arXiv:1011.3618v2 [astro-ph.CO] 24 Jan 2011
Mon. Not. R. Astron. Soc. 000, 1–10 (2011)Printed 25 January 2011(MN LATEX style file v2.2)
Recovering modified Newtonian dynamics by changing inertia
120/F, Building 128, Nanhuxiyuan, Chaoyang District, Beijing, China
Milgrom’smodifiedNewtoniandynamics(MOND)has donea great jobon accountingfor the
rotation curves of a variety of galaxies by assuming that Newtonian dynamics breaks down
for extremely low acceleration typically found in the galactic contexts. This breakdown of
Newtonian dynamics may be a result of modified gravity or a manifest of modified inertia.
The MOND phenomena are derived here based on three general assumptions: 1) Gravita-
tional mass is conserved;2) Inverse-squarelaw is applicable at large distance; 3) Inertial mass
depends on external gravitational fields.
These assumptions not only recover the deep-MOND behaviour, the accelerating expan-
sion of the universe is also a result of these assumptions. Then Lagrangian formulae are de-
veloped and it is found that the assumed universal acceleration constant a0is actually slowly
varying by a factor no more than 4. This varying ‘constant’ is just enough to account for the
mass-discrepancy presented in bright clusters. It is also found that an inevitable result of the
above three assumptions is that the speed of light is varying in gravitational field, which is
partly discussed within the solar system in other two papers.
Key words: gravitation – dark matter – cosmology: theory.
The modified Newtonian dynamics (MOND), originally proposed by Milgrom (1983) as an alternative to the cold dark matter paradigm to
account for the rotation curves of spiral galaxies, has extended its success to dwarfs, low surface brightness galaxies (LSB) and ellipticals
(see Sanders & McGaugh 2002, for a review). When confronting with clusters, especially rich clusters, MOND shows some drawback.
On the cluster scale, MOND still needs dark matter, which is what MOND was particularly devised to eliminate. To overcome this dif-
ficulty, neutrinos were speculated to be responsible (Sanders 2003; Angus et al. 2007; Gentile, Zhao & Famaey 2008). Neutrinos with
mass ∼ 2eV, marginally allowed by current most accurate neutrino mass measurement, contributing negligibly to galaxies’ mass budget,
could be dynamically significant in clusters of galaxies. Though this hypothesis is successful in some aspects, it is still controversial (e.g.
Pointecouteau & Silk 2005; Angus, Famaey & Buote 2008).
Despite this drawback, MOND has drawn much attention because of its impressive success compared with the standard cold dark
matter paradigm, which is facing with some difficulties, especially on the galactic scale. MOND is a phenomenological theory that may
be interpreted in different ways. First of all, it may indicate a breakdown of Newtonian gravity (Bekenstein & Milgrom 1984) where the
standard Poisson equation is replaced by ∇ · [µ(|∇ϕ|/a0)∇ϕ] = 4πGρ, and a0 (∼ 1.2 × 10−8cms−2), introduced by MOND, is
a new acceleration constant below which dynamics and/or gravity become significantly non-Newtonian. Milgrom (2002) reviewed this
interpretation which relates the gravity to a potential flow. Although this is a field need more investigation, the lack of profound physical
foundation makes this interpretation less attractive. A second interpretation is that gravitational constant increases when accelerations are
lower than a0(Bekenstein 2004). This relativistic extension of MOND can mimic MOND’s behaviour at low acceleration extreme, but it is
still a subject of debate (e.g. Reyes et al. 2010).
A third interpretation is what Milgrom (1983) proposed that the Newtonian dynamics may break down at low accelerations. Instead of
the usual F = ma, Milgrom (1983) suggested a modified dynamics
F = mgµ(a/a0)a,
µ(x ≫ 1) ≈ 1,µ(x ≪ 1) ≈ x,
where mgis the gravitational mass of the body moving in the field. This relation is equivalent to
a ≈ (aNa0)1/2
in the deep-MOND regime, where aNis the acceleration derived from Newtonian dynamics. It is just this simple relation that works remark-
ably well in reproducing the dynamics of a variety of galaxies with quite different morphologies and luminosities.
