 Newtonian limit

The Newtonian limit of general relativity describes slowly moving matter in a weak, approximately static gravitational field. With metric signature $(-,+,+,+)$ and $x^0=ct$, identify the Newtonian gravitational potential $\Phi$ byNeglecting time derivatives and terms of higher order in particle velocity divided by the speed of light, the spatial geodesic equation becomesThis recovers Newtonian gravitational acceleration rather than merely matching a metric coefficient.

For nonrelativistic matter, $T_{00}\simeq \rho c^2$ dominates the stress-energy tensor. With negligible anisotropic stress, the weak-field Einstein tensor gives $G_{00}\simeq 2{\nabla }^2\Phi /c^2$. Matching $G_{\mu \nu }=\kappa T_{\mu \nu }$ to the Poisson equationtherefore fixes $\kappa =8\pi G/c^4$. Setting $c=1$ gives the commonly written $h_{00}=-2\Phi$ and $\kappa =8\pi G$. Relativistic pressure, rapid time variation or strong gravitational fields requires terms omitted in this limit.