= Solution
The <Einstein tensor> is $G^\mu{}_\nu=R^\mu{}_\nu-\tfrac12\delta^\mu_\nu R$. Thus $G^0{}_0=-R_{00}-R/2$, where the time raising introduces the minus sign. Substitution cancels the $\Phi$ and time-derivative terms in that component and gives
$$
\boxed{\begin{aligned}
G^0{}_0&=-\frac{2\nabla^2\Psi}{c^2}=-\frac{8\pi G\rho}{c^2},\\
G^i{}_j&=\frac{2\partial_0^2\Psi\delta^i_j+\nabla^2(\Phi-\Psi)\delta^i_j+\partial^i\partial_j(\Psi-\Phi)}{c^2}=0.
\end{aligned}}
$$
The right sides follow from the <Einstein field equations> $G^\mu{}_\nu=(8\pi G/c^4)T^\mu{}_\nu$ with zero <cosmological constant>. The full equations also contain the momentum constraint $G^0{}_i=-2\partial_0\partial_i\Psi/c^2=0$ for dust at rest to this order. Specifying arbitrary time-dependent potentials without that constraint would not give a solution of all the <Einstein field equations>.
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