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I'm wondering if there are some important restrctions on the 'applicability' of first order perturbation theory.

I know there's a way to deduce Schwarzschild's solution to Einstein's field equations that assummes one can decompose the 4D metric ##g_{\mu\nu}## as Minkowski ##\eta_{\mu\nu}## + perturbation ## h_{\mu\nu} = \epsilon \gamma_{\mu\nu} ## ignoring terms of order ##\epsilon^2## in subsequent calculations.

1. Why can we do so if Schwarzschild's (exterior) solution is supposed to work even in the vecinity of big black holes where one could think the metric wouldn't be just Minkowski + a small perturbation.

2. If I'm working in coordinates that force the spacetime to be conformally-flat (like in FLRW(##k=0##) using cartesian coordinates and conformal time) is it possible to decompose ##g_{\mu\nu}## as ## \eta_{\mu\nu} + h_{\mu\nu}## ?, even more, given that ## g_{\mu\nu} ## is conformally flat, ie: ## g_{\mu\nu}=\Omega\eta_{\mu\nu} ##, then ## h_{\mu\nu} = (\Omega-1)\eta_{\mu\nu} ## ?

Thanks in advance for any idea you can comment below.