That doesn't mean there's anything wrong with the model. It's just GIGO.
Of course, it's not actually disappearing, just moving, but the original point was about detecting sharp changes in the gravitational waves. A quick Google search tells me that gravitational wave red shifting is a thing, and I imagine that with blackholes it's a very important phenomenon and area of study. And I would guess that there can also be interesting second-order effects that such a blackhole's movements have on the propagation of gravitational and electromagnetic waves from other objects.
Yes.
> with blackholes it's a very important phenomenon
Black holes can lense gravitational radiation emitted by background systems.
Most background systems we are likely to detect soon will involve black holes. But these are black holes in some sort of mutual orbit, rather than black holes simply moving across some system of celestial coordinates.
For black holes that are moving linearly at near the speed of light, the black hole's effect on the metric elongates like a pencil, with the field weak outside and growing strong towards the centre of the "lead" or graphite. This is similar to Lorentz-contracting the near region around the black hole, and one can generalize a bit and say that as the boost between an observer and any object increases, the object thins. In the ultra-ultrarelativistic limit, the object and all the strengthening-towards-infinity field values around it become infinitely thin.
As one's speed relative to a black hole gets very close to c, the black hole becomes quite easy to model as an exceptionally high-energy massless particle.
You get this effect when your small space capsule whizzes by our galaxy's central black hole at speeds near that of light too, and your small momentary perturbation basically affects the black hole not at all. Because Lorentz contraction is reciprocal, whizzing a black hole -- even a large one -- at ultrarelativistic speeds past the International Space Station is going to have very little effect on it.
We model this with the https://en.wikipedia.org/wiki/Aichelburg%E2%80%93Sexl_ultrab... metric of General Relativity and usually some gauge fixing and small perturbations.
Tossing a large black hole past the ISS at low speeds compared to light will really mess up the neighbourhood of the solar system, but your space capsule can pretty safely manage a slow-compared-to-light hyperbolic orbit around a large black hole without much problem (ignoring any accretion disc and twin "paradox" issues).
If the sun instantaneously vanished, we would see it disappear at the same instant as its gravitational effect stops, 8 minutes after the actual event occurred. For those 8 minutes while the light and gravitational information are in transit, the Earth will continue to revolve around a visible (though now nonexistant) sun.
In the same way as if the sun suddenly jerked ten million miles to the south, we would see it move at the same instant as its gravitational force vector changed, 8 minutes after the actual event occurred, but that's harder to keep in your head.
See also: https://physics.stackexchange.com/questions/100893/is-einste...
These are usually probed by adding test masses of some sort, letting them evolve along available trajectories. Some such test masses are pointlike, neutral, and nearly massless; others are some sort of classical or quantum field. In most cases, the goal is to keep T^{\mu\nu} negligible.
One can alternatively be lead by the stress-energy tensor, and may be tempted to call T^{\mu\nu} the matter tensor in that case. One typically chooses some vacuum background -- Minkowski space, usually, but any background can be used -- and then uses perturbation theory to capture how the chosen matter alters that background curvature. This is very common in cosmology.
> Except for the cosmological constant, but that's different
No, it's not different; one has flexibility to move the cosmological constant into the RHS for calculational convenience without having to change its interpretation as part of the background curvature: https://en.wikipedia.org/wiki/Lambdavacuum_solution