# MOND-related formula

1. Apr 24, 2007

### Jonathan Scott

In GR calculations below the cosmological scale, it is
conventional to assume that a sufficient distance from
the central object, space becomes flat. However, when
that object is a galaxy or similar, it seems to me that it
might be better to assume that the boundary is more
like the 3-D equivalent of a cone, constituting a fraction
(m/M) of the total solid angle needed to close the
universe, where m is the local mass and M is the mass
of the universe.

If one assumes that the area of an enclosing sphere
has been decreased by a factor (1-m/M), then linear
dimensions have been decreased by approximately
the square root of that, (1-m/2M). That forms the
cosine of the angle by which the "conical" space
diverges from being flat, so the sine and hence the
angle relative to flat space is approximately sqrt(m/M).
The curvature of the "cone" is 1/r times this, and it
seems plausible that for slow-moving objects this
could give rise to an additional acceleration relative
to flat space of c^2/r sqrt(m/M). Note that this
formula was only reached by somewhat imprecise
analogies, so the above is not actually a rigorous
derivation, and even if the analogies are valid it
could still hide factors of 2 or similar. However,
the resulting formula seems quite promising.

In MOND, when the acceleration due to normal
gravity becomes low enough, a different term
in the acceleration comes into effect which is of
the form sqrt(G m a_0)/r. This matches the
above formula if the MOND acceleration parameter
a_0 is equal to c^4/GM. The experimental value of
a_0 is around 1.2*10^-10 ms^-2 so this matches the
formula if the mass of the universe is approximately
10^54 kilograms. This is certainly around the right
order of magnitude, which seems very interesting,
given that this formula was derived from an idea
relating to the shape of space and the closure of the
universe, unlike MOND itself which is (as far as I
know) purely empirical at present.

In this case, the extra acceleration would merely be
added to the Newtonian acceleration, which in the
MOND formalism is formally equivalent to using an
interpolation function of the following form, assuming
my calculations were correct:

mu(x) = (sqrt(1+1/4x)-sqrt(1/4x))^2

(I found it quite surprising that the above
expression is equal to x when x is small, as I
would not have guessed that at first glance).

Does anyone know whether this interpolation
function (based on adding the accelerations together)
is considered viable with current galaxy data?

This formula c^2/r sqrt(m/M) has another curious
feature, which is however far from cosmological.
I found this when I was investigating under what
conditions the MOND and Newtonian accelerations
are comparable. Specifically, consider the
acceleration at the surface of a particle of mass
m and Compton radius r = hbar/mc, and consider
when it is equal to the "conical space" acceleration:

Gm/r^2 = c^2/r sqrt(m/M)

Moving some factors of c and r around we get

Gm/rc^2 = sqrt(m/M)

If we substitute the Compton radius expression
for r, we get:

Gm^2/(hbar c) = sqrt(m/M)

Squaring and rearranging this, we get

m^3 = ((hbar c/G)^2)/M

or

m = cube root of (((hbar c/G)^2)/M)

If we use M = 10^54 kg from matching the MOND
result, this gives

m = approx 34 MeV/c^2

That is, the mass for which these two acceleration
expressions are equal is around 65 times the mass
of the electron, around the right order of magnitude
for all common particles.

No, I don't know whether either of the above results
(MOND or particle) is physically meaningful, but I
just thought they both seemed rather interesting.

2. Jun 26, 2007

### Jonathan Scott

I've subsequently found that if MOND effects simply involved adding in the MOND acceleration, using the previously mentioned interpolation function, then anomalous results attributable to MOND would probably have already been detected in solar system experiments, as the effect would have been stronger than the known "Pioneer anomaly", and it might well also have been detected in Cavendish-type laboratory experiments to measure G (as mentioned in another thread). However, I would still be interested to know of any specific evidence which definitely rules out such local MOND effects.

It appears that the proponents of MOND maintain that the MOND effect only "switches on" when the overall potential gradient due to all fields is less than the critical acceleration. However, it is unclear how stars could then be affected by MOND when almost all of the component particles within the star are within a gravitational field which far exceeds the MOND threshold.