Recovering modified Newtonian dynamics by changing inertia
ABSTRACT Milgrom's modified Newtonian dynamics (MOND) has done a great job on
accounting for 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) Gravitational mass is
conserved; 2) Inverse-square law is applicable at large distance; 3) Inertial
mass depends on external gravitational fields. These assumptions not only
recover the deep-MOND behaviour, the accelerating expansion of the universe is
also a result of these assumptions. Then Lagrangian formulae are developed and
it is found that the assumed universal acceleration constant a0 is 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.
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.
3 LAGRANGIAN FORMALISM
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.
3.1 Comments 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,