= Static scalar perturbation of Minkowski spacetime
{title2=$ds^2=(1+2A)dt^2-(1+2C)\delta_{ij}dx^idx^j$}
For static scalar perturbations with no mixed time-space or spatial shear terms, the displayed <metric tensor> uses signature $(+---)$. With the reversed convention in <curvature-sign convention in Killing derivative identities>, its first-order <Einstein tensor> is $\delta G_{00}=2\Delta C$, $\delta G_{0i}=0$, and $\delta G_{ij}=\partial_i\partial_j(A+C)-\delta_{ij}\Delta(A+C)$. These follow by substituting the metric into the first-order <Christoffel symbols> and <Ricci tensor>, discarding products of perturbations. A static source with the <dust stress-energy tensor> obeys $\Delta C=-4\pi G\rho$ and the decaying spatial equation makes $A=-C$. For a point mass the <Newtonian potential of a point mass> gives $A=-GM/r$, $C=GM/r$. The spatial potential is necessary to determine light propagation; a time potential alone is insufficient.
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