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ARTICLE A suggestion for MOND Peter Rastall Abstract: Modified Newtonian dynamics (MOND) has had considerable success in describing motion in galaxies. It uses a single force which falls off inversely with the distance at large distances and inversely with the square of the distance at smaller distances. We present an alternative theory with two forces, one the traditional Newtonian inverse-square, and the other that falls off inversely with distance at large distances. This obvious possibility has been avoided because of fear that the second force would be incompatible with observed planetary motions. However, the nonlinear field equation that governs this force is shown to reduce its strength near stars. The theory is derived from a Lagrangian density with two scalar potentials. It is nonrelativistic, but nevertheless agrees with the “classical tests of relativistic gravity” and can be used to calculate the bending of light in gravitational fields. Possible applications to interactions in galactic clusters and to anomalous planetary motions are noted. PACS No.: 04.50.Kd. Résumé : La dynamique de Newton modifiée (MOND) connaît beaucoup de succès dans sa description des mouvements dans les galaxies. Elle utilise une simple force qui diminue inversement avec la distance a ` grande distance et inversement au carré de la distance a ` courte distance. Nous proposons une variation de cette théorie avec deux forces, l’une est la force de Newton traditionnelle en inverse du carré de la distance et l’autre décroît inversement avec la distance a ` grande distance. Cette évidente possibilité avait été laissée de côté par crainte que la seconde force serait incompatible avec le mouvement connu des planètes. Cependant, nous montrons que l’équation de champ non linéaire qui gouverne cette force réduit son intensité près des étoiles. La théorie dérive d’une densité lagrangienne avec deux potentiels scalaires. Elle est non relativiste, mais agrée néanmoins avec le test classique de la gravité relativiste et peut être utilisée pour calculer la déviation de la lumière dans les champs gravita- tionnels. Nous notons des applications possibles aux interactions dans les amas galactiques et aux mouvements planétaires anomaux. [Traduit par la Rédaction] 1. Introduction It has long been known that the motions in galaxies cannot be explained by the Newtonian gravitational attraction of the visible matter [1, 2]. To account for the missing force, several kinds of dark matter have been postulated, but nothing has been found in sufficient quantity. Another possibility is that the Newtonian law should be changed. One must of course take care that a new law does not significantly perturb motions in the solar system, which have been accurately measured. M. Milgrom discovered in the early 1980s that the motion of stars in the outer regions of galaxies can be explained if the grav- itational force falls off inversely with distance, rather than in- versely with the square of the distance as in the Newtonian theory [3, 4]. He postulated that the force becomes inverse-square at smaller distances. In what follows, it is assumed that two forces act: the usual Newtonian inverse-square, and another that behaves like Mil- grom’s at large distances. For simplicity, one considers a spheri- cally symmetric galaxy of mass M and a star of mass m that moves outside this galaxy with speed v in a circular orbit of radius r. The gravitational force f that acts on the star is directed towards the centre of the galaxy and has magnitude f f mMGr 2 mAr 1 (1.1) where G = 6.67 × 10 −11 m 3 s −2 kg −1 is the Newtonian gravitational constant and A is a constant that depends on M. The acceleration, a, of the star has magnitude v 2 r −1 = MGr −2 + Ar −1 . For large r, the r −2 term is negligible and v 2 = A. Hence v is constant, independent of r (one says that the galactic rotation curve is flat). It is found that for a large number of galaxies v = (GMa 0 ) 1/4 , where a 0 = (1.2 ± 0.2) × 10 −10 ms −2 according to [4], and hence A (GMa 0 ) 1/2 KM 1/2 (1.2) where K =(Ga 0 ) 1/2 = 8.9 × 10 −11 m 2 s −2 kg –1/2 . Note that K, like a 0 , is the same for all the galaxies. (If c = 3 × 10 8 ms −1 is the speed of light, then K −2 c 4 has the dimensions of mass and is about 10 54 kg, which is roughly the mass of the observable universe.) We recall that in traditional MOND the gravitational attraction a that is produced by a point mass M at a distance r is given by (a/a 0 )a = GMr −2 , where a 0 is the constant that appears in (1.2) and is a function such that (x) approaches 1 for large x and (x) approaches x for small x. This is a radical change from the princi- ples of Newtonian mechanics. The purpose of the present paper is to show that the introduction of the function is unnecessary and that standard Newtonian mechanics can be preserved at the cost of introducing an additional force. The force (1.1) on the star can be written as f =−m, where MGr 1 A ln r (1.3) Because the Laplacian of the first term vanishes, one has 2 = Ar –2 , valid for the case of spherical symmetry and where the mass Received 19 April 2014. Accepted 22 May 2014. P. Rastall. Department of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T1Z1, Canada. E-mail for correspondence: [email protected]. Pagination not final (cite DOI) / Pagination provisoire (citer le DOI) 1 Can. J. Phys. 92: 1–3 (2014) dx.doi.org/10.1139/cjp-2014-0216 Published at www.nrcresearchpress.com/cjp on 29 May 2014. Can. J. Phys. Downloaded from www.nrcresearchpress.com by SUNY AT STONY BROOK on 11/03/14 For personal use only.

