Frequently Asked Questions

Q: What is the modification?
A: MOND can be interpreted as either a modification of gravity through a change to the Poisson equation, or as a modification of inertia through a breaking of the equivalence of inertial and gravitational mass.
The modification occurs at very small accelerations. Above a critical acceleration a₀ (the one parameter of the theory), everything is normal. Below a₀, the effective acceleration approaches a = (g_N a₀)^1/2, where g_N is the normal Newtonian acceleration. The two regimes are joined smoothly by an interpolation function mu(x) with the asymptotic property mu(x) → 1 for x » 1 and mu(x) → x for x « 1, where x = a/a₀.
In the case of a modification of the Law of Inertia, a=F/m becomes a=F/[m*mu(x)], and the effective acceleration can be found from

,

Q: Isn't it rather drastic to modify such fundamental Laws?
A: Sure. Just as it is drastic to fill the universe with non-baryonic cold dark matter consisting of new fundamental particles we don't actually know to exist. The data drive us to one of these extremes. The question, of course, is which is a better approximation of Nature?

Q: What is mu(x)?
A: The interpolation function is not specified theoretically, but there is a fairly narrow range of empirically acceptable mu(x). Possible forms which are acceptable to galaxy data are
mu(x) = x(1+x²)^-1/2
mu(x) = x/(1+x), and
mu(x) = 1-e^-x.

Q: What is the acceleration scale?
A: a₀ = 1.2 x 10^-10 m s^-2, i.e., about one Angstrom per second per second. This is one part in 10¹¹ of what we feel on the surface of the earth. The precise value depends on the distance scale to galaxies, so perhaps it would be better to say a₀ = 1.2 x 10^-10 m s^-2 h₇₅², where h = H₀/75 is the Hubble Constant (the expansion rate of the universe) in units of H₀ = 75 km s^-1 Mpc^-1. (Currently, most measurements report values in the neighborhood of H₀ = 72 km s^-1 Mpc^-1.)

Q: Why an acceleration based modification?
A: We usually think first in terms of a modification at some length scale: galaxies are big, so maybe gravity is different on large scales. This does not work. But there are other scales which are different about galaxies. One of them is the very low centripetal acceleration experienced by stars orbiting within galaxies. This is just as far removed from our laboratories as is the size scale of galaxies. MOND was motivated by two observations: 1) the asymptotic flatness of rotation curves and 2) the slope of the Tully-Fisher relation (M ~ V_flat⁴). These two things lead to an acceleration scale:

Q: MOND fits the rotation curves of spiral galaxies well, but it was designed to do that. So do such data provide any test?
A: Yes.
That MOND was "designed" to fit rotation curves and is therefore guaranteed to do so is a common misconception.
Both MOND and dark matter were invented to explain the presence of mass discrepancies in astronomical data, especially the flat rotation curves of spiral galaxies. It is often asserted that MOND is ad hoc (and therefore bad), but this is also true of dark matter. Once invoked, dark matter can be distributed in any way necessary to explain just about anything. It is difficult to test, and does not constitute a falsifiable theory. In contrast, once a modified force law is specified, there is no freedom to adjust its predictions. MOND can not fit any arbitrary rotation curve: it is tied down by the observed mass distribution. However, it does fit real data. I find it remarkable that of the infinite variety of things rotation curves might plausibly do were they caused by a Newtonian disk + dark matter halo, they in fact do the one (and only one) thing allowed by MOND.
Fundamental Clue or Big Cosmic Accident? (Is God subtle or malicious?)
MOND's detractor's often assert it is just some sort of fluke, so I guess they prefer to believe that God is malicious.

