Evidence Based Science: This Matter of Gravity

This is a very complicated issue. The inverse square law, a fundamental part of the theory of gravity, is under investigation. While it works just fine for calculating orbits, launching space craft and building bridges, it isn't exactly correct it seems. Was Newton wrong about his famous Law of Gravity?

An interesting article here, a new theory.

Outer space: A matter of gravity

by John D. Barrow
When Isaac Newton reflected upon his law of gravitation — the famous "inverse square law" which states that the force of gravitational attraction between two masses whose centres are separated by a distance r is inversely proportional to r² — he realised there was a big gap in his arguments. Filling that gap delayed the publication of his momentous Principia until 1687. He had been assuming that the gravitational force exerted by a sphere is exactly the same as that exerted by an idealised point of the same mass located at its centre — let's call it the spherical property. He had assumed that this was always true, but maybe it wasn't. It certainly made life easier to assume it to be true. Planets could be thought of as mere points feeling the gravitational pull of a point-like Sun as they traced out their oval orbits in his notebook, just as they did around the solar system.
Eventually, in the first part of the Principia, Newton was able to show that his assumption had been a good one, but along the way he discovered some other things that are at least as surprising. If the law of gravitational force between two masses separated by a distance r is proportional to rⁿ then the only values of n for which the spherical property holds are n = -2, which gives the inverse square law, and n = +1, giving the so-called "harmonic law". The other possible laws with different values of n wouldn't allow us to replace a sphere by just a single point of any mass and get the same results. Newton was no doubt relieved to discover that his intuition had been right all along.

Newton studied the n = +1 law because it was easy to solve but he discarded it as being of no scientific interest for the study of gravity because it required forces between masses to increase as their separation grew in space. He didn't know of any natural forces that behaved like that. Gravity certainly didn't. Nearly 125 years later, Pierre Laplace went a step further and showed (what Newton likely knew) that the most general possible force law that has the spherical property is just the sum of any n = -2 law and any n = +1 law. So a mass m is accelerated by another mass M by a gravitational force equal to
Gravitational Force = -GMm/r² + mλr,
where G is the Newtonian gravitational constant. Here, I have equipped one of the constants of proportionality, -GMm which gives Newton's law of gravity, with a minus sign because gravity is attractive. The other, λ, will be repulsive if λ is positive, and so it appears with a plus sign. The second term (λr) is strange and, like Newton before him, Laplace ignored it in matters of gravity....
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Testing the gravitational inverse-square law

If the universe contains more than three spatial dimensions, as many physicists believe, our current laws of gravity should break down at small distances

Nothing seems more certain than the "fact" that there are three dimensions of space. But can we be sure that there are only three dimensions? Imagine a tightrope walker balancing on a cable high above the ground. To the tightrope walker the cable is effectively a 1D object, because he only needs one coordinate to specify his position as he walks back and forth. But an ant, for instance, sees the cable as a 2D object, because it can crawl along and also around the cable.

Torsion pendulum

Today, increasing numbers of physicists are seriously questioning whether we are like tightrope walkers, unaware of the true number of dimensions in space. New ideas from theoretical physics suggest that the best way to discover the actual dimensionality of space is to study how the gravitational attraction between two objects depends on the distance between them.
The whole story on this here