Modified Newtonian dynamics

"MOND" redirects here. For other uses, see Mond.

In physics, Modified Newtonian dynamics (MOND) is a hypothesis that proposes a modification of Newton's law of gravity to explain the galaxy rotation problem. When the uniform velocity of rotation of galaxies was first observed, it was unexpected because Newtonian theory of gravity predicts that objects that are farther out will have lower velocities. For example, planets in the Solar System orbit with velocities that decrease as their distance from the Sun increases.

MOND was proposed by Mordehai Milgrom in 1983 as a way to model this observed uniform velocity data. Milgrom noted that Newton's law for gravitational force has been verified only where gravitational acceleration is large, and suggested that for extremely low accelerations the theory may not hold. MOND theory posits that acceleration is not linearly proportional to force at low values.

MOND stands in contrast to the more widely accepted theory of dark matter. Dark matter theory suggests that each galaxy contains a halo of an as yet unidentified type of matter that provides an overall mass distribution different from the observed distribution of normal matter. This dark matter modifies gravity so as to cause the uniform rotation velocity data.

Overview: Galaxy dynamics

Observations of the rotation rates of spiral galaxies began in 1978. By the early 1980s it was clear that galaxies did not exhibit the same pattern of decreasing orbital velocity with increasing distance from the center of mass observed in the Solar System. A spiral galaxy consists of a bulge of stars at the centre with a vast disc of stars orbiting around the central group. If the orbits of the stars were governed solely by gravitational force and the observed distribution of normal matter, it was expected that stars at the outer edge of the disc would have a much lower orbital velocity than those near the middle. In the observed galaxies this pattern is not apparent. Stars near the outer edge orbit at the same speed as stars closer to the middle.

Figure 1 - Expected (A) and observed (B) star velocities as a function of distance from the galactic center.

The dotted curve A in Figure 1 at left shows the predicted orbital velocity as a function of distance from the galactic center assuming neither MOND nor dark matter. The solid curve B shows the observed distribution. Instead of decreasing asymptotically to zero as the effect of gravity wanes, this curve remains flat, showing the same velocity at increasing distances from the bulge. Astronomers call this phenomenon the "flattening of galaxies' rotation curves".

Scientists hypothesized that the flatness of the rotation of galaxies is caused by matter outside the galaxy's visible disc. Since all large galaxies show the same characteristic, large galaxies must, according to this line of reasoning, be embedded in a halo of invisible "dark" matter.

The MOND Theory

In 1983, Mordehai Milgrom, a physicist at the Weizmann Institute in Israel, published three papers in Astrophysical Journal to propose a modification of Newton's law of gravity. A pedagogical introduction to MOND can be found in Bekenstein, who characterizes MOND as follows: "Relativistic MOND as here described has developed from the ground up, rather than coming down from the sky: phenomenology, rather than pure theoretical ideas."

Actually, Milgrom provided several interpretations of his proposal, one being a modification of Newton's second law of motion. However, this proposed interpretation is inconsistent with conservation of momentum, requiring some unconventional physical assumptions to regain plausibility. A second interpretation, as a modification of the law of gravity, requires that the acceleration due to gravitational force does not depend simply upon the mass m, but upon the form m/μ(a/a₀), where μ is some function approaching the value one for large arguments and a/a₀ for small arguments, and a is the acceleration caused by gravity and a₀ is a natural constant, a₀ ≈ 10⁻¹⁰ m/s². The centripetal accelerations of stars and gas clouds at the outskirts of spiral galaxies tend to be below a₀.

The exact form of µ is unspecified, only its behavior when the argument a/a₀ is small or large. As Milgrom proved in his original paper, the form of µ does not change most of the consequences of the theory, such as the flattening of the rotation curve.

In the everyday world, a is much greater than a₀ for all physical effects, therefore µ(a/a₀)=a and F=ma as usual. Consequently, the change in Newton's law of gravity is negligible and Newton could not have seen it.

