Gravitational radiation from a perturbed metric other than Minkowski

You can linearize general relativity on any background. With an appropriate choice of gauge the resulting of equations of motion for the perturbation closely resemble a wave equation on a curved background.

This is explored in great depth in the literature. The most prominent spacetimes to expand around are:

1. FLRW cosmological spacetimes. to study gravitational waves in cosmology.
2. Black hole space times.

The first is important in cosmology. I will not say much about this as it is not my area of expertise.

In the second case it turns out solving directly for the metric perturbation is often not the most convenient thing. Instead one often solves for the perturbations of the Weyl scalars instead. It turns out that in the case of a black hole background, the equations of motion for the (perturbed) Weyl scalar $psi_4$ completely decouples from the other components of the curvature tensor. Moreover, it turns out that the metric perturbation can be recovered completely once you know $psi_4$. Top things off, $psi_4$ is gauge invariant. One can thus solve a single (complex) PDE, instead of a set of coupled PDEs for a rank-2 tensor field.

The equation of motion for $psi_4$ is known as the Teukolsky equation, and is closely related to the equation of motion for a massless scalar field.

The Teukolsky equation (and black hole perturbation theory in general) is used for studying gravitational waves in two main ways:

1. Linear perturbation theory can be used to study how a perturbed black hole (e.g. just after a merger) settles down to a stationary state. The modes of the linear field during ringdown are known as quasinormal modes. Here is a topical review regarding their calculation by Berti, Cardoso, and Starinets: 0905.2975

2. Black hole perturbation theory can also be used to study the evolution of a black hole binary under the emission of gravitational waves in the limit that one of the components of the binary is much heavier than the other. In this case the evolution of the binary and the resulting gravitational wave form can be systematically expanded in powers of the mass-ratio. This is known as "small mass-ratio" or "gravitational self-force" approach. Here is a recent review by Barack and Pound: 1805.10385.

   Despite this approach to modelling black hole binaries being primarily at small mass-ratio systems (such as extreme mass-ratio inspirals), it provides surprisingly accurate results even for equal mass binaries. (See 2006.12036)