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 precessions). In the GINGER experiment, there could be enough, 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. Keywords: gravitation, gravitic field, dark matter gives the following field equations (HOBSON et al., 2009) = (with 2 ): 1. Overview 1 2 2 8 −=∆0 ; = 2 ( ) General relativity implies the existence of two gravitational With: ℎ� ℎ� − 4 components. In addition of the gravity field, there is a gravitic field 1 = ; ; = ; = ( ) (together giving what is called the gravitomagnetism) just like the 2 magnetic field in electromagnetism. The new gravitic field can be Theℎ� generalℎ −solution ℎ of theseℎ ≡ ℎ equations ℎ is: ℎ ℎ� −ℎ measured by its precession effect, known as Lense-Thirring effect. 4 ( | |, ) ( , ) = Several experiments have validated this effect for the Earth | | − ⃗ − ⃗ ⃗ 3 gravitic field, NASA's LAGEOS satellites or Gravity Probe B (ADLER, In the approximationℎ� ⃗ of− a source4 � with low speed, one has:⃗ ⃗ − ⃗ 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 one has: (RUGGIERO, 2015) with an expected accuracy of around 1%. 4 ( ) ( ) = | | ⃗ 3 In (LE CORRE, 2015), a solution is proposed to explain the dark At this step, by proximityℎ� ⃗ with− electromagnetism,4 � ⃗ one traditionally ⃗ − ⃗ 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, 2008 ) for the 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 4 4 = ; = ; = 0 gravitic field could be measured by its precession effect. Such a 00 0 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 : detection. And even, in the most advantageous case, the accuracy ( ) ( ) of 1% (as expected in GINGER experiment) could be enough to | | ⃗ 3 detect it. (⃗) ≡(−) � ⃗ ( ) ( ) ( ) ⃗=− ⃗ | | | | First, I recall the theoretical idealization used in this article and in ⃗ ⃗ 3 −1 ⃗ ⃗ 3 ⃗ ≡ − 2 � ⃗ − � ⃗ (LE CORRE, 2015). With a new constant ⃗defined− ⃗ by: ⃗ − ⃗ = 2. Gravitation in linearized general relativity This definition gives ~7.4 × 102 very small compare to . −1 −28 From general relativity, one deduces the linearized general The field equations () can be then written (Poisson equations): relativity in the approximation of a quasi-flat Minkowski space 4 = 4 ; = = 4 ( ) ( = + ; | | 1). With following Lorentz gauge, it −1 ∆ ∆ 2 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: = ⃗ �⃗ . �����⃗ = ; = �⃗ 4 3. Gravitic field: a way to measure it = 0 ; = 0 ; ⃗ −����������⃗ �⃗ ������⃗ ��⃗ = 4 ; = 4 ������⃗⃗ �⃗ Just like for the electromagnetism, this gravitic field implies a −1 ������⃗�⃗ p phenomenon of precession. It is known as the Lense-Thirring With relations ( ),⃗ one− has: − ȷ��⃗ effect. Instead of taking into account only the own gravitic field of 2 4 = = = = ; = ; = 0 ( ) the earth, we are also going to take into account the hypothetical 00 11 22 33 0 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 going to recall what the equations in the general relativity are for 1 ~ the Lense-Thirring effect. And secondly, we will use it to test our 2 2 2 solution by calculating the contribution to the precession effect It then leads2 to the− movement