Predicted rotation curve

Far away from the center of a galaxy, the gravitational force a star undergoes is, with good approximation:

$F={\frac {GMm}{r^{2}}}$

with G the gravitation constant, M the mass of the galaxy, m the mass of the star and r the distance between the center and the star. Using the new law of dynamics gives:

$F={\frac {GMm}{r^{2}}}=m\mu {\left({\frac {a}{a_{0}}}\right)}a$

Eliminating m gives:

${\frac {GM}{r^{2}}}=\mu {\left({\frac {a}{a_{0}}}\right)}a$

Assuming that, at this large distance r, a is smaller than a₀, $\mu {\left({\frac {a}{a_{0}}}\right)}={\frac {a}{a_{0}}}$. This gives:

${\frac {GM}{r^{2}}}={\frac {a^{2}}{a_{0}}}$

Therefore:

$a={\frac {\sqrt {GMa_{0}}}{r}}$

Since the equation that relates the velocity to the acceleration for a circular orbit is $a={\frac {v^{2}}{r}}$, one has:

$a={\frac {v^{2}}{r}}={\frac {\sqrt {GMa_{0}}}{r}}$

and therefore:

$v={\sqrt[{4}]{GMa_{0}}}$

Consequently, the velocity of stars on a circular orbit far from the center is a constant, and does not depend on the distance r : the rotation curve is flat.

The proportion between the "flat" rotation velocity to the observed mass derived here is matching the observed relation between "flat" velocity to luminosity known as the Tully-Fisher relation.

At the same time, there is a clear relationship between the velocity and the constant a₀. The equation v=(GMa₀)^1/4 allows one to calculate a₀ from the observed v and M. Milgrom found a₀=1.2×10⁻¹⁰ ms⁻². Milgrom has noted that this value is also

"... the acceleration you get by dividing the speed of light by the lifetime of the universe. If you start from zero velocity, with this acceleration you will reach the speed of light roughly in the lifetime of the universe."

Retrospectively, the impact of assumed value of a>>a₀ for physical effects on Earth remains valid. Had a₀ been larger, its consequences would have been visible on Earth and, since it is not the case, the new theory would have been inconsistent. (Reference Needed)

Consistency with the observations

According to the Modified Newtonian Dynamics theory, every physical process that involves small accelerations due to gravity will have an outcome different from that predicted by the simple law F=ma. Therefore, astronomers need to look for all such processes and verify that MOND remains compatible with observations, that is, within the limit of the uncertainties on the data. There is, however, a complication overlooked up to this point but that strongly affects the compatibility between MOND and the observed world: in a system considered as isolated, for example a single satellite orbiting a planet, the effect of MOND results in an increased velocity beyond a given range (actually, below a given acceleration, but for circular orbits it is the same thing) that depends on the mass of both the planet and the satellite. However, if the same system is actually orbiting a star, the planet and the satellite will be accelerated in the star's gravitational field. For the satellite, the sum of the two fields could yield acceleration greater than a₀, and the orbit would not be the same as that in an isolated system.

For this reason, the typical acceleration of any physical process is not the only parameter astronomers must consider. Also critical is the process's environment, which is all external forces that are usually neglected. In his paper, Milgrom arranged the typical acceleration of various physical processes in a two-dimensional diagram. One parameter is the acceleration of the process itself, the other parameter is the acceleration induced by the environment.

This affects MOND's application to experimental observation and empirical data because all experiments done on Earth or its neighborhood are subject to the Sun's gravitational field, and this field is so strong that all objects in the Solar system undergo an acceleration greater than a₀. This explains why the flattening of galaxies' rotation curve, or the MOND effect, had not been detected until the early 1980s, when astronomers first gathered empirical data on the rotation of galaxies.

Therefore, only galaxies and other large systems are expected to exhibit the dynamics that will allow astronomers to verify that MOND agrees with observation. Since Milgrom's theory first appeared in 1983, the most accurate data has come from observations of distant galaxies and neighbors of the Milky Way. Within the uncertainties of the data, MOND has remained valid. The Milky Way itself is scattered with clouds of gas and interstellar dust, and until now it has not been possible to draw a rotation curve for the galaxy. Finally, the uncertainties on the velocity of galaxies within clusters and larger systems have been too large to conclude in favor of or against MOND. Indeed, conditions for conducting an experiment that could confirm or disprove MOND may only be possible outside the Solar system. A couple of near-to-Earth tests of MOND have been proposed though: one involves flying the LISA Pathfinder spacecraft through the Earth-Sun saddlepoint; another involves using a precisely controlled spinning disk to cancel out the acceleration effects of Earth's orbit around the Sun, and Sun's orbit around the galaxy; if either of these tests are carried out, and if MOND holds true, then they should feel a slight kick as they approach the very low acceleration levels required by MOND.

