I'm studying linearized gravity, for which the weakness of the gravitational field is expressed as ability to decompose the metric into the (flat) Minkowski metric plus a small perturbation, i.e. $$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},\;\;\;\;\; \left|h_{\mu\nu}\ll 1\right|.$$ Let's restrict to coordinates in which $\eta_{\mu\nu}$ takes its canonical form, $\eta_{\mu\nu}=\mathrm{diag}\left(-1,+1,+1,+1\right)$. The assumption that $h_{\mu\nu}$ is "small" allows us to ignore anything that is higher than first order in this quantity. My teacher said that we thus obtain $$g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu},$$ where $h^{\mu\nu}=\eta^{\mu\rho}\eta^{\nu\sigma}h_{\rho\sigma}$.
I understand why we can raise and lower indices with $\eta$ (due to perturbative order of corrections), but - and I know it's silly - I can't see why there should be a relative minus sign in the expression for $g^{\mu\nu}$.
