THE PHENOMENOLOGICAL VERSION OF MODIFIED NEWTONIAN

THE PHENOMENOLOGICAL VERSION OF MODIFIED NEWTONIAN DYNAMICS FROM THE RELATIVITY PRINCIPLE OF MOTION ´ JAUME GINE Departament de Matem` atica, Universitat de Lleida, Av. Jaume II, 69. 25001 Lleida, Spain E.mail: [email protected]

Abstract. In this paper we deduce the first phenomenological version of Modified Newtonian dynamics(MOND) proposed by Milgrom from the relativity principle of motion and assuming the proven fact of the accelerated expansion of the universe. It is found a new form of µ(x) in the Milgrom formula for the second Newton law. Moreover, it is established a relation between the Modified Newtonian dynamics (MOND) and the Deceleration Parameter.

1. Introduction The Modified Newtonian dynamics (MOND) theory was introduced by Milgrom to modify the Newton’s second law as an alternative to the dark matter introduced to solve the galaxies rotation curves problem. The galactic rotation curve problem is a discrepancy between the interpretation of the observed luminance to mass ratio of matter in the disk portions of spiral galaxies and the luminance to mass ratio of matter in the cores of galaxies, see [48, 49, 50, 51]. This discrepancy is currently thought to betray the presence of dark matter that permeates the galaxy and extends into the galaxy’s halo. Dark matter was postulated by Fritz Zwicky [52] in 1934 to account for evidence of “missing mass” in the orbital velocities of galaxies in clusters. An alternative explanation is a modification of the Newton’s second law or the laws of gravity, such as MOND. 1991 Mathematics Subject Classification. Primary 83F05. Secondary 83C05. Key words and phrases. Modified Newtonian Dynamics, Cosmology, cosmic acceleration. The author is partially supported by a MCYT/FEDER grant number MTM200800694 and by a CIRIT grant number 2009SGR 381. 1

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In particular Milgrom allowed for the inertia term not to be proportional to the acceleration of the object but rather to be a more general function of it. More precisely, it has the form mi µ(a/a0 ) a = F, where µ(x ≫ 1) ≈ 1, and µ(x ≪ 1) ≈ x and a = |a|, replacing the classical form mi a = F. Here mi is also the inertial mass of a body moving in an arbitrary static force field F with acceleration a, see [33]. For accelerations much larger than the acceleration constant a0 , we have µ ≈ 1, and Newtonian dynamics is restored. In fact, this formulation of the MOND theory, the original one, has been known for a long time to bare serious problems. The Milgrom formulation does not lend itself to a Lagrangian formulation and therefore one is not able to identify a conserved energy and momentum. To solve this problem Bekenstein and Milgrom propose in [6] a nonrelativistic potential theory for gravity which differs from the Newtonian one. The theory involves a modification of the Poisson equation and can be derived from a Lagrangian. Therefore the total momentum, angular momentum, and energy of an isolated system are conserved. The significant difference with the previous model of Milgrom is that the modification is not in the Newton’s second law but in the Newton’s law of gravitation. In later works Bekenstein [3, 4, 5] developed a relativistic formulation which has a MOND and Newtonian limits under the proper circumstances. An important feature of MOND is the following. The internal dynamics of a gravitating system s embedded in a larger one S is affected by the external background field E of S even if it is constant and uniform, thus implying a violation of the strong equivalence principle: it is the so-called External Field Effect (EFE). Milgrom [33] originally introduced EFE in order to explain that the amount of dark matter detected in certain open star clusters in the Galactic neighborhood of the solar system was very low, although their internal accelerations were 5 or 10 times smaller than a0 . The Galactic acceleration felt by such open clusters is just of the order of a0 . The first, preliminary attempts to look at EFE in the Oort cloud were made by Milgrom in [33, 34]. More detailed analysis on EFE in the Oort cloud is made by Iorio in [28, 53, 54]. EFE was adapted to the planetary regions of the Solar System, where the field is strong, see [37]. Some implications were discussed in [9, 29, 31]. Finally it should be mentioned that several studies of MOND were performed in the solar system, see [7, 9, 24, 26, 28, 29, 31, 53, 37, 40, 54, 42, 44].

