Hilbert stress-energy tensor ($T_{ab}=\frac{2}{\sqrt{-g}}\frac{\delta S_m}{\delta g^{ab}}(+---)$)

The metric variational stress tensor is defined by the matter action's response to a metric change. With signature $(+---)$ and the corresponding usual matter-action signs, the displayed convention means $\delta S_m=\frac{1}{2}\int \sqrt{-g}{T}_{ab}\delta g^{ab}=-\frac{1}{2}\int \sqrt{-g}{T}^{ab}\delta g_{ab}$. Fix the independent matter variables while varying the metric. Sign conventions must be matched to the action; copying a mostly-plus stress formula into this convention can reverse its energy density.