The modified dynamics can be interpreted as a modification of inertia, which, if applied properly, may solve a lot of puzzles faced with
modern physics. Modification of inertia is not a new idea since the time of Mach who challenged Newton’s idea about inertia. Since the
theorization of Unruh radiation (Unruh 1976) and the discovery of Casimir effect (Casimir 1948), their relation to inertia and gravity has
frequently been speculated by several authors (e.g. Puthoff 1989; Haisch, Rueda & Puthoff 1994). Though Casimir effect is a reality both
theoretically and experimentally, what does it mean for and how to apply it to cosmology remains a subject of debate. Puthoff’s gravity,
based on Casimir effect, has the flaw of missing experimental support. As for Unruh radiation, it is not clear what this radiation means for
cosmology and whether it is related to inertia.
In this paper, I propose a new approach, based on some general speculations, to modification of inertia. By this modification of inertia,
(2) is successfully reproduced. Lagrangian formulae are developed and its profound indications are discussed then.
2 ASSUMPTIONS AND RESULTS
In this section, three assumptions are proposed: 1) Gravitational mass is conserved; 2) Inverse-square law is applicable at large distance; 3)
Inertial mass depends on external fields.
If we make a comparison between gravity and electromagnetic interaction, we immediately realize that gravitational mass is analogous
to electric charge. It is the electric charge, as a source, who produces electric field. Electric charge cannot be created and destroyed, it is just
a being. Yang-Mills gauge theory, the foundation for standard model in particle physics, is based on the conservation of charge. As a source
of gravitational field, it is unphysical that gravitational mass is not conserved.
In electromagnetic field, the inverse-square law is directly related to the zero-mass-ness of photons. Coulomb’s law has been tested from
∼ 2 × 1010m down to 10−18m, a magnitude span of 28 orders (Adelberger, Heckel & Nelson 2003). The astronomical tests of Newton’s
gravity has been mainly confined within solar system. The most accurate astronomical tests are lunar-laser-ranging studies of the lunar orbit,
which don’t show a deviation of gravity from Newton’s law. On galactic scale, a Yukawa-like gravity has been proposed by many authors
(see, e.g. Sanders & McGaugh 2002, and references therein) to account for the mass discrepancy in galaxies. However, Milgrom (1983)
pointed out that this is inconsistent with the empirical Tully-Fisher law (Tully & Fisher 1977). This line of arguments indicate that gravity is
not Yukawa-like from submillimeter scale to at least galactic scale. Large scale structure of the Universe favors a Newton’s gravity law even
on cosmic scale. Therefore it is quite safe to assume that inverse-square law is accurate on the scale of our interest in this paper.
The equivalence of gravitational mass to inertial mass, upon which general relativity is based, has been tested experimentally with
very high accuracy (Will 2009). But these tests are confined within solar system, no direct test is available on the galactic scale. Unlike
the gravitational mass, inertial mass doesn’t associate with any physical field. Physically, because of the association with gravitational field,
gravitational mass should be conserved, but this is not necessarily the case for inertial mass. Inertia is one object’s ability to keep its original
state of motion. However, how does one object know its original motion state if no reference is available?
Mach speculated that inertia is the result of the object’s motion relative to the mean mass distribution of the Universe as a whole. In
other words, inertia is meaningless if no mass is there other than the object itself. Let’s consider another situation: If the mass is distributed
uniformly throughout the Universe but one distinct object. Whatever the state of motion of this distinct object, the state of the Universe will
keep the same. This is indicative of the dependence of inertia on the mass distribution of the Universe.
It has been long that Milgrom found that external field plays a role on the internal dynamics of open clusters where the internal field
is well below the critical acceleration a0 and therefore should be in deep-MOND regime. However, the dynamics of these systems does
not show any evidence of dark matter. Milgrom realized that external field, if significantly above the transitional acceleration between
Newtonian dynamics and MOND, could play a role and make the dynamics of these open clusters Newtonian. In addition, according to
Mach, gravitational field, being the result of the mass distribution of the Universe, therefore endows the object with inertia. This indicates
that if no external gravitational field exists, the object could be in random states of motion, i.e. the inertial mass is zero. With the increase
of external field, inertial mass increases accordingly. The climbing-up of inertia, however, does not continue infinitely. Having the external
field, i.e. the gradient of the potential, as a reference, the object has a sense of its past motion state. The stronger the external field, the more
sense it has about its original motion state. But if the external field is strong enough, increasing the field’s strength will not increase its sense
of past motion state because it just has enough “information” about its original state of motion. As a result, we set the inertial mass mIin the
strong field limit to be its gravitational mass mg, as we know from the dynamics of solar system.
