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A real scalar field is minimally coupled to a metric tensor when its kinetic term uses the metric contraction of first derivatives and its scalar potential depends only on the field, without an explicit scalar curvature coupling. For metric signature $(-,+,+,+)$ the displayed Lagrangian and volume density $\sqrt{-g}$ give $\square \varphi -V^{\prime }(\varphi )=0$. Its stress-energy tensor is $T_{ab}={\nabla }_a\varphi {\nabla }_b\varphi -g_{ab}[\frac{1}{2}(\nabla \varphi {)}^2+V]$. The scalar stress-energy divergence identity explains its conservation on the field equation.

The metric dependence of the minimally coupled scalar field Lagrangian is $\partial L/\partial g^{\alpha \beta }=-\frac{1}{2}{\nabla }_{\alpha }\varphi {\nabla }_{\beta }\varphi$. Part (ii) therefore gives the stress-energy tensorUse metric compatibility and the symmetric Hessian of a scalar field to computeThis is the scalar stress-energy divergence identity, an instance of the diffeomorphism Noether identity for a scalar field. On the scalar equation of motion it gives stress-energy conservation, $\boxed{{\nabla }_{\mu }{T}^{\mu }{}_{\alpha }=0}$.