Gravitational waves energy source in linearized theory

Question

By linearizing the metric in the following way (approach in most textbooks):

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\text{ with } |h_{\mu\nu}|\ll 1$

and choosing the transverse-traceless gauge a wave equation for the perturbation $h_{\mu\nu}^{TT}$ is obtained (with $\Box=\eta_{\mu\nu}\partial^{\mu}\partial^{\nu}$:

$\Box h_{\mu\nu}=0$

Therefore possible solutions for the perturbations $h_{\mu\nu}$ are waves with $+$- resp. $\times$-polarization. These waves have an effect on test masses eg. deforming a ring according to the polarization of the wave. So by introducing little springs between the test masses the gravitational wave must carry energy (stored in the springs by friction). But what is the source of this energy?

In Gravitational Waves Vol. I by Michele Maggiore the stress-energy tensor of GWs $T_{\mu\nu}^{\text{GW}}$ is derived by linearizing the metric like this:

$g_{\mu\nu}=g_{\mu\nu}^B+h_{\mu\nu}$

Where $g_{\mu\nu}^B$ is the curved background. This makes sense to me, because GWs themselves are sources of curvature.

I'm confused by the first approach: The GWs seem to carry energy, but in the field equations $T_{\mu\nu}$ is set to zero. So where is this energy coming from? How should the waves be interpreted physically?

EDIT:

After thinking about it again: $T_{\mu\nu}$ is set to zero because we look at a region in spacetime, where there is no matter/energy density and describe how a the perturbation $h_{\mu\nu}$ behaves (like a wave). But the perturbation must be caused somewhere by some source. So the wave is coming from a region with a source (similar to the vacuum wave solution in electrodynamics).

I'm still a bit confused what the physical interpretation of these waves is.

Answer 1

Someone else can probably answer this more authoritatively and with better technical facility, but here is my rough idea of what is going on. I think you have two separate sets of issues to deal with in this interpretation.

One set of issues has to do with the fact that linearized gravity is incomplete and inconsistent. For example, linearized gravity can't describe bound systems, so it can't describe a realistic source such as two stars orbiting each other.

The other issues arise because of the way energy is represented in GR. GR (a) doesn't have a globally conserved, scalar measure of mass-energy that applies to all spacetimes, nor (b) does it have a local measure of the energy in the gravitational field. Issue a arises from the fact that energy-momentum is a vector, and you can't compare vectors that came from different places except through parallel transport, which is path-dependent. Issue b comes from the equivalence principle. Because of these issues, you should not expect to have a local energy density, such as a term in the stress-energy tensor, that represents the energy of the gravitatioal waves.

In asymptotically flat spacetimes, we do have various conserved measures of the total mass-energy of the spacetime, such as the Bondi energy and the ADM energy. The ADM energy, for example, does include energy being radiated away to null infinity by gravitational waves.

For a simple way to convince yourself that gravitational waves really do carry energy, consider that they manifest themselves in experiments as tidal forces, exactly like any other tidal forces, such as the ones that create the earth's ocean tides. Such tidal forces are certainly capable of doing work.