Changes

I know I said I wouldn't get re-involved with this debate, but let me just make a few points.  Re: the red shift - yes it's true a massive object is slightly red-shifted due to the energy light loses to its gravitational field, but this effect is no stronger for distant objects than near ones, and the effect is extremely minuscule for all but the strongest gravitational fields.  The red shift of distant cosmic object is a result of their incredible speed away from us, or at least their speed away from us at the time the light was emitted.

I know I said I wouldn't get re-involved with this debate, but let me just make a few points.  Re: the red shift - yes it's true a massive object is slightly red-shifted due to the energy light loses to its gravitational field, but this effect is no stronger for distant objects than near ones, and the effect is extremely minuscule for all but the strongest gravitational fields.  The red shift of distant cosmic object is a result of their incredible speed away from us, or at least their speed away from us at the time the light was emitted.

Second, 2 and 2.000001 would not be close enough to "match up" the geometry of space with the mathematics of geometry, here is why.  The equation for the surface area for a perfect sphere isn't an approximation, or a guess, it's exact.  Therefore, the imaginary spheres of different radii we imagine around a point mass would, if made manifest, actually have area<math>4/pi r^2 \ </math>, and nothing else, just as the circumference of a circle is precisely <math>2\pi r \ </math>.  On this, there can be no debate - it's math, not physics.  

Second, 2 and 2.000001 would not be close enough to "match up" the geometry of space with the mathematics of geometry, here is why.  The equation for the surface area for a perfect sphere isn't an approximation, or a guess, it's exact.  Therefore, the imaginary spheres of different radii we imagine around a point mass would, if made manifest, actually have area<math>4 \pi r^2 \ </math>, and nothing else, just as the circumference of a circle is precisely <math>2\pi r \ </math>.  On this, there can be no debate - it's math, not physics.  

So lets move to physics now.  Newtons formula for the acceleration caused on a test particle a distance r from a point mass was <math>a_g = G /frac{m_1}{r^2} \ </math>.  (Marking the mass of the test particle <math>m_2 </math> gives, by <math>F=ma </math> the more familiar <math>F= G \frac{m_1 m_2}{r^2} \ </math>)  Basically, what his formula for acceleration is saying is that no matter how far you are from a point mass, there is a constant effect due to that point mass' gravity, but spread out over the area of the sphere which has the radius that is the test particles distance from the point mass.  Since these spheres grow larger and larger as the test particle moves away, the acceleration grows smaller.

So lets move to physics now.  Newtons formula for the acceleration caused on a test particle a distance r from a point mass was <math>a_g = G \frac{m_1}{r^2} \ </math>.  (Marking the mass of the test particle <math>m_2 </math> gives, by <math>F=ma </math> the more familiar <math>F= G \frac{m_1 m_2}{r^2} \ </math>)  Basically, what his formula for acceleration is saying is that no matter how far you are from a point mass, there is a constant effect due to that point mass' gravity, but spread out over the area of the sphere which has the radius that is the test particles distance from the point mass.  Since these spheres grow larger and larger as the test particle moves away, the acceleration grows smaller.

Now, moving to physics, let's ignore all theories of relativity, just for a moment, since the question was, after all, asked about Newtonian gravity.  If the force of gravity <math>a_g \propto /tfrac{1}{r^{2+x}}, x>0 \ </math> where x is some tiny positive number, say, 0.00001, then there are either two possibilities: you have the part of this which is due to the spreading out of the effect over the spheres, and another component, <math> G \frac{m}{r^x} </math>, which would mean the gravitational constant is not constant at all but a function of distance, OR, the surface areas of the spheres different than expected.   

Now, moving to physics, let's ignore all theories of relativity, just for a moment, since the question was, after all, asked about Newtonian gravity.  If the force of gravity <math>a_g \propto \tfrac{1}{r^{2+x}}, x>0 \ </math> where x is some tiny positive number, say, 0.00001, then there are either two possibilities: you have the part of this which is due to the spreading out of the effect over the spheres, and another component, <math> G \frac{m}{r^x} </math>, which would mean the gravitational constant is not constant at all but a function of distance, OR, the surface areas of the spheres different than expected.   

The former of these possibilities is what must be true if the exponent is indeed 2.0001, because the latter implies that space is negatively curved far away from the point mass, where the increased exponent would result in larger-than-Euclidean surface areas, and positively curved close in to the point mass, where the increased exponent would result in smaller surface areas - this would mean that the formula would work only for this one point mass, and no others in the universe; that this point mass occupied a place of privilege, the only part of the universe where space would be positively curved.  Keep in mind this isn't the ST curvature of GR, which causes the effect we call gravity, and is itself caused by mass; this would be a curvature of 3-space independent of the mass distribution and having no properties that might be interpreted as a force.  

The former of these possibilities is what must be true if the exponent is indeed 2.0001, because the latter implies that space is negatively curved far away from the point mass, where the increased exponent would result in larger-than-Euclidean surface areas, and positively curved close in to the point mass, where the increased exponent would result in smaller surface areas - this would mean that the formula would work only for this one point mass, and no others in the universe; that this point mass occupied a place of privilege, the only part of the universe where space would be positively curved.  Keep in mind this isn't the ST curvature of GR, which causes the effect we call gravity, and is itself caused by mass; this would be a curvature of 3-space independent of the mass distribution and having no properties that might be interpreted as a force.