I'm studying General relativity, and I want to clarify the qualitative nature of how gravitational waves propagate.
Simple is best, so I want to imagine a single binary black hole system orbiting in the $x,y$ plane at $(0,0,0)$ forever at the same radius.
Now the quadrupole formula:
$$\overline h_{ij}(\mathbf x,t) = \frac{2G}{|\mathbf x|c^6}\partial_t^2\int y^iy^jT_{00}(\mathbf y, t_r) d^3y$$
$$t_r := t - |\mathbf x|/c$$
$$\overline h_{\mu \nu} := h_{\mu \nu} - \eta_{\mu \nu}\eta^{\sigma \tau}h_{\sigma \tau}$$
tells us that anywhere in space, we have the same perturbation of the Minkowski metric $h_{\mu\nu}$, just scaled by $\frac1{|\mathbf x|}$ and appropriately delayed.
I imagine this situation like the binary system is a small particle oscillating in a block of jelly (gelatin dessert, not jam) with the whole block wobbling in the same plane, and the wobbling diminishing asymptotically.
Where I start to doubt this visualisation though, is when I here that gravitational waves are transverse. Specifically, it seems like gravitational radiation is propagating in the $x$ direction, which conflicts in my head with the fact that the metric is perturbed in this direction.
Is my picture of gravitational waves in some sense accurate? What does it mean to say that gravitational waves are transverse?
Edit: This animation seems to me to conclude that the image I have in my head is wrong, namely, it has a perturbation of the metric in the $z$ direction. I simply cannot reconcile this with the quadrupole formula, which gives no perturbation in the $z$ direction.
