 Solution

Write $\mathcal{L}=-\frac{1}{2}{g}^{\alpha \beta }{\partial }_{\alpha }\varphi {\partial }_{\beta }\varphi -V$. Vary the covariant metric while holding the scalar fixed. The supplied determinant and inverse-metric variations giveBy the definition $\delta S=\frac{1}{2}\int \sqrt{-g}{T}^{\mu \nu }\delta g_{\mu \nu }$, lowering its indices yields the stress-energy tensor of a canonical scalar field:The sign in the determinant variation is positive for varying $g_{\mu \nu }$ and would be negative for varying the inverse metric; matching that convention is essential.