  Maxwellian approximations to linear general relativity are revisited in light
of relatively recent results on the degrees of freedom in the linear
gravitational field. The well-known Maxwellian formalism obtained in harmonic
coordinates is compared with a Maxwellian formalism obtained under a coordinate
choice where each of the metric components corresponds to each of the
coordinate-invariant degrees of freedom of the linear gravitational field. The
coordinate freedom of general relativity can be exploited to cast the field
equations into Maxwellian form, but such forms can be mere mirages of the
coordinate choice -- mirages such as vector gravitational waves. A coordinate
choice that yields perfectly-Maxwellian field equations, will yield a force
equation that is not Lorentzian. If field definitions are chosen to obtain
Lorentz-like terms in the force equation, then Maxwellian forms are compromised
in the field equations. Many treatments of gravito-electromagnetism will make
inconsistent ordering choices between the field equations and force equations,
or else truncate terms of relevant order from the force equation. Often such
mistakes reflect an attempt to force exact Maxwellian analogs simultaneously in
both the field equations and the force equation, with the result that terms
dropped are as large as those kept.
