The following question uses the analogy between EM (electromagnetism) and GM (gravitomagnetism).
In order to force two like electric charges nearby, some work has to be done. This implies that the following integral for the total energy increases: $$\int \frac{\epsilon_0 |E|^2}{2} dV \propto \int |E|^2 dV$$
Now, as EM's and GM's laws are analogous, the following similar integral must also increase in GM ($\vec{E}$ is just multiplied by some constant to obtain $\vec{g}$): $$\int |g|^2 dV$$
The problem is, however, that two like mass-charges attract. Thus total energy must decrease: $$\int K|g|^2 dV \propto \int |g|^2 dV$$ implying $K < 0$ - all gravitational energy is negative. I strongly suspect that the constant for gravitomagnetic field will also be negative.
Question: How can a system emit positive energy gravitational waves if the energy density of these waves should be negative-definite? I suspect the problem might have something to do with the inconsistency of GM, as I have read that like charges can't attract each other when the mediator is a spin-1 particle (gives rise to first-order tensor (vector) fields).
It should be noted that I am aware that the energy density of the curvature of metric is not that well-defined. However, I can still define some energy density in GM, even though it indeed might not be "real" energy in the general-relativistic sense. I can just treat the energy as a convenient mathematical tool without any physical interpretation, however due to analogies, theorems (such as energy conservation) still apply (so our radiating system should still gain energy).
