# Can we correctly invoke energy conservation to explain why binary systems do collapse in GR?

I've learned that binary systems emits gravitational waves, so there is energy leaving the system in the form of waves and so the radius of the binary system must decrease as to maintain the conservation of energy. The explanations go a little more accurate but the idea is the one above. I'm in very trouble to believe this. Sure, using this leads to correct calculations even done by Einstein in the past century, but what I kind of guess is that there is another explanation in the context of General Relativity dynamical spacetime that can predict the same behaviour without bringing conservation of energy principle.
My reason goes like:

1. Imagine a Sun-Earth system, one very heavy body and a not heavy body, very far from the heavy one. The small body will have a trajectory due to the geodesics generated by the heavy one (the small one gives small correction to the binary system curvature supposedly), so the small one does Not have a local trajectory due to it's energy, it have a trajectory defined only by its initial conditions and the geometry generated by the big body. Then the energy radiated by gravitational waves does not seems to me to be connected directly to the big-small bodies relation, so it does not seems to be reasonable to invoke conservational principles.

2. Imagine a system like (1) but the small body with a very very large trajectory, like the system [SUN-object in a distance ten times the radius of Pluto]. It is reasonable to look for a small part of its path and it will look very much like it is in flat spacetime, since at such distance Sun will give almost no curvature to spacetime. Now, it still emits a wave-front of radiation by simply moving straight, that will be measured by some experiment nearby. So even the gravitational wave do have only one peak like a soliton it will constantly radiate energy. But theoretically, the system will decrease in radius of translation over time, but wait, what is locally the difference from a small part of this path in this system to a body moving in flat spacetime? So a moving body in a flat region of spacetime always emits a soliton like wavefront of gravitational wave (like a ""Cherenkov cone""), so it emits energy constantly, but its energy resembling it's movement Don't decrease, because it is only dictated by the local curvature and the initial velocity.

I accept easy to mathematical answers/helps, because this thought really bothers me.

$$E_{\rm bondi} = Mc^2 - k^\mu p_\mu,$$
where $$M$$ is the mass of the larger body, $$p_\mu$$ is 4-momentum of the smaller body, and $$k^\mu$$ is time translation Killing vector, normalized such that $$k^\mu k_\mu=-1$$ in the asymptotically flat region.
The quantity $$E= - k^\mu p_\mu$$ is called the geodesic energy of the smaller object. Through the GSF formalism one can also prove that the rate of change of $$E$$ averaged over an orbital timescale is equal to the average energy flux to infinity and absorbed by the bigger object (see e.g. gr-qc/0702054)