Maxwellian mirages in general relativity

The tensor field equations of general relativity, in their linear approximation, exhibit a well-known Maxwellian form quite similar to the vector field equations of electromagnetism. Indeed, electric-like and magnetic-like 3-forces arise in the geodesic equation from the 4 time components of the metric perturbation, as if they were an electromagnetic-like potential 4-vector. The gravitomagnetic force is famously known as frame dragging, or the Lense-Thirring effect. Yet the analogy between linear gravity and electromagnetism breaks down, both in the field equations and in the equations of motion. Because the Einstein equations provide only 6 independent equations in the metric potentials, 4 coordinate conditions are needed to close the system of equations. Yet the 6 independent equations map to 6 independent degrees of freedom of the gravitational field, and the choice of coordinates will in general mix the gravitational degrees of freedom among the metric potentials. Gravitoelectromagnetic waves are implied in harmonic coordinates, but they are actually a coordinatedependent mirage, and do not appear in transverse coordinates. The harmonic coordinate choice, based on identifications in the field equations, leads to Maxwellian gravito electromagnetic fields, but these fields do not appear in the force equation in a way one would expect from the Lorentz force law. If one chooses gravito-electromagnetic fields based on identifications in the force equation, then the force equation takes a form quite similar to the Lorentz force, but the fields do not appear in the field equations in strict Maxwellian form. In this way, the coordinate freedom of general relativity can lead to Maxwellian mirages in the field equations.