Solution
= Solution
In the static limit, the supplied linearized Riemann tensor gives
$$
R_{00}=\delta^{ij}R_{i0j0}=-\frac12\nabla^2h_{00}.
$$
The trace of the <Einstein field equations> is $R=-\kappa T$. Here $T\simeq-\rho$, so $R=\kappa\rho$. The $00$ field equation is therefore
$$
R_{00}-\frac12\eta_{00}R=R_{00}+\frac12R=\kappa\rho,
$$
which yields $R_{00}=\kappa\rho/2$. Combining the two expressions for $R_{00}$ gives
$$
\boxed{-\nabla^2h_{00}=\kappa\rho.}
$$