1. INTRODUCTION. A solution for the dark matter mystery based on Euclidean relativity. Frédéric LASSIAILLE 2009 Page 1 14/05/2010. Frédéric LASSIAILLE

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1 Frédéric LASSIAILLE 2009 Page 1 14/05/2010 Frédéric LASSIAILLE lmimi2003@hotmail.com A soltion for the dark matter mystery based on Eclidean relativity The stdy of this article sggests an explanation for the dark matter mystery. This explanation is based on a modification of Newton s law. This modification is condcted from an Eclidean vision of relativity. Concerning the mystery of the velocity of the stars inside a galaxy, the stdy calclates a theoretical crve which is different from the one coming from Newton s law. This theoretical crve is very close to the measred one. Concerning the mystery of the velocity of a galaxy inside its grop, the explanation is more direct. For this mystery, the stdy calclates a greater vale for G, the gravitational constant. 1. INTRODUCTION The theoretical aim of this article is to address an isse yielded by general relativity. This isse may be explained by: global space's shape determination. This absolte space-time is the general relativity space-time, and this space deformation in space-time mst be in conformity with Newton s law at least for long distances. The adopted point of view is an Eclidean relativity. An Eclidean mathematical context is sed, with 4 dimensions (three of space, x, y, z, and one of time: ct). This for restricted relativity. For general relativity of corse we se the same and we extend it overall with a tensor. Except that here locally it is an Eclidean metric sed to represent space-time. Within this mathematical framework the physics principles are exactly those of relativity: isotropy of space, inertial frames of reference with reciprocities between those inertial frames, constancy of the speed of light in each inertial frames, moving from restricted relativity to general relativity by covariance along the geodetic trajectories of the inertial frames, etc This for the restricted version, and of corse for the general one: deformation of space-time by energy, expression of a force (gravitational ) by a space-time deformation. There exists however a difference in the physics principles, between relativity and or approach. Indeed, here constancy of space-time distance is not an aim, in a global representation of space-time. In other words, we do not se Minkowski s representation since we wish to remain in Eclidean representation. Hence, the basic principle of an invariant space-time length when changing inertial frames is left off. The method sed consists to postlate first that Lorentz s eqations are simply a conseqence of a space-time deformations by energy. In other words we try to express the general relativity deformation principle, in the context of restricted relativity. Once this done, physics inconsistencies are fond. Of corse we try to elde them. This leads to finally postlate the existence of indivisible particles, from which matter is made of. Back to or first postlate, we find a space time determination, coherent with Lorentz s eqations. This will be or final determination of the shape of global space inside space-time. Frédéric LASSIAILLE 2009 Page 1

2 Frédéric LASSIAILLE 2009 Page 2 14/05/ RETRIEVING LORENTZ S EQUATIONS Lorentz s physics context is sed. Let s point it ot. There are two inertial frames, R (O, x, y, z, ct) and R' (O', x', y', z', ct'), in niform rectilinear motions at the v speed one compared to another, along Ox axis. X is increasing along Ox axis, and x is decreasing along O'x' axis. In this case, only x, ct, and x', ct' dimensions are important. At t = x = 0 there is also t' = x' = 0. In order to find Lorentz s eqations within this physics framework, and since or representations are Eclidean, it is necessary to sppose that O'x' axis rocked with an α angle compared to Ox axis, with sin(α) = v/c. See figre 1. In the same way it is necessary to have O coordinates eqal to: (x = vt, and ct = v² t/c). Conversely nder these conditions the reader will be able to calclate that Lorentz s eqations are fond. See figre 1. Figre 1: Lorentz s eqations in the context of Eclidean relativity. On the basis of this Eclidean model of restricted relativity, we are tempted to sppose a coherent physics postlate. This postlate is the following. POSTULATE 1 << Any particle with a non nll mass m, moving with v speed along Ox axis, x increasing, compared to an inertial frame R (O x y z ct), deforms space-time arond it with a rotation of the 0x0ct plan arond the 0y0z axis, with an α angle between Ox and Ox, sch as sin(α) = v/c. Dring the displacement of this particle from O to A(x = vt, ct),a vacm appeared inside space-time. The location of this vacm is the (O, O, H) triangle, sch as: O coordinates are O (vt, v²t/c), H coordinates are H(vt, 0)>>. In the borderline case of a photon, with v = c, the swing becomes maximm: α = π/2, and the vacm is the (O, A, H) triangle. Figre 2 below represents the effect of postlate 1. A P particle is moving at v speed in the inertial reference frame R, parallel to Ox axis, and in the direction of x increasing. At the t instant, the particle is located coordinates x and ct in R (A point). The inertial frame R' centered on O' is attached to the particle. Hence R is also moving niformly along Ox axis. It is the same case as the one of figre 1, except that here exists the P particle on A point. On figre 2, the space line rocked with the α angle, locally in A. Frédéric LASSIAILLE 2009 Page 2

3 Frédéric LASSIAILLE 2009 Page 3 14/05/2010 Figre 2: Postlate 1. On the other hand, far from A point this line of space has becomes parallel to Ox axis. This is indeed the only realistic possibility! It is difficlt to imagine the movement of a particle deforming the entire niverse this way along Ox axis. With this postlate, the eqations of Lorentz now express a local deformation of space-time, cased by the energy of the moving particle (postlate 1 above). This respect of Lorentz s eqations is only local to the particle. In other words this respect is done only for low vales of x' and ct' (and also for R sch as O is close to A at the t instant). It will be noticed that this re-discovery of Lorentz s eqations works thanks to the positioning of O' (ths the positioning of A) indicated previosly: (vt, v²t/c) for O. It is known that this position of O explains, noticeably, the asymmetry of the twin paradox in relativity. This position corresponds to a real vacm which appears inside space-time. In addition it works thanks to the space-time swing indicated in the postlate above. This swing modifies with the cos(α) ratio, eqal to [1 - sin²α] = [1 - v²/c²], each temporal vales (in ct) and space (in x). However this swing is only local, not global. Restricted relativity is retrieved with the help of a simple deformation of space in space-time. Nevertheless, it is necessary to distingish position of space in space-time, on one hand, and representation of space-time in sch or sch reference frame, on the other hand. Indeed, space line in space-time became deformed only locally within the particle. It allows localizing space-time events in a comprehensive bt complicated view. This vision is complicated becase space-time is not Eclidean any more bt Riemannian with an Eclidean local base. Ths space lines have the shapes of crves and are no more simple straight lines like Ox axis. Otherwise O'x' and O ct axis are straight lines. They represent space-time locally bt does not represent it overall any more. Broadly the representation of space-time which is done by the inertial frame R', allows to see spacetime in a legal way in the sense of relativity. This representation is the only one respecting, locally, the general physics principles of restricted relativity: in particlar the constant speed of light in each inertial frame. Frédéric LASSIAILLE 2009 Page 3