ℎ00 equations:− � ℎ0 − ℎ0� generated by our gravitic field that explain the dark matter. ~ + 4 = + 4 2 ⃗ 3.1. Gravitic field and precession effect 2 − ����������⃗ ⃗ ∧ �������⃗��⃗� ⃗ ⃗ ∧ �⃗ From relation ( ), 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 studied in several papers. In general relativity, it leads to a 2 2 2 2 In a quasi-�Minkowski2 � space , one has: − � − 2 � �� � precession of . For example, with (ADLER, 2015), one can write = = . the following equations: We retrieve the known expression (HOBSON et al., 2009) with our 1 1 − −��⃗ ����⃗ = + + ( + ) definition of : 2 4 ⃗̇ 2 ����������⃗ �⃗ ⃗ 2 8 . 2 Which lead� �to define� a� geodetic ∧ vector⃗� field and������⃗ ℎa� “∧gravito - = 1+ 1 magnetic” vector field : 2 2 2 ��⃗ ����⃗ 2 �Ω����⃗ � 2 � − − � − 2 � �� � 1 1 = + Ω�������⃗ ; = ( + ) Remark: Of course, one retrieves all these relations starting 2 4 with the parameterized post-Newtonian formalism. From These�Ω��� �⃗expressions� � use2 �� ��the�������⃗ PPN ∧ formalism.⃗� Ω�������⃗ As seen before,������⃗ ℎ�⃗ for (CLIFFORD M. WILL, 2014) one has: general relativity, one has: 1 ( ) ( ) = 1 ; = 1 = (4 + 4 + ) ; ( ) = 2 | | It leads to: ⃗ ⃗ 3 The0 gravitomagnetic− field1 and its⃗ acceleration2 � contribution ⃗ 3 1 ⃗ − ⃗ = ; = are: 2 2 = ; = In our notationΩ���:� �⃗ 2 ����������⃗ ∧ ⃗ Ω�������⃗ ������⃗ ℎ�⃗ And in the case of general relativity0 (that is our case): = ; = ����⃗ ∇��⃗ ∧ � ���⃗� ����⃗ ⃗ ∧ ����⃗ 4 = 1 ; = 0 ℎ�⃗ One then has ��⃗ �⃗ ��⃗ It then gives: ������⃗ 1 = 2 = 4 ; = 4 And with our definition: Ω�������⃗ �⃗ 0 − ����⃗ ∇��⃗ ∧ �− ���⃗� 3.2. Measure of the dark matter ( ) ( ) = = = ( ) | | ⃗ ⃗ 3 In our solution, around the Earth, represents the addition of two One then has−: 2 � ⃗ ⃗ ⃗ − ⃗ terms, the own gravitic field of the earth and the external = 4 ; = 4 = 4 �⃗ uniform gravitic field : ����⃗ 0 ����⃗ =��⃗4 ���⃗ ��⃗ ���⃗ − ∇ ∧ �− � ∇ ∧ � � 0 = + = 4 ����⃗ ∇��⃗ ∧ ��⃗ In the same way, the Lense�-⃗Thirring����⃗ effect����0⃗ is then composed With the following definition of gravitic field: ����⃗ ������⃗ ��⃗ of the own Earth gravitic field term and of a new Ω�������_⃗ = 4 supplementary term of “dark matter” _������� ����⃗ ����⃗ Ω = 2 + 2 = + One then retrieves our previous�⃗ relations: Ω������_��������⃗ _ = ; = = 4 The term is the traditional frame-dragging precession: _ Ω�������⃗ ����⃗ ����0⃗ �Ω����������⃗ Ω��������������⃗ A last remark: The interest of our notation is that the field �⃗ ������⃗ ��⃗ ����⃗ ⃗ ∧ ����⃗ ⃗ ∧ �⃗ �Ω����������⃗ 3 = = ; = equations are strictly equivalent to Maxwell idealization (in 4 2 _ particular the speed of the gravitational wave obtained from ℎ����⃗ ⃗ ⃗ �����⃗ � 2 3� �⃗ ∧ ⃗� �Ω����������⃗ 2 � 3 − 5 �⃗ ∙ ⃗�� these equations is the light celerity). Only the movement In the Gravity Probe B experiment, the expected values 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�����⃗_� �Ω���������������⃗� 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�������������⃗� �����0⃗� _ = 0.4 / In fact, from the sample of galaxies studied in (LE CORRE, 2015), �Ω��������������⃗� one obtains the following possible interval for : 10 .