Derivations

Starting out one has to recognize the similarity of a moving mass and a moving charged particle, mass currents versus the more familiar electron charge currents.

Gravitomagnetic effects should exist, whether they are observable in all cases or not. The strength of the interactions are much different than the more familiar electromagnetic analogs.

Gyroscopes generally do not react with each other on a classical scale. The LAGEOS experiments define the difficulty of observing classical scale phenomena. Angular momentum interactions on a quantum mechanical scale are a little more obvious.

Consider the spin-orbit interaction of an electron in a hydrogen atom. Quantum mechanics handles things related to spins and orbits of particles on a quantum scale fairly well. Gravitomagnetism must correspond with these interactions.

Consider the universe as a sphere of matter with a uniform density and we are near the center, i.e. r << R. We have a gyro that we want to describe.

Define a standard vector potential integral with a relative current density distribution. The scalar gravitomagnetic potential may be easier to solve, on a practical note. The problem is similar to a rotating electric charge distribution in which the electric charges cannot move. The trick in understanding this is to realize that is it the relative rotation that defines this universal gravitomagnetism.

Neglect Lienard-Weichert type terms and retarded potentials related to the speed of gravitomagnetic interactions to be consistent with observation. Newton and Laplace have pointed out long ago that the speed of the inertial terms were much greater than light. When we are talking about currents we are talking about these inertial forces and description of nature must correspond with observation.

Find the scalar magnetic potential on a line (z axis of symmetry). Expand the potential in z. Find the full solution with zonal harmonics (r^n * Pn(cos?) or (r^-(n+1) * Pn(cos?) where Pn(cos?) are the Legendre polynomials.

Azimuthal symmetry is a requirement of this method, and since we are taking an average value for mass density, it is met for the model.

Take the gradient and find the universal gravitomagnetic field H and the flux density B.

Invert the curl operator (carefully) and get the corresponding vector potential A.

Normalize to what you see in classical mechanics, establishing the Newtonian limits. This is similar to the correspondence principle in quantum mechanics. This way you can go back and forth between classical mechanics and gravitomagnetics.