4 Frédéric LASSIAILLE 2009 Page 4 14/05/ LUMINOUS POINTS However at this stage a problem of coherence arises since a particle is constited of smaller particles. Indeed, how to ensre that space-time deformation generated by the movement of a big particle, composed by a heap of smaller particles, can rise from the deformations of these smaller particles? To ensre this coherence a soltion consists in spposing that matter is made p of a restricted grop of very small indivisible particles. These small particles mst be conceived in sch a way that they can explain spacetime deformations generated by any other composed particle. For this explanation a simple operation mst calclate the final deformation generated by the large particle, from the small particles it is made of. Ths defined, this operation mst allow, by constrction, calclation of the shape of absolte crves of space, starting from the positions and energies of these small indivisible particles. At the same time, this operation mst be, of corse, compliant with postlate 1 and Lorentz s eqations. From there the second postlate arises, which follows. POSTULATE 2 << Each particle consists of a certain nmber of smaller particles, called the lminos points. These lminos points are moving constantly at the c speed, inside the first particle, and with respect to any inertial frame of reference >>. From this postlate, it is possible to determine in a single way any deformation generated by any particle. For that, we apply postlate 1 to these lminos point particles. For these lminos points the α angle is eqal to its limit vale π/2. The shape of space is ths at any time the reslt of sccessive combinations of these small deformations cased by all these lminos points. What remains to be specified is the way of combining those varios deformations. This will be specified by the postlate 3 which follows. After that, it will be possible to check that the α angle calclated starting from postlates 2 and 3 is well given by the formla of postlate 1. For that let s retrn to Lorentz s eqations. A first mathematical observation is essential. The conservation eqation of the energy of restricted relativity is fond by qantifying the lminos point trajectories lengths, inside P particle. It is what we will see. The particle attached to the O' point is modelled as consisting of only one lminos point. Conseqently the obtained model is the one described by figre 3. It can be check that the reasoning remains valid in the general case of a particle made p of several lminos points. When the P particle moves from O point (on the figre) to A point, along OA segment, the lminos point contained follows a trajectory having a << V >> shape, that is.: a) First stage: displacement at the speed +c along Ox, (milked in fat on the figre). b) Second stage: displacement at the speed -c along Ox (milked in fat on the figre). For the first stage, L1 is the displacement length, and L2 (positive) is the displacement length of the second stage. If x is the position of the A point, we can write: x = vt = L1 - L2. This x position is also the coordinate of P in R at this time t. Indeed at t, in R, the position of P coincides with A point. Frédéric LASSIAILLE 2009 Page 4

5 Frédéric LASSIAILLE 2009 Page 5 14/05/2010 C t A O L2 X L1 Figre 3: Lminos point trajectory in a moving particle. Hence: ct = L1+ L2 vt = L1- L2 (1) In addition we can write: (L1 + L2)² (L1- L2)² = + 2 L1L2 2 2 (2) This last eqation is nothing more than relativistic eqation of energy: E² = E + E 2 2 mvt m with E = mc²/ [1 - v²/c²], Emvt = mvc/ [1 - v²/c²], and Em = mc². To obtain the eqivalent eqation for the energy densities, each term of this eqation is divided by the vale (L1 + L2)²/2 which is the vale of the total energy of the particle: (L1- L2)² 1 = + oper(l1, L2)² (L1+ L2)² (3) With: oper(l1, L2) = 2 L1L2 L1 + L2 The introdced operator is the relationship between the algebraic average and the arithmetic mean. It is eqal to the relativistic coefficient [1 - v²/c²]. Indeed, from (1): 2L1/ct = 1 + v/c, and 2L2/ct = 1 - v/c, and then [1 - v²/c²] = 2 [L1 L2] / (L1 + L2). Frédéric LASSIAILLE 2009 Page 5

6 Frédéric LASSIAILLE 2009 Page 6 14/05/2010 Let s remind that eqation (3) is also written: 1 = sin²(α) + cos²(α) swing of postlate 1. where α is the angle of the space-time Finally, this last stdy of Lorentz s eqations led s to retain an operator. We will se this operator to postlate, finally, the mode of determination of general relativity absolte spacetime. By constrction, this determination will be compatible with restricted relativity. Frédéric LASSIAILLE 2009 Page 6

7 Frédéric LASSIAILLE 2009 Page 7 14/05/ RELATIVISTIC OPERATOR This is done by the following postlate which follows. It only generalizes the previos observation done abot Lorentz s eqations. POSTULATE 3 << Space shape in space-time is given at any point by the ratio of the infinitesimal space lengths, ds along space line, and dx its length projected on Ox axis. This ratio is eqal at any point to the relativistic operator applied to the two following vales: a) L1: sm of the heights of vacm of space-time deformations propagated in Ox direction, x increasing, b) L2: sm of the heights of vacm of space-time deformations propagated in Ox direction, x decreasing >> That is to say: dx 2 L1L2 = ds L1 + L2 Where L1 and L2 are the 2 above-mentioned sms. This operator is neither linear nor associative. Bt this doesn t matter since it is calclated once, at any point of space. Bt it mst be checked that this operator yields the same reslt when calclated in different inertial frames. That is to say that, for postlate 3, the choice of the inertial frame has no incidence on the final calclated space s slope. In fact this is the case becase of the Lorentz s eqations properties, and also becase this operator s vale is eqal to [1 v²/c²]. However, this can be also proven by direct calclation. This is done on appendix 5. It is written above that L1 and L2 are the sms of the heights of vacm of the propagated deformations. It is necessary to describe how these space-time deformations are propagated. The mechanism is very similar to waves propagated by the movement of a boat over a water srface. Let s consider figre 1. The initial deformation relates to the OxOct plan. The propagations of this deformation in space-time are carried ot on remaining space dimensions, i.e. Oy and Oz, more generally on any Or direction, half-line based on O and contained in the Oyz plan. The form of these propagated deformations is each time exactly the same as the initial deformation. The initial deformation was done on OxOct plan (space-time swing represented figre 1). Now the propagated deformation is the same bt it relates to the OrOct plan in place of OxOct plan. The height of this propagated deformation attenates progressively as r increases. The attenation law, g, will be given frther. At any time, the lminos point ths emits this deformation. Therefore, like in the case of the boat, the finally overall propagated deformation is the envelope of all these propagations of deformation. (In the case of a boat this envelope has a << V >> shape which is the final shape of these waves over the water srface). Above, the initial space-time deformation is connected to the existence of a space-time vacm. What does become this vacm dring the propagation of this deformation? It is propagated too. It relates to 3 dimensions y, z, and ct. Finally the L1 sm is calclated for all the propagated and received deformations. Each of these deformations is generated by a niqe lminos point. An niqe deformation dl1 received is sch as dl1² = k d²v where k is a constant and d²v the vacm propagated with the space-time deformation. In the same way for L2, except that this time the deformations come from the opposite sense of propagation (on the same axis). Let s stdy the overall reslt of all the propagated deformations which are received in the same M point at the same time. The postlate 3 above express the length ratio along Ox axis only. Bt at one M point there are nmeros sch directions coming to. Hence it is necessary to calclate the relativistic operator of postlate 3 for each space Frédéric LASSIAILLE 2009 Page 7

