 Einstein equation in the sequential-derivative curvature convention

With signature $(+---)$, let $R^a{}_{bcd}={\partial }_d{\Gamma }^a{}_{bc}-{\partial }_c{\Gamma }^a{}_{bd}-{\Gamma }^a{}_{pc}{\Gamma }^p{}_{bd}+{\Gamma }^a{}_{pd}{\Gamma }^p{}_{bc}$ and $R_{ab}=R^c{}_{acb}$. This curvature is the negative of the convention in which $[{\nabla }_c,{\nabla }_d]X^a=R^a{}_{bcd}{X}^b$. The physical Einstein field equations are $G_{ab}+\Lambda g_{ab}=-8\pi G{T}_{ab}/c^4$. In a static weak field $R_{00}=-{\nabla }^2\Phi /c^2$; the trace-reversed equation then reproduces the Poisson equation ${\nabla }^2\Phi =4\pi G{\rho }_{\mathrm{mass}}$. Metric variation of $\int \sqrt{-g}(R-2\Lambda )/(2\kappa )+S_m$ gives the same sign when $\delta S_m=\int \sqrt{-g}{T}_{ab}\delta g^{ab}/2$.