MOND-related Formula in GR Calcs at Cosmological Scale

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In summary, the conversation discusses the idea of assuming a conical boundary instead of a flat boundary when calculating GR for galaxies and the potential for a new acceleration term in MOND. The resulting formula from this idea matches the MOND acceleration parameter and the mass of the universe. The conversation also touches on the viability of a specific interpolation function in relation to galaxy data and the interesting result that this formula also matches the acceleration at the surface of particles. The possibility of local MOND effects and evidence ruling them out is also mentioned.
  • #1
Jonathan Scott
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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.
 
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  • #2
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.
 
  • #3


Thank you for sharing your thoughts and calculations on the possible relationship between MOND and the shape of space at the cosmological scale. It is certainly an intriguing idea to consider the shape of space as a factor in understanding the behavior of gravity in galaxies. However, as you mentioned, these are not rigorously derived formulas and may hide other factors that could affect their validity.

Regarding the viability of the interpolation function you proposed, it would be difficult to say without further analysis and comparison with observational data. However, it is always valuable to explore new ideas and potential connections between different theories, so your contributions are certainly appreciated.

As for the second result you mentioned, the mass for which the MOND and Newtonian accelerations are equal, it is interesting to consider but may not have a physical significance. It is always important to critically evaluate and test any new ideas or hypotheses, and I encourage you to continue your investigations in this area. Thank you again for sharing your thoughts and calculations.
 

1. What is MOND-related formula in GR calcs at cosmological scale?

MOND-related formula in GR calcs at cosmological scale refers to the Modified Newtonian Dynamics (MOND) theory applied in the context of General Relativity (GR) calculations at a large scale, such as in cosmology. It is a proposed modification of the laws of gravity that aims to explain the observed discrepancies between the predictions of GR and the observed dynamics of galaxies and clusters of galaxies.

2. How does MOND-related formula differ from Newton's law of gravity?

MOND-related formula differs from Newton's law of gravity in that it introduces a new acceleration scale, known as the "Milgrom acceleration," below which the law of gravity deviates from the standard inverse-square law. This modification is proposed to account for the observed discrepancies in the rotation curves of galaxies, which cannot be explained by Newton's law of gravity.

3. What is the current status of MOND-related formula in the scientific community?

The status of MOND-related formula in the scientific community is still debated. While it has been successful in explaining some observations, such as the rotation curves of galaxies, it has not yet been supported by strong empirical evidence. Many scientists continue to explore and test MOND-related formula, but it has not been widely accepted as a replacement for GR.

4. How is MOND-related formula applied in cosmological scale calculations?

MOND-related formula is applied in cosmological scale calculations by modifying the standard equations of GR to include the extra acceleration scale and the effects of MOND. This modified framework can be used to calculate the dynamics of large-scale structures, such as galaxies and galaxy clusters, and compare them to observations.

5. Can MOND-related formula be tested or proven?

While MOND-related formula has been successful in explaining some observations, it is challenging to test or prove it definitively. The main reason is that MOND-related formula is not a complete theory, and it is not clear how it can be reconciled with other fundamental theories, such as quantum mechanics. Additionally, the observed discrepancies that MOND-related formula aims to explain can also be attributed to other factors, such as dark matter. Therefore, further research and observations are needed to fully understand the validity and implications of MOND-related formula.

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