= Metric variation of a scalar-curvature coupling
{title2=$E_{ab}=\Phi G_{ab}+\nabla_a\nabla_b\Phi-g_{ab}\Box\Phi$}
With the <curvature sign convention> in which $\delta R_{bc}=\nabla_c\delta\Gamma^a{}_{ba}-\nabla_a\delta\Gamma^a{}_{bc}$, the inverse-metric <variation> of $\int\sqrt{-g}\,\Phi R$ is $\int\sqrt{-g}\,E_{ab}\delta g^{ab}$ plus a <boundary term>, for a <scalar> multiplier held fixed in that variation. The <Palatini identity> expresses the <curvature> variation as divergences of $\delta\Gamma$. Two <integrations by parts> move the <derivatives> onto $\Phi$, yielding the displayed terms. For constant $\Phi$ they vanish. The opposite <curvature> convention instead has $E_{ab}=\Phi G_{ab}+g_{ab}\Box\Phi-\nabla_a\nabla_b\Phi$.
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