8 Frédéric LASSIAILLE 2009 Page 8 14/05/2010 direction. The final deformation is then obtained. The only qestion is: what is the mode of contribtion of the deformations of all these directions in order to obtain the reslt? This reslt will be the reslting space-time deformation at M point. It will be ths necessary to generalize the above operator with a second more generic operator, which will take into accont each space directions. The reslt given by this second generalized operator mst be the famos final space-time deformation at M point. It will be probably sefl to se a mathematical base like the qaternion for that. In this article this complexity will not be seen becase fortnately not necessary. We ths fond relativity starting from postlates 1, 2, and 3. The α angle of the postlate 1 rotation is calclated, by applying postlates 2 and 3. Overall, we obtained a way for calclating space shape inside space-time. We can now stdy Newton s law. Frédéric LASSIAILLE 2009 Page 8

9 Frédéric LASSIAILLE 2009 Page 9 14/05/ FIRST MODIFICATION OF NEWTON S LAW The stdied case is a particle of mass M isolated in a space filled niformly with an energy density constant and weak in front of M. The particle coordinates are x = y = z = 0 which are those of the O point in or sal inertial frame R of reference. The stdied case being invariant by any rotation of center O, only the Ox axis with x>0, and the axis of time Oct, are important. How does evolve the local slope tan(α) of space, along Ox axis? The postlate 3 above is applied. Let s consider a space-time P point in which comes at least one deformation from a lminos point pertaining to the M mass. We sppose P x-coordinate positive strict that is: x>0. The M mass propagates on P point the following deformations: L1 m = g(x) An attenation fnction which will be given frther. (4) Propagation in the direction of x increasing. L2 m = 0 No deformation propagated in the direction of x decreasing, coming from M, becase x>0. The srronding niverse with constant energy density propagates on P point the following deformations: Hence: L1 Propagation in the direction of x increasing. (5) L2 = L1 = L Now application of postlate 3 is possible: Same thing in the direction of x decreasing becase space time is spposed isotropic. L1 = L1 + L1 m = L + g(x) L2 = L2 + L2 = L m dx ds = oper(l + g(x), L ) Let s calclate: dx ds = 2 (L + g(x)) L L + g(x) + L = g(x) 1+ L g(x) 1+ 2L Let s write: e = g(x) L (6) Frédéric LASSIAILLE 2009 Page 9

10 Frédéric LASSIAILLE 2009 Page 10 14/05/2010 dx ds = 1+ e (7) e 1+ 2 Approximation: x great, ths g(x) low in front of L. Therefore e<<1. After a limited development of e at order 2: Which makes finally, sing eqation (6): dx e² 1 - ds 8 dx g(x)² cos( α ) = 1 (8) ds 8L ² Where α is the angle between: 1) the tangent to the reqired crve of space in space time : ct = f(x), and 2) the Ox axis. This crve is the searched space crve. In addition let s apply the formla of the expression of a force, to an m mass moving. The relativistic eqation is the following one: F = dv mv dx v² (1 - ) c² 3/2 Now let s take the case of a particle with a negligible mass at rest. It is with this particlar case that is applied the principle of general relativity: the trajectory of this particle will follow a space-time geodetic. Moreover, if we sppose the particle located infinitely far at t=0, then we have, for any x, v = c tan(α). From where: F d tan( α) tan( α) = mc² (9) 3/2 dx (1 tan²( α)) With tan(α) = v/c. Otherwise, we can write: tan(α)² = 1/cos(α)² - 1 1/(1 - g(x)²/(8l²))² - 1 sing eqation (8) (1 + g(x)²/(4l²)) - 1 = g (x)²/(4l²) tan(α) g(x)/(2l) (10) In addition the reqired Newton s eqation is: R F = - mc² (11) x 2 Written with R = MG/c², the Schwarzschild ray. G is niversal gravitation constant. Let s identify eqations (9) and (11): m c² d/dx[tan(α)] tan(α)(1 - tan(α)²)(- 3/2) = - m c² R / x² d/dx (g(x)) g(x) (1 - v²/c²)(- 3/2) = - 4 L² R / x² sing eqation (10) d(g(x))/dx g(x) - 4 L² R / x² (12) Becase for x large, and a particle at rest when infinitely far, there is v<<c. Frédéric LASSIAILLE 2009 Page 10

11 Frédéric LASSIAILLE 2009 Page 11 14/05/2010 Therefore, for x large, the soltion of the differential eqation (12) is the following: g(x) = L 8R x (13) This eqation can be jstified from a physics point of view, sing the model of this docment, and conservation of space-time propagated vacm. However, in this docment, the possible jstifications of this assmption will be left off, and only its conseqences will be stdied. We will now postlate that this eqation (13) is correct not only for long distances bt for any vales of x. Therefore, now we can modify eqation (9), with the help of eqation (10), and the help of eqation (13), which is now spposed always correct. Let s remind: dx/ds = [1 + e] / (1 + e/2) (eqation (7)) = cos(α) becase α is the angle between the tangent to the reqired crve, and Ox axis. tan(α)² = 1/cos(α)² - 1 trigonometric eqation. = (1 + e/2)²/(1 + e) - 1 sing eqation of cos(α) jst above. tan(α) = [(1 + e/2)²/(1 + e) - 1] (14) with e = [8R /x] according to (6) and (13). (15) The eqation (9) ths becomes, sing (14): F = mc² d/dx{ [(1 +e/2)²/(1 + e) - 1]} [(1 + e/2)²/(1 + e) - 1] (1 - v²/c²)(- 3/2) From where finally by sing the eqation (15), and writing p = [8R]: F = mc² d/dx{ [(1 + ½ p/ x)²/(1 + p/ x) - 1]} [(1 + ½ p/ x)²/(1 + p/ x) - 1] {2 - (1 + ½ p/ x)²/(1 + p/ x)}(-3/2) After calclations carried ot on appendix 2, eqation (B): F = - 2R 1 + mc²r x x² 8R 8R 2R 1 + (1 + - ) x x x 3/2 (16) This eqation (16) is an exact eqation, and is not approximated for long distances. After a limited development of e ntil order 2, carried ot on appendix 2, we get: mc²r 2R 19R F - ( ) (17) x² x x Of corse, Newton s eqation is retrieved for the particlar case of long distances: F = - 0 mc²r x² However, there is the addition of an x(-5/2) term for the other cases: Frédéric LASSIAILLE 2009 Page 11

