𝑐 = = . the following equations:𝜇𝜇 We retrieve the known𝑖𝑖 expression𝑗𝑗 (HOBSON𝑖𝑖 et al., 2009) with our 𝑆𝑆 1 1 𝐻𝐻𝑖𝑖𝑑𝑑𝑥𝑥 −𝛿𝛿𝑖𝑖𝑖𝑖𝐻𝐻 𝑑𝑑𝑥𝑥 −𝐻𝐻��⃗ �𝑑𝑑𝑑𝑑��⃗ = + + ( + ) definition of : 2 4 𝛼𝛼 ⃗̇ 2 ��⃗ �⃗ ⃗ 𝑖𝑖 2 8 . 2 Which 𝑆𝑆lead� �to𝛾𝛾 define� a� 𝑔𝑔geo𝑔𝑔𝑔𝑔𝑔𝑔detic𝜑𝜑 ∧ vector𝑣𝑣⃗� field𝛾𝛾 𝛼𝛼 and�𝑟𝑟𝑟𝑟𝑟𝑟��⃗ ℎa� “∧gravito𝑆𝑆 - = 1𝐻𝐻+ 1 𝑐𝑐 magnetic” vector field : 𝐺𝐺 2 𝜑𝜑 2 2 𝐻𝐻��⃗ �𝑑𝑑𝑑𝑑��⃗ 𝜑𝜑 𝑖𝑖 2 �Ω��⃗ 𝑑𝑑𝑑𝑑 � 2 � 𝑐𝑐 𝑑𝑑𝑑𝑑 − 𝑐𝑐𝑑𝑑𝑑𝑑 − � − 2 � ��𝑑𝑑𝑥𝑥 � 1 1 𝑐𝑐 𝑐𝑐 𝑐𝑐 = + Ω��𝐿𝐿𝐿𝐿��⃗ ; = ( + ) Remark: Of course, one retrieves all these relations starting 2 4 𝛼𝛼 with the parameterized post-Newtonian formalism. From These�Ω�� 𝐺𝐺⃗expressions�𝛾𝛾 � use2 �𝑔𝑔� ��the�𝑔𝑔𝑔𝑔𝑔𝑔��⃗ PPN𝜑𝜑 ∧ formalism.𝑣𝑣⃗� Ω��𝐿𝐿𝐿𝐿��⃗ As seen𝛾𝛾 𝛼𝛼 before,�𝑟𝑟𝑟𝑟𝑟𝑟��⃗ ℎ�⃗ for (CLIFFORD M. WILL, 2014) one has: general relativity, one𝑐𝑐 has: 1 ( ) ( ) = 1 ; = 1 = (4 + 4 + ) ; ( ) = 2 | | It leads to: 𝐺𝐺 𝜌𝜌 𝑦𝑦⃗ 𝑢𝑢𝑖𝑖 𝑦𝑦⃗ 3 𝛾𝛾 𝛼𝛼 The𝑔𝑔0 𝑖𝑖gravitomagnetic− 𝛾𝛾 field𝛼𝛼1 𝑉𝑉and𝑖𝑖 𝑉𝑉 𝑖𝑖its𝑥𝑥⃗ acceleration2 � contribution𝑑𝑑 𝑦𝑦⃗ 3 1 𝑐𝑐 𝑥𝑥⃗ − 𝑦𝑦⃗ = ; = are: 2 2 = ; = In our notationΩ��:� �𝐺𝐺⃗ 2 𝑔𝑔��𝑔𝑔𝑔𝑔𝑔𝑔��⃗ 𝜑𝜑 ∧ 𝑣𝑣⃗ Ω��𝐿𝐿𝐿𝐿��⃗ �𝑟𝑟𝑟𝑟𝑟𝑟��⃗ ℎ�⃗ 𝑐𝑐 𝚤𝚤 And in the case of𝑔𝑔 general relativity0𝑖𝑖 (that𝑔𝑔 is our𝑔𝑔 case): = ; = �𝐵𝐵��⃗ ∇��⃗ ∧ �𝑔𝑔 𝑒𝑒��⃗� 𝑎𝑎��⃗ 𝑣𝑣⃗ ∧ 𝐵𝐵��⃗ 4 = 1 ; = 0 ℎ�⃗ One then has ��⃗ �⃗ ��⃗ It then gives: 𝐻𝐻 𝑘𝑘 �𝑟𝑟𝑟𝑟𝑟𝑟��⃗ 𝐻𝐻 𝛾𝛾 𝛼𝛼1 = 2 = 4 ; = 4 And with our definition: 𝚤𝚤 �Ω��𝐿𝐿𝐿𝐿��⃗ 𝑘𝑘�⃗ 𝑔𝑔0𝑖𝑖 − 𝑉𝑉𝑖𝑖 𝐵𝐵��𝑔𝑔⃗ ∇��⃗ ∧ �− 𝑉𝑉𝑖𝑖𝑒𝑒��⃗� 3.2. Measure of the dark matter ( ) ( ) = = = ( ) | 𝑗𝑗| 𝑗𝑗 𝐺𝐺 𝜌𝜌 𝑦𝑦⃗ 𝛿𝛿𝑖𝑖𝑖𝑖𝑢𝑢 𝑦𝑦⃗ 3 In our solution, around the Earth, represents the addition of two One