The parameter etha in equation (17) is necessary to smooth the mathematical artifacts of the singularities at the distance R/2 and -R/2 along the Z direction, which are the center of mass of the individual hemispheres in the rest frame and V(r) is the Newtonian gravity from the rotating spherical shell.
The above functional expression is the three dimensional dipole gravity potential which is responsible for the jets and the dark matter problem. It is noted that this dipole term doesn't exist in the traditionally known theory of general relativity. The jets and the dark matter problem are two different sides of the same coin of dipole gravity. One can not be explained without the other. Numerical simulation and the various curve fitting will start from the equation above. The information related to the rotational frequency and the longitudinal asymmetry of the object is contained in the parameter representing the gravitational dipole moment dz/2, where dz is given by the total mass M times delta rc.
where R is the radius of the rotating spherical shell and c the speed of light for a slow limit of rotational frequency(v/c<<1). For a fast rotational frequency, the following expression can be used for the shift of the center of mass which applies to the rotating spherical shell.
However, in most of the applications, it would be difficult to tell if the core of the rotating galaxy can be represented by a spherical shape or by something totally different. Regardless of the detailed shape, the rotational frequency and the asymmetry(asymmetricity) of the geometrical configuration of the source is reflected in the gravitational dipole moment dz. Hemisphere is only one of the many possible geometrical shapes that can create gravitational dipole moment. The importance of the dipole gravity potential may be in the overall topological consistency, when it comes to the explanation of the large scale cosmological problems, which means if the functional form has all the necessary coordinations to predict the correct results.
