Imagine binary neutron stars inspiral, they generate gravitational wave which carries energy away and causing the pair to become ever closer. Since all wave must carry energy I wonder where do gravitational wave draws their energy from? I know mass is not a conserved quantity so i think it should be the gravitational potential energy converted into kinetic energy but how can vacuum spacetime gains this kinetic energy? Is this perhaps due to moving observer seeing different result compared to nearby observer?
In simple terms, if you think of the geometry as a field on a given spacetime, it does not simply consist of a pair of lumps at the position of each neutron star as one would intuitively expect in newtonian dynamics.
Instead, it is a smooth configuration extended to the whole spacetime. Using the ADM formalism, we can actually define a unique energy (ADM energy) corresponding to this geometry configuration.
When the two Neutron Stars get closer, although their "mass" remain constant, the ADM energy of the field is radiated away in the form of GWs. We can say that before the neutron stars collide, Gravitational waves feed directly from the "energy" of the "geometry field" itself.
Another way to approach the concept is as follows:
The “missing” mass/energy was indeed radiated away as gravitational waves. But that isn't due to mass escaping from a black hole – it comes from the potential energy in the positions and spins of the two black holes.