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Page 1: A suggestion for MOND

ARTICLE

A suggestion for MONDPeter Rastall

Abstract: Modified Newtonian dynamics (MOND) has had considerable success in describing motion in galaxies. It uses a singleforce which falls off inversely with the distance at large distances and inversely with the square of the distance at smallerdistances. We present an alternative theory with two forces, one the traditional Newtonian inverse-square, and the other thatfalls off inversely with distance at large distances. This obvious possibility has been avoided because of fear that the second forcewould be incompatible with observed planetary motions. However, the nonlinear field equation that governs this force is shownto reduce its strength near stars. The theory is derived from a Lagrangian density with two scalar potentials. It is nonrelativistic,but nevertheless agrees with the “classical tests of relativistic gravity” and can be used to calculate the bending of light ingravitational fields. Possible applications to interactions in galactic clusters and to anomalous planetary motions are noted.

PACS No.: 04.50.Kd.

Résumé : La dynamique de Newton modifiée (MOND) connaît beaucoup de succès dans sa description des mouvements dans lesgalaxies. Elle utilise une simple force qui diminue inversement avec la distance a grande distance et inversement au carré de ladistance a courte distance. Nous proposons une variation de cette théorie avec deux forces, l’une est la force de Newtontraditionnelle en inverse du carré de la distance et l’autre décroît inversement avec la distance a grande distance. Cette évidentepossibilité avait été laissée de côté par crainte que la seconde force serait incompatible avec le mouvement connu des planètes.Cependant, nous montrons que l’équation de champ non linéaire qui gouverne cette force réduit son intensité près des étoiles.La théorie dérive d’une densité lagrangienne avec deux potentiels scalaires. Elle est non relativiste, mais agrée néanmoins avecle test classique de la gravité relativiste et peut être utilisée pour calculer la déviation de la lumière dans les champs gravita-tionnels. Nous notons des applications possibles aux interactions dans les amas galactiques et aux mouvements planétairesanomaux. [Traduit par la Rédaction]

1. IntroductionIt has long been known that the motions in galaxies cannot be

explained by the Newtonian gravitational attraction of the visiblematter [1, 2]. To account for the missing force, several kinds ofdark matter have been postulated, but nothing has been found insufficient quantity. Another possibility is that the Newtonian lawshould be changed. One must of course take care that a new lawdoes not significantly perturb motions in the solar system, whichhave been accurately measured.

M. Milgrom discovered in the early 1980s that the motion ofstars in the outer regions of galaxies can be explained if the grav-itational force falls off inversely with distance, rather than in-versely with the square of the distance as in the Newtonian theory[3, 4]. He postulated that the force becomes inverse-square atsmaller distances.

