A Solution for the Dark Matter Mystery based on Euclidean Relativity

Transcription

1 Long Beach 2010 PROCEEDINGS of the NPA 1 A Solution for the Dark Matter Mystery based on Euclidean Relativity Frédéric Lassiaille Arcades, Mougins 06250, FRANCE lumimi2003@hotmail.com The study of this article suggests an elanation for the dark matter mystery. This elanation is based on a modification of Newton s law. This modification is conducted from an Euclidean vision of relativity. Concerning the mystery of the velocity of the stars inside a galay, the study calculates a theoretical curve which is different from the one obtained using Newton s law. This theoretical curve is very close to the measured one. Concerning the mystery of the velocity of a galay inside its grou, the elanation is more direct. For this mystery, the study calculates a greater value for G, the gravitational constant. 1. Introduction This article aims to address an issue yielded by general relativity, which may be elained by global sace's shae determination. This absolute sace-time is general relativity sacetime, and this sace deformation in sace-time must be in conformity with Newton s law at least for long distances. The adoted oint of view is an Euclidean relativity. An Euclidean mathematical contet is used, with 4 dimensions (three of sace,, y, z, and one of time: ct). This is for restricted relativity. For general relativity of course we use the same and we etend it overall with a tensor. Ecet that here locally it is an Euclidean metric used to reresent sace-time. Within this mathematical framework the hysics rinciles are eactly those of relativity: isotroy of sace, inertial frames of reference with recirocities between those inertial frames, constancy of the seed 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 course for the general one: deformation of sace-time by energy, eression of a force (gravitational ) by a sace-time deformation. There eists however a difference in the hysics rinciles, between relativity and our aroach. Indeed, constancy of sace-time distance is not an aim, in a global reresentation of sace-time. We do not use Minkowski s reresentation, and the Lorentz s invariant length is left off. The method consists to ostulate first that Lorentz s equations are simly a consequence of a sace-time deformations by energy. In other words we try to eress the general relativity deformation rincile, in the contet of restricted relativity. Once this is done, hysics inconsistencies are found. Of course we try to elude them. This leads to finally ostulate the eistence of indivisible articles, from which matter is made of. Back to our first ostulate, we find a sace time determination, coherent with Lorentz s equations. This will be our final determination of the shae of the global sace inside sace-time. 2. Retrieving Lorentz s Equations Lorentz s hysical contet is used. Let us oint it out. There are two inertial frames, R (O,, y, z, ct) and R' (O', ', y', z', ct'), in uniform rectilinear motions at the v seed one comared to another, along O ais. X is increasing along O ais. Here, only dimension, ct, and ', ct', are imortant. At t = = 0 there is also t' = ' = 0. In order to find Lorentz s equations within this hysics framework, and since our reresentations are Euclidean, it is necessary to suose that O'' ais rocked with an angle comared to O ais, with sin( = v/c. See figure 1. In the same way it is necessary to have O coordinates equal to: ( = vt, and ct = v² t/c). Conversely under these conditions the reader will be able to calculate that Lorentz s equations are found. Fig. 1. Lorentz s equations in the contet of Euclidean relativity On the basis of this Euclidean model of restricted relativity, we are temted to suose a coherent hysics ostulate. This ostulate is the following. Postulate 1. Any article with a non null mass m, moving with v seed along O ais, increasing, comared to an inertial frame R (O y z ct), deforms sace-time around it with a rotation of the 0-0ct lan around the 0y0z ais, with an angle between O and O, such as sin( ) = v/c. During the dislacement of this article from O to A(=vt, ct),a vacuum aeared inside sace-time. The location of this vacuum is the (O, O, H) triangle, such as: O coordinates are O (vt, v²t/c), H coordinates are H(vt, 0).

