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The Galactic Dark Matter As Relativistic Necessity Nicolas Poupart The Galactic Dark Matter As Relativistic Necessity Nicolas Poupart To cite this version: Nicolas Poupart. The Galactic Dark Matter As Relativistic Necessity. 2016. hal-01399780 HAL Id: hal-01399780 https://hal.archives-ouvertes.fr/hal-01399780 Preprint submitted on 20 Nov 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. Distributed under a Creative Commons Attribution| 4.0 International License The Galactic Dark Matter As Relativistic Necessity Nicolas Poupart, Independent Scholar (2016) 12269 rue Lévis, Mirabel, Québec, Canada (J7J 0A6) (450) 939-2167 [email protected] Introduction Since the assumption of the existence of galactic dark matter (dark mass) by Vera Rubin to explain the flatness of galactic rotation curves1,2,3, no convincing explanation about the nature of this dark mass has been made. Attempts to explain this missing mass by an invisible form of ordinary baryonic matter was largely refuted by the programs MACHO4, EROS5 and AGAPE6, it is the same for explanations using ordinary non-baryonic matter. Numerous detection attempts of exotic particle that could explain the missing mass, have also all been unsuccessful. Similarly, the new CERN accelerator appears to confirm that the physics is limited to the standard model and the existence of an exotic particle is less and less probable7,8,9. It also seems extremely difficult, even impossible, to explain this phenomenon with the current theory of gravitation either the Newtonian gravity10,11 (NG) or general relativity (GR). An alternative is to modify gravity so as to adapt it to regime change at galactic level. On the contrary, such a project is exposed to the prodigious adequacy of the GR to the phenomenological reality12,13 and the physical existence of dark matter14,15. The explanation proposed in this article is of an entirely different nature. The dark mass is not some form of actual matter, it is also not an epiphenomenon caused by gravitation. The dark mass is actually a necessary consequence of relativistic mechanics (RM), that is to say, the combination of classical mechanics (CM) and special relativity (SR). It will be demonstrated that this relativistic mass is necessary if a body like a galaxy can physically collapse into a compact ball. The explanation proposed ignores the forces of physics and therefore is a purely mechanical explanation. The existence of a compact ball state The theorem of the Schwarzschild radius limit can be simply derived from NG and the postulate of the light speed as a maximum speed. Indeed, simply pose that escape velocity V = (2GM/Rs) is equal to c which gives Rs = 2GM/c2. It is also known that the same equation can be derived with GR. The fundamental axiom of our demonstration is that a galaxy can contract in a compact ball with a radius proportional to the mass thus Ra = aM. The only constraint is that Ra radius is much smaller than that of the compacted galaxy, which limits the choice of a. To simplify calculations, we'll set b = a c2/2 2 and so Ra = 2bM/c [D1]. We consider two models for the dynamics of this compact ball: 1) the rigid ball in relativistic rotation at constant angular velocity v(r) = r and 2) the homogeneous energy ball at constant linear velocity (r) = v/r. The rigid ball requires the existence of a bonding force, non existent in RM only, while a simple force of friction would produce a homogenization of speeds. In the case of the rigid ball, the spin is defined as a = /max and in the case of the homogeneous energy ball a = (r)/max(r). The spin value is a constant and by posing = (R) and max = max(R) then in both cases a = (r)/max(r) = (R)/max(R) = /max. 1/6 The energy and angular momentum of the ball in relativistic rotation First calculate the mass M of a rigid ball of radius