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In this paper we deduce the first phenomenological version of Modified Newtonian dynamics(MOND) proposed by Milgrom from the relativity principle of motion and assuming the proven fact of the accelerated expansion of the universe. It is found a new form of µ(x) in the Milgrom formula for the second Newton law which is more consistent with the recent observational data. Of course the proposal made have the absence of energy conservation. However, following the reasonings given in this paper it will be possible to construct a formulation similar to Bekenstein one. In [17] was made a first approximation to the problem and was deduced the form µ(x) in the context of the Mach’s principle, formula given by ( ) a (1) mi a = F. a + a0 This simple form of µ(x) yields very good results in fitting the terminal velocity curve of Milky Way and others, see [12]. However, it has been shown that such Milgrom theory, while solving a few difficulties, gives rises to other fresh problems, see for instance [13, 40]. The Milgrom formula (1) with the constant acceleration a0 is also present in the inner solar system where it is dramatically inconsistent with the motion of the inner planets. In fact, it fails completely in the strong gravity regime where a ≫ a0 , and thus cannot be valid in the Solar system, see [40]. Hence, one must argue that the MOND acts in a very different way for bound objects like planets. This has already been seen pointed out by Milgrom, see [36]. In [19] it is shown that the Pioneer anomaly can be explained using the formula (1). The Pioneer anomaly [1, 2] consists of an unexpected, almost constant and uniform acceleration directed approximately towards the Sun aP = (8.74 ± 1.33) × 10−10 ms−2 first detected in the analyzed data of the spacecraft Pioneer 10 and Pioneer 11 after they passed the threshold of 20 Astronomical Units. However, the recent new data of the Pioneer anomaly suggest that it is variable and environment dependent rather than a fixed value and still is not clear its direction with the possibility that the Pioneer anomaly be Earth directed, see [46, 47]. The effects of Pioneer anomaly are nondetected on the major bodies of the solar system and in several papers is studied its gravitational origin [14, 20, 21, 22, 23, 25, 27, 43, 45]. The problem that Pioneer anomaly does not show up in the motion of the major bodies of the Solar system is, actually, a problem which should actually be tackled by all the proposed attempts to explain Pioneer anomaly itself with unconventional gravitational mechanisms. Meanwhile there exits other works where it is studied its non-gravitational

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origin, see [8, 39, 55, 56, 57, 58, 59]. In [17] it is also shown that the acceleration constant a0 that appears in the Milgrom theory must be a0 = cH0 where H0 is the actual value of the Hubble constant. In [18] it is given another way to find the relation a0 = cH0 in this case using the scale factor of the universe. The fact that a0 is of order cH0 was pointed out by Sanders and McGaugh, see [41] where some cosmological issues are studied. Later, this relation it was suspected by Milgrom and as he said may be a clue of the origin of MOND, see [36, 37]. The forms to deduce the relation a0 = cH0 tell us the relation of the MOND theory with the global dynamics of the universe. However there exits works where it was argued that Pioneer anomaly is not of cosmological origin, see instance [32, 38]. We recall that the value cH0 is related to the acceleration in the Hubble horizon, see [11]. The relation of MOND with cosmology from different contexts is recently analyzed in [62, 63]. An interesting application of MOND to anthropic principle arguments can be found in [64]. The value of a0 firstly presented by Milgrom was a0 ≈ 2×10−10 cm s−2 which turn to be of the same order as cH0 = 5 × 10−10 cm s−2 , see [33]. Recent value of a0 is around the value a0 ≈ 1.2 × 10−10 cm s−2 even far from the value of cH0 . This value of a0 was obtained in [61]. The difference between a0 and cH0 is also confirmed by the recent determined values of H0 and is not still explained, see [60]. The new form of µ(x) in the Milgrom formula presented in this work does not explain this difference but it can explain some of the problems that has previous expression (1).

