 Static scalar perturbation of Minkowski spacetime

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.