Inertia, Mach's Principle and Gravitomagnetism

Gravitomagnetism is required to correspond to classical mechanics.

In the unrational form, flux density B is defined to be ?.

The source term for inertia is defined to be the integral over all space of the relative current distribution with respect to the mass in question. The propagation velocity is constrained to be consistent with observation of inertial forces and there will be no Lienard-Weichert terms. This integral is a vector potential and is one definition of Mach's principle.

Once you have this vector potential, A you can invert the curl and find B noting that the vector potential must reduce to v and the flux density must reduce to ? in their respective Newtonian limits.

The product of mass and the time derivative of the vector potential of a test object on a curved path is equivalent to Newton's F=ma in the Newtonian limit. Remember that a is dv/dt and A reduces to v in the Newtonian limit.

On a general curved path of radius r, with a velocity v, ?=v/r=B in the Newtonian limit. B is the curl of A, and ? is the curl of v.

The mv x B term reduces to the familiar radial term m?r x ? in the Newtonian limit, holding r constant.

The mdA/dt term reduces to the familiar tangential term F=mrd?/dt, in the Newtonian limit, holding r constant and with the constraint that the dot product of the unit vectors of ? and d?/dt is 1 or -1.

Think of the universe as a big ball of matter that you are moving inside. This is a definition of the source of inertia forces or gravitomagnetism. Compare this to charged particles moving inside a large solenoid magnet. While the classical interaction between moments may be weak, integration over the universe accounts for for the centrifugal-coriolis term.