By these three assumptions, we will derive the MOND relation (2) in next section.
MOND as a result of changing inertia
Now we consider one particle’s motion under the gravitational interaction of a massive object at large separation. The particle moves inward
by converting potential energy to kinetic energy: F = dp/dt = (dmI/dt)v + mI(dv/dt), where p = mIv and mI, of course, is its
inertial mass, as usual. Because F = −GMmgr/r3, where M is the gravitational mass of the massive object, we get
dtv = −GMmg
For radial motion, we have
here the primes denote derivatives relative to the distance r. Substituting u for 1/r, the above equation is reduced to
To move further on, we have to figure out, under the guide of the third assumption proposed in above section, how mIis shaped by external
field. In general, based on the assumption that mI ∝ mg, an inertial mass function of the form
mI = ν
is expected, where ν (x) is a function of gN/aIonly. As is well known, mIis a constant in the strong field case, and by the third assumption,
mI → 0 in the weak-field extreme. We are interested in the weak-field extreme case. A plausible assumption for the weak field case is
mI ∝ (gN)αmg, or
mI(gN ≪ aI) =
where gN = GM/r2= GMu2, the Newtonian gravitational field. Here we introduce a new constant, aI, which is related to Milgrom’s
constant a0, as can be seen in the next section. In order to smoothly bridge the strong field case and weak field case, we expect 0 < α < 1,
which means mI increases rapidly when gN ≪ aIbut ceases to climb up when gN ≃ aI. The index α has to be fixed phenomenologically.
As can be easily checked, if we set α = 1/2, the desired MOND behaviour is recovered, i.e.
mI(gN ≪ aI) =
aI = v4
and v0the particle’s asymptotic velocity. Because ν (x) ≈ 1 when x ≫ 1, a simple but quite plausible assumption is
ν (x) =
1 + x
for all x. It should be stressed that, inthis model, mI/mgisinfluenced only by external fields, not by any other factors, including theparticle’s
dynamical quantities, e.g. its velocity. That is to say, mIis a true scalar. Because of this behaviour of inertial mass, the particle’s Lagrangian
can be expressed as equation (13).
If the particle is on circular orbits, therefore a constant inertial mass, the equation of motion, by (3), is reduced to
mggN = mIa,
a = (gNaI)1/2,
i.e. the recovery of equation (2) if aI = a0 is recognized. This equation, however, cannot be applied to any other orbits other than circular
ones due to the variation of the particle’s inertia. In general, equation (3) should be used. This indicates that the original MOND prescription
is most suitable to describe the dynamics of spiral galaxies where internal motion is almost perfectly circular.
Now let us consider how to escape the gravity of a massive object. Substituting equation (8) into equation (5) we find, for a radial
du+ v2= v2
This equation tells us that if the particle is moving away from the massive object and has a velocity v = v0, then dv/du = 0 and the particle
keeps moving with a constant velocity. Therefore the particle is able to escape the gravitational pull of a massive object by itself.
If v < v0, then dv/du > 0 and the particle decelerates until reach a maximal distance. Of particular interest is the case of v > v0,
where dv/du < 0 and the particlekeeps accelerating and the acceleration is increasing. It is straightforward to check that its velocity satisfies
the following equation
Obviously, if no other field’s disturbance, the velocity will increase infinitely so that v ≫ v0and
v = Ar.
This is the direct result of the decreasing inertia while the particle moves away from the gravitational field. This equation is formally same as
Hubble’s law. But please bear in mind that Hubble’s law describes velocity field, equation (12), on the other hand, describes one particular
particle’s velocity change with distance. I will defer the discussion of this equation to Section 3.3.
It is clear that, since the particle’s mass is dependent of the strength of gravity, gravity’s apparent effect is not always attractive, but
takes on different aspects dependent of the particle’s motion state. If the particle is bound to the massive object, the gravity’s apparent effect
is attraction. On the other hand, if the particle is in an unbound state, the gravity’s effect is repulsion, as indicated by (12), the particle’s
velocity is not decreasing but increasing as it moves away from the gravitational field.
In above section we just assume that inertial mass depends on the external field and then extend Newtonian dynamics in a minimum manner.