12 Frédéric LASSIAILLE 2009 Page 12 14/05/2010 R F F mc² x 3/2 0 5/2 The gravitational force is weaker than the Newton s force, for small distances. However, this does not explain the mystery of velocity crve for the stars in the galaxies. In fact, this correction appears for relativistic vales only (v close to c). Frédéric LASSIAILLE 2009 Page 12

13 Frédéric LASSIAILLE 2009 Page 13 14/05/ STAR SPEED MYSTERY In order to obtain the explanation of the stars speed in a galaxy it is necessary to take into accont these stars masses. For this, the P point is spposed located in the middle of these stars i.e. inside the stdied galaxy. We se the same model as above, bt now M is the galactic center mass, and x is the distance from the galactic center. We will approximate that the deformation propagated by these stars and received in P is roghly proportional to the matter density of srronding stars arond P. We will sppose that this density of matter in a galaxy evolves following an 1/x² law. Confrontation with experimental vales, rather than this 1/x² law, will be done later on, in appendix 6 and in [5]. It is possible to add this additional term to L1 (see eqation (5)). The srronding stars propagate on P the following deformations: L1 = L2 = s s q x Where q is a constant Hence, a matter density following an 1/x² rle implies that the corresponding symmetric contribtion L1s = g(y) follows an 1/x rle. (y is the distance between P and a stdied star, g is the fnction of eqation (13)). That is becase we now sppose, as a postlate, the vale of each lminos point contribtion, g(x, y), having the form: g(x, y) = [8R/y] = [8MG/(c²y)] = k [8(1/x²)G/(c²y)]. (k is a constant). This is eqation (13) above. Hence we obtain the factor M and then [1/x²] = 1/x. The expressions of the qantities of deformations L1 and L2 received in the P point become: L1 = L1 + L1 m + L1 s = L + g(x) + x q L2 = L2 + L2 m + L2 s = L x q It is cleaner to write that in a homogeneos way, sing the reslt of the preceding stdy: r 8R L1 = L (1 + + x x ) r L2 = L (1 + ) x r is the ray from which the gravitational effect of srronding stars is noticed. The eqation (15), giving e, changes in the following way. e = 8R x r 1 + x received asymmetrical propagations received symmetrical propagations (18) The other eqations remain nchanged: Frédéric LASSIAILLE 2009 Page 13

14 Frédéric LASSIAILLE 2009 Page 14 14/05/2010 cos( α ) = oper(l1, L2) = 1 + e e F mc² d tan( α) tan( α) dx (1 tan²( α)) = (19) 3/2 v = F x m The last eqation above explains the centrifgal force in the galaxies. This new Newton s law (expression of F, eqation (19)) is complicated. It can be calclated by compter. The last program of appendix 1 carries that ot. The red crve, which follows, represents the evoltion of the calclated speed v. We sppose r = 1 kpc. This vale has been adjsted in order to obtain the best possible red crve. The Milky Way vale is sed for the galactic center mass. The red crve represents the speed of a star in this spposed galaxy. This star is located between 1 and 15 kpc ( 10**19 m and 4 10**20 m) from the galactic center. X-coordinate represents this distance between the star and the galactic center (mins 1kpc = 3 10^9 m; that is to say that the vale x=0 corresponds to a distance of 1 kpc between the star and the galactic center). The nit sed for x is the meter. The ordinate, y, represents the speed, in m/s. The ble crve represents the speed reslting from traditional Newton s law. Frédéric LASSIAILLE 2009 Page 14

15 Frédéric LASSIAILLE 2009 Page 15 14/05/2010 Figre 4: Theoretical crves of the speed of the stars in the galaxy, taking in accont their masses. The variation of the speed of stars in a galaxy is explained. In any case the explanation is good for distances x in the Milky Way ranging between 3 and 15 kpc (see on the figre between 10 ** 20 and 4 * 10 ** 20 meters). The figre 4 is false qantitatively. Indeed, the maximm speed indicated for stars is approximately 1000 m/sec. However, measred speeds are actally of 220 km/sec for the Milky Way, which is 5 times lower. That s becase is necessary to se the measred evoltion of the density of matter. Refer to appendix 6, and [5], for confrontation with experimental vales, with other galaxies. For distances higher than 15 kpc, the two crves fit exactly each other (this is visible on figre 4 bt mch more visible with another plot in the program. For this, se the program in appendix 1 with other bondaries for plotting, in order to see clearly this fitting of the two crves). This recalls that, ot of the vicinity of the stars, this correction of Newton s law has no effect. Conversely, this effect takes place in the galaxies becase of the presence of a strong density of srronding matter, inside the galaxies. This great srronding matter density is coming from the stars of the galaxy. As we have seen, this is adding a third term for the calclation of the relativistic operator which drives space-time shape. For this last reason, this Newton s law correction explains the weak variation of speeds of stars in a galaxy. Finally, for distances lower than 3 kpc, the theoretical crve obtained is of the same type as the measred one. Or correction of Newton s law explains solid natre of galactic center. Refer to appendix 1 for more precise details abot figre 4. Frédéric LASSIAILLE 2009 Page 15

16 Frédéric LASSIAILLE 2009 Page 16 14/05/ GALAXY SPEED MYSTERY Let s stdy now the case of galaxy grops. We sppose that they are moving arond each other, like in the Coma or Virgo galaxy grops. Hence we start from the classical Newton s law model of chapter 5: the M mass, now representing a galaxy, and the srronding matter modelled by a niform density of matter. Therefore, the eqations stay nchanged. Let s remind them: dx/ds = [1 + e] / (1 + e/2) with e = L1/L e²/8 as seen above dx/ds = cos(α) = 1 / [ 1 + tan²(α)] 1 - ½ tan²(α) tan(α) e/2 comes from jst above F = mc² d/dx(tan(α)) tan(α) (1 - tan(α)²)(- 3/2) mc²/4 e de/dx (1 - tan(α)²)(- 3/2) sing tan(α) above mc²/4 e de/dx for long distances Hence F is, very roghly, proportional to: L1 - L2 asymmetric contribtions e² = ( )² = ( )² L2 symmetric contribtions Here the asymmetric contribtions come from the M mass. The symmetric contribtions come from the oter space only. However, in the case of or old Newton s law, this is different: asymmetric contribtions are seen the same way (coming from the sn). Bt symmetric contribtions are mch greater. That is becase they come not only from oter space, bt also from the Milky Way itself. As a conclsion, for eqal asymmetric contribtions, symmetric contribtions are mch greater inside the Milky Way than otside any galaxy. Therefore, the gravitational force, and with it, the G constant itself, is mch greater, otside any galaxy, than, like s, inside the Milky Way. Another way to condct this stdy is to calclate the G constant. This is done on appendix 4, and gives the following reslt. G = 2 c E(cT/d) 1 8 ρ d² ( )² n n = 1 This formla may be written in a more generic way: G = 4 p ( )² p c 8e x p (20) Obtained the same way as in appendix 4. Frédéric LASSIAILLE 2009 Page 16

