Weak gravitational field ($|\varphi |/c^2\ll 1$)

A weak gravitational field is a regime in which the metric differs little from a suitable flat reference metric. In a slow-motion, approximately stationary situation, $g_{tt}\simeq 1+2\varphi /c^2$ for signature $(+---)$, with Newtonian potential $\varphi$. Proper-clock rates then differ fractionally by $\Delta \varphi /c^2$. Small metric perturbations do not by themselves imply every velocity is nonrelativistic; slow motion is a separate approximation when deriving Newtonian dynamics.