General Relativity admits general curved vacuum metrics (vacuum meaning: no matter anywhere), and many of them are useful theoretical approximations to real astrophysical systems. Famous ones include the Schwarzschild and Kerr metrics (both of which have T^{\mu\nu} = 0, where T is the stress-energy tensor), de Sitter and anti-de Sitter space, and Minkowski space. Useful ones include vacuum pp-waves, used in studying gravitational radiation from the perspective of an observer at large distance from the source. There's even the Sexl ultraboost, which can approximate ultrarelativistic motion between a black hole and a low-mass observer.
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
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