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For action $S=\int \sqrt{-g}[-\frac{1}{2}(\nabla \Phi {)}^2-\xi R{\Phi }^2]d^{4}x$, metric variation defines $\delta S=\frac{1}{2}\int \sqrt{-g}{T}^{ab}\delta g_{ab}{d}^{4}x$ and gives the displayed tensor. The contracted Bianchi identity and the scalar derivative commutator yield ${\nabla }^a{T}_{ab}=(\square \Phi -2\xi R\Phi ){\nabla }_b\Phi$, so the stress-energy tensor is conserved on the field equation.

A trace anomaly is a quantum contribution to the trace of a stress-energy tensor which was traceless on shell in the classical theory. Running renormalized couplings can supply such a contribution through their beta functions; curvature-dependent contributions can occur as well. Classical stress-energy tensor improvement alone does not remove quantum scale dependence.

In four spacetime dimensions, the trace reversal of a stress-energy tensor has the displayed form, where $T=g^{\mu \nu }{T}_{\mu \nu }$. Its trace is $-T$, so applying trace reversal twice returns the original tensor. Taking the trace of the Einstein field equations without a cosmological constant gives $R=-\kappa T$; substitution yields $R_{\mu \nu }=\kappa {\overline{T}}_{\mu \nu }$. In a general dimension $d>2$, solving for the Ricci tensor instead uses $T_{\mu \nu }-g_{\mu \nu }T/(d-2)$.

In a region with vanishing stress-energy tensor, the Einstein field equations reduce to $R_{\mu \nu }=0$ when the cosmological constant vanishes.

A vacuum spacetime has vanishing matter stress-energy tensor. In $D>2$ dimensions, tracing the Vacuum Einstein equations gives $R=2D\Lambda /(D-2)$ and hence $R_{ab}=2\Lambda g_{ab}/(D-2)$; in four dimensions this reduces to $R_{ab}=\Lambda g_{ab}$. With zero cosmological constant it is Ricci flat, although its full Riemann curvature tensor can be nonzero: a vacuum Kerr black hole and vacuum gravitational waves illustrate that distinction.

A virial current $V^{\mu }$ represents the trace of a conserved stress-energy tensor as a divergence. Then the dilation Noether current $D^{\mu }=x_{\nu }{T}^{\mu \nu }-V^{\mu }$ is conserved, since its divergence is $T^{\mu }{}_{\mu }-{\partial }_{\mu }{V}^{\mu }$. This establishes scale invariance; a traceless stress-energy tensor improvement is the extra condition needed for conformal currents.