17 Frédéric LASSIAILLE 2009 Page 17 14/05/2010 In this << >> sm above, we do no take into accont the lminos points pertaining to the M and m masses which are concerned by Newton s law itself (F = - GMm/x²). This sm is done for each lminos point, p, along an infinite half-line. This half-line is centered on the location in which we want to calclate G. That is to say that we take into accont each lminos points in the niverse, except those of the interacting masses for Newton s law. ep is the energy of each lminos point. xp is the distance of each lminos point from the location in which we calclate G. On eqation (20), G is inversely proportional to ( [8 ep / xp] )². This vale differs greatly from the cases inside, and, otside the Milky Way. Let s note also: S = i p 8e x p p for the lminos points inside the Milky Way S = o p 8e x p p for the lminos points otside the Milky Way Then for or case, inside the Milky Way, we can write: 8e p = S i + So (21) p xp Now for the stdied case, which is otside any galaxy, we can write: 8e p = So (22) p xp If we note G the G constant located otside any galaxy, then we have: G' (S + S )² S = = ( 1+ )² G S S i o i 2 o o Coming from eqations (20), (21), and (22). For an example, if we need 10 more mass in order to explain the measred galaxies speeds, (this factor 10 was measred for some galaxy grop), then we dedce from or explanation: G /G= 10, and then Si/So = 2,1. Let s approximate very roghly in order to appreciate this vale. If we note Si = [8ei/xi], and So = [8eo/xo], (with ei, xi, eo, and xo the corresponding pin pointed energies and distances corresponding to those Si and So sms) then we have: ei x x e i o o 5 This vale order represents the energy ratio, divided by the distances ratio, between the Milky Way, and the oter space, and along a half-line. (This half-line might be represented physically by an infinitely small solid angle). Therefore, this ratio corresponds also to a ratio of linear matter density, along a solid angle, between the two cases (Milky Way and oter space). Of corse the linear matter density is mch greater inside the Milky Way than in the oter space. A ratio of << 5 >> for this seems possible at first glance. Frédéric LASSIAILLE 2009 Page 17

18 Frédéric LASSIAILLE 2009 Page 18 14/05/2010 As an intermediate conclsion, or modelling of relativity explains the dark matter mystery for the velocity of the galaxies inside a heap of galaxies. This explanation is based on a great increase of the G constant in sch cases. Also, this explanation is the same for the mystery of light beam deviation in the vicinity of a galaxy. Moreover we may conclde at this point that or stdy explains each dark matter mystery. 8. UNIVERSE FINITE AGE Finally, the stdy above proves a finite niverse age (spposing a constant density of matter in the niverse, throgh time and space). Indeed, the << ρ/ x >> sm, calclated on the whole volme of the niverse does not converge if this volme is infinite. (ρ is a constant proportional to the density of matter). This sm expresses the vale of L, the finite coefficient which was sed in preceding calclations. This coefficient represents the contribtion of the energies of oter space to the deformation received on P point. If the niverse is infinite, then this vale L is infinite, and the relativistic operator is eqal to 0 everywhere, which is a contradiction. On the other hand, it is not possible to calclate the vale of this finished age. Indeed, for that it shold be necessary to know the vale of the ρ parameter. The stdy of this docment does not make it possible to calclate this vale. Frédéric LASSIAILLE 2009 Page 18

19 Frédéric LASSIAILLE 2009 Page 19 14/05/ CONCLUSION As a conclsion, this new modelling of space-time retrieves general and restricted relativity. Nevertheless, it is more than a simple Eclidean version of relativity, as shown by postlate 1 and 2. It is enogh to add a 3 rd postlate in order to explain dark matter mysteries, which are: star s speed inside galaxies, explained by or Newton s law correction, speed of the galaxies themselves, inside their grop, and deformation of light trajectories in the vicinity of a galaxy. Overall, this third postlate condcts a modification of Newton s law. This modification is conceived in order to find exactly Newton s law in the specific case of pin pointed masses and long distances. It ses the sal modelling of Newton s law, with two specific masses concerned, a gravitational mass M, and an attracted mass m. It finds by constrction the law of Newton, for long distances between m and M. However, a correction of Newton s law is seen in the case of short distances. This first correction occrs in fact for relativistic speeds (speed v of the m mass close to c). Nevertheless, the important reslt of this correction of Newton s law occrs when the galaxy s stars are introdced in the model. After this, a strong difference appears between the calclated force and Newton s law. The crve of the speed of stars in the galaxies, obtained with this corrected Newton s law, is very close to experimental measrements. This theoretical crved similarity with measred one remains even valid for the solid part of the galaxy (i.e. nearest to the galactic center, in which measred speeds are almost proportional to their distance to the center). That is for stars speed dark matter mystery. For the other dark matter mystery, which concerns the case of galaxy grops, the explanation is more direct. This is explained by a different vale of G, between or case inside the Milky Way, and the case otside any galaxy. Here again, the third speaker, which decrease the G constant, is the stars of or galaxy. That is for the practical reslts yielded by this modelling. Moreover, and from a theoretical point of view, this stdy finds a way for space-time determination. At any point in space-time, this determination is based on the matter density distribtion throgh the whole niverse. It remains also to check if the model described in this article is coherent with other actal physics theories (electromagnetism, qantm mechanics, etc). As an answer to this last qestion one will notice that this model is in conformity with [4]. Frédéric LASSIAILLE 2009 Page 19

20 Frédéric LASSIAILLE 2009 Page 20 14/05/ APPENDIX APPENDIX 1 COMPARISON BETWEEN NEWTON S LAWS 1) Speed of stars in a galaxy, WITHOUT taking into accont the stars This crve is shown on figre 5. In ble is the law of Newton. In red is the law of Newton corrected withot taking into accont of the effect of stars. The formla v = [Fx/m], where F is the gravitational force, is sed. The m mass is left in the programs eqations below only for comprehension, since in fact it is always simplified. X-coordinate (in meters) is the distance which separates the star from the center of the galaxy (mins R). The y ordinate represents the speed v (in m/s) of the star in its movement arond the galactic center. The distances lie between R (Schwarzschild s ray) and 40 R. They are ths weak compared to the total size of the galaxy. A difference appears however between the two crves. On the other hand, this difference becomes nimportant for distances ranging between 1 and 15 kpc. Figre 5: Theoretical crves of the speed of the stars in the galaxy, withot taking in accont their masses. Program MAPPLE having posted this figre: Digits:= 20; m:=1; M:= 7 * 10**36 ; G:= * 10**(-11); c:= 3 * 10**8; Frédéric LASSIAILLE 2009 Page 20