To formulate a self-consistent theory we need to develop a set of Lagrangian formulae. As usual, we write down the Lagrangian:
where φ is the scalar gravitational field that is determined by Poisson equation ∇2φ = 4πGρ with ρ the gravitational mass density. The
Euler-Lagrange equations of motion
∂ ˙ xi
dt= −mg∇φ +1
Compared with equation (3) we find that equation (14) contains an additional term
depends on external field, so is a function of position. It turns out that this term plays a key rule in accounting for the dynamics of dwarfs,
spiral galaxies and clusters in a consistent and self-contained way. Applying p = mIv to the left hand side of equation (14) gives:
2v2∇mI, stemming from the fact that inertial mass
mIa = mggN+1
2v2∇mI− (∇mI· v)v.
For the deep-MOND case, substituting equation (8) into above equation yields
This equation is the general motion equation for the deep-MOND regime, including the non-spherically symmetric systems.
As above, let us first consider the circular motion. In this case the above equation, after applying equation (8), reduces to:
a = (aIgN)1/2
From this equation we can easily find the stable circular velocity:
MOND as a result of changing inertia
vc = (4GMaI)1/4.
However, only the sufficiently virialized systems can attain to this high circular velocity. For those systems less virialized, the objects on
the outskirts of the systems will follow quasi-circular orbits and gradually spin up. This indicates that Milgrom’s constant is actually slowly
varying according to:
a0 = aI
that is to say, a0is varying in a narrow range aI ? a0 ? 4aIand older systems tend to have higher values of a0.
To find the value of aI, an easy but reliable way is to carefully select a large enough sample of well studied galaxies so that the scatter of
a0is moderate and MOND works quite well for this sample. I select the galaxies from Begeman, Broeils & Sanders (1991), Sanders (1996)
and Sanders & Verheijen (1998). The reasons for selecting these galaxies are three-fold. First, these galaxies are among the highest quality
of observational data and it was demonstrated that MOND can account for the data with a relatively high precision. Secondly, these galaxies
are all spiral galaxies for which MOND is most successful. In addition, as indicated above, a0evolves with galaxies. To reduce the scatter of
a0, we should select galaxies with comparable properties. Finally, these three samples contain several galaxies, e.g. NGC 2841, that are quite
controversial within the context of MOND. It is quite desirable if a varying a0 can settle down these issues. The resulting sample contains
63 spiral galaxies, but 9 galaxies from Sanders & Verheijen (1998) are eliminated from the list adopted to calculate a0because of the reason
In Table 1 the calculated values of a0 are listed for every galaxy in the sample. aI is found by requiring the average value of a0 =
1.21 × 10−8cms−2:
aI = 0.667 × 10−8cms−2.
Several points should be mentioned. The accelerations at the last measured points of the rotation curves (almp) of NGC 3949, NGC 3953,
NGC 4085, UGC 6973 are in the Newtonian regime and therefore eliminated from the calculation of aI. In light of the found value of aI, we
expect that the galaxies with almpcomparable to or larger than aIwill be quite dynamically different from other galaxies, these galaxies are
NGC 3877, NGC 3972, NGC 4051, NGC 4217, NGC 4389, with almp> 0.7 × 10−8cms−2. As a result, these five galaxies are eliminated
from the calculation of aI. But it should be mentioned that, if these five galaxies are included in the galaxies responsible for calculating aI,
aIwould be slightly smaller. By this selection criterion, the resulting scatter of a0, ±0.30, around the mean value is moderate, as desired.
