 Newtonian normalization of the Einstein field equations

Assume a metric theory in four dimensions, a Levi-Civita connection, covariant stress-energy conservation and field equations linear in second metric derivatives. The contracted Bianchi identity selects $G_{ab}=R_{ab}-g_{ab}R/2$. Its trace-reversed equation is $R_{ab}=\kappa (T_{ab}-g_{ab}T/2)+\Lambda g_{ab}$. In the convention $R_{00}\simeq -{\nabla }^2\phi$ and signature $(+---)$, nonrelativistic matter gives $R_{00}\simeq \kappa \rho /2$. The Poisson equation ${\nabla }^2\phi =4\pi G\rho$ fixes $\kappa =-8\pi G$. Reversing the curvature convention reverses this displayed coupling sign without changing the physical theory. The strong equivalence principle alone does not establish uniqueness without the stated additional assumptions.