In what follows, it is assumed that two forces act: the usualNewtonian inverse-square, and another that behaves like Mil-grom’s at large distances. For simplicity, one considers a spheri-cally symmetric galaxy of mass M and a star of mass m that movesoutside this galaxy with speed v in a circular orbit of radius r. Thegravitational force f that acts on the star is directed towards thecentre of the galaxy and has magnitude f

f � mMGr�2 � mAr�1 (1.1)

where G = 6.67 × 10−11 m3s−2kg−1 is the Newtonian gravitationalconstant and A is a constant that depends on M.

The acceleration, a, of the star has magnitude v2r −1 = MGr −2 +Ar −1. For large r, the r −2 term is negligible and v2 = A. Hence v isconstant, independent of r (one says that the galactic rotationcurve is flat). It is found that for a large number of galaxies v =(GMa0)1/4, where a0 = (1.2 ± 0.2) × 10−10 ms−2 according to [4], andhence

A � (GMa0)1/2 � KM1/2 (1.2)

where K = (Ga0)1/2 = 8.9 × 10−11 m2s−2kg–1/2. Note that K, like a0, is thesame for all the galaxies.

(If c = 3 × 108 ms−1 is the speed of light, then K−2c4 has thedimensions of mass and is about 1054 kg, which is roughly themass of the observable universe.)

We recall that in traditional MOND the gravitational attractiona that is produced by a point mass M at a distance r is given by�(a/a0)a = GMr −2, where a0 is the constant that appears in (1.2) and� is a function such that �(x) approaches 1 for large x and �(x)approaches x for small x. This is a radical change from the princi-ples of Newtonian mechanics. The purpose of the present paper isto show that the introduction of the function � is unnecessary andthat standard Newtonian mechanics can be preserved — at thecost of introducing an additional force.

The force (1.1) on the star can be written as f = −m��, where

� � �MGr�1 � A ln r (1.3)

Because the Laplacian of the first term vanishes, one has �2� =Ar –2, valid for the case of spherical symmetry and where the mass

Received 19 April 2014. Accepted 22 May 2014.

P. Rastall. Department of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T1Z1, Canada.E-mail for correspondence: [email protected].

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Can. J. Phys. 92: 1–3 (2014) dx.doi.org/10.1139/cjp-2014-0216 Published at www.nrcresearchpress.com/cjp on 29 May 2014.

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Page 2: A suggestion for MOND

density of the sources vanishes. In Sect. 2, a field equation withoutthese restrictions will be found.

2. Field equationsThe potential � is written as � = � + � and the Lagrangian

density for � and � is

L � �,i �,i � 8G� � (�,i �,i)3/2 � 6�� (2.1)

where � is a constant; is the mass density of the sources of thefields; �,i = ��/�xi, etc.; the xi are rectangular cartesian coordi-nates; and the summation convention applies to repeated Latinsuffixes with the range {1, 2, 3}.

The field equations that follow from (2.1) are

�,ii � 4G (2.2)

�,j �,j �,ii � �,i �,j �,ij � 2�(�,i �,i)1/2 (2.3)

Equation (2.2) is the usual Poisson equation for the Newtonianpotential �. In vector notation, (2.3) is

| �|2 2� �1

2 (| �|2) · � � 2�| �| (2.4)

or equivalently

div(| �| �) � 2� (2.5)

The Lagrangian density (2.1) is similar to that used in ref. 5 for theMOND theory, but without the � terms.

In the special case of spherical symmetry, (2.4) becomes

� ′′ � r�1�′ � �|�′|�1 (2.6)

where �′ = d�/dr, etc. If = 0, (2.6) gives �′ = Ar −1 with A constant,in agreement with (1.3).

