In GR calculations below the cosmological scale, it is(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# MOND-related formula

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