Trace-reversed Einstein field equations in D dimensions

For $D>2$, tracing and substituting back gives

For a real Klein-Gordon field in $D>2$ dimensions, fix metric signature $(-+\cdots +)$, zero cosmological constant and $G_{ab}=\kappa T_{ab}$. Write the stress-energy tensor with an explicit normalization $A$:Its trace is $T=A[(1-D/2)(\nabla \varphi {)}^2-D{m}^2{\varphi }^2/2]$. Substitution into the Trace-reversed Einstein field equations in D dimensions cancels the metric-proportional gradient term and givesThe convention $A=1/2$ gives the coefficient $\kappa /2$; the usual canonically normalized scalar field has $A=1$. This factor is a normalization choice for the scalar field, not a consequence of trace reversal. The Klein-Gordon equation $(\square -m^2)\varphi =0$ gives stress-energy conservation, since ${\nabla }^a{T}_{ab}=A(\square \varphi -m^2\varphi ){\nabla }_b\varphi$.^[ref]