One must show that A is proportional to the square root of themass of the source. Applying the divergence theorem to a ball ofradius R centred at the origin and assuming that r2(r) vanishes asr ¡ 0 gives

|�′(r)|�′(R) � � �

2�R�2I(R) (2.7)

from (2.5), where I(R) = �4r2dr, with integration from 0 to R, isthe total mass in the ball of radius R. If (r) = 0 for r > R, then I(R) =M, the total mass of the sources. If �′ is assumed to be non-negative, �′(R) = (�/2)1/2R −1M1/2. When r > R, one has �′ = Ar −1 andhence

A � � �

2�1/2

M1/2 � KM1/2 (2.8)

where K = (Ga0)1/2, as in (1.2), to agree with the observations ofgalactic motion, and � = 2K2 = 5.03 × 10−20 m4s−4kg−1.

Instead of 6�� in (2.1), one might have chosen another functionof �. However, this spoils the simple M1/2 dependence of (2.8).

3. Near the SunThe field equations are assumed to be generally valid. They can

therefore be used to find the effect of the � field on planetary mo-tions. For simplicity, we assume a spherically symmetric Sun of mass

M = 2 × 1030 kg. The force per unit mass exerted by the � field at adistance r from the Sun’s centre has magnitude �′(r) = KM1/2r−1

and that exerted by the � field has magnitude � ′(r) = GMr−2. At theorbit of Mercury, radius 5.8 × 1010 m, one has �′(r) = 2.17 × 10−6 ms−2

and the ratio of these forces is Kr/GM1/2 = 5.5 × 10−5. This is to becompared with GM/rc2 = 2.6 × 10−8, which is roughly the ratio ofthe post-Newtonian perturbation force to the Newtonian forceexerted by the � field. The force exerted by the � field is thereforethree orders of magnitude greater than the observed perturbationforce on Mercury.

The conclusion is clear: the � field is not in agreement with theobserved planetary motions. This was of course obvious to Mil-grom, and is why he chose a force field that did not have an r −1

dependence at small distances.But this cannot be the whole story. For simplicity again, sup-

pose that the galaxy has 1011 stars of mass M, distributed in aspherically symmetric manner. According to our previous result,each of them produces a � field proportional to M1/2, and the total� field of the galaxy might therefore be expected to be 1011/2 asgreat [1011M1/2 = 1011/2(1011M)1/2]. But we know from (2.8) that thecorrect factor is 1, not 1011/2. The assumption that one can addtogether the contributions of the individual stars is incorrect —what prevents it is the nonlinearity of the field equations. Wemust show how the fields of the stars are melded together to givethe total field of the galaxy. We may expect that this will diminishthe fields near the individual stars by large factors.

So much for hand-waving! To show how the � field is repressed,one considers a large number of stars uniformly distributed in athin, spherical shell of radius R and mass MS. One makes the usualassumption that the stars can be represented by a fluid of density. Because (r) and I(r) vanish inside the sphere, it follows from(2.7) that so does �′(r) (we are assuming that � and �′ are nons-ingular inside the shell). Outside the shell, one finds as in (2.8) that�′ = Ar–1,, where A = KMS

1/2. If MS = 1038 kg (about 0.5 × 108 solarmasses) and R = 3.5 × 104 ly = 3.3 × 1020 m (roughly the Sun’sdistance from the centre of the galaxy), one has �′(R) = 2.7 ×10−12 ms−2. Compare this with 2.17 × 10−6 ms−2, the naively calcu-lated value of �′ because of the Sun at the orbit of Mercury. If wenow abandon the fluid description and consider the stars in asmall part of the shell, this means that they produce no � fieldtowards the inside of the shell and that the field that they producetowards the outside depends on the total mass of the shell, and issix orders of magnitude less than that found by the naive calcula-tion.

The last paragraph considered a thin, spherical shell of stars. Tounderstand better how the � field of an individual star is re-pressed, one should solve the � field equation for the galaxy andan individual star, both idealized as spherically symmetric balls offluid. (One can consider the two cases of the star inside or outsidethe galaxy.) Using these solutions, one can check whether theforce on the star is −m��G, where �G = �G + �G is the potentialdue to the galaxy. The essential condition is probably that themass of the star is very small compared with the mass of thegalaxy.

