# Another bullet for MOND?

1. Jan 15, 2014

### Chronos

This paper appears to further the case for DM vs MOND http://arxiv.org/abs/1401.3162, A simple model linking galaxy and dark matter evolution.

2. Jan 16, 2014

### Jonathan Scott

I'd like to point out that the empirical MOND rule doesn't need such a model - it already works perfectly well for galaxies. It is only DM which needs some additional explanation to make the ratios and distribution work, to catch up with MOND.

It would however be interesting if this leads to an explanation of how DM manages to reproduce the MOND rule.

Where MOND has a long way to go is on the theoretical side, in that I'm not convinced that the usual version (with a cut-off) is even self-consistent, let alone plausible.

3. Mar 13, 2015

### TEFLing

MOND can be construed as claiming, that at large distances, i.e. long range, gravity switches character, from 1/r2 to 1/r

Does that have the simple interpretation, that gravity as we know it, is composed of two separate phenomena... One of which varies in strength as 1/r2, the other as 1/r ?

Fg = GMm/r2 + G' Mm/r

So you could explain MOND, if you could explain what two separate phenomena, explained each of the two terms?

Relatedly, what GR like equation, would have a 1/r weak field limit? The standard GR equation has a 1/r2 limit... What modified form of the GR equation would generate weak field 1/r like forces ?

4. Mar 13, 2015

### Jonathan Scott

You should probably have started a new thread, but I'll just try to give a quick answer anyway.

The MOND acceleration term is proportional to the square root of the source mass divided by the radius (which has the obvious and rather unfortunate effect that it is not linearly proportional to the source mass, so in the original MOND model the forces are not equal in both directions, although there are various ways of patching this up). One way of expressing this is as $\sqrt{a_0 \, GM/r^2}$ where $a_0$ is the universal MOND acceleration parameter.

For purposes of computing galactic rotation curves, you can simply add the MOND acceleration to the Newtonian acceleration, without any cut-off. However, if MOND were a simple additional acceleration added to Newtonian gravity, then it would be expected to be just about detectable in solar system and laboratory gravitational observations, so the theory comes with an idea that it only takes effect below a certain "cut-off" acceleration threshold, with some sort of arbitrary rule that makes it undetectable in laboratories but significant in galactic rotation curves. This idea has a problem, in that even when a star or star system has an average acceleration below the MOND threshold, just about every atom of the star or star system will be experiencing gravitational accelerations from the local masses well in excess of that threshold, and so far I have not seen any plausible explanation of how the system as a whole can behave differently from its parts. However, as the MOND acceleration is not linear in the source mass and does not add up in a vector-like way, the fact that this extra acceleration it is not observed in the solar system could be due to other factors rather than a cut-off.

In partial answer to your question, one example of a idea which is based on GR-like concepts but which leads to a MOND-like effect is to assume that at sufficient distance from a large mass space does not tend asymptotically to being flat (as is usually assumed in the Schwarzschild solution) but rather it becomes like a higher-dimensional "cone" with a solid angular deficit proportional to the enclosed amount of matter as a fraction of the universe, so that overall matter exactly closes the universe spatially. That matches MOND's experimental acceleration parameter if the mass of the universe is about $10^{54}$kg, which seems about right. However, that idea is too speculative to discuss any further here.

5. Mar 13, 2015

Staff Emeritus

6. Mar 14, 2015

### TEFLing

Are there any versions of MOND which posit a transition from 1/r2 to 1/r behavior of the form

Fg = GMm/r2 + G' Mm/r

With the former dominating at close range and the latter at galactic scales?

7. Mar 14, 2015

### Jonathan Scott

No, that wouldn't be MOND if it did.

A key point of MOND is that it describes different galaxies, with different masses, using the same acceleration constant, but that only works if the additional force is assumed to be proportional to the square root of the source mass, as I've already mentioned. If I assume you intend M to represent the source mass, this gives a formula of the form $$F_g = G Mm/r^2 + G' \sqrt{M} m/r$$ where $G' = \sqrt{a_0 G}$.

8. Mar 14, 2015

### ChrisVer

You can as well rewrite the $F=ma$ for very small accelerations (comparable to some characteristic one) with $F= m \frac{a^2}{a_0}$ for the characteristic acceleration of $a_0 \approx 10^{-8}~cm ~s^{-2}$ (empirical value from the rotation of galaxy halos).

This would result in the $G \frac{Mm}{r^2} = F = m \frac{a^2}{a_0}$ Or $a= \sqrt{Ga_0} \sqrt{\frac{M}{r^2}}= \sqrt{GMa_0} \frac{1}{r}$. Thus (for the centrifugal motion) the velocities would be like:
$\frac{u^2}{r} = \sqrt{GMa_0} \frac{1}{r} \Rightarrow u = (Ga_0)^{1/4} M^{1/4}(r)$

The motivation is mainly that the Newton's law has only been tested within the Solar System and not to the very weak accelerations going on in the galactic halos.

From this point of view, MoND seems to modify the acceleration (replaces the $a$ with $\frac{a^2}{a_0}$) rather than the gravitational law.