3.1Comments on Selected Individual Galaxies
A quick glance of the result for a0 suggests that large and high surface brightness galaxies tend to have a high a0. Several comments are
presented below for those values significantly deviated from the standard value:
NGC 2841. – This is the most controversial spiral galaxy for MOND. Begeman, Broeils & Sanders (1991) found that, if a Hubble
distance (9.5Mpc) was used, MOND obviously fails to account for the rotation curve. On the other hand, if the Tully-Fisher distance
was used, MOND is in well agreement with the observed data. However, the Tully-Fisher distance is twice as the Hubble distance, a seemly
unacceptable result.Subsequently, Bottema et al. (2002) found adistanceof 14.1Mpc toNGC 2841 based onCepheid method. Thisdistance
goes half way between the Hubble distance and Tully-Fisher distance. Asa result, the situation isalleviated somewhat, but the predicted curve
still deviates systematically from the observed data. Table 1 gives this galaxy’s a0a value of 1.48×10−8cms−2, the largest in this sample if
not include the values not adopted for calculating aI, which is more than 22% larger than the standard value. This value will further alleviate
the situation and make MOND in principle compatible with the data. But it should be pointed out that even with this high value, the data still
can not be comfortably matched by the prediction. Here is a caveat. We find that, for this galaxy, almp= 0.66×10−8cms−2∼ aI. Because
aIis the transition acceleration beyond which the modified inertia transits to Newtonian inertia, we expect the value a0calculated in this way
is only qualitatively correct. If otherwise almp ≪ aI, the value listed in Table 1 should be exact. But NGC 2841 evolved beyond this stage
because it has a much higher almp. Quantitatively, a value of 1.87 × 10−8cms−2for a0will do the work if the Cepheid distance is used. If
we adopt equation (10) as the inertial mass dependence on external gravitational field, we find a value of 1.86 × 10−8cms−2(∼ 2.77aI)
for a0, which is just what is needed for this galaxy to bring MOND prediction in accordance with the observation. As in the general case,
when almp∼ aI, the actual a0depends on the ratio of rotation velocity to its stable circular velocity. When rotation velocity approaches its
stable circular velocity, a0will have a value of 2.45 × 10−8cms−2(∼ 3.66aI). This implies 2.77aI ? a0 ? 3.66aI when almp∼ aI. We
find that the maximum value (3.66aI) for a0when almpis in the vicinity of aIis less than the corresponding value (4aI) when almp≪ aI,
indicating that the Newtonian dynamics begin to take over when acceleration enters the Newtonian regime. The oscillating rotation curve,
rather than a perfect flat rotation curve, of this galaxy on the outskirts may have some implication of this subtlety.
DDO 154. – A value of 0.9 × 10−8cms−2, significantly less than the average value, is given for this dwarf and gas-rich galaxy.
Nonetheless, this low value can be largely attributed to the low rotation velocity at the last measured points. This is a quite controversial case
for MOND. Milgrom & Braun (1988) demonstrated that this galaxy is among the most acute test of MOND. Shortly, Lake (1989), however,
showed that a substantially small value of a0 is necessary to account for the observed curve. Milgrom (1991) disputed Lake’s conclusion,
arguing that a different distance to this galaxy be more reliable. But if the ideas presented in this paper are correct, a small value of a0 is
inevitable because of the declining nature of the rotation curve at the outermost points. The internal acceleration of DDO 154 is everywhere
less than a0 so that MOND should play a dominant role. But MOND cannot account for the declining nature of the rotation curve. In the
spirit of this paper, however, the declining nature of gas-rich dwarf galaxies can be naturally accounted for. The main point is that gas-rich
galaxy is newly formed object with a less relaxed outer part and a fairly relaxed inner part. Consequently, the inner part should have a higher
value of a0 than its outer part. This indicates that a0 is varying not only from galaxy to galaxy, but also within galaxies. For sufficiently
relaxed galaxies, a0could be treated as a constant without loss of accuracy, but for gas-rich galaxies, the variation of a0is maximized.
NGC 3893. – Table 1 gives a0 a value of 1.47 × 10−8cms−2. This high value is largely the result of the high almp. But because this
galaxy has a disturbed velocity field, the result should not be taken seriously.
NGC 3992. – Table 1 gives a0 a value of 1.48 × 10−8cms−2or 1.86 × 10−8cms−2if equation (10) is used, an apparently not
necessary adjustment for the standard value if the good agreement between predicted and observed rotation curve is noticed. However, as
mentioned by Sanders & Verheijen (1998), the fitted value of mass-to-light ratio of the stellar disk is unusually large compared with other
values in that sample. If, however, an a0with above value is used, the near-infrared mass-to-light ratio will be reduced by a factor of 1.22 or
1.54, a value compatible to other galaxies in the sample.
NGC 4010. – Table 1 gives a0 a value of 1.44 × 10−8cms−2and even higher if equation (10) is used. This high value of a0 seems
unexpected for such a low surface brightness galaxy. However, this galaxy has a very extended disk and the scale length is much larger
compared with other galaxies in its mother cluster (Tully & Verheijen 1997). As a result, this galaxy has a much larger internal acceleration,
so is a0. This galaxy is an example that a0 does not actually directly depend on surface brightness. Since young galaxies with outstanding
star formation activities tend to have a high surface brightness even if its internal acceleration is low. On the other hand, some old galaxies
devoid of star formation tend to have a low surface brightness because of their old star population. In addition, we find that the rotation curve
of this galaxy begins to decline at large radii, another example that a0is varying within one galaxy.
