This article deals with the problem of the motion of stars in galaxies. By
using the Newton's theory combined with a gravitational time dilatation for the
weak gravitational field, it is possible to give a solution without using the
dark matter

Moulay, Emmanuel

English

arXiv

This article deals with the problem of the motion of stars in galaxies. By using the Newton's theory combined with a gravitational time dilatation for the weak gravitational field, it is possible to give a solution without using the dark matter

HAL-UNILIM

The motion of stars in galaxies and the gravitationaltime dilatationEmmanuel MoulayTo cite this version:Emmanuel Moulay. The motion of stars in galaxies and the gravitational time dilatation. 2010.<hal-00473533v2>HAL Id: hal-00473533https://hal.archives-ouvertes.fr/hal-00473533v2Submitted on 21 May 2010HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.The motion of stars in galaxies and thegravitational time dilatationEmmanuel Moulay ∗Xlim (UMR-CNRS 6172) - De´partement SICUniversite´ de Poitiers - FranceAbstractThis article deals with the problem of the motion of stars in galaxies. By usingthe Newton’s theory combined with a gravitational time dilatation for the weakgravitational field, it is possible to give a solution without using the dark matter.1 IntroductionSince the observations of Zwicky in [12], we wonder why the motion of stars in galax-ies do not corroborate the gravitational effects. The main explanation is the dark matterwhich is hypothetical matter of the standard cosmological model Λ-CDM that is unde-tectable by its emitted radiation, but whose presence can be inferred from gravitationaleffects on visible matter [9]. However, such a theory raises some problems [3]. Thereexists an alternative theory to the dark matter which is the Modified Newtonian dynam-ics MOND of Milgrom who has developed a change in the Newton’s law [6–8]. Acomparison between these two approaches is developed in [2] by Blanchet who hasalso proposed a new kind of dark matter: the dipolar dark matter [3].The main goal of this paper is to provide another solution for the problem of the motionof stars in galaxies by using a new gravitational time dilation for the weak gravitationalfield combined with the Newton’s theory.The paper is organized as follows. The problem of the motion of stars in galaxies isrecalled in Section 2 and the solution involving a new gravitational time dilation forthe weak gravitational field is addressed in Section 3. Finally, a conclusion is given inSection 42 The problem of the motion of stars in galaxiesThe galaxies are characterized by a very weak gravitational field. So, such astrophysi-cal systems must be correctly described by the Newton’s theory of gravitation.∗E-mail: emmanuel.moulay@univ-poitiers.fr1At a distance d of the center of a galaxy, the Newton’s gravitational force FN a starundergoes is given by:FN (d) = GMg Msd2(1)where G is the gravitational constant, Mg the mass of the galaxy and Ms the mass ofthe star. Using the Newton’s law of dynamics gives the following acceleration for thestar:a(d) = GMgd2m.s−2. (2)Since the equation that relates the velocity to the acceleration for a circular orbit isgiven by:a =v2d, (3)we have the following velocity for a star located at the distance d from the center of agalaxy:v(d) =√G Mgdm.s−1. (4)It seems to depend on d. Figure 1: Galaxy rotation curveInstead of decreasing asymptotically to zero as the effect of gravity wanes, theobserved distribution curve remains flat, showing the same velocity at increasing dis-tances from the bulge.3 The solution of the gravitational time dilationAt this stage, either we suppose that the previous calculus is correct or not. In thispaper, we postulate that the Newton’s dynamics are correct. We also know that therestrictions on variations in the gravitational constant are strict [11]. Nevertheless, we2want to provide a solution for the problem of the motion of stars in galaxies withoutusing the assumption of the dark matter.The Newton’s theory is only an approximation of relativity. Something happens inrelativity which is not taken into account: this is the gravitational time dilation. Wemean the gravitational time dilation due to the changing gravitational field, not therelativistic time dilatation due to the velocity of the stars which is not significant. Letus recall that the relativistic time dilation induced by the velocity is given by:∆T =√1−v2c2∆T ′ (5)where c is the speed of light and v the relative velocity between the observer whoseflow of time is given by ∆T ′ and the moving clock whose flow of time is given by∆T [4, Section 2.4].Due to the huge value of d, we know that the gravitational field is very weak.The problem of the motion of stars in galaxies could indicate that the gravitationaltime dilatation cannot be neglected for the weak gravitational field. However, it israther difficult to evaluate this gravitational time dilation due to the changing weakgravitational field which leads to the motion of stars in galaxies. Indeed, gravitationaltime dilation effects for the solar system and the Earth have been evaluated from thestarting point of an approximation to the Schwarzschild solution to the Einstein’s fieldequation. It is given for all d, d′ > RS by:∆T (d) =√1− Rsd1− Rsd′∆T (d′) (6)whereRS = 2GMc2 is the Schwarzschild radius [4, Page 219]. If we use the Schwarzschildsolution to describe the motion of stars in