21 Frédéric LASSIAILLE 2009 Page 21 14/05/2010 R:= M*G/c**2; # Initialisations. e:= sqrt(8*r/x) ; # Famos relativistic coefficient. cos2:= (1 + e) / ((1 + e/2)**2) ; # Sqare of the relativistic operator. tg:= sqrt( 1/cos2-1) ; # Slope of space inside space-time. F:= - m * c**2 * diff(tg,x) * tg * (1 - tg**2)**(-3/2) ; # Or Newton s law. FN:= m * c**2 * R / x**2 ; # Classical Newton s law. v:= sqrt( F * x / m ) ; # Centrifgal force and tangential speed v. vn:= sqrt( FN * x / m ) ; # The same for classical Newton s law. plot([v(x), vn(x)], x=r..40*r, color=[red,ble], style=[line,line], nmpoints=1000); # Plotting the 2 crves from R to 40 R. 2) Speed of stars in a galaxy, TAKING into accont the stars This crve has been shown on figre 4. In ble is Newton s law. In red is or corrected Newton s law, corrected taking into accont the effect of the stars. The program ses the formla v = [Fx/m], where F is the gravitational force. It ses also r = 1 kpc. This vale was adjsted in order to obtain the best possible red crve. X-coordinate (in meters) is the distance which separates the star from the center of the galaxy. The y ordinate (in m/s) represents the speed v of the star in its movement arond the center of the galaxy. The distances considered lies between 1 and 15 kpc, they are those of the stars in the Milky Way. This time the difference between the two crves is very clear. Program MAPPLE having posted this figre: Digits:= 30; m:=1; M:= 7 * 10**36 ; G:= * 10**(-11); c:= 3 * 10**8; R:= M*G/c**2; kpc:= 3.08 * 10**19 ; # Initialisations. r:= 1 * kpc; # Vale fitted progressively in order to obtain the best possible red crve. e:= sqrt(8*r/x) / (1 + r/x) ; # Famos relativistic coefficient. cos2:= (1 + e) / ((1 + e/2)**2) ; # Sqare of the relativistic operator. tg:= sqrt( 1/cos2-1 ) ; # Slope of space inside space-time. F:= - m * c**2 * diff(tg,x) * tg * (1 - tg**2)**(-3/2) ; # Or Newton s law. FN:= m * c**2 * R/(x**2) ; # Classical Newton s law. v:= sqrt( F * x / m ) ; # Centrifgal force and tangential speed v. vn:= sqrt( FN * x / m ) ; # The same for classical Newton s law. plot([v(x),vn(x)], x=1*kpc..15*kpc, color=[red,ble], style=[line,line], nmpoints=1000); # Plotting from 1 to 15 kpc. 3) Comparison with the measred speed of the stars of the galaxies See figre 7 below. The shape of the red crve of figre 6 is similar to the one below. Frédéric LASSIAILLE 2009 Page 21

22 Frédéric LASSIAILLE 2009 Page 22 14/05/2010 Figre 6: Crve of the measred speed of stars in a galaxy. Frédéric LASSIAILLE 2009 Page 22

23 Frédéric LASSIAILLE 2009 Page 23 14/05/2010 APPENDIX 2 FIRST NEWTON S LAW CORRECTION This appendix contains the calclations sed to retrieve the first correction of Newton s law of chapter 5 (the case of a planetary system). F = mc² d/dx{ [(1 + ½ p/ x)²/(1 + p/ x) - 1]} [(1 + ½ p/ x)²/(1 + p/ x) - 1] {2 - (1 + ½ p/ x)²/(1 + p/ x)}(-3/2) 1) Calclation of derived formla: d/dx{ [(1 + ½ p/ x)²/(1 + p/ x) - 1] } = ½[ (1 + ½ p/ x)²/(1 + p/ x) - 1](-1/2) (f g fg )/g² = ½[ (1 + ½ p/ x)²/(1 + p/ x) - 1](-1/2) (f g fg )/g² f = (1 + ½ p/ x)² f = 2(1 + ½ p/ x) p/2 (-1/2) x(-3/2) = - p/2 (1 + ½ p/ x) x(-3/2) g = (1 + p/ x) g = p (-1/2) x(-3/2) = - p/2 x(-3/2) (f g fg )/g² = [ - p/2 (1 + ½ p/ x) x(-3/2) (1 + p/ x) - (1 + ½ p/ x)² (- p/2) x(-3/2) ] / (1 + p/ x)² = (p/2) (1 + ½ p/ x) x(-3/2) [ - (1 + p/ x) + (1 + ½ p/ x) ] / (1 + p/ x)² = (p/2) (1 + ½ p/ x) x(-3/2) (-½) p/ x / (1 + p/ x)² = - (p/4) (1 + ½ p/ x) x(-3/2) p / x (1 + p/ x)² = - (p²/4) (1 + ½ p/ x) x(-3/2) / [(1 + p/ x)² x] d/dx{ [(1 + ½ p/ x)²/(1 + p/ x) - 1 ] } = ½[ (1 + ½ p/ x)²/(1 + p/ x) - 1](-1/2) [ - (p²/4) (1 + ½ p/ x) x(-3/2) ] / [(1 + p/ x)² x] = - (p²/8) [ (1 + ½ p/ x)²/(1 + p/ x) - 1](-1/2) [ (1 + ½ p/ x) x(-3/2) ] / [(1 + p/ x)² x] F = - mc² p²/8 { [ (1 + ½ p/ x)²/(1 + p/ x) - 1](-1/2) [ (1 + ½ p/ x) x(-3/2) ] / [(1 + p/ x)² x] } [(1 + ½ p/ x)²/(1 + p/ x) - 1] {2 - (1 + ½ p/ x)²/(1 + p/ x)} (-3/2) 2) Simplification: Let s s say: e = p/ x. Then: F = - m c²p²/8 x [ (1 + ½ e)² (1 + e)(-1) - 1](-1/2) (1 + ½ e) x(-3/2) ((1 + e)(-2)) [(1 + ½ e)² ((1 + e)(-1)) - 1] [2 - (1 + ½ e)² ((1 + e)(-1))] (-3/2) F = - m c²p²/8 x (1 + ½ e) x(-3/2) ((1 + e)(-2)) Great simplification here [2 - (1 + ½ e)² ((1 + e)(-1))] (-3/2) F = - m c²p²/8x² (1 + ½ e) ((1 + e)(-2)) [2 - (1 + ½ e)² ((1 + e)(-1))] (-3/2) (A) 1 + ½ e F = - mmg / x² (1 + e)² [2 - (1 + ½ e)² /(1 + e)](3/2) 1 + ½ e F = - mmg / x² (1 + e)² [(2(1+e) - (1 + ½ e)²) / (1 + e)](3/2) Frédéric LASSIAILLE 2009 Page 23

