Gravitational waves energy source in linearized theory

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.

• In acoustics, to write the equation of sound propagation in a medium, what one can do is to look at a slice of fluid and assess the forces on it, algebraically. That's because you describe how the fluid is allowed to move in a region with fluid only. The conclusion you get from an equation derived that way is "the form of free propagating wave obeys..." and nothing is said about the hypothetical source, nor the presence of such waves. Here it is quite the same, in the sense of describing how space-time may behave in absence of matter-energy content, at linear order. – Naptzer Sep 6 '18 at 21:37
• You're correct that $T_{\mu \nu} = 0$ because GW is propagating through empty space (like sunlight for EM waves). I can't vouch for more technical expertise, but wiki says: "the second time derivative of the quadrupole moment (or the l-th time derivative of the l-th multipole moment) of an isolated system's stress–energy tensor must be non-zero in order for it to emit gravitational radiation." It's good to draw an analogy with EM, where accelerating charges radiate. So just look for the place where Maggiore discusses quadrupole moment. – Avantgarde Sep 9 '18 at 2:21