2. The computation of µ(x) First we consider a test particle at rest and at the same place that the observer. In the inertial frame of the test particle and the observer we have that the most distant matter of the universe which is at the distance RU (the causal connected radius of the universe) is accelerated with the acceleration a0 = cH0 far away, see Fig. 1. In fact in this inertial frame each body of the universe goes away from the test particle with certain acceleration due to the expansion of the universe. Farther away from the test particle most is the acceleration that feels any body up to a0 for the most distant matter of the universe. This is the observed behavior of the accelerated expansion of the universe.


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Now we apply the relativity principle of motion. Using this principle of motion we can think that is the most distant matter of the universe which is at rest (respect to the radial acceleration in direction to the test particle, of course there can be other non-radial accelerations and local movements that now we do not consider). In this case the test particle feels in any radial direction the constant acceleration a0 , see Fig. 2. In fact we can consider that at each point of the universe we have applied a radial acceleration vector of modulus a0 and direction toward the test particle. In each closer point of the test particle acts an acceleration closer to a0 and direction toward the test particle. In these closed points this acceleration cannot can be appreciated because it is negligible compared with the local dynamics. The test particle suffers this acceleration a0 in any radial direction and consequently is at rest (because initially it was at rest), see Fig 2. We call this new frame of reference S and in it the most distant matter of the universe is at rest (respect to any radial direction) and also the test particle is at rest. Therefore, in these two frames of reference (the original inertial frame and the frame S) the test particle is at rest. This relativity principle of motion is used by Einstein to established the special relativity where is applied to velocity movements but it can be applied to accelerated movements. It is also used by Einstein in the equivalence principle where we can think that is the acceleration g of the gravitation that acts on a person in an elevator or is the elevator that is accelerating with an acceleration g in an upward direction. Now we consider a particle in the Hubble horizon, a particle that feels the constant acceleration a0 far away. The equation of movement

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of this particle respect to the observer is mi a0 = Fe −

Gmg MU , RU2

where Fe is the force that induce the accelerated expansion of the universe (in the cosmological standard models due to the dark energy and recently due to the entropic force, see [11]) and RU and MU represent the mass and radius of the universe and RU = c/H0 where H0 is the actual value of the Hubble constant. For an accelerated expanding universe we can establish the following principle: When an observer describes the movement equation of a particle which is not at the same place that the observer, this particle suffers the acceleration expansion of the universe. Applying the relativity principle any observer is never at rest in an expanding universe except when the observer is beside the particle who is analyzing. In the classical Newtonian dynamics when we say that a particle is at rest is respect an absolute space first introduced by Newton. This absolute space were defined by the most distant stars that we thought that were at rest. However now we know that we are in accelerated expanding universe and in order to apply correctly the classical Newtonian dynamics we owe to place in the frame of reference S where the most distance matter is at rest and we have correctly defined the absolute space. This gives the answer the question of what frame of reference we must use the original inertial one or the frame of reference S (where the distant matter of the universe is at rest respect to the radial direction). The original one is not a inertial frame in “stricto sensus” because the test particle and the observed are at rest respect nothing, the rest must be respect anything. When we made a description of a local movement is not necessary to apply the relativity principle and we can use the original inertial frame. Moreover the accelerate expansion of the universe is very low comparable respect to any local movement. However when we consider a non-local movement (for instance the movement of a star) we must to apply it. When this happens we must to use the frame of reference S. We will see that we must use a modified inertial mass mim . Now we consider a star of gravitational mass mgs and inertial mass mis and consider an observer located in a third body (in our case our planet). First, we consider an inertial frame of reference respect to the observer. In this frame of reference, the distant matter of the universe is accelerated with the acceleration a0 far away, like in Fig. 1. The star feels also a negligible acceleration (because the star is relatively close) due to the expansion of the universe far away in the direction fixed by straight


