Modified Newtonian dynamics

For this reason, the typical acceleration of any physical process is not the only parameter one must consider. Also critical is the process' environment, that is all external forces that are usually neglected. In his paper, Milgrom arranged the typical acceleration of various physical process in a two-dimensional diagram (see figure 3). One parameter is the acceleration of the process itself, the other parameter is the acceleration induced by the environment.

How does this affect our discussion of MOND's application to the real world? Very simply: all experiments done on Earth or its neighborhood are subject to the Sun's gravitational field. This field is so strong that all objects in the Solar system undergo an acceleration greater than a₀. That's why MOND effects have escaped detection.

Therefore, only the dynamics of galaxies and larger systems need to be examined to check that MOND is compatible 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 our Galaxy. Finally, the uncertainties on the velocity of galaxies within clusters and larger systems has been too large to conclude in favor of or against MOND.

Is it possible to design an experiment that would confirm MOND predictions, or rule it out? Unfortunately, conditions for conducting this experiment can be found only outside the Solar system. However, the Pioneer and Voyager probes are currently traveling beyond Pluto and perhaps they have already reached this zone. To check that, let's calculate the radius of the gravitational sphere of influence of the Sun, inside which a probe undergoes an acceleration greater than a₀.

We have seen above that the equation relating the acceleration a to the distance r from the Sun is

<math>\frac{GM}{r^2}=\mu\left(\frac{a}{a_0}\right)a<math>

So, for a=a₀, assuming µ(a/a₀)=µ(1)=1, with G=6.67 10^-8 in cgs units and M (the mass of the Sun)=2 10³³ g, we get r=1.05 10¹⁷ cm. This is roughly 0.034 parsecs or 0.1 light years, over 100 times the distance between Voyager 1, the most remote probe, and the Sun. It is therefore doubtful that an experiment could be accurate enough to test the departure from Newton's second law. Perhaps µ(1) is less than 1, but it is very likely greater than 0.2. Consequently, experiments on MOND will have to wait for the next age of space exploration.

In search for observations that would validate his theory, Milgrom noticed that a special class of objects, the low surface brightness galaxies (LSB) are of particular interest: the radius of a 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 essentially flat, and with a relation between the flat velocity and the mass of the LSB identical to that of brighter galaxies.

Since then, many such LSBs have been observed, and while some astronomers have claimed their data invalidated MOND, others have been able to confirm their predictions. At the time of this writing, the debate is still hot, and scientists are waiting for more accurate observations.

The mathematics of MOND

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

<math>\nabla^2 \Phi_N = 4 \pi G \rho<math>

(where <math>\Phi_N<math> is the gravitational potential and <math>\rho<math> is the density distribution) is modified as

<math>\nabla\cdot\left[ \mu \left( \frac{\left\| \nabla\Phi \right\|}{a_0} \right) \nabla\Phi\right] = 4\pi G \rho<math>

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

<math>

\nabla \cdot \left[ \frac{\left\| \nabla\Phi \right\|}{a_0} \nabla\Phi - \nabla\Phi_N \right] = 0 <math>

and that simplifies to

<math> \frac{\left\| \nabla\Phi \right\|}{a_0} \nabla\Phi - \nabla\Phi_N = \nabla \times \mathbf{h} <math>

The vector field <math>\mathbf{h}<math> 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

<math>

\mathbf{g}_M = \mathbf{g}_N \sqrt{\frac{a_0}{\left\| \mathbf{g}_N \right \|}} <math>

where <math>\mathbf{g}_N<math> is the normal Newtonian field.

Discussion and Criticisms

One reason why some astronomers find MOND difficult to accept is that it is an effective theory, not a physical theory. As an effective theory, it describe the dynamics of accelerated object with an equation, without any physical justification. This approach is completely different from Einstein's, who assumed that some fundamental physical principles were true (continuity, smoothness and isotropy of space-time, conservation of energy, principle of equivalence) and derived new equations from these principles, including the famous E=mc² and the less famous but extremely powerful G=8πT. For many, MOND lacks a physical ground, some new fundamental principle about matter, vacuum, or space-time that would lead to the modified equation F=mµ(a/a₀)a.

Attempts in this direction have essentially been modifications of Einstein's theory of gravitation. When one looks at the equation F=mµ(a/a₀)a, the value of a, and the parameter of µ seem to depend on m as well as F. However, for the gravitational force, F also depends on m. Therefore, a change in Newton's second law can be a change of the gravitational force or a change of inertia. The two are indistinguishable. Note that this is not true, for example, for the electromagnetic force: moving in the same weak electromagnetic field, two particles with the same charge but with different masses would follow fundamentally different trajectories. With the same charge, the F term in the MOND equation is the same for the two particles. However, with a different value for m, one could have a MONDian trajectory and not the other, even though they are subject to the same force. However, in interstellar space, gravity is the main acting force, and since no experiment could be performed on Earth to check whether MOND is a new theory of inertia or a new theory of gravity, physicists have concentrated their effort on the later. Until now, they have achieved only partial success, coming up with some more complicated version of Einstein's theory of gravitation. Although it doesn't look like a big trouble (after all, relativity is much more complicated than Newton's law of gravity) one must remember that each and every relativistic theory of gravitation proposed since 1915, and the first appearance of Einstein's theory, has been ruled out or abandoned since. One way or another, only the simplest form of Einstein's theory has passed the many tests physicists put it on. Future will tell if this new one stands or falls. The most successful relativistic version of MOND, which was proposed in 2004, is known as "TeVeS" for Tensor-Vector-Scalar and was proposed by black hole physicist and Milgrom associate Bekenstein. It is currently receiving scholarly review in the field.

In the eyes of astronomers, MOND is just an alternative to the more widely accepted theory of dark matter. As new data is coming from telescopes, MOND as well as dark matter is sometimes invalidated and sometimes supported, and no clear-cut observation has yet helped decide which theory is the one. Toward this goal, supporters of MOND have concentrated their effort on specific areas:

• To look for new predictions of MOND that could be tested. For example, the dynamics of satellites of our Galaxy could be distorted by MOND effects, in a way difficult to explain with a dark matter halo.
• To obtain the relativistic extension of MOND, that would incidentally help understand how light is bent by galaxies' gravitational field, one feature MOND cannot explain.
• Alternatively, to establish MOND as a theory of inertia and find its fundamental principles, although progress in this direction has been pretty small.

Another problem with MOND is that it violates the principle of least astonishment (also known as the principle of maximum boredom). This principle is that the explanation which is the least astonishing and which is the most boring is usually (but not always) the right one. Any modifications in Newton's laws can also be explained in terms of distributions of dark matter, and the second explanation is more boring in that it requires fewer changes to what we think we know. However, proponents of MOND point out that some of the candidates for dark matter such as WIMPs are as astonishing as asserting that gravity behaves differently over long distances than we normally think.

Beside MOND, another theory that tries to explain the mystery of the rotational curves is Nonsymmetric Gravitational Theory proposed by John Moffat.

External links

• Preprints related to MOND (http://arXiv.org/find/physics,astro-ph/1/abs:+mond/0/1/0/all/0/1)
• Mordehai Milgrom: Does Dark Matter Really Exist?, Scientific American, August 2002
• MOND - A Pedagogical Review (http://nedwww.ipac.caltech.edu/level5/Sept01/Milgrom2/Milgrom_contents.html)de:Modifizierte Newtonsche Dynamik