= Ricci curvature sourced by a Klein-Gordon scalar field
{c}
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 $A$:
$$
T_{ab}=A\left[\nabla_a\phi\nabla_b\phi-\frac12g_{ab}\big((\nabla\phi)^2+m^2\phi^2\big)\right].
$$
Its <trace> is $T=A[(1-D/2)(\nabla\phi)^2-Dm^2\phi^2/2]$. Substitution into the <Trace-reversed Einstein field equations in D dimensions> cancels the metric-proportional gradient term and gives
$$
R_{ab}=\kappa A\left[\nabla_a\phi\nabla_b\phi+\frac{m^2}{D-2}g_{ab}\phi^2\right].
$$
The 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> $(\Box-m^2)\phi=0$ gives <stress-energy conservation>, since $\nabla^aT_{ab}=A(\Box\phi-m^2\phi)\nabla_b\phi$.<General Relativity by David Tong>[Sections 4.5.2–4.5.3, equations (4.195) and (4.197); general-D trace reversal worked out here]{ref}
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