= Einstein metric condition in f(R) gravity
{