 Trace coefficient fixed by stress-energy conservation

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 $\alpha \ne 1/4$,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.