Gravitomagnetic Variables and Transformations
Gravitomagnetic theory is based upon two fundamental ideas: (1) The quantities for mass/mass interactions are related by Maxwellian-Lorentzian equations, with appropriate substitutions, and (2) The gravitational field vector (g), the gravitomagnetic field vector (O), and the gravitational mass (m) are mathematically imaginary. An important distinction between the two regimes is that q is considered to be invariant, whereas mass is assumed to have the celebrated dependence on speed: m=gm0.
Relevant substitutions are:
G is substituted for 1/4pe0,
g is substituted for E,
O is substituted for B.
Hence the equivalent of Maxwell's equations are:
div g=4pGrm (where rm is the density of matter),
curl O=(4pG/c2)Jm+(1/c2)dg/dt (where Jm is the current density of matter).
Useful transformations are:
gy'=g(gy-vOz) (where v is the speed of frame K' relative to frame K),
The equivalent of the Lorentz force law is:
F=m[g+(u x O)], where u is the velocity of the particle being acted upon. Note that since m, g and O are mathematically imaginary, a left hand rule must be used to determine the direction of F.
Again it is worth emphasizing that, unlike q in electromagnetic problems, m is presumably a function of particle speed.