In quantum mechanics we start with a Hamiltonian $H_0$ for which we know the exact eigenstates and energy eigenvalues. We perturb it by a known term $H$, and then attempt to compute (approximately) the new eigenstates and eigenvalues.

In general relativity, my understanding is we start with a metric $g_{\mu \nu}$, and perturb it by a known $h_{\mu \nu}.$ But in my lecture notes (http://arxiv.org/pdf/0804.2595.pdf), the lecturer shows how to compute $h_{\mu \nu}$. I thought we perturbed a system by a known quantity; can someone clarify the regular procedure of perturbation theory in general relativity, and what typical 'goals' are?

The only alternative I see is that we perturb a known solution $g_{\mu \nu}$ by an unknown perturbation $h_{\mu \nu}$, state how we would like the stress-energy $T_{\mu \nu}$ to change, and then try and compute $h_{\mu \nu}$ such that it does. Could this be the correct interpretation?

In quantum mechanics, we start with a Hamiltonian $H_0$ for which we know the exact eigenstates and energy eigenvalues. We perturb it by a known term $H$, and then attempt to compute (approximately) the new eigenstates and eigenvalues.

In general relativity, my understanding is we start with a metric $g_{\mu \nu}$, and perturb it by a known $h_{\mu \nu}.$ But in my lecture notes (https://arxiv.org/abs/0804.2595), the lecturer shows how to compute $h_{\mu \nu}$. I thought we perturbed a system by a known quantity; can someone clarify the regular procedure of perturbation theory in general relativity, and what typical 'goals' are?

The only alternative I see is that we perturb a known solution $g_{\mu \nu}$ by an unknown perturbation $h_{\mu \nu}$, state how we would like the stress-energy $T_{\mu \nu}$ to change, and then try and compute $h_{\mu \nu}$ such that it does. Could this be the correct interpretation?