Linearized curvature of a two-potential weak-field metric ($d{s}^2=-(1+2\Phi /c^2)(d{x}^0{)}^2+(1-2\Psi /c^2)d{\mathbf{x}}^2$)

For metric signature $(-,+,+,+)$, $x^0=ct$ and small dimensionless potentials, the metric perturbation is $h_{00}=-2\Phi /c^2$, $h_{ij}=-2\Psi {\delta }_{ij}/c^2$. Substitution into the linearized Ricci tensor and scalar formula givesThe contraction is $R=(4{\nabla }^2\Psi -2{\nabla }^2\Phi -6{\partial }_0^2\Psi )/c^2$, so $G^0{}_0=-2{\nabla }^2\Psi /c^2$. These signs use $R^{\mu }{}_{\nu \alpha \beta }{V}^{\nu }=[{\nabla }_{\alpha },{\nabla }_{\beta }]V^{\mu }$. Static zero-anisotropic-stress boundary conditions give $\Phi =\Psi$ and the Newtonian limit; without those assumptions the two potentials and their time derivatives must be retained.