= Solution
{c}
For the <Newtonian gravitational potential> $\Phi$, the weak static <metric tensor> has $g_{00}\simeq-1-2\Phi/c^2$. The <Newtonian limit> derived above consequently gives
$$
R_{00}\simeq-\frac1{c^2}\nabla^2\Phi
=\frac12C\rho c^2,
\qquad
\nabla^2\Phi=-\frac12C c^4\rho.
$$
Matching the <Poisson equation for Newtonian gravity>, $\nabla^2\Phi=4\pi G\rho$, fixes
$$
\boxed{C=-\frac{8\pi G}{c^4}.}
$$
Here $G$ is the <gravitational constant> and $c$ is the <speed of light>. This negative coupling agrees with the source formula sheet. The familiar positive coupling uses the opposite <curvature sign convention> for the <Ricci tensor>; the physical <Poisson equation for Newtonian gravity> is the same.
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