There is an issue I'm having regarding the Einstein field equation I would like clarification on... Given that: $$R_{\mu v} -\frac{1}{2}g_{\mu v}R=kT_{\mu v}$$ Why can't the $g_{\mu v}R$ term be converted to $R_{\mu v}$? turning the field equations into: $$R_{\mu v} -\frac{1}{2}R_{\mu v}=kT_{\mu v}$$ $$\frac{1}{2}R_{\mu v}=kT_{\mu v}$$
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5$\begingroup$ Because $g_{\mu\nu}R$ is not the same thing as $R_{\mu\nu}$, in general. Why do you think they're the same? If you address this, perhaps we can better clarify your doubt. $\endgroup$– peek-a-booCommented Jun 28, 2022 at 22:57
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$R = g^{\mu\nu} R_{\mu\nu}$ does NOT mean $R_{\mu\nu} = R g_{\mu\nu}$.