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line that link the star with the observer. If the star is moving with acceleration a respect to center of mass of its galaxy with gravitational Mgc , the equation of motion of the star in the inertial frame of the observer is Gmgs Mgc ˆ (2) mis a = Rc , Rc2 where Rc is the distance between the star and the center of mass of its ˆ c is fixed by the straight galaxy and the direction of the vectors a and R line that link the star with the center of mass of its galaxy. Now, applying the relative movement (the relativity principle for accelerations), we consider the frame of reference S where the distant matter of the universe is at the rest (respect to the radial direction) and the observed is also at rest due to the cancelation of all forces in the observer place, see Fig. 2. In this frame of reference the star feels an effective acceleration given by ( ) Robs ˆ (3) aes = a0 1 − RU , RU where RU is the radius of the universe, the direction of the unitary ˆ U is the straight line that link the star and the observer and vector R Robs is the distance between the star and the observer. Notice that from equation (3) the direction of the acceleration vector aes is the ˆ U . If the star had been at a distance RU then aes = 0 and same that R while more closer is the star more bigger is aes up to a0 . In fact we have made a linear approximation for the behavior of a function that we know that must be zero at RU and must be closer to a0 for a point closer to the observer. Hence, the equation of movement of the star in this frame of reference S is now ( ) Gmgs Mgc ˆ Robs ˆ (4) mis (a + aes ) = Rc + mis a0 1 − RU . Rc2 RU The second term is an inertial force that appears because the observer is not jointly with the star and applying the relativity principle we consider the frame of reference S where the distant matter of the universe is at rest respect to the radial direction. In the original inertial frame of the observer and in the frame of reference S the observer is at rest. However, both frames of reference are distinguishable attending to the distant matter because in the first one the distant matter is accelerated far away and in the second is at rest respect to any radial direction toward the observer. Moreover, in the first one we ignore the expansion of the universe and in the second one this fact is reflected.

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For a local description it is sufficient the first frame of reference, but for a more global description of the movements we need to introduce the effects of the accelerated expansion of the universe in our equations, which is done reasoning in the second frame of reference S and in order to have a well defined absolute space of Newton. Is the same that happens when we describe some local movements in the Earth, for some local movements we can think that we are in an inertial frame of reference, but for others movements we need to introduce the rotation movement of the Earth (for instance the coriolis effect). When we measure the acceleration of a star in a galaxy we ignore this inertial force of equation (4) and we have the equation Gmgc Mgc ˆ (5) mis aobs = Rc , Rc2 where aobs is the observed acceleration which really is aobs = a + aes . If we compare with (4) we have two options: • To modify the Newton’s second law i.e., the inertia replacing into (5) the inertial mass mis by a modified inertial mass mis m . • To modify the Newton’s law of gravitation by adding the additional term (the inertial force). In this work we have take the first option to obtain a modification of the Milgrom formula. Nevertheless, it is also possible to modify the Newton’s law of gravitation in order to obtain a similar theory of Bekenstein and Milgrom one (see [6]) but now with the corresponding modifications due to the expansion of the universe. Therefore to compare with equation (4) we must introduce a modified inertial mass and hence we obtain the equation Gmgc Mgc (6) mis m (|a + aes |) = , Rc2 where mis m is the modified inertial mass of the star. Hence if we compare equation (2) with equation (6) (equations of motion respect the original inertial frame reference and the frame of reference S) we obtain mis |a| = mis m (|a + aes |), and we have the proportional formula mi mis = sm . |a + aes | |a| Therefore, in general, the movement equation of a body with an acceleration a respect to an observer in an expanding universe is given by mim a = F,


