= Trace coefficient fixed by stress-energy conservation
{title2=$R_{\mu\nu}-\alpha g_{\mu\nu}R=-\kappa T_{\mu\nu}$}
In four dimensions, the <contracted Bianchi identity> gives $(1/2-\alpha)\partial_\mu R=-\kappa\nabla^\nu T_{\mu\nu}$ for the displayed equation. Its trace is $(1-4\alpha)R=-\kappa T$. For $\alpha\ne1/4$,
$$
\nabla^\nu T_{\mu\nu}=\frac{1/2-\alpha}{1-4\alpha}\partial_\mu T.
$$
Compatibility with covariant <stress-energy conservation> for unrestricted conventional matter therefore requires $\alpha=1/2$, yielding the <Einstein tensor>. Other coefficients impose constant trace on conserved sources. The exceptional $\alpha=1/4$ equation instead requires $T=0$; conservation additionally makes $R$ constant. A trace-free equation sourced by $T_{\mu\nu}-g_{\mu\nu}T/4$ is a different system and does not impose vanishing matter trace.
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