2 2 Lassiaille: A Solution for the Dark Matter mystery based on Euclidean Relativity Vol. 1, No. 1 In the borderline case of a hoton, with v = c, the swing becomes maimum: 2, and the vacuum is the (O, A, H) triangle. Figure 2 below reresents the effect of ostulate 1. A P article is moving at v seed in the inertial reference frame R, arallel to O ais, and in the direction of increasing. At the t instant, the article is located coordinates and ct in R (A oint). The inertial frame R' centered on O' is attached to the article. Hence R is also moving uniformly along O ais. It is the same case as the one of figure 1, ecet that here eists the P article on A oint. On figure 2, the sace line rocked with the angle, locally in A. Fig. 2. Postulate 1 On the other hand, far from A oint this line of sace is arallel to O ais. This is indeed the only realistic ossibility! It is difficult to imagine the movement of a article deforming the entire universe this way along O ais. With this ostulate, the equations of Lorentz now eress a local deformation of sace-time, caused by the energy of the moving article (ostulate 1 above). This resect of Lorentz s equations is only local to the article. In other words this resect is done only for low values of ' and ct' (and also for R such as O is close to A at the t instant). It will be noticed that this re-discovery of Lorentz s equations works thanks to the ositioning of O' (thus the ositioning of A) indicated reviously ( vt, v² t c for O ). It is known that this osition of O elains, noticeably, the asymmetry of the twin arado of relativity. This osition corresonds to a real vacuum which aears in sace-time. In addition it works thanks to the sace-time swing indicated in the ostulate above. This swing modifies with the cos( ) ratio, equal to [1 - sin² ] = [1 - v²/c²], each temoral values (in ct) and sace (in ). However this swing is only local, not global. Restricted relativity is retrieved with the hel of a simle deformation of sace in sace-time. Nevertheless, it is necessary to distinguish osition of sace in sace-time, on one hand, and reresentation of sace-time in such or such reference frame, on the other hand. Indeed, sace line in sace-time became deformed only locally to the article. It allows localizing sace-time events in a comrehensive but comlicated view. This vision is comlicated because sace-time is not Euclidean any more but Riemannian with an Euclidean local base. Thus sace lines have the shaes of curves and are no more simle straight lines like O ais. Otherwise O'' and O ct ais are straight lines. They reresent sace-time locally but does not reresent it overall any more. Broadly the reresentation of sace-time which is done by the inertial frame R', allows to see sace-time in a legal way in the sense of relativity. This reresentation is the only one resecting, locally, the hysics rinciles of restricted relativity: in articular the constant seed of light in each inertial frame. 3. Luminous Points However at this stage a roblem of coherence arises since a article is constituted of smaller articles. Indeed, how to ensure that sace-time deformation generated by the movement of a big article, comosed by a hea of smaller articles, can rise from the deformations of these smaller articles? To ensure this coherence a solution consists in suosing that matter is made u of a restricted grou of very small indivisible articles. These small articles must be conceived in such a way that they can elain sace-time deformations generated by any other comosed article. For this elanation a simle oeration must calculate the final deformation generated by the large article, from the small articles it is made of. Thus defined, this oeration must allow, by construction, calculation of the shae of absolute curves of sace, starting from the ositions and energies of these small indivisible articles. At the same time, this oeration must be, of course, comatible with ostulate 1 and Lorentz s equations. From there the second ostulate arises, which follows. Postulate 2. Each article consists of a certain number of smaller articles, called the luminous oints. These luminous oints are moving constantly at the c seed, inside the first article, and with resect to any inertial frame of reference. From this ostulate it is ossible to determine in a single way any deformation generated by any article. For that, we aly ostulate 1 to these luminous oint articles. For these luminous oints the angle is equal to its limit value 2. The shae of sace is thus at any moment the result of successive combinations of these small deformations caused by all these luminous oints. What remains to be secified is the way of combining those various deformations. This will be secified by the ostulate 3 which follows. After that, it will ossible to check that the angle calculated from ostulates 2 and 3 is well given by the formula of ostulate 1. For that let us return to Lorentz s equations. A first mathematical observation is essential. The energy conservation equations of restricted relativity are found by quantifying the luminous oint trajectories lengths, inside P article. It is what we will see. The article attached to the O' oint is modelled as consisting of only one luminous oint. Consequently the obtained model is the one described by figure 3. It can be checked that the reasoning remains valid in the general case of a article made u of several luminous oints. When the P article moves from O oint (on the figure) to A oint, along OA segment, the luminous oint contained follows a trajectory having a V shae, that is: 1) First stage: dislacement at the seed +c along O, (milked in fat on the figure). 