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where mim is the modified inertial mass. Introducing its value respect to the real inertial mass we obtain ( ) |a| (7) mi a = F. |a + ae | In the case that the acceleration of the body a would be much bigger than ae , i.e., a >> ae , we have a/(|a + ae |) → 1 and we obtain the classical Newtonian form mi a = F. On the other hand, in the case ae >> a, we have that a/(|a + ae |) → a/ae and substituting this value we get a mi a = F. ae In short, the new founded µ(x) in the Milgrom formulation is µ(a/a0 ) = |a|/(|a + ae |) where ( ) Robs ae = a0 1 − , RU where Robs is the distance to the object and RU is the radius of the universe. For local objects ae ∼ a0 and for far away objects ae ∼ 0. Moreover, in the formula (7) we have a vectorial sum of accelerations and depending if the vectors are quasi-collinear or are perpendicular the result gives different values. In the case of a star we have a vectorial sum of the own acceleration of the star around the center of mass of the galaxy and the acceleration ae in a direction to us (the observer). Statistically this sum is always almost perpendicular if we look stars at the edges of galaxies. In the case of the inner planets of the solar systems, as they are local objects we have that ae ∼ a0 , but the sum of the accelerations varies periodically as the planet completes its orbit. In this case the vectors can be quasi-collinear or almost perpendicular depending of the relative position of the Earth and the planet. For the Pioneer craft in its unbounded orbit, as a local object ae ∼ a0 but in this case the sum of accelerations is always quasi-collinear and almost constant at this distance and we obtain the result given in [19]. In fact, the form of µ(x) obtained in this work explains that the Pioneer anomaly it is variable and environment dependent and attending to the results presented it must be Earth directed. 3. MOND and the Deceleration Parameter Until a few years for most cosmologists the words “modern cosmology” meant the same as in the thirties: Friedmann cosmology, see [15, 16]. It is now an inflationary universe model with an inflationary phase of pre-Friedmannian expansion, where a range of 10−32 s the size of the universe increases by 1055 times. Friedmann model describes

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a homogeneous isotropic universe filled by matter (with the energymomentum tensor) and the metric of Friedmann-Robertson-Walker ( ) dr2 2 2 2 2 2 2 2 2 ds = −dt + R(t) + r dθ + r sin θ dϕ , 1 − kr2 where R(t) is the scale factor and k is a constant representing the curvature of the space. Einstein’s equations in the Friedmann’s model for the metric of Friedmann-Robertson-Walker have the form ( )2 R˙ 8πG kc2 Λc2 H 2 (t) ≡ = ρ+ 2 + , R 3 a 3 and

( ) ¨ R 4πG 3p Λc2 ˙ H +H = ρ+ 2 + =− , R 3 c 3 where ρ is the energy density ρ = ρm + ρR , k is the curvature sign, p is the pressure and Λ is the cosmological constant. The density parameter Ω is defined as the ratio of the actual (or observed) density ρ to the critical density ρc of the Friedmann universe. The critical density is the watershed between an expanding and a contracting Universe. An expression for the critical density is found by assuming Λ zero and setting the normalized spatial curvature k equal to zero. When the substitutions are applied to the first of the Friedmann equations we find: 3H 2 ρc = . 8πG The density parameter (useful for comparing different cosmological models) is then defined as: ρ 8πGρ Ω≡ = . ρc 3H 2 According to the ΛCDM model, there are important components of Ω due to baryons, cold dark matter and dark energy. The spatial geometry of the universe has been measured by the WMAP satellite to be nearly flat, meaning that the spatial curvature parameter k is zero. Hence, the first Friedmann equation is often seen in a form with density parameters as ( )2 R˙ (8) = H 2 = H02 (ΩR R−4 + ΩM R−3 + Ωk R−2 + ΩΛ ). R 2

Here ΩR is the radiation density today, ΩM is the matter density today (dark plus baryonic), Ωk = 1 − Ω is the spatial curvature density, and ΩΛ is the cosmological constant or vacuum density. Now we assume


