In one of the FGT YouTube videos of February 2008 it was mentioned that unlike a tug of war where the tension in the rope is the same throughout when the rope is stationary, gravity is one mass acting independently on another. In the case of two masses of spherical shape the force is known to be F_{12} = Gm_{1}m_{2}/r_{12}^{2}, which we can equate to m_{1}a or m_{2}a, depending on whether we want to calculate the acceleration of m_{1} or m_{2}, and where r_{12} is the distance between the centers of the two masses. With planetary masses that are close to spherical, and the objects they pull, this formula has been used most reliably, while it has been known for a long time by learned mathematical and scientific people that shape matters, and using center of mass with the standard physics book formula can cause increased error in some instances.

As an analysis that has already been done, let us use Kline Chapter 16, Sections 5, 6, and 7. Starting with Section 5, “Gravitational Attraction of Rods”, the example given is that of a “rod 6 feet long and of mass 18 pounds which is uniformly distributed” and “so thin that we think of it as extending in one dimension only. Three feet from one end of the rod and along the line of the rod is a small object of mass 2 pounds which we shall regard as located at one point.” Kline first calculates the force that the rod exerts on the 2-pound mass as though the entire mass of the rod were concentrated at its center, according to the standard F_{12} = Gm_{1}m_{2}/r_{12}^{2} formula, which comes out as equaling G poundals. Then he does the calculation properly using an integration over the rod, showing that the force that the rod exerts on the 2-pound mass is actually (4/3) G poundals. In Section 6, “Gravitational Attraction of Disks” [1], again a difference from the standard formula is shown which for comparison includes Exercise 1 from that section. Finally, in Section 7 it is shown that the standard physics book formula can be used with spheres.

Since the title of the referenced book includes “An Intuitive and Physical Approach”, intuition may tell us that, since the rod was integrated along its length to obtain a correct answer, every piece of a nearly one dimensional rod in the limit of a Riemann sum exerts a gravitational force independently on a separate point mass.

With General Relativity, unproven to date, a mass produces surface curvature in space, which may contact a series of points on other surfaces, due to another mass, through a tensor product [2] that reduces to a force vector. This may be interpreted as space-time surface interaction, and not necessarily complete independent action. At least two multi-million dollar projects are built and running ([3], [4]) and at least one is being developed [5] in attempts to prove General Relativity.

Special Relativity, on the other hand, E = mc^{2}, has long been proven correct, and the gravitational theory presented on this web site would not work without it. Some physicists in their writings seem to make a transition from Special to General Relativity as though the two are somehow linked, and in reality they bear no relationship to each other.

[1] Kline, Morris, Calculus, An Intuitive and Physical Approach, John Wiley and Sons, Inc., 1967, 1977; Dover (1998) unabridged republication.

[2] Rainich, George Yuri, Mathematics of Relativity, John Wiley and Sons, Inc., 1950, Chapter 4

[3] http://www.ligo.caltech.edu/

[4] http://www.nasa.gov/mission_pages/gpb/index.html

[5] http://lisa.nasa.gov/