 Solution

Let $\mathcal{L}=-\frac{1}{2}{g}^{\alpha \beta }{\partial }_{\alpha }\varphi {\partial }_{\beta }\varphi -V$. In the metric variation specified here, the covariant metric tensor $g_{\mu \nu }$ is varied while the scalar field is held fixed. Differentiating the inverse and the determinant givesThus $\delta \mathcal{L}=\frac{1}{2}{\partial }^{\mu }\varphi {\partial }^{\nu }\varphi \delta g_{\mu \nu }$, and the action varies asThe functional derivative therefore gives $T^{\mu \nu }={\partial }^{\mu }\varphi {\partial }^{\nu }\varphi +g^{\mu \nu }\mathcal{L}$. Lowering both indices yields the stress-energy tensor of a canonical scalar field:The determinant variation has a positive sign because the variable is $g_{\mu \nu }$; varying the inverse metric tensor instead would reverse that intermediate sign.