Linearised Gravity and the GEM Analogy

The move itself

Write the metric as flat spacetime plus a small perturbation, keep only terms first order in that perturbation, choose the harmonic gauge, and the field equations of General Relativity collapse into a set of wave equations whose structure is the same as Maxwell's. Out of that rearrangement fall two derived fields: a gravitoelectric field that reproduces Newtonian attraction, and a gravitomagnetic field sourced by mass currents rather than mass alone. The whole of gravitoelectromagnetism is that rearrangement and nothing more.

What makes the analogy respectable is that it is not a resemblance somebody noticed. It is an algebraic consequence of the linearisation, and its predictions have been measured. Frame dragging around a rotating mass — the gravitomagnetic term — has been checked to the percent level by satellite gyroscopes and by laser ranging to orbiting retroreflectors. When GEM says a spinning body drags inertial frames, it is not speaking figuratively.

Constitutive parameters, and where they stop being parameters

It is tempting, once the equations look like Maxwell's, to complete the parallel by defining a gravitational permittivity and permeability — an ε_g and a μ_g — so that a region of spacetime can be discussed as a medium with a response. Formally this is legitimate: they are combinations of G and c, arranged so that the wave speed comes out right, and they are useful bookkeeping for problems posed as circuits.

The difficulty is that in electromagnetism ε and μ are properties of matter that vary from material to material, which is what makes optics a design discipline. In gravity the corresponding quantities are constants of nature. There is no gravitational glass. When a paper speaks of engineering ε_g, what is being engineered is the effective response of a mass distribution inside the linear regime, not a property of spacetime. That distinction is the whole chapter.

The band

GEM holds where the field is weak, the sources move slowly compared with light, and the geometry is close enough to flat that second-order terms can be dropped without changing the answer. Inside that band it is not an analogy at all; it is an approximation with a controlled error. Outside it, the analogy fails in a specific and instructive way: Maxwell's equations are linear and General Relativity is not. Gravity gravitates. The moment the field is strong enough that its own energy is a significant source, the superposition that makes circuit reasoning possible is gone, and nothing in the GEM formalism warns you that it has happened.