2) Second stage: dislacement at the seed -c along O (milked in fat on the figure). For the first stage, L 1 is the dislacement length, and

3 Long Beach 2010 PROCEEDINGS of the NPA 3 L 2, (ositive), is the dislacement length of the second stage. If is the osition of the A oint, we can write: = vt = L 1 - L 2. This osition is also the coordinate of P in R at this time t. Indeed at this time t, in R, the osition of P coincides with A oint. C t Hence: O L1 A L2 Fig. 3. Luminous oint trajectory in a moving article ct L L vt L L (1) ( L1 L2)² ( L1 L2)² 2 L1 L 2 (2) 2 2 This last equation is nothing more than relativistic equation of energy: 2 2 E E E (3) ² c m with E mc² 1 v² c², Ec mvc 1 v² c², and Em mc². To obtain the equivalent equation for the energy densities, each term of this equation is divided by the value ( L1 L 2) ² / 2 which is the value of the total energy of the article: ( L 1 L2)² 1 oer( L1, L2)² ( L L )² L1 L2 with oer( L1, L2) (5) L1 L2 The introduced oerator is the relationshi between the algebraic average and the arithmetic mean. It is equal to the relativistic coefficient 1 v² c². Finally, this last study of Lorentz s equations led us to define an oerator. We will use this oerator to ostulate, finally, the mode of determination of general relativity absolute sace-time. By construction, this determination will be comatible with restricted relativity. 4. Relativistic Oerator This is done by the following ostulate. It only generalizes the revious observation done about Lorentz s equations. X (4) Postulate 3. Sace shae in sace-time is given at any oint by the ratio of the infinitesimal sace lengths, ds along sace line, and d its length rojected on O ais. This ratio is equal at any oint to the relativistic oerator alied to the two following values: L 1 : sum of the heights of vacuum of sace-time deformations roagated in O direction, increasing, L 2 : sum of the heights of vacuum of sace-time deformations roagated in O direction, decreasing. That is to say: d 2 L1 L2 (6) ds L L 1 2 where L 1 and L 2 are the 2 above-mentioned sums. This oerator is neither linear nor associative. But this doesn t matter since it is calculated once, at any oint of sace. But it must be checked that this oerator yields the same result when calculated in different inertial frames. That is to say that, for ostulate 3, the choice of the inertial frame has no incidence on the final calculated sace s sloe. This is the case because of the constancy of the seed of light through Lorentz s transformations. It can be also roven by direct calculation. It is written above that L 1 and L 2 are the sums of the heights of vacuum of the roagated deformations. It is necessary to describe how these sace-time deformations are roagated. The mechanism is very similar to waves roagated by the movement of a boat over a water surface. Let us consider Figure 1. The initial deformation relates to the O-Oct lan. The roagations of this deformation in sace-time are carried out on remaining sace dimensions, i.e. Oy and Oz, more generally on any Or direction, half-line based on O and contained in the Oyz lan. The form of these roagated deformations is each time eactly the same as the initial deformation. The initial deformation was done on O-Oct lan (sace-time swing reresented figure 1). Now the roagated deformation is the same but it relates to the (O+Or)-Oct lan in lace of O-Oct lan. The height of this roagated deformation attenuates rogressively as r increases. The attenuation law, g, will be given further. At each moment, the luminous oint thus emits this deformation. Therefore, like in the case of the boat, the finally overall roagated deformation is the enveloe of all these roagations of deformation. (In the case of a boat this enveloe has a V shae which is the final shae of these waves over the water surface). Above, the initial deformation of sace-time is connected to the eistence of a sace-time vacuum. What does become of this vacuum during the deformation roagation? It is roagated too. It relates to 3 dimensions y, z, and ct. Finally the L 1 sum is calculated for all the roagated and received deformations. Each of these deformations is generated by a unique luminous oint. A unique deformation dl 1 received is such as dl 1 = 2 k d²v where k is a constant and d²v the vacuum roagated with the sace-time deformation. The same holds for L 2, ecet that the deformations come from the oosite sense of roagation (on the same ais). Let us study the overall result of all the roagated deformations which are received in the same M oint at the same mo-

