= Einstein equation in the sequential-derivative curvature convention
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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> $\nabla^2\Phi=4\pi G\rho_{\rm mass}$. Metric variation of $\int\sqrt{-g}(R-2\Lambda)/(2\kappa)+S_m$ gives the same sign when $\delta S_m=\int\sqrt{-g}T_{ab}\delta g^{ab}/2$.
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