__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=gm_{0}.

Relevant substitutions are:

G
is substituted for 1/4pe_{0},

**g
** is substituted for **E**,

**O**
is substituted for **B**.

Hence the equivalent of Maxwell's equations are:

div
**g**=4pGr_{m}
(where r_{m} is the density of matter),

curl
**g**=-d**O**/dt,

div
**O**=0,

curl
**O**=(4pG/c^{2})**J**_{m}+(1/c^{2})d**g**/dt
(where **J**_{m} is the current density of matter).

Useful transformations are:

g_{x}'=g_{x},

g_{y}'=g(g_{y}-vO_{z})
(where v is the speed of frame K' relative to frame K),

g_{z}'=g(g_{z}+vO_{y}),

O_{x}'=O_{x},

O_{y}'=g(O_{y}+(v/c^{2})g_{z}),

O_{z}'=g(O_{z}-(v/c^{2})g_{y}).

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.