> My mental model of this is that it perturbs the aether ("spacetime") through which it travels, radiating waves which carry energy out of the system. Is there a different one?
Yes, and I'll get to that in my third last paragraph below.
Two isolated compact objects in mutual orbit generate a spacetime-filling "tumbling barbell" dynamical metric (as opposed to static or stationary; the two weights on the ends of the notional bar eventually collide), and gravitational radiation is simply a piece of the dynamical spacetime.
Let's slice the spacetime into spacelike hypervolumes indexed by time, and call our sources B and b, and our observer O.
Let's take two times, t_| and t_-. Pipes or hyphens are the not-really-there bar of the barbell connecting the weighted ends; in the schematic below - and | have essentially identical spatial length.
t_-:
B---b O
t_|:
B
|
| O
|
b
In a full 3+1 formalism of General Relativity we'd calculate the geodesics generated by B and b. More on that later. However, since O is weakly stressed and moving very slowly compared to c, and B and b are moving slowly with respect to c, we are firmly in the weak field limit and can
linearize [1].
In effect we can think pretty Newtonian: what attractive force does O feel? If O is a planet with its equator is on the extended line of the barbell at t_- then observers in the upper hemisphere will measure a gravitational attraction pointing equator-wards. Conversely, at t_| the same observers will measure a gravitational attraction pointing away from the equator.
Since Bb is tumbling (suppose it's clockwise about the middle hyphen or vertical bar in this case), the attraction sweeps equatorwards and anti-equatorwards, and we'll also see the times when the positions of B and b are essentially reversed in the two time-slices above.
The frequency of the changes from anti-equatorwards to equatorwards (in our schematic, that's the maximum stretch/squeeze) is the detection frequency, the periodic change in amplitude depends on how circular or elliptic the tumbling is, the amplitude grows as the orbit decays, and the amplitude itself depends on the masses and the distances among the objects.
Feynman's "sticky bead" [2] visualization is probably helpful here. If we replace our polar observers on O, and O itself, with a long pole (perpendicular to O's equator, which means we are perpendicular to the "bar" at t_-) with a couple of beads able to slide north-and-south, then as we approach t_- we expect the north bead to slide south, and the south bead to slide north. As we approach t_- we expect the north bead to slide north, and the south bead to slide south. The beads oscillate north-and-south matching the Bb orbit. If there is friction between the pole and the beads, heat is generated.
If we make the pole vanish and let the beads sit in space at the same positions they would be on the pole, we can see that their behaviour is simply geodesic; over short timescales the deviation from purely timelike geodesic motion is neglibible (over long timescales they may crash into each other or get drawn into the Bb system, since tumbling barbell metrics are collapsing spacetimes). The geodesics are curved in a patch of local Cartesian coordinates with the origin at the centre of the pole at some moment in time.
In the full GR picture the heat from the "sticky beads" on the pole is because the pole pulls the beads off their geodesics. Moving objects out of free-fall (i.e., off geodesics) requires work. This is a local phenomenon, General Relativity being a local theory.
However, the full GR picture is a bear to work with, and as there are several reasonable choices for slicing the 4-spacetime into 3+1 and as at astronomical distances post-Newtonian corrections are small, linearized gravity is the formalism of choice, and that choice drives vocabulary (and intuitions, especially in experiment design) somewhat. In effect, our BbO schematic system is similar to a https://en.wikipedia.org/wiki/Cavendish_experiment apparatus.
Finally, "... carry energy out of the system ..." depends on the system. In the full GR theory in our toy model above, the system is the entire spacetime. We could measure an energy at spacelike or lightlike infinity and see that it's constant [3]. However, if we measure in a region of spacetime encircling Bb but not O (or our sticky beads, or freely floating beads) then energy is clearly leaving that region. But what's special about that region, physically? Nothing. General Relativity does its thing, with moving masses generating curvature (which you can represent as perturbations of a static background), and curvature determining geodesic motion versus accelerated motion (which you can represent as moving sources dumping momentum into spacetime, and spacetime dumping momentum into matter, and that's about as close to an aether as you can get, but it is UNLIKE the luminiferous aether in crucial ways [4]).
- --
[1] https://en.wikipedia.org/wiki/Linearized_gravity which is where perturbations of a non-dynamical background enter formally
[2] https://en.wikipedia.org/wiki/Sticky_bead_argument
[3] We can rely on the https://en.wikipedia.org/wiki/Peeling_theorem to let us use Bondi mass M_B; cf. http://www2.phys.canterbury.ac.nz/ACGRG5/talks/Scholtz~Marti...
[4] http://www-history.mcs.st-andrews.ac.uk/Extras/Einstein_ethe... - pay close attention to the clause after the comma in the second-last sentence there. Einstein's main point is that flat spacetime is simply a special case of a general curved spacetime where the curvature is dominated by the sources (matter) or the absence thereof.
Also, it's important to remember that nobody's splitting up of 4-spacetime into 3+1 space+time is "most right"; in our toy model above we have chosen a particular foliation and have not introduced any pesky relativistic observers. That we can use a background (especially flat spacetime) and perturb against that does not mean the background is physically privileged.