then𝐻𝐻𝑖𝑖 has−: 𝛿𝛿𝑖𝑖𝑖𝑖𝐻𝐻 2 � 𝑑𝑑 𝑦𝑦⃗ 𝑉𝑉𝑖𝑖 𝑥𝑥⃗ 𝑐𝑐 𝑥𝑥⃗ − 𝑦𝑦⃗ terms, the own gravitic field of the earth and the external = 4 ; = 4 = 4 𝑘𝑘�⃗ uniform gravitic field : 𝚤𝚤 𝑗𝑗 𝚤𝚤 𝑘𝑘��𝐸𝐸⃗ 0𝑖𝑖 𝑖𝑖 ��𝑔𝑔⃗ =��⃗4 𝑖𝑖��⃗ ��⃗ 𝑖𝑖𝑖𝑖 ��⃗ 𝑔𝑔 − 𝐻𝐻 𝐵𝐵 ∇ ∧ �− 𝐻𝐻 𝑒𝑒 � ∇ ∧ � 𝛿𝛿 𝐻𝐻 𝑒𝑒 � 0 = + = 4 𝑘𝑘��⃗ ∇��⃗ ∧ 𝐻𝐻��⃗ In the same way, the Lense�-⃗Thirring��𝐸𝐸⃗ effect��0⃗ is then composed With the following definition𝑔𝑔 of gravitic field: 𝑘𝑘 𝑘𝑘 𝑘𝑘 �𝐵𝐵��⃗ �𝑟𝑟𝑟𝑟𝑟𝑟��⃗ 𝐻𝐻��⃗ of the own Earth gravitic field term and of a new Ω��𝐿𝐿𝐿𝐿��_⃗ = 4 supplementary term of “dark matter” _��𝐿𝐿𝐿𝐿�� 𝐸𝐸⃗ ��𝑔𝑔⃗ Ω 𝐵𝐵 = 2 + 2 = + One then retrieves our previous𝑘𝑘�⃗ relations: Ω��𝐿𝐿𝐿𝐿��_��𝐷𝐷𝐷𝐷��⃗ _ = ; = = 4 The term is the traditional frame-dragging precession: _ �Ω��𝐿𝐿𝐿𝐿��⃗ �𝑘𝑘��𝐸𝐸⃗ �𝑘𝑘��0⃗ �Ω��𝐿𝐿𝐿𝐿��𝐸𝐸⃗ �Ω��𝐿𝐿𝐿𝐿��𝐷𝐷𝐷𝐷��⃗ A last remark: The interest of our notation is that the field 𝑘𝑘�⃗ �𝑟𝑟𝑟𝑟𝑟𝑟��⃗ 𝐻𝐻��⃗ 𝑎𝑎��𝑔𝑔⃗ 𝑣𝑣⃗ ∧ 𝐵𝐵��𝑔𝑔⃗ 𝑣𝑣⃗ ∧ 𝑘𝑘�⃗ �Ω��𝐿𝐿𝐿𝐿��𝐸𝐸⃗ 3 = = ; = equations are strictly equivalent to Maxwell idealization (in 4 2 _ particular the speed of the gravitational wave obtained from ℎ��𝐸𝐸⃗ 𝐺𝐺 𝐺𝐺 𝐽𝐽⃗ 𝑟𝑟⃗ 𝐻𝐻��𝐸𝐸⃗ � 2 3� �𝑟𝑟⃗ ∧ 𝐽𝐽⃗� �Ω��𝐿𝐿𝐿𝐿��𝐸𝐸⃗ 2 � 3 − 5 �𝑟𝑟⃗ ∙ 𝐽𝐽⃗�� these equations is the light celerity). Only the movement In the Gravity Probe𝑐𝑐 𝑟𝑟 B experiment, the expected𝑐𝑐 𝑟𝑟 value𝑟𝑟s were: equations are different with the factor “4”. But of course, all = 6606 / the results of our study could be obtained in the traditional = 39 / �Ω��𝐺𝐺⃗_� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 ��Ω��𝐿𝐿𝐿𝐿��𝐺𝐺𝐺𝐺��𝐺𝐺⃗� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑦𝑦𝑒𝑒𝑒𝑒𝑒𝑒 2