In search of observations that would validate his theory, Milgrom noticed that a special class of objects, the low surface brightness galaxies (LSB), is of particular interest: the radius of an LSB is large compared to its mass, and thus almost all stars are within the flat part of the rotation curve. Also, other theories predict that the velocity at the edge depends on the average surface brightness in addition to the LSB mass. Finally, no data on the rotation curve of these galaxies was available at the time. Milgrom thus could make the prediction that LSBs would have a rotation curve which is essentially flat, and with a relation between the flat velocity and the mass of the LSB identical to that of brighter galaxies.

Since then, the majority of LSBs observed has been consistent with the rotational curve predicted by MOND.

An exception to MOND other than LSB is prediction of the speeds of galaxies that gyrate around the center of a galaxy cluster. Our galaxy is part of the Virgo supercluster. MOND predicts a rate of rotation of these galaxies about their center, and temperature distributions, that are contrary to observation.

Computer simulations show that MOND is generally very precise at predicting individual galaxy rotation curves, of all kinds of galaxies: spirals, ellipticals, dwarfs, etc. However, MOND and MOND-like theories are not so good at predicting galactic cluster-scale, or cosmological scale structures.

A test that might disprove MOND would be to discover any of the theorized Dark Matter particles, such as the WIMPs.

A recent proposal is that MOND successfully predicts the local galactic escape speed of the Milky Way, a measure of the mass beyond the galactocentric radius of the Sun.

Lee Smolin and co-workers have tried unsuccessfully to obtain a theoretical basis for MOND from quantum gravity. His conclusion is "MOND is a tantalizing mystery, but not one that can be resolved now."

The mathematics of MOND

In non-relativistic Modified Newtonian Dynamics, Poisson's equation,

$\nabla ^{2}\Phi _{N}=4\pi G\rho$

(where $\Phi _{N}$ is the gravitational potential and $\rho$ is the density distribution) is modified as

$\nabla \cdot \left[\mu \left({\frac {\left\|\nabla \Phi \right\|}{a_{0}}}\right)\nabla \Phi \right]=4\pi G\rho$

where $\Phi$ is the MOND potential. The equation is to be solved with boundary condition $\left\|\nabla \Phi \right\|\rightarrow 0$ for $\left\|\mathbf {r} \right\|\rightarrow \infty$. The exact form of $\mu (\xi )$ is not constrained by observations, but must have the behaviour $\mu (\xi )\sim 1$ for $\xi >>1$ (Newtonian regime), $\mu (\xi )\sim \xi$ for $\xi <<1$ (Deep-MOND regime). In the deep-MOND regime, the modified Poisson equation may be rewritten as

$\nabla \cdot \left[{\frac {\left\|\nabla \Phi \right\|}{a_{0}}}\nabla \Phi -\nabla \Phi _{N}\right]=0$

and that simplifies to

${\frac {\left\|\nabla \Phi \right\|}{a_{0}}}\nabla \Phi -\nabla \Phi _{N}=\nabla \times \mathbf {h} .$

The vector field $\mathbf {h}$ is unknown, but is null whenever the density distribution is spherical, cylindrical or planar. In that case, MOND acceleration field is given by the simple formula

$\mathbf {g} _{M}=\mathbf {g} _{N}{\sqrt {\frac {a_{0}}{\left\|\mathbf {g} _{N}\right\|}}}$

where $\mathbf {g} _{N}$ is the normal Newtonian field.

Discussion and criticisms

An empirical criticism of MOND, released in August 2006, involves the Bullet cluster, a system of two colliding galaxy clusters. In most instances where phenomena associated with either MOND or dark matter are present, they appear to flow from physical locations with similar centers of gravity. But, the dark matter-like effects in this colliding galactic cluster system appears to emanate from different points in space than the center of mass of the visible matter in the system, which is unusually easy to discern due to the high energy collisions of the gas in the vicinity of the colliding galactic clusters. MOND proponents admit that a purely baryonic MOND is not able to explain this observation. Therefore a “marriage” of MOND with ordinary hot neutrinos of 2eV has been proposed to save the hypothesis. Beside MOND, two other notable theories that try to explain the mystery of the rotational curves are Nonsymmetric Gravitational Theory proposed by John Moffat, and Conformal gravity by Philip Mannheim. Another recent analysis , show that Newton's Law's predict the measured velocities of stars in nearby galaxies, without the use of MOND or of halo's of dark matter. The calculation above of figure 1 curve B is more in line with calculations using Kepler's Law's. When Newton's Law's are used, the curve looks like curve B. So, the MOND calculation is not required. The reason that the velocity is uniform is due to the exponential increase in mass (i.e. number of Sun's) closer to the galactic center of galaxies. The exponential mass increase is confirmed by luminosity measurements of these same galaxies.