An exactly similar calculation would serve to calculate the �field of two galaxies, each idealized as a spherically symmetricball of fluid. It seems unlikely that the force on one of them can beexpressed in terms of the fields produced by the other — andindeed, for a nonlinear field, this statement makes no sense. Wenote that traditional MOND does not predict the correct forcesbetween galaxies: perhaps the present approach will be moresuccessful. The completion of these ideas is left as an exercise forthe reader (the author is now too frail to handle nonlinear partialdifferential equations).

The � field of the Sun will not be completely repressed. It willbe interesting to see if it accounts for the remaining small anom-alies in solar system motions [6].

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Page 3: A suggestion for MOND

4. Gravitational calculationsThe search for dark matter has so far been fruitless. It appears

likely that the explanation of galactic motions will require a newgravitational force law and the abandonment or modification ofthe Newtonian and Einsteinian theories. This does not, however,imply that we shall lose the ability to calculate gravitational ef-fects, at least in the nonrelativistic domain. It has been known formany years that the Newtonian theory of gravitation, with analmost trivial generalization, can account for all the so-called“classical tests of relativistic gravity”. These are the gravitationalredshift, the anomalous perihelion advance of Mercury, and thedeflection of light in a gravitational field (and one might add theradar time delay) [7, 8]. An outline of this generalized Newtoniantheory, which has been called gravitostatics, is in Sect. 1 of ref. 9; alonger account is in ref. 10.

The theory described in this paper is easily incorporated intogravitostatics: one simply interprets the potential � as � + �, as inSect. 2. Chapter 3 of ref. 10 gives a Lagrangian for the motion ofparticles, including photons, which enables the calculation oflight deflection and gravitational lensing.

5. ConclusionIt is not necessary to modify Newtonian dynamics in a radical

way, as originally proposed in MOND. One has only to introducean additional force, which is produced by a nonlinear field. The

nonlinearity represses the short-range effects of this force onplanetary motions.

The theory proposed is nonrelativistic. It is sufficient for deal-ing with galactic motions, gravitational lensing, etc. It would notbe difficult to produce a relativistic generalization (see ref. 11), butthis may be premature. It may be advisable to first explore furtherthe properties of the additional force, and to try to account for themotion of galaxies in clusters.

References1. J. Oort. Bull. Astron. Soc. Neth. 6, 249 (1932).2. F. Zwicky. Helv. Phys. Acta, 6, 110 (1993).3. M. Milgrom. Astrophys. J. 270, 365 (1983). doi:10.1086/161130; Ibid. 371 (1983).

doi:10.1086/161131; Ibid. 384 (1983). doi:10.1086/161132.4. M. Milgrom. Mon. Not. R. Astron. Soc. 437, 2531 (2014). doi:10.1093/mnras/

stt2066.5. J.D. Bekenstein. In Proceedings of the Second Canadian conference on gen-

eral relativity and relativistic astrophysics: University of Toronto, 14–16 May1987. Edited by C.C. Dyer, B.O.J. Tupper, and A.A. Coley. World Scientific,Singapore. p. 68. 1988.

6. J.D. Anderson and M.M. Nieto. In Relativity in fundamental astronomy, Pro-ceedings IAU Symposium No. 261. Edited by S.A. Klioner, D.K. Seidelmann,and M.H. Soffel. p.189. 2009.

7. P. Rastall. Can. J. Phys. 57(7), 944 (1979). doi:10.1139/p79-133.8. P. Rastall. Can. J. Phys. 38(8), 975 (1960). doi:10.1139/p60-108.9. P. Rastall. 2004. arXiv:astro-ph/0402339v1.

10. P. Rastall. Postprincipia: gravitation for physicists and astronomers. WorldScientific, Singapore. 1991.

11. J.D. Bekenstein. Phil. Trans. Roy. Soc. London, A 369, 5003 (2011). doi:10.1098/rsta.2011.0282.

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