Gravitational forces at very large distances
David Noel
<davidn@aoi.com.au>
Ben Franklin Centre
PO Box 27, Subiaco, WA 6008, Australia.
The standard formula for the force of gravitational attraction between two bodies is
F=G (M1*M2/D2)
where F is the force, G is the gravitational constant, M1 and M2 are the masses of the two bodies, and D is their separation.
Consider the case where the Universe has very-large-scale uniformity, and M1 is some local central reference-point object, say our Sun. For a very distant shell of interstellar/intergalactic material, say a spherical shell 100 light-years thick and with its central surface 100 million light-years from the Sun, the formula gives a force Fa = G (Ms*Mx/ L2), where Mx is the total mass of the shell, Ms is the Sun's mass, and L is a linear distance of 100 million light-years.
Of course the standard formula applies strictly to two point masses. If one of the masses is a spherical shell with the other a tiny sphere floating within it, the net gravitational force on the sphere is zero. Newton showed that this net force of zero applies wherever the sphere might be within the shell.
For example, if the Earth was hollow and consisted of a uniform spherical shell, the net gravitational force on an object within it would be everywhere zero, right from the centre of the Earth up to just within the inner surface of the shell.
Now consider the interstellar/intergalactic case for a greater distance, say 2L. The mass of the bigger spherical shell would be its volume times its average density. For a shell of the same thickness, 100 light-years, its volume would increase as the square of its radius r, being equal to its surface area, 4*pi*r2, times its standard thickness (100 light-years). The mass of this larger shell would therefore be 4Mx.
Inserting these terms in the formula gives Fb = G (Ms*4Mx / 4L2), exactly equal to Fa. In other words, the gravitational attraction of a shell of interstellar /intergalactic material of given thickness (not radius) is exactly the same however far away it may be, and does not decrease with distance. Because this attraction is spread out evenly over the spherical shell, its net result on the contained body is zero.
Because of this independence with distance, and because of Newton's finding of a force independent of position within a spherical shell, it follows that within a completely uniform Universe, there is no net attractive or repulsive gravitational force acting on any body.
This finding only breaks down when uniformity is lost, that is when the scale of examination becomes small or local enough that individual bodies (stars, galaxies, galactic clusters) must be taken into account.
A consequent deduction is that the Universe must be infinite, or else some parts of it would not have very-large-scale uniformity and so the zero net gravitational force would no longer apply. In such a case, the unbalanced section of the Universe would be expected to condense down toward the side of greater net gravity.