4 4 Lassiaille: A Solution for the Dark Matter mystery based on Euclidean Relativity Vol. 1, No. 1 ment. The ostulate 3 above eresses the length ratio along O ais only. But at one M oint there are numerous such directions coming to. Hence it is necessary to calculate the relativistic oerator of ostulate 3 for each sace direction. The final deformation is then obtained. The only question is: What is the mode of contribution of the deformations of all these directions in order to obtain the result? This result will be the resulting sace-time deformation at M oint. It will be thus necessary to generalize the above oerator with a second more generic oerator, which will take into account each sace directions. The result given by this second generalized oerator must be the famous final sacetime deformation at M oint. It will be robably useful to use a mathematical base like the quaternion for that. In this article this comleity will not be seen because fortunately not necessary. We thus found relativity starting from ostulates 1, 2, and 3. The angle of the ostulate 1 rotation is calculated, by alying ostulates 2 and 3. Overall, we obtained a way for calculating sace shae inside sace-time. We can now study Newton s law. 5. First Modification of Newton s Law The studied case is a article of mass M isolated in a sace filled uniformly with an energy density constant and weak in front of M. The article coordinates are y z 0, which are those of the O oint in our usual inertial frame R of reference. The studied case being invariant by any rotation of center O, only the O ais with >0, and the ais of times Oct, are imortant. How does evolve the local sloe tan( of sace, along O ais? The ostulate 3 above is alied. Let us consider a sacetime P oint to which comes at least one deformation from a luminous oint ertaining to the M mass. We suose P - coordinate ositive strict that is 0. The M mass article roagates on P oint the following deformations, with roagation directions given: L1m g increasing (7) L 2m 0 decreasing (8) There is an attenuation function given further for the first equation, and no deformation roagated in the direction of decreasing, coming from M, because 0. The surrounding universe with constant energy density roagates on P oint the following deformations, with roagation directions given: L 1u increasing (9) L2u L1u L u decreasing (10) because sace-time is assumed isotroic. Hence: L1L1u L1 mlu g( ) (11) L2L2u L2mL u (12) d oer( Lu g( ), Lu) ds (13) which is an alication of ostulate 3. After calculations: d g ( )² cos( ) 1 ds L (14) 8 2 u In addition let s aly the formula of the eression of a force, to an m mass moving. The traditional relativistic equation is the following one: dv mv F d (15) v² 3/2 (1 ) c² This is a very classical relativity result. Now let us take the case of a article with a negligible mass at rest. It is with this articular case that is alied the rincile of general relativity: the trajectory of this article will follow a sace-time geodetic. Moreover, if the article is located infinitively far at t = 0, then we have, for any, v c tan, where is the sloe angle of the curve ct = f() required. This curve is the searched sace curve. From where: d tan( ) tan( ) F mc ² (16) d 3/2 (1 tan²( )) For large, F mmg ², which is Newton s equation. After calculation, we have, for large: 8R g ( ) Lu (17) MG with R (18) c² This equation (17) can be justified from a hysical oint of view, using the model of this document, and conservation of sace-time roagated vacuum. However, in this document, the ossible justifications of this assumtion will be left off, and only its consequences will be studied. We will now ostulate that this equation (17) is correct not only for long distances but for any values of. Therefore, now we can modify equation (16), with the hel of equation (13), and equation (14), and the hel of equation (17) which is now suosed always correct. After calculations, the Newton s law is corrected, but this correction aears for relativistic values only ( v c tan close to c). Those values aear for short distances, that is, for stars in the vicinity of the galactic center. 6. Star Seed Mystery In order to obtain the elanation of the stars seed in a galay it is necessary to take into account these stars masses. For this, the P oint is suosed located in the middle of these stars i.e. inside the studied galay. We will aroimate that the deformation roagated by these stars and received in P is roughly roortional to the matter density of surrounding stars around P. We will suose that this density of matter in a galay evolves following a 1/² law. ( is the distance from the galactic center). It is ossible to add this additional term to L 1u (equation (9)). The surrounding stars roagate on P the following deformations: q L1s L2s (19) where q is a constant.