Recently Swaters, Sanders & McGaugh (2010) study a sample of 27 dwarf and low surface brightness galaxies within the framework of
MOND and find that there are some systematics that cannot be reconciled easily: MOND curve predicts higher rotation velocities in the outer
regions and lower in the central regions for low surface brightness dwarfs; higher rotation velocities in the central regions and lower in the
outer regions for high surface brightness galaxies (UGC 5721, UGC 7323, UGC 7399, UGC 7603, UGC 8490). This intriguing phenomenon
forces Swaters et al. to try a fit with a0 as a free parameter. By allowing a0 vary, Swaters et al. find a trend that lower surface brightness
galaxiestend to havelower a0. Thisisjust what equation (19) expected if lower surface brightness galaxies tend tobe lessrelaxed. If equation
(19) is applied to this sample of systems, an average value of 1.06 × 10−8cms−2is found, which is significantly lower than the commonly
used value. Swaters, Sanders & McGaugh (2010), however, find an even lower average value of 0.7 × 10−8cms−2for a0 in this sample,
a value that is just above aI. Though the calculated value of a0 is significantly lower than the commonly used value, it is still significantly
higher than the fitted average value in this sample. Furthermore, the fitted values for the galaxies in this sample have an enormously large
scatter that is completely out of the reach for (19) to account for. This enormous scatter, on the one hand, is a consequence of the internal
dynamics of the systems; on the other hand, is a consequence of observational uncertainties. If such a large scatter of a0is necessary for this
sample, then it will be a challenge for the ideas presented in this paper.
3.2 Dynamical Mass of Clusters of Galaxies
It has been quite controversial with the applicability of MOND to clusters. The & White (1988) first noted that MOND is inconsistent with
the dynamics of Coma cluster unless the MOND acceleration parameter a0is4 times larger than the value inferred fromspiral galaxy rotation
curves. Sanders (1994), based on a small sample of X-ray-emitting clusters, concluded that MOND mass is consistent with the detected gas
mass. Subsequently, based on a much larger sample of X-ray-emitting clusters, Sanders found that MOND over-predicted the mass by a
factor of 2 than the detected gas mass (Sanders 1999). In light of this finding, the mass over-prediction was already implied by Sanders
(1994) though with a less statistical significance.
This mass over-prediction flaws MOND since MOND was particularly speculated to eliminate non-baryonic dark matter. To rem-
edy MOND, Sanders (2003) suggested that neutrinos, with mass of 2eV, is responsible for the mass discrepancy in rich clusters. This
mass of neutrinos, having a negligible effect on the galaxy scale, could have a significant dynamical effect on cluster scale. Unfortunately,
Angus, Famaey & Buote (2008) showed that neutrinos are incompatible with the observed mass distribution within X-ray bright groups and
clusters. Consequently, Angus et al. proposed that sterile neutrinos would close the cluster mass problem. This conclusion contradicted the
finding by Milgrom (1998) who found that MOND is compatible with galaxy groups. Unfortunately, by studying the gravitational lensing
of clusters, Natarajan & Zhao (2008) conclude that even sterile neutrinos are far from enough to close the mass budget of clusters.
However, if the inertia is allowed to vary, as presented above, no dark matter is needed. Of the 93 clusters studied by Sanders (1999), the
internal accelerations are of ∼ 0.6×10−8cms−2, that is, in the vicinity of aI. Section 3.1 shows that for systems with internal accelerations
comparable to aI, a much higher effective a0should be used. It can be checked that for these 93 clusters, a0approaches the maximum value
?2.02 × 1.21 × 10−8cms−2?. This high value is just enough to eliminate the mass discrepancy in bright clusters. In this respect, we
say that the dynamics of bright clusters are quite similar to that of NGC 2841. This also implies that not all clusters would have the mass
MOND as a result of changing inertia
discrepancy problem within the framework of MOND. Those clusters with much lower internal accelerations will have similar dynamics as
But the mass discrepancy problem is not completely closed because in some cases MOND over-predicts masses by a factor of 3 or even
higher. If this phenomenon is ubiquitous and no conventional interpretations, e.g. dark baryons or neutrinos, are reliable, then there really
exists a dark matter problem for MOND.