Let’s evaluate the order of magnitude of the external gravitic field (our dark matter) around the Earth. From (LE CORRE, 2015), an average value is: . _ = 2 ~2 × 10 It then gives −16 5 −1 ��Ω��𝐿𝐿𝐿𝐿��𝐷𝐷𝐷𝐷��⃗� �𝑘𝑘��0⃗� 𝑠𝑠 _ = 0.4 / In fact, from the sample of galaxies studied in (LE CORRE, 2015), �Ω��𝐿𝐿𝐿𝐿��𝐷𝐷𝐷𝐷��⃗� 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑦𝑦𝑦𝑦𝑎𝑎𝑟𝑟 one obtains the following possible interval for : 10 . < < 10 . 𝑘𝑘0 If we assume that these−16 galax62 ies can be represent−16 3 ative of our own �𝑘𝑘��0⃗� galaxy, the expected discrepancy should be in the following interval (in / ): 0.3 < < 0.6 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦_

�Ω��𝐿𝐿𝐿𝐿��𝐷𝐷𝐷𝐷��⃗� 3.3. Discussion

_ represents around 1% of _ . But until now, is only known with a precision of 10%. We need �Ω��𝐿𝐿𝐿𝐿��𝐷𝐷𝐷𝐷��⃗� _ �Ω��𝐿𝐿𝐿𝐿��𝐺𝐺𝐺𝐺��𝐺𝐺⃗� to have a better accuracy on this kind of experiments to hope to �Ω��𝐿𝐿𝐿𝐿��𝐺𝐺𝐺𝐺��𝐺𝐺⃗� detect this discrepancy. GINGER experiment should have a precision of 1%. It could be enough to detect a discrepancy. 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 (LE CORRE, 2015) we have seen that at this distance the gravitic field of the galaxy could be of the same magnitude. Therefore the expected value should be around 2 times greater than _ . One also have to take into account the unknown direction�Ω��𝐿𝐿𝐿𝐿��𝐷𝐷𝐷𝐷�� ⃗of� , implying that the magnitude of the effect could be reduced. Furthermore the effect �𝑘𝑘��0⃗ of precession _ could be spread on the two components and and then decrease the discrepancy. ��Ω��𝐿𝐿𝐿𝐿��𝐷𝐷𝐷𝐷��⃗� �Ω��𝐺𝐺⃗� �Ω��𝐿𝐿𝐿𝐿��⃗� 3.4. Conclusion

In the better case, a precision of 1% could reveal a discrepancy in the measure of the expected precession effects. The next generation of experiments (as GINGER) will have such an accuracy. In our solution the expected discrepancy should be in the following interval (in / ) 0.3 < _ < 0.6. But without any detection, a higher accuracy will be required 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 �Ω��𝐿𝐿𝐿𝐿��𝐷𝐷𝐷𝐷��⃗� to definitively declare that this solution is irrelevant. In particular if the direction of is very disadvantageous.

𝑘𝑘��0⃗ References

ADLER R. J., ”The three-fold theoretical basis of the Gravity Probe B gyro precession calculation”, 2015 RUGGIERO M. L., "Sagnac Effect, Ring Lasers and Terrestrial Tests of Gravity", Galaxies 3, 84-102, 2015 CLIFFORD M. WILL, “The confrontation between general relativity and experiment”, arXiv:1403.7377v1, 2014 HOBSON et al. “Relativité générale”, ISBN 978-2-8041-0126-8, 2009 December LE CORRE S., "Dark matter, a new proof of the predictive power of General Relativity", ArXiv:1503.07440, 2015 MASHHOON B., “Gravitoelectromagnetism: A brief review”, arXiv:0311030v2, 2008 April 17