5 Long Beach 2010 PROCEEDINGS of the NPA 5 Hence, a matter density following a 1/² rule imlies that the corresonding symmetric contribution L 1s = g(y) follows a 1/ rule. (y is the distance between P and a studied star, g is the function of equation (17)). The eressions of the quantities of deformations L 1 and L 2 received in the P oint become, written in a homogeneous way, and using the receding result: r 8R L1 Lu (1 (20) r L2 Lu (1 (21) With R = MG/c², and M is the mass of the galactic center. r is the ray from which the gravitational effect of surrounding stars is noticed. We get a ratio of received asymmetrical roagations to received symmetrical roagations: 8R e r 1 1 e cos( ) oer( L1, L2) e 1 2 (22) (23) d tan( ) tan( ) F mc ² (24) d 3/2 (1 tan²( )) F v (Centrifugal force) (25) m This new Newton s law (eression of F, equation (24)) is comlicated. It can be calculated by comuter. That is done in [4], and yields figure 4. The red curve (down) reresents the evolution of the calculated seed v. It uses: r = 1 kc. This value has been adjusted in order to obtain the best ossible red curve. It uses the Milky Way value for the galactic center mass. X- coordinate reresents, in meter, the distance between the star and the Milky Way center (minus 1 kc = m). It varies between 1 and 15 kc. The ordinate, y, reresents the seed, in meter/seconds. The blue curve (to) reresents the seed resulting from traditional Newton s law. Fig. 4. Theoretical seed rofile for the stars in a galay, calculated with a 1/² matter density curve The variation of the seed of stars in a galay is elained. Hence we retrieve the global shae of a galay seed rofile. More recisely, a rogram calculating the NGC 7541 seed rofile yields the following curve (Figure 5). Fig. 5. Theoretical seed curve of the NGC 7541 galay This curve has been comuted in [5], by using the matter density rofile available in [2]. The corresonding measured seed rofile of NGC 7541 is also found in [2]. The shae of this measured curve is the same as the theoretical curve of Figure 5. X-coordinate is in kc, and y ordinate is a relative seed. The seed value scale has been fitted and seed values have been increased with a fitted constant value. For some galaies, it is ossible to calculate the eact values of the theoretical seed, for the maimums of the seed rofiles. For NGC 3310, the measured value in [1] for the first maimum is around 175 km/s. The corresonding theoretical value is in the range [20, 700] km/s. For NGC 1068, the measured value in [1] is 210 km/s, the theoretical value is 80 km/s. For these calculations, the masses of the galactic centers are equal to ^10 sun mass and 10^10 sun mass, for NGC 3310 and NGC 1068, resectively, and the etragalactic gravitational constant is suosed to be 10 times greater than the normal one. 7. Galay Seed Mystery Let us study now the case of galay grous. We suose that they are moving around each other, like in the Coma or Virgo galay grous. From the above equations we have, after calculations: 4 c G (26) 8e ( )² This sum is done for each luminous oint,, along an infinite half-line. This half-line is centered on the location in which we want to calculate G. e is the energy of each luminous oint.