R in relativistic rotation around a central axis. (r) is the relativistic density at the distance r from the axis of rotation and 0 the inert density on the axis of 2 2 rotation (no relativistic expansion of the mass). The relativistic density is (r) = 0/[1-v(r) /c ]. The height of the basic cylinder at the distance r from the axis of rotation is h(r) = 2[R2-r2] and its volume is given by V(r) = 2rh(r) dr. Therefore, taking into account the relativistic mass change by integrating from 0 to R : 2 2 2 2 2 2 2 3 2 2 M = (r) V(r) = (r) 2rh(r) dr = 20R 2[R -r ]/[1-v(r) /c ] d[r /2R ] = 20R [1-r /R ]/[1- v(r)2/c2] d[r2/R2]. 1) In the case where the ball is rigid and rotates at constant angular speed max = c/R (maximum speed 3 2 2 2 2 2 2 3 2 2 at the equator) we have M = 20R [1-r /R ]/[1-(rmax) /c ] d[r /R ] = 20R [1-r /R ]/[1- 2 2 2 2 2 2 3 2 2 3 3 3 c r /c R ] d[r /R ] = 20R d[r /R ] = 20R . So M = 20R [L1]. Because M0 = 40R /3 (density by the volume of the ball) therefore M = 3M0/2. The energy gain Mr = M-M0 when a = 1 is maximum Mr = M0/2 [T1]. 2) In the case where the ball has a homogeneous energy and rotates at the constant linear velocity 3 2 2 2 2 2 2 3 2 v = r(r) we have M = 20R [1-r /R ]/[1-(r(r)) /(rmax(r)) ] d[r /R ] = 20R /(1-a ) (1- 2 2 2 2 3 2 2 2 2 3 2 3 r /R ) d[r /R ] = 40R /(1-a ) r(1-r /R )/R dr = 40R /3(1-a ) [L2]. Since that M0 = 40R /3 2 2 (density by the volume of the ball) then M = M0/(1-a ). The energy gain Mh = M-M0 = M0(1/(1-a )-1) [T2] tends to infinity with the spin of the ball (a = 1) and is zero for a nul spin (a = 0). It is now possible to calculate the relativistic moment of inertia of the ball : I = r2 dm = r2(r) V(r) = 2 4 2 2 2 2 4 4 5 2 2 2 2 4 4 r (r) 2rh(r) dr = 20R 2[R -r ]/[1-v(r) /c ] d[r /4R ] = 0R [1-r /R ]/[1-v(r) /c ] d[r /R ]. 1) In the case where the ball is rigid and rotates at constant angular speed max = c/R (maximum speed 5 2 2 2 2 4 4 5 2 2 2 2 2 2 at the equator) we have I = 0R [1-r /R ]/[1-(rmax) /c ] d[r /R ] = 0R [1-r /R ]/[1-c r /c R ] 4 4 5 5 2 d[r /R ] = 0R . Thus I = 0R and by [L1] Ir = ½MR . 2) In the case where the ball has a homogeneous energy and rotates at the constant linear velocity 5 2 2 2 2 4 4 5 2 2 2 v = r(r) we have I = 0R [1-r /R ]/[1-(r(r)) /(rmax(r)) ] d[r /R ] = 0R /(1-a ) (1-r /R ) 4 4 5 2 3 2 2 4 5 2 5 2 d[r /R ] = 0R /(1-a ) 4r (1-r /R )/R = 80R /15(1-a ). So I = 80R /15(1-a ) and by [L2] 2 Ir = 2MR /5. The relativistic angular momentum of the maximum speed rotating rigid ball is given by Jr = Ir = ½MR2 [T3], which is not very different from the non-relativistic angular momentum J = 3MR2/5. 2 2 This calculation is not exact because Jr should vary from 3MR /5 for a spin of a = 0 to ½MR for a spin of a = 1. The relativistic angular momentum of the rigid ball of homogenous energy is Jh = 2 Ih = 2MR /5 [T4], this calculation is, on the other hand, exact. The dark mass production 2 Consider a rigid ball in relativistic rotation with a radius Ra, thus by [T3] J = I = ½MRa [L3]. Since a = /max and max = c/Ra then = ac/Ra and by [L3] J = ½acMRa [L4]. By [D1] and [L4] we obtain 2 16 Jbr = abM /c [T5], which is exactly the calculated angular momentum by Kerr if b = G. Thus, the moment of inertia of the Kerr black hole, regardless of its spin, is that of a rigid ball of which the surface rotates at the speed of light. Using the same calculation with the ball of homogeneous energy is 2 obtained by [T4] and [D1] Jbh = 4abM /5c = 4Jbr/5 [T6]. 2/6 For virtually any rigid body I = jMR2 such as j is a constant characterizing the shape of the body : j = 1 for the ring, j = 1/2 for the disc, j = 2/3 for the sphere, and j = 3/5 for the ball. Since = V/R, it 2 follows that Jj = I = (jMR )(V/R) = jMRV [T7]. 2 If such a body is compacted into a ball with a radius Ra then by [T5] Jf = abM /c and by [T7] Ji = jMRV.
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