Linearized Ricci tensor and scalar
= Linearized Ricci tensor and scalar
To first order around flat spacetime,
$$
R_{\mu\nu}
=\frac12\left(
\partial_\rho\partial_\mu h^\rho{}_{\nu}
+\partial_\rho\partial_\nu h^\rho{}_{\mu}
-\mathop{\Box}h_{\mu\nu}
-\partial_\mu\partial_\nu h
\right),
$$
and
$$
R=\partial_\mu\partial_\nu h^{\mu\nu}-\mathop{\Box}h,
\qquad h=\eta^{\mu\nu}h_{\mu\nu}.
$$