Q: But wait, there is a fundamental objection to an acceleration scale. The atoms within stars have acceleratoins in the Newtonian regime. Since the star's constituents are Newtonian with a > a₀, the star itself can never be in the MOND regime.
A: This comes down to when you can consider an object to be a billiard ball, as we are so fond of doing in physics. Is a star an unremitingly Newtonian object because of the microscopic motion of its constiuent atoms? Or can it be treated as a point mass in the much larger gravitational field of the entire galaxy?
It would be quite a thing if the accelerations of atoms within stars could affect their motion in galaxies. There is a matching condition between the Newtonian and MOND regimes. This effectively defines the location at which you are in one regime or the other - and whether things interior to that regime can be treated as a single billiard ball for the purpose of calculating its motion, e.g the barycenter of the solar system about the galaxy. The solar system is Newtonian out to about a tenth of a light year; all the substructure we are familiar with is well within this radius and matters not at all to the galactic motion.
If quantum mechanics has taught us anything, it is that there is a regime where substructure matters, and another where the billiard-ball approximation is quite good. Whether this analogy extends to MOND or not I expect will be debated for many years to come. It seems not unreasonable that it might.

Q: Wouldn't MOND effects show up in precision solar system tests?
A: They may have already. Whether they should or not depends on the interpolation function mu(x). The accelerations in the inner solar system are millions of times higher than the MOND acceleration scale, so all that can be tested here is the (slight) deviation of the interpolation function from unity. This might be perceptible if mu(x) approaches unity slowly in the limit x » 1, but would not be if it converged rapidly. For example, if mu(x) = 1-e^-x, then the deviation one is looking for is of order one part in e¹⁰⁸.

MOND and General Relativity

Q: Is there a complete relativistic theory incorporating MOND?
A: There have been suggestions, such as Bekenstein's Phased Coupled Gravity and Sanders's Stratified Scalar-Tensor Theory. It is not obvious that either of these are satisfactory, but they do at least demonstrate that it is possible to construct theories which encompass both GR and MOND.
This is a hard problem, especially conceptually. It took a long time to go from the inverse square law to the Poisson equation, and from special relativity to GR. If both GR and MOND are correct (as appears to be true empirically), then there must be a grander theory encompassing both. But don't expect to derive it on the back of an envelope in a few minutes.

Q: What about the Equivalence Principle?
A: The inertia interpretation of MOND breaks the symmetry of the equivalence of gravitational and inertial mass, which is one of the foundations of GR. This makes it especially challenging to come up with a relativistic theory. Experimenters have long searched for violations of the equivalence without success, so it seems inviolable. However, these experiments are all performed in acceleration fields eleven orders of magnitude greater than the acceleration scale of MOND. Accelerations are not relative. Recall the Einstein thought experiment of an observer in an elevator. The observer may not be able to tell whether the weight that he feels is due to being stationary on the surface of the Earth, or because he is being accelerated at 1 gee out in space. But he knows very sell he is being accelerated! You can do the Eotvos experiment here on Earth to arbitrary accuracy and never discover MOND. Moreover, such experiments look for composition-dependent differences: wood vs. gold, or some such. There is nothing in MOND that is composition dependent. Indeed, the galaxian data which support MOND are the only data which test the Equivalence Principle in the relevant regime. One fair interpretation is that it breaks at small accelerations. We do not have a satisfactory understanding of the origin of inertial mass. This phenomenology may be a big clue.

Q: MOND can't actually be true, can it?
A: Empirically it works. No empirical piece of evidence clearly contradicts^* it. Its predictions come true where those of dark matter based theories fail. Could be.

* - This was true when I wrote this web page in the late '90s. Now (2013) I would say there are two pieces of apparently contradictory evidence: the acoustic power spectrum of the cosmic microwave background (CMB), and rich clusters of galaxies. Both of these are further discussed on the main MOND page. In brief, the CMB isn't really relevant [yet] as MOND provides no definitive prediction for it. It is very well fit by the standard ΛCDM cosmology, but we've given ourselves enough free parameters to guarantee a fit, so I'm not sure how impressed we should be with that. Rich clusters show a residual mass discrepancy - one still needs unseen mass, even with MOND! This is a clear problem for the idea, though it is conceivable that some normal matter remains undetected (we do have a missing baryon problem, after all). Even if this does indeed falisify MOND, it does not alleviate the burden of explaining the observed phenomenology: if MOND is wrong, why does it work as well as it does so often?

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