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Originally Posted by papageno
I explained it to Lunatik.
If G depended on positions, the mass of an object would not be affected, but the gravitational force would be.
A different force would give a different acceleration, because the inertial mass has not changed.
But this problem is no more exotic than a variable dielectric constant in electromagnetism (which gives us refraction, and lenses).
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I see this really as a question referring to our units of measure, what we call kilograms. Can the same kilograms be used if G is different from what we know it to be as a universal constant?
In yours you said:
"The Equivalence Principle says that the gravitational mass equals dynamical (a.k.a. inertial) mass. It has nothing to do with the value of G."
Granted, given that G is universally the same, it has nothing to do with it, though G is part of the function describing Newton's formula for gravitation, as per yours above:
F = G * (m*M) / r^2 , which is related to Newton's second law:
F = M * a
Now, this equivalence can be also shown as:
F = M * a = M * (G*m) / r^2, where by default
a = (G*m) / r^2
which also means:
G = (r^2 * a) / m
Now assume that both
a and
r^2 are fixed, same values, but G is greater, viz. G1 = 10G. So we have:
G1 = (r^2 * a) / m1, except now of necessity, m1 = 1/10th of m, if G1 = 10G.
However the mass
had not changed, same mass (same atomic composition and volume), so the mass did not suddenly shrink to a tenth of its original form. What changed instead was that the
measures in kilograms had changed, to where now the kilograms are 10 times greater than the kilograms used earlier, to match up with G ten times Newton's G.
Can you see how this could be a problem? Though for now, given that G is universal, we don't have a problem. But if it were discovered that G is different, something might have to be adjusted in the measure of our (Earth derived) kilograms.
(That said, I still think that the answer above, kg1 = 10kg is wrong, but I'm not sure of what the right answer is. I suspect
a is in fact not fixed as assumed, for a variable G. Hypothetically, the real answer may be more like kg1 = 100 kg, if G1 = 10G, or its squared. It may take 10 times as much acceleration to move the same mass in 10G, so
a is not fixed, but rather
a1 = 10X a. But I don't know this.)
So you can see why I am frustrated, and I don't like my own answers!
There must be a better way to see this.
Interesting if this might not apply as well to a " variable dielectric constant in electromagnetism", since it might impact how light bends around stars, which would impact gravitational lensing. :-?
Actually, now that I re-read this, I can almost begin to appreciate the frustration Galileo must have had trying to prove why the Earth is not standing still with the heavens going around, but instead it is spinning.
