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.