After a discussion of circular motion, let us now turn to radial motion. In this case equation (15) reduces to:
a = (aIgN)1/2
We find the conclusion is not changed much for radial motion by an application of Lagrangian formula. The only difference is a change of
the asymptotic velocity, defined by equation (18) other than the value v0determined by equation (9).
Equation (12) reminds us of the Hubble’s law. But does it really imply Hubble’s law. By its low magnitude (H0 = 72kms−1Mpc−1)
compared to the typical galaxies’ asymptotic rotation velocities (≃ 200kms−1) and their typical sizes (? 100kpc), if it is really Hubble’s
law, we expect that Hubble’s law is not the result of mass inhomogeneity on the galactic scale. On contrary, Hubble flow is a result of mass
inhomogeneity on cosmic scale. Obviously, if the mass distribution is completely homogeneous, galaxies would not feel the existence of
an external gravitational field, resulting in zero inertial masses and therefore random motions. But actually every galaxy is embedded in an
ubiquitous gravitational field. A fluctuation of mass density on cosmic scale gives riseto the small gravity on largescale, those cosmic objects
unbound by thisinhomogeneity of mass distributionacceleratingly expand. If δM isthemass inhomogeneity caused by adensity perturbation
∆ = δρ/ρ on the cosmic scale, we expect the Hubble recession velocity exceeds δv0 = (4GδMaI)1/4so that an accelerating expansion
is guaranteed. This indicates that the accelerating expansion of the Universe can be most readily observed on scales that the clustering of
galaxies becomes negligible. The two-point correlation function of galaxies shows that these scales are r ≃ 5h−1Mpc. So the accelerating
expansion of the Universe should be free of peculiar contamination beyond these scales. The accelerating expansion of the Universe seems to
imply an increase of the universe energy, or in modern parlance, the existence of dark energy. But above analysis shows that the accelerating
expansion of Universe does not violate conservation of energy because equation (12) is derived based on the conservation of energy. All
difference is the dependence of inertia on gravity. Although galaxies’ velocities are becoming larger and larger, their inertial masses are
dropping dramatically, resulting in the conservation of total energy.
Although we recover MOND’s relation (2) in the case of circular orbits, the ideas presented above are quite different from that of
MOND. In MOND, gravity’s effect is always attractive, the moving particles in the gravitational field are all feeling the same acceleration as
long as they locate at the same place in the field. In fact MOND, including the relativistic MOND theory proposed by Bekenstein, complies
with weak equivalence principle. This is however not the case for the ideas presented here. Just as we already find, gravity is attractive
in the case of bound state, and repulsive in an unbound configuration. Even for particles in the same type of state, say bound state, their
accelerations could be quite different because of their different radial velocities, which determine at what rate the moving particles’ inertial
masses change. As a consequence, all particles moving from large distance towards the massive object tend to converge to the same velocity,
i.e. the asymptotic velocity v0 = (4GMaI)1/4. If the particles’ velocities are less than v0, the attraction effect of the gravity increases their
velocities. On the other hand, if the particles’ velocities are larger than v0, the repulsion effect of the gravity decrease their velocities. Only
in the Newtonian regime is the gravity always attractive.
In addition, since the gravitational potential is not infinite, as is in MOND, the moving body can escape the gravitational pull easily
without resorting to other nearby massive galaxies’ assist as described by Famaey, Bruneton & Zhao (2007). This is yet another advan-
tage of the changing inertia proposal over the traditional MOND. In many cases, by finely tuning the parameters, the predictions of CDM
model are barely discernible from that of MOND. But globular clusters are outliers, which are believed contain no dark matter. Conse-
quently Baumgardt, Grebel & Kroupa (2005) suggested a test of gravitational theories by studying velocity dispersion of globular clusters.
Recently these studies are carried out by several groups, finding an inconsistency between MOND and observations (Moffat & Toth 2008;
Haghi H. et al. 2009; Sollima & Nipoti 2010). The problem is that MOND systematically overpredicts the velocity dispersions. This can
be understood as follows: Because of the infinite escape velocity intrinsic in MOND, at any given radius the velocity distribution should be
large. This is no longer the case for the changing inertia proposal since at any radius the escape velocity is bound.