galaxies, such a gravitational time dilatationis not significant. However, if the Schwarzschild solution is an exact solution of theEinstein’s field equation which correctly describes the strong gravitational field out-side a spherical non-rotating mass, we cannot be sure that the same is true for the weakgravitational field of galaxies. We have no relevant way for evaluating this gravita-tional time dilation in galaxies, due in particular to the special distribution of matter ingalaxies and above all, due to the fact that the Einstein’s gravitational field has neverbeen tested for the weak gravitational field.Let d≫ 0, d′ ≫ 0 such that a(d)≪ a0, a(d′)≪ a0 with a0 ∼ 1.2×10−10m.s−2the constant given in the MOND theory. Suppose that the gravitational time dilation,induced by the weak gravitational field for stars located at d, d′ in galaxies, is asfollows:∆T (d) =√dd′∆T (d′). (7)We draw the reader’s attention to the fact that the value of√dd′is rather weak becausethis is the ratio of the huge distances from the center of the galaxy which is taken intoaccount. Equation (7) is consistent with the fact that the time flow slows down whenthe gravitational field decreases, i.e. when the distance to the center of the galaxy3increases. Indeed, one second at the distance d of the center of the galaxy corresponds√d′dseconds at the distance d′ of the center of the galaxy. For an observer located atd, the observed velocity at d′ satisfies:vobs(d′) = v(d′)√d′d= v(d) (8)which is in accordance with the observations given by Figure 1. Indeed, the observer,located at the distance d ≫ 0, observers the same velocity for the stars located at alldistances d′ ≫ 0.Example 1 Let us consider the motion of the Sun and the one of another star into theMilky Way. We suppose that the star is 10 times closer to the galactic center than theSun. Recall that the distance from the Sun to the galactic center is now estimated atd′su = 26, 000 ly, so the distance from the star to the galactic center is dst = 2, 600 ly.The gravitational time dilatation (7) is given by:∆T (dst) =√110∆T (d′su) ≈ 0.316 ∆T (d′su). (9)This example shows that the problem of the motion of stars in our galaxy can be solvedby using a gravitational time dilation of about 3 times with respect to the flow of timeof the weak gravitational field where the Sun orbits the Milky Way.Remark 2 We can infer that if the gravitational time dilatation of the weak gravita-tional field is responsible for the problem of the motions of stars in galaxies, the samecould be true for the problem of the dark energy which is the hypothetical form of en-ergy that tends to increase the rate of expansion of the universe [9]. In such a case,the weak gravitational field becomes weaker in the intergalactic space and the grav-itational time dilatation between the galaxies increases. If the flow of time elapsesquickly in the intergalactic space, it implies that the rate of expansion between thegalaxies increases.This article raises the problem of finding a relativistic theory for the weak grav-itational field which would be consistent with the gravitational time dilation (7), butalso with the other observations such as gravitational lenses, considering that the Ein-stein’s field equation is the one of the strong gravitational field. We can mention thetensor-vector-scalar gravity (TeVeS) developed by Bekenstein and Sanders in [1,10] asan attempt to define such a theory. Finally, the Weyl tensor equation is also an appro-priate tool for developing a relativistic theory for the weak gravitational field and alsoa complete Riemannian theory of gravitation [5, Section 4.1].4 ConclusionEven if the Λ-CDM model is the main theory, we cannot rule out other possibilities, aslong as we have no direct evidence of the existence of dark matter. The gravitationaltime dilatation for the weak gravitational field is one of the possibility to explain themotion of stars in galaxies.4References[1] J.D. Bekenstein. Relativistic gravitation theory for the modified newtonian dy-namics paradigm. Phys. Rev. D, 70(083509), 2004.[2] L. Blanchet. Gravite´ modifie´e ou matie`re modifie´e? l’Astronomie, 14(19), 2009.arXiv:0902.1712.[3] L. Blanchet and A. Le Tiec. Dipolar dark matter and dark energy. Phys. Rev. D,80, 2009.[4] O. Grøn and S. Hervik. Einstein’s general theory of relativity: with modernapplications in cosmology. Springer, 2007.[5] S. W. Hawking and G. F. R. Ellis. The Large Scale Structure of Space-Time.Cambridge University Press, 1975.[6] M. Milgrom. A modification of the newtonian dynamics as a possible alternativeto the hidden mass hypothesis. Astrophys. J., 270(2):365–370, 1983.[7] M. Milgrom. A modification of the newtonian dynamics: implications for galax-ies. Astrophys. J., 270(2):371–383, 1983.[8] M. Milgrom. A modification of the newtonian dynamics: implications for galaxysystems. Astrophys. J., 270(2):384–389, 1983.[9] L. Papantonopoulos. The Invisible Universe: Dark Matter and Dark Energy.Lect. Notes Phys. 720, Springer, 2007.[10] R.H. Sanders. A tensor-vector-scalar framework for modified dynamics and cos-mic dark matter. Mon. Not. Roy. Astr. Soc., 363(2):459–468, 2005.[11] J. P. Uzan. The fundamental constants and their variation: observational andtheoretical status. Rev. Mod. Phys., 75:403–455, 2003.[12] F. Zwicky. Die rotverschiebung von extragalaktischen nebeln. Helv. Phys. Acta,6:110–127, 1933.5

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