Stress-energy tensor of a canonical scalar field ($T_{\mu \nu }={\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -g_{\mu \nu }[(\nabla \varphi {)}^2/2-V]$)

For signature $(+---)$ and $S_{\varphi }=\int \sqrt{-g}[(\nabla \varphi {)}^2/2-V(\varphi )]$, the convention $T_{\mu \nu }=2(\delta S_{\varphi }/\delta g^{\mu \nu })/\sqrt{-g}$ gives the displayed tensor. Varying the inverse metric yields the derivative product, while varying the volume element yields the term proportional to $g_{\mu \nu }$. A homogeneous field has energy density ${\dot{\varphi }}^2/2+V$ and pressure ${\dot{\varphi }}^2/2-V$.