24 Frédéric LASSIAILLE 2009 Page 24 14/05/ ½ e F = - mmg / x² (1 + e)² [(2+2e e - ¼ e²) / (1 + e)](3/2) 1 + ½ e F = - mmg / x² (1 + e)² [(1+ e - ¼ e²) /(1 + e)] (3/2) 1 + ½ e F = - mmg / x² [1 + e] (1+ e - ¼ e²)(3/2) With e = p/ x = 2 [2MG] / c x: 1 + [2MG] / c x F = - mmg / x² [1 +2 [2MG] / c x] (1+ 2 [2MG] /c x - ¼(2 [2MG]/(c x))²)(3/2) 1 + [2R] / x F = - mmg / x² (B) [1 + 2 [2R] / x] ( [2R]/ x - 2R/x )(3/2) R Schwarzschild length: R = MG/c². 3) Limited Taylor development: From (A) eqation: F = - mc²p²/8x² (1 + ½ e) (1-2 e + (-2)(-3)/2 e²) [2 - (1 + e +1/4 e²) (1 e + e²)] (-3/2) F = - mc²p²/8x² (1 + ½ e) (1-2e + 3e²) [2 - (1 - e + e² + e(1 - e + e²) +1/4 e²)] (-3/2) F = - mc²p²/8x² ((1 + ½ e) - 2e(1 + ½ e) + 3e²) [2 - (1 - e + e² + e e² + 1/4 e²)] (-3/2) F = - mc²p²/8x² (1 3/2 e + 2e²) [1-1/4 e²] (-3/2) F = - mc²p²/8x² (1 3/2 e + 2 e²) (1 + 3/8 e² ) F = - mc²p²/8x² (1 + 3/8 e² - 3/2 e + 2 e²) With e = p/ x, that is: F = - mc²p²/8x² (1-3/2 p/ x + 19/8 p²/x) F = - mc²p²/8x² ( 1 (3p/2)/ x + (19p²/8)/x ) Frédéric LASSIAILLE 2009 Page 24

25 Frédéric LASSIAILLE 2009 Page 25 14/05/2010 APPENDIX 3 COSMOLOGICAL VALUES These are the cosmological vales, sed in this docment: G = 6, **(-11) m3 kg(-1) s(-2) niversal gravitation constant 1 year light = **8 m = **15 m 1 parsec = 3.26 year-light = **15 m = **15 m = **16 m 1 kpc = 1000 parsec = **19 m M0 = 2 10**30 kg sn mass M = **6 M0 mass of the Milky Way center = 7 10**36 kg. Frédéric LASSIAILLE 2009 Page 25

26 Frédéric LASSIAILLE 2009 Page 26 14/05/2010 APPENDIX 4 CALCULATION OF G The aim of this appendix is to calclate the vale of G, the niversal constant of gravitation. We will se L, vale which has been sed in this docment (eqation (5)). L is representing the contribtion of oter space matter, for the calcls of relativistic operator. Let s remind that the relativistic operator allows the determination of space shape, inside space-time, for each point of space. The Lp contribtion of each lminos point, sed for the calclation of L final vale, is given by eqation (13) applied in the case of a lminos point: Lp = k [Rp/x] Where Rp = mp G/c², with mp = ep /c², and ep is the energy, spposed constant, here, of each lminos point. k is a constant whom vale is not of great importance, becase it will always be simplified later on where calclating relativistic operator. x is the distance between the lminos point and the P point, P point where this contribtion Lp is received. L is the sm of each Lp contribtion, for each lminos point along a half-line. This half-line is centered on P point. On this P point we wish to calclate the shape of space in space-time. That is to say: L = k Lp On the half-line Here we will make important approximations. We will now forget the Milky Way in this sm. Becase the aim here is to obtain a formal simple eqation. Moreover, the sm will be roghly evalated here: we will consider that matter is reglarly distribted to form a grid of lminos points, with a distance of d between each lminos points. With sch hypothesis, the eqation of L becomes: E(cT/d) E( x ) is the first decimal nmber smaller than x. L = k [Rp /(nd)] d is the mean distance between two lminos points. n = 1 T is the age of the niverse. E(cT/d) = k [mp G / (c²nd)] (A) n = 1 We also get, if ρ is the niverse matter density: ρ = (1/d)3 mp Actal mass in a cbic meter. Then: mp = ρ d3 (B) Then, replacing mp by this vale in eqation (A): Frédéric LASSIAILLE 2009 Page 26

27 Frédéric LASSIAILLE 2009 Page 27 14/05/2010 L = k [ρ d² G / (c²n)] = k d/c [ρ G] 1/ n (C) Same way, for an m mass, the Lm contribtion is: Lm = k [R/x] With R = mg/c². (D) Other way, we have, coming from eqation (13): Lm = L [8R/x] (E) Then sing (D) and (E): k [R/x] = L [8R/x] That is: k [R/x] = k d/c [ρ G] 1/ n [8R/x] Using (C). This eqation is, eliminating k, R and x and sing (B): G = d c² E(cT/d) 1 8 m p ( )² n n = 1 (F) We note that G is proportional with d and inversely proportional with mp. Matter density in the niverse leads the symmetric contribtions for relativistic operator. We have written: e = Lasymmetric / Lsymmetric. Therefore, a great symmetric part leads to a weaker e, and then leads to increase the relativistic operator oper(l1, L2), as we saw from eqation (7). Conversely, a low asymmetric part leads to decrease it. Of corse the vale of this relativistic operator leads directly the vale of the attracting force, and leads the vale of G. The weaker the relativistic operator dx/ds, the greater the generated force. Finally, conseqently, the more is srronding niverse matter density, the weaker is G. Let s rewrite G from (B) and (F): G = 2 c E(cT/d) 1 8 ρ d² ( )² n n = 1 (G) T ρ d c G Universe age. Universe average mean matter density. Universe average mean distance between 2 lminos points. Speed of light. Gravitation niversal constant. We dedce from there, the vale of d. This mst yields the nmber of lminos points in each particle (photon, electron, qark, etc..). Frédéric LASSIAILLE 2009 Page 27

