flat rotation curves and 'foam' large scale struct

If G is the true constant and Ge is the adjusted value:

Ge = c ^ 2 / ( c ^ 2 / G + m / r )
Ge = G / ( 1 + G * m / ( r * c ^ 2 ) )

Write for simplicity
k = m / ( r * c ^ 2 )

Ge = G / ( 1 + G * k )
1 / Ge = ( 1 + G * k ) / G
1 / Ge = 1 / G + k

And assume
B = 1 / G
Be = 1 / Ge

Then
Be = B + k

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Let's look at the gravitational field (omitting the unity-vector)

g(r) = Ge * m / r ^ 2
g(r) = m / ( Be * r ^ 2 )
g(r) = m / ( ( B + k ) * r ^ 2 )
g(r) = m / ( B * r ^ 2 + m * r / c ^ 2 )

This gives the familiar result when m is small and r is big.
When m is big or r close to zero it returns:

g(r) = c ^ 2 / r

Note that there's no trace of G in there...

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Now for gravitational time dilation (GTD)

GTD = SQR( 1 / ( 1 - 2 * Ge * m / ( r * c ^ 2 ) )
GTD = SQR( 1 / ( 1 - 2 * m / ( Be * r * c ^ 2 ) )
GTD = SQR( 1 / ( 1 - 2 * m / ( ( B + k ) * r * c ^ 2 ) )

GTD = SQR( 1 / ( 1 - 2 * m / ( B * r * c ^ 2 ) + m )

So the Schwarzschild radius is at

1 = 2 * m / ( B * r * c ^ 2 + m )
2 * m = B * r * c ^ 2 + m
m = B * r * c ^ 2

r = G * m / c ^ 2

Hmmm, that's exactly half of the unadjusted value we're used to.

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I'm not sure where this leads, but I hope it helps.

Be = B + 2 * k
would be a lot more interesting.

GTD = SQR( 1 / ( 1 - 2 * m / ( B * r * c ^ 2 ) + 2 * m )

The Schwarzschild radius can now be found at

r = 0

Hehe, the escape velocity is

ve = SQR( 2 * Ge * m / r ) = SQR( 2 * m / ( B * r + 2 * m / c ^ 2 ) )

With m to infinity that gives

ve = c

[ I corrected a 'small' error shortly after posting ]
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So that's when your adjustment is

Ge = c ^ 2 / ( c ^ 2 / G + 2 * m / r )

Dear Michiel,

Thanks for your suggestion that G(effective) = c^2/(c^2/G +2m/r)

instead of c^2/(c^2/G + m/r). It is true that either formula would reduce the effective value of G for masses of high m/r ratio.

Your formula leads to a zero schwarzchild radius, which would be good to avoid singularities. The original formula was derived in a straightforward way from the conjecture in www.gravity.uk.com - but maybe a proper relativistic treatment would lead to the extra factor 2 which you suggest.

At this stage maybe we could try to encourage more physicists to try to find evidence for , or against this conjecture - the advantages are clear : a natural solution of the flatness problem,
avoidance of singularities
explanation of flat rotation curves, and foam
appearance of large scale structure.

So here is a challange to all open minded physicists out there - can anyone find evidence against the conjecture?? A case of champagne (or a years supply of Playboy magazine, if it's Stephen Hawking), to anyone who can.

John Hunter.

Hello John,

A case of champagne would be good. A working modification to the law of gravity would be even better.

Where could an extra factor 2 arise? I don't have an answer to that, but finding it is probably easier then building a bridge between both peaks of Kilimandjaro.

In our solar system there is no example of a high enough m/r ratio to have an obvious effect. Especially since the gravitational constant is not known to great accuracy. The center of a galaxy is the place to look for significant effects, in my view.

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I have also been looking for a third-order term in the denominator of the gravitational field formula. Meta Model suggests such a term (limited range of gravity). As far as I know there are no constants with the right dimensions, so the only term I could think of is:

r ^ 3 / ( G * dg )

where dg is simply the distance at which the term takes over from r ^ 2 / G

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First I'll do a 2-body computer simulation of the modified G and see what happens.

Dear Michiel,

I agree that it is difficult to test the conjecture using solar system tests. The value of G is not well known, but the product GM (for the earth for example) is much better known.

One test is by Lunar Laser Ranging. Because the earth-sun distance changes over a year, the conjecture predicts a change in the product GM, which in turn would cause an annual 6cm change in the earth-moon distance.

This has been measured by Lunar Laser Ranging, which could be regarded as evidence for the conjecture. However some 'experts' believe this is a time dilation effect of General Relativity, so its not clear which interpretation is correct.

For galaxies, the m/r ratio approaches c^2/G at the centre of the galaxy, it may be possible that a reduction of G there leads to the jets emitted by active galactic nuclei (AGNs).

LISA may measure much larger than expected changes in arm lengths, if the conjecture is correct, due to solar oscillations changing the sun's GM value.

I would be interested to hear what happens in your computer simulation using a modified G.

All the best,

John Hunter.

Dear John,

Men, not robots, have placed reflectors on the moon. That's how we know the moon's orbit is about 3.8 cm wider each year. There are two ways this could happen:

1) The gravitational force is not in line with the centre of the attracting body
2) The gravitational force differs between the inward and outward journey (for eccentric orbits only).

In the case of our moon it's tidal forces that transfer energy from the earth's rotation to kinetic energy of the moon's orbit. The gravitational field of the earth, as seen by the moon, causes the moon to speed up. It's a matter of energy conservation. The earth-sun distance, and certainly not the variation of that distance, are of any significance here.

A modified G does not change the direction of the gravitational force nor does it discriminate between radial velocities. So orbits will keep their size, although true elliptical orbits will be impossible. My first simulations show this, but I have to admit there's no correction for relativity yet, while relativity is probably the dominant factor.

See you soon.

By the way, if there's anyone with a proposal for 2) please start a new thread.