The first term is the Newtonian gravity and the second term is dipole gravity where the information regarding the rotational frequency and the geometrical shape of the object are included in the parameter "delta r" which is the relativistic dynamic shift of the center of mass. The direction of the shift vector of the center of mass is toward the direction where it moves away from the stationary state.

Note that the traditional interpretation of general relativity did not allow us to include this information regarding the rotational frequency. So, general relativity could not be used to explain any of the fast rotating cosmological objects. And most of the problems related to the fast rotating cosmological objects including the spiral galaxy was begging for explanation. This is one of the reasons we have MOND and other phenomenological theories to explain them.

For a rotating full sphere, we have to add another hemisphere in the opposite direction to the one above. Since the potential function is a scalar quantity, we can add as many units of dipoles as far as their coordinates are adjusted with respect to the one single coordinate system.

If you fold two dipoles in the exact opposite direction, at the identical center of mass, the dipole effect will be canceled. However, remember that the center of mass of the two hemispherical dipoles in a single sphere are separated by the distance r (in the rest state).

**So, they do not cancel in a rotating sphere.**

We can now see that the poles of the fast rotating spheroidal condensed objects can be strong repulsive(antigravity) gravity centers.

This effect can cause the phenomenon of the hollow poles and the rings of the Saturn, as well as the observed jets from the fast rotating neutron stars and also from the fast rotating black holes.

This effect can also cause the stationary communication satellites to oscillate back and forth from the equatorial plane of the Earth which has been observed and by now a well known fact due to the GPS misalignment problem.

The equatorial plane of the fast rotating cosmological body is a shallow potential dip of dipole gravity which causes the debris and the satellites to oscillate back and forth around this potential dip. As the debris lose kinetic energy due to the collision and other frictional effect, they will permanently settle down at the equatorial plane to form the layers of the rings.

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