28 Frédéric LASSIAILLE 2009 Page 28 14/05/2010 APPENDIX 5 INVARIANCE OF THE RELATIVISTIC OPERATOR 1) Aim The aim of this appendix is to check, directly with calclation, that the final space s shape is the same when calclating in another inertial frame. That means checking that the choice of the inertial frame, made by postlate 3, has no incidence on its final reslt. That is to say, also, that the relativistic operator is a space-time invariant throgh inertial frames. What mst be proven exactly is that the vale of this operator in a R reference frame, oper(l1, L2), is always eqal to [1 v²/c²], where v is the speed of the particle inside R. This invariance is a direct conseqence of constancy of speed of light throgh Lorentz s transformation. Hence, the calclations which are done from figre 3 are always the same whatever R reference frame is sed, becase the slope of L1 and L2 segments are eqal to 1 and -1, respectively. This last property comes from the fact that the speed of light is always the same. Nevertheless, we will prove this by direct calclation. 2) Modeling Let s take two inertial frames R (O x y z ct) and R (O x y z ct ), like in the constrction of classical Lorentz s eqations. We sppose then O point moving at constant speed v along Ox axis, parallel to O x axis, and we sppose O(0,0) in R when and where O (0,0) in R. We se also the same model as the one sed in figre 3. We draw L1 and L2 trajectories inside R inertial frame. We mst also draw those trajectories in R, this time noted L1 and L2. We sppose a P particle at a constant w speed along Ox axis in R, and w speed compared to R. At t=0 in R we sppose P particle position at x=0. At t in R we sppose P particle position at x sch as l1 + l2 = ct and l1 l2 = wt. Hence this is figre 3 drawing. We have also a second figre, constrcted from figre 3 and replacing L1 trajectory by L1, L2 by L2, and R(O x y z ct) by R (O x y z ct ). Also we have the eqations of Lorentz s transformation from R to R : x = γ ( x vt ) with γ = 1 / [ 1 v²/c²] (A) ct = γ ( -(v/c)x + ct ) (B) 3) From L1 to L1 At the end of the L1 segment, the (x, ct) coordinates in R are the following: (l1 l1). In R those coordinates become (l1, ct1 ) sch as: l1 = γ l1 (1 v/c) sing eqation (A) with x = ct = l1 (C) Bt also we have: l1 = (ct/2 ) (1 + w/c) sing eqation (1) located at the beginning of this docment, after replacing v by w in these eqations (particle s speed in R). Hence, (C) eqation is modified : l1 = γct/2 (1 + w/c) (1 v/c) (D) Using (B) eqation we retrieve the same vale. Hence finally: Frédéric LASSIAILLE 2009 Page 28

29 Frédéric LASSIAILLE 2009 Page 29 14/05/2010 ct1 = l1 of corse, constancy of light speed throgh Lorentz s transformation 4) From L2 to L2 The (x, ct) coordinates of the A point, located at the end the L2 segment inside R frame, are: (l1-l2, l1+l2). These coordinates in R became (x2, ct2 ) sch as: x2 = γ ( l1-l2 vt ) sing (A) eqation ct2 = γ [ -(v/c)( l1-l2) + ct ] sing (B) eqation x2 = γ (w v) t sing éqation (1) (E) ct2 = γ ct (1 vw/c² ) idem Then we can calclate l2 : l2 = x1 x2 = γct/2 (1 + w/c) (1 v/c) - γ (w v) t sing (D), (E), and x1 = l1 = γct/2 (1 + v/c) (1 w/c) (F) The same way: l2 = ct1 ct2 sing now the eqation for time, = γct/2 (1 + w/c) (1 v/c) - γ ct (1 vw/c² ) this yields of corse = γct/2 (1 + v/c) (1 w/c) the same reslt. 5) Calclating the relativist operator in R Let s write again the reslts above: l1 = γct/2 (1 v/c) (1 + w/c) l2 = γct/2 (1 + v/c) (1 w/c) We can now calclate the relativistic operator in R frame: oper(l1, l2 ) = [ l1 l2 ] / (l1 + l2 )/2 [ (1 v/c) (1 + w/c) (1 + v/c) (1 - w/c) ] = [(1 v/c) (1 + w/c) + (1 + v/c) (1 - w/c)] / 2 [ 1 v²/c²] [ 1 w²/c²] = (G) 1 vw/c² We mst se the relativistic speed composition formla, becase we are in the case of two Lorentz s transformations to compose: v + w w = vw /c² Hence, the (G) eqation becomes: [ 1 v²/c²] [ 1 ((v + w )/( 1 + vw /c²))²/c² ] oper(l1, l2 ) = [(v + w )/( 1 + vw /c²)]v/c² Which is the expected reslt. = [ 1 w ²/c²] Frédéric LASSIAILLE 2009 Page 29

30 Frédéric LASSIAILLE 2009 Page 30 14/05/2010 APPENDIX 6 VALIDATION 1) Aim This appendix validates the soltion for dark matter mystery of the actal docment. For this, a program is calclating the crve for the star s velocities arond the center of the NGC3310, NGC1068, NGC157, and NGC7541 galaxies. This program is available in [5], as well as details abot this validation. The reslt is the following: o o exactly same shape between measred crves and theoretical crves. Speed vales 50 times and 21 times greater than the measred ones. Those greater speed vales might be explained by a greater galactic center masses than the measred ones. This calclation is based on corrected Newton s law of chapter 6. It ses experimental data coming from [1,2,3]. More details abot this validation are available in [5]. Frédéric LASSIAILLE 2009 Page 30

31 Frédéric LASSIAILLE 2009 Page 31 14/05/2010 2) Reslts NGC3310 Figre 7:Theoretical speed crve of the NGC3310 galaxy. The nit of the x ordinate on the figre above is the kpc. The nit of the y ordinate is m/s. Figre 8: Measred speed crve of the NGC3310 galaxy. Frédéric LASSIAILLE 2009 Page 31

32 Frédéric LASSIAILLE 2009 Page 32 14/05/2010 NGC1068 Figre 9: Theoretical speed crve of the NGC1068 galaxy. The nit of the x ordinate on the figre above is the kpc. The nit of the y ordinate is m/s. Figre 10: Measred speed crve of the NGC1068 galaxy. Frédéric LASSIAILLE 2009 Page 32

33 Frédéric LASSIAILLE 2009 Page 33 14/05/2010 NGC157 Figre 11: Theoretical speed crve of the NGC157 galaxy. The nit of the x ordinate on the figre above is the kpc. The nit of the y ordinate is m/s. Figre 12: Measred speed crve for NGC157 (above in plain line). Frédéric LASSIAILLE 2009 Page 33

34 Frédéric LASSIAILLE 2009 Page 34 14/05/2010 NGC7541 Figre 13: Theoretical speed crve of the NGC7541 galaxy. The nit of the x ordinate on the figre above is the kpc. The nit of the y ordinate is m/s. Figre 14: Measred speed crve for NGC7541. Frédéric LASSIAILLE 2009 Page 34