= Hilbert stress-energy tensor
{c}
{title2=$T_{ab}=\frac{2}{\sqrt{-g}}\frac{\delta S_m}{\delta g^{ab}}\quad(+---)$}
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 $\delta S_m=\tfrac12\int\sqrt{-g}\,T_{ab}\delta g^{ab}=-\tfrac12\int\sqrt{-g}\,T^{ab}\delta g_{ab}$. 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.
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