Tensor-vector-scalar gravity

Main article: Tensor-vector-scalar gravity

Tensor-Vector-Scalar gravity (TeVeS) is a proposed relativistic theory that is equivalent to Modified Newtonian dynamics (MOND) in the non-relativistic limit, which purports to explain the galaxy rotation problem without invoking dark matter. Originated by Jacob Bekenstein in 2004, it incorporates various dynamical and non-dynamical tensor fields, vector fields and scalar fields.

The break-through of TeVeS over MOND is that it can explain the phenomenon of gravitational lensing, a cosmic phenomenon in which nearby matter bends light, which has been confirmed many times.

A recent preliminary finding is that it can explain structure formation without cold dark matter (CDM), but requiring ~2eV massive neutrinos. [4] [5]. However, other authors (see Slosar, Melchiorri and Silk [6]) claim that TeVeS can't explain cosmic microwave background anisotropies and structure formation at the same time, i.e. ruling out those models at high significance.

See also

• Dark matter
• Cold dark matter
• Lambda-CDM model
• Tensor-vector-scalar gravity
• Scalar-tensor-vector gravity
• Nonsymmetric gravitational theory
• Dark Fluid
• Pioneer anomaly

Further reading

• Bertone, Gianfranco (2010). Particle Dark Matter: Observations, Models and Searches. Cambridge University Press. p. 762. ISBN 13: 9780521763684. {{cite book}}: Check |isbn= value: invalid character (help); Cite has empty unknown parameter: |coauthors= (help)
• Mordehai Milgrom MOND: time for a change of mind?, August 2009
• Mordehai Milgrom: Does Dark Matter Really Exist?, Scientific American, August 2002
• Slosar, Melchiorri, & Silk: Did Boomerang hit MOND?, Physical Review D, November 2005
• Mordehai Milgrom: Do Modified Newtonian Dynamics Follow from the Cold Dark Matter Paradigm?, Astrophysical Journal, May 2002
• David Lindley: Messing around with gravity, Nature, 15 October 1992
• Bekenstein, Jacob D.: Modified Gravity vs Dark Matter: Relativistc theory for MOND, JHEP Conference Proceedings, 2005
• Massey, Richard; Rhodes, Jason; Ellis, Richard; Scoville, Nick; Leauthaud, Alexie; Finoguenov, Alexis; Capak, Peter; Bacon, David; Aussel, Hervé (2007). "Dark matter maps reveal cosmic scaffolding". Nature. 445 (7125): 286–290. doi:10.1038/nature05497. PMID 17206154.
• A. Yu. Ignatiev, Is Violation of Newton's Second Law Possible?, Phys. Rev. Lett. 98, 101101 (2007).
• Modified Newtonian dynamics om arxiv.org(I)Authority
• Modified Newtonian dynamics om arxiv.org(II)Review

External links

• MOND: time for a change of mind?
• Preprints related to MOND
• MOND - A Pedagogical Review
• The MOND Pages: A great resource for MOND related information
• Literature relating to the Modified Newtonian Dynamics (MOND)
• Alternatives to Dark Matter and Dark Energy
• Alternatives to Dark Matter
• Dark matter doubters not silenced yet
• Vacuum Modified Gravity as an explanation for flat galaxy rotation curves
• Another MOND is possible
• TeVeS
  • J.D. Bekenstein, Phys. Rev. D70, 083509 (2004), Erratum-ibid. D71, 069901 (2005) (arXiv:astro-ph/0403694), original paper on TeVeS by Jacob D. Bekenstein
  • J.D. Bekenstein and R.H. Sanders, A Primer to Relativistic MOND Theory, arXiv:astro-ph/0509519
• STVG
  • Gravity theory dispenses with dark matter (New Scientist)
  • Scalar-tensor-vector gravity theory JW Moffat (Journal of Cosmology and Astroparticle Physics) 6 March 2006
  • Scalar-Tensor Gravity Theory For Dynamical Light Velocity M. A. Clayton, J. W. Moffat (arXiv) Sun, 31 Oct 1999 22:09:24 GMT
• Relativistic MOND
  • Einstein's Theory 'Improved'? (PPARC)
  • Original paper by Jacob D. Bekenstein (arXiv)
  • Refining the MOND Interpolating Function and TeVeS Lagrangian (Journal of Astrophysics Letters)
  • Refining MOND interpolating function and TeVeS Lagrangian (arXiv)