= Solution
For a slowly moving particle in a static weak field, the spatial part of the <geodesic equation> reduces to
$$
\frac{d^2x^i}{dt^2}=-\Gamma^i{}_{00}.
$$
To first order in the perturbation,
$$
\Gamma^i{}_{00}
=-\frac12\delta^{ij}\partial_jh_{00},
$$
and hence
$$
\frac{dv^i}{dt}=\frac12\delta^{ij}\partial_jh_{00}.
$$
Newtonian acceleration is $-\delta^{ij}\partial_j\Phi$, so $h_{00}=-2\Phi$. Substitution into part (a), followed by the <Poisson equation> $\nabla^2\Phi=4\pi G\rho$, gives
$$
\kappa\rho=-\nabla^2h_{00}=2\nabla^2\Phi=8\pi G\rho.
$$
Thus the <Newtonian limit of general relativity> fixes
$$
\boxed{h_{00}=-2\Phi,\qquad \kappa=8\pi G}.
$$
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