In the gravitoelectromagnetism approximation of GR, we have equations analogous to Maxwell's equations with some sign changes. As pointed out in another post of mine, this leads to unstable run-away solutions (the gravitoelectromagnetism correspond to the case where both $\mu_{0}$ and $\epsilon_{0}$ are negative). In the post, I said,
Now suppose both $\epsilon_{0}$ and $\mu_{0}$ were negative. Then consider a loop of wire and you begin to drive the current in some direction more and more. Let's say the current is counterclockwise from the top view. Since $\mu_{0}$ is negative, the "reverse" of Ampere's law will create a magnetic field such that it goes into the loop from the top view. As current increases, so will $\vec{B}$, creating an electric field that points counterclockwise from the top view. By the force law written above, this will drive charges to go counterclockwise around the loop even more, increasing the current further. This again leads to a cascade effect.
A reply confirmed this point, saying,
Finally, flipping both of $ϵ_0$ and $μ_0$ does not lead to runaway solutions in "vacuum" (i.e., due solely to the field energy.) In this case, the above integral is "negative definite" rather than "positive definite". But these fields can also (for example) do work on electric charges, giving them mechanical kinetic energy; and the kinetic energy of these charges is always positive definite. So in this case, there still exists the possibility of runaway solutions in which charged objects get very large kinetic energies (with a positive energy of great magnitude) and the fields become arbitrarily large (with a negative energy of great magnitude.)
My question is, if the above is true, then how can the gravitoelectromagnetism approximation possibly be valid? We know, for example, that rotating planets don't destabilize themselves, but my "loop argument" can be used to argue that this is inherently an unstable system that must drive itself to higher and higher kinetic energies.
Are the quotes above just flat-out nonsense? I want to know if the posts that I linked to are actually true or if they're simply wrong.