6 6 Lassiaille: A Solution for the Dark Matter mystery based on Euclidean Relativity Vol. 1, No. 1 is the distance of each luminous oint from the location in which we calculate G. Let us write: i 8e S o 8e S (27) Si is for the luminous oints inside the Milky Way. So is for luminous oints outside the Milky Way. Then for our case, inside the Milky Way, we can write: 8 e Si So (28) Now for the studied case, which is outside any galay: 8 e So (29) If we note G the G constant located outside any galay: G' ( S )² i So S i )² (30) G 2 So So For eamle, if we need 10 more mass in order to elain the measured galaies seeds, then we deduce G /G= 10, and Si/So = 2,1. Let us aroimate very roughly in order to areciate this value. If we note Si = [8ei/i], and So = [8eo/o], (with ei, i, eo, and o the corresonding in ointed energies and distances corresonding to those Si and So sums) then we have: ei o 5 (31) i eo This value order reresents the energy ratio, divided by the distances ratio, between the Milky Way, and the outer sace, and along a half-line. (This half-line might be reresented hysically by an infinitely small solid angle). Therefore, this ratio corresonds also to a ratio of linear matter density, along a solid angle, between the two cases (Milky Way and outer sace). Of course this linear matter density is much greater inside the Milky Way than in the outer sace. A ratio of 5 for this seems ossible at first glance. Therefore our modelling of relativity elains the dark matter mystery for the velocity of the galaies inside a hea of galaies. Also, this elanation is the same for the mystery of light beam deviation in the vicinity of a galay. 8. Conclusion As a conclusion, this new modelling of sace-time retrieves general and restricted relativity. Nevertheless, it is more than a simle Euclidean version of relativity, as shown by ostulate 1 and 2. It is enough to add a 3 rd ostulate in order to elain dark matter mysteries, which are: 1) star s seed inside galaies, elained by our Newton s law correction, 2) seed of the galaies themselves, inside their grou, and deformation of light trajectories in the vicinity of a galay. Overall, this third ostulate conducts a modification of Newton s law. This modification is conceived in order to find eactly Newton s law in the secific case of in ointed masses inside an homogeneous universe, and long distances. However, immediately a correction of Newton s law is noticed in the case of short distances. This first correction occurs in fact for relativistic seed. Nevertheless, the imortant result of this correction of Newton s law occurs when the galay s stars are introduced in the model. After this, a strong difference aears between the calculated force and Newton s law. The shaes of the star seed rofiles, obtained with this corrected Newton s law, are very close to eerimental measurements. The seed values can be calculated also and are close to the measured ones. That is for stars seed dark matter mystery. For the other dark matter mystery, which concerns the case of galay grous, the elanation is more direct. This is elained by a different value of G, between our case inside the Milky Way, and the case outside any galay. Here again, the third seaker, which decreases the G constant, is the stars of our galay. That is for the ractical results yielded by this modelling. Moreover, and from a theoretical oint of view, this study finds a way for sace-time determination. At any oint in sace-time, this determination is based uon the matter density distribution through the whole universe. It remains also to check if the model described in this article is coherent with other actual hysics theories (electromagnetism, quantum mechanics, etc). As an answer to this last question one will notice that this model is in conformity with a unifying theory called hysics theory of the three elements. [3] References [.1.] G. Galletta, and E.Recillas-Cruz, The large scale trend of rotation curves in the siral galaies NGC 1068 and NGC 3310 (1982), htt://adsabs.harvard.edu/full/1982a&a g. [.2.] G.A. Kyazumov, The velocity field of the galay NGC 7541 (1980), htt://adsabs.harvard.edu/full/1980sval k&am. [.3.] F. Lassiaille, Three elements theory (1999), htt://lumi.chezalice.fr/3elt.df. [.4.] F. Lassiaille, A solution for the dark matter mystery based on Euclidean relativity (33 ages, Feb 2010), htt://lumi.chezalice.fr/anglais/mystmass.df. [.5.] F. Lassiaille, Validation of dark matter elanation (May 2010), htt://lumi.chez-alice.fr/anglais/calcul_ngc3310.df.