MOND has been well established as a serious alternative to the standard dark matter model for about three decades owing to its remarkable
success in accounting for the dynamics of a variety of galaxies with quite different luminosities, morphologies. However, when confronting
with dwarfs and clusters, MOND is controversial. There exists cumulated evidence that the acceleration parameter a0, assumed to be univer-
sal, is varying in the sense that low surface brightness galaxies tend to have low a0. For bright clusters, a factor of 2 mass over-prediction is
well established. Again, this over-prediction can be accommodated by an a02 times larger than the usually adopted value.
With above observational evidence, a motivation to theoretically account for the phenomena is prompt. This paper postulates that the
MOND phenomenology can be accounted for by three assumptions: 1) Gravitational mass is conserved; 2) Inverse-square law is applicable
at large distance; 3) Inertial mass depends on external fields. The first two assumptions are quite general and should not find objection. The
third assumption, given by equation (8) appropriate at the low acceleration regime, is key to reproduce the phenomenology of MOND. It is
found that the inertia modified in this way exactly recover the formulae suitable to MOND in the circular motion case.
Byaconsideration ofLagrangian formulae, however, itisfound thattheusual relation(2) isreplaced by(17).A comparison of equations
(2) and (17) shows that a0, a proposed universal constant, is actually varying in a narrow range, aI ? a0 ? 4aI, if the internal accelerations
of the systems in question (or the external field) are much lower than aI. A scrutiny of equations (15) and (10) indicates that the effective a0
not only depends on the orbital velocity but also on the external field. This varying a0 is just enough to eliminate the over-prediction factor
of 2 found in bright clusters and the lower value in low surface brightness galaxies. Because of this nature of a0, a0 not only varies from
system to system, but also varies within one galaxy. This phenomenon can readily account for the decline of rotation curves found for many
low surface brightness galaxies.
Above statement suits only for circular motion. For radial motion, situation is quite different. It turns out that there exists a critical
velocity, vc defined by equation (18), for every galaxy. If the object in radial motion is moving slower than this critical velocity, gravity
appears to be attractive, as usual. But for a motion faster than this critical velocity, gravity appears to be repulsive in the sense that the
object begins to accelerate. This immediately reminds us of the accelerating expansion of the universe. By a careful inspection of the cosmic
parameters, if the modified inertia is correct, it is realized that the accelerating expansion of the universe must be the result of inhomogeneity
of the universe on cosmic scales.
In general, besides circular motion and radial motion, equations (15) and (10) should be used to account for the dynamics of systems
in question. These two equations indicate that the traditional MOND prescription is only appropriate to describe circular motions. It is found
that a0 is not a fundamental constant, its value is an indication of the system’s age and relaxation index. In general, high surface brightness
objects tend to have high value of a0. But this is not appropriate for all systems. a0 is actually determined by the orbital velocities and
external gravitational fields. A more accurate value of aIshould be determined by these two equations. But owing to the success of MOND
in spiral systems, the true value of aIshould not deviate far from the value given by equation (20).
A basic assumption of this paper is that inertial mass decreases when external gravitational field becomes weak. In particular when the
external field vanishes, the inertia of the object should vanish too. A zero inertia will immediately result in a random motion of the object,
i.e. its velocity could be indefinitely large, including the speed of light. This violates our belief that the speed of light c is a constant in
gravitational field. This issue is partly addressed in the context of the solar system in other two papers (Wang 2011a,b, i.e. Paper II and Paper
III). It is found that a changing speed of light is consistent with the observed anomalies in the solar system.
I am grateful to Robert H. Sanders for his encouragement in this field. I also wish to thank Moti Milgrom for his help during the year the
ideas discussed here were conceived. I thank HongSheng Zhao for his critical reading of the manuscript and helpful comments.
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Table1.ThesampleselectedforthecalculationofaI.Calculatedvaluesofa0arelistedoncolumn 2.Reference: (1)Begeman, Broeils & Sanders
(1991); (2) Sanders (1996); (3) Sanders & Verheijen (1998). Nine galaxies from Sanders & Verheijen (1998), NGC 3877, NGC 3949, NGC
3953, NGC 3972, NGC 4051, NGC 4085, NGC 4217, NGC 4389, UGC 6973, are eliminated from the calculation of aIbecause of the rea-
son presented in text. The data for UGC 6446 are taken from Swaters, Sanders & McGaugh (2010) to utilize the most recent update, but the
differences are small.