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that ΩΛ = 0 and taking into account that today we are in the matterdominated universe, equation (8) takes the form ( )2 R˙ (9) = H 2 = H02 (ΩM R−3 + Ωk R−2 ). R The derivative of this equation gives the cosmic acceleration ( ) ΩM 2 ¨ . R = −H0 2R2 which is independent of the the spatial curvature density Ωk . If we define, as usual, the deceleration parameter as ( ) ¨ ¨ RR H˙ + H 2 H˙ R q=− =− 2 =− =− 1+ 2 , H R H2 H R˙ 2 from the range of values of ΩM ≈ 0.2−0.3 we obtain that q0 = ΩM /2 ≈ 0.15, that is a deceleration of the universe. This result is in contradiction with standard candle observations of type Ia supernovae which give a cosmic acceleration of q0 ≈ −0.5. Hence, we must to introduce the dark energy changing equation (9) by ( )2 R˙ (10) = H 2 = H02 (ΩM R−3 + Ωk R−2 + ΩΛ ). R In this case taking into account the range of values of ΩΛ = 0.7 − 0.8 we obtain that q0 = ΩM /2 − ΩΛ ≈ −0.5 in agreement with the experimental data, see [65]. Using the Hubble law to obtain the recessional acceleration and introducing the deceleration parameter we obtain ( ) dvr dH 2 (11) ar (t) = = + H D = −qH 2 D = −qHvr , dt dt where vr is the speed of recession and not the radial velocity of the proper motion of the particle. We note that the inclusion of such peculiar velocity may explain two astrometric anomalies, as pointed out in [30]. Hubble law is applied for close distances. We assume the same behavior at first order for largest observable distance. Then, evaluating equation (11) for the objects receding from us at a rate faster than the speed of light we obtain that their recessional acceleration is a0 = −q0 H0 c. If we compare with the results given by the Modified Newtonian dynamics (MOND) theory (using different arguments (see [17, 18]) and with the recently results using entropic forces [11], we

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have a0 = H0 c) that q0 = −1. Moreover, the recessional acceleration at large distances is the constant acceleration of the MOND theory. 4. Final comments In the section 2 we have seen that from the cosmic acceleration we can fix the form of µ(x) in the first phenomenological version of the Modified Newtonian dynamics. In the previous section we see the converse, the Modified Newtonian dynamics implies, in some sense, the cosmic acceleration of the universe. The question still open is: why this cosmic acceleration? Is the dark energy the answer or is a consequence of the real cosmological model? Are the entropic forces the correct answer? Notice that the value q0 = −1 is not the same that gives the observations, but this value is the extremal value that occurs for distances larger than the radius of the Hubble sphere rHS . This value of q0 = −1 is also the value obtained for the universe model dominated by vacuum energy density. References [1] J.D. Anderson, P.A. Liang, E.L. Lau, A.S. Liu, M.M. Nieto & S.G. Turyshev, Indication, from Pioneer 10/11, Galileo and Ulysses Data, of an apparent weak anomalous, long-range acceleration, Phys. Rev. Lett. 81 (1998), 2858–2861. [2] J.D. Anderson, P.A. Laing, E.L. Lau, A.S. Liu, M.N. Nieto & S.G. Turyshev, Study of the anomalous acceleration of Pioneer 10 and 11, Physical Review D 65 (2002), no. 8, 082004. [3] J.D. Bekenstein, Relativistic gravitation theory for thr modified Newtonian dynamics paradigm, Phys. Rev. D 70 (2004), no. 8, 083509. [4] J.D. Bekenstein, Erratum: Relativistic gravitation theory for the modified Newtonian dynamics paradigm [Phys. Rev. D 70, 083509 (2004)], Phys Rev D 71 (2005), no. 6, 069901. [5] J.D. Bekenstein, An alternative to the dark matter paradigm: relativistic MOND gravitation, In: JHEP conference proceedings of science; 2005. [6] J. Bekenstein & M. Milgrom, Does the missing mass problem signal the breakdown of Newtonian gravity?, Astrophysical Journal 286 (1984), 7–14. [7] J. Bekenstein & J. Magueijo, Modified Newtonian Dynamics Habitats Within the Solar System, Physical Review D 73 2006, no. 10, 103513. ´ramos, Thermal analy[8] O. Bertolami, F. Francisco, P.J.S. Gil & J. Pa sis of the Pioneer anomaly: A method to estimate radiative momentum transfer Physical Review D 78 (2008), no. 10, 103001. [9] L. Blanchet & J. Novak, External field effect of modified Newtonian dynamics in the Solar system, Mon. Not. Roy. Astron. Soc. 412 (2011), no. 4, 2530-2542. [10] T.M. Davis & C.H, Linewater, Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the universe Publications of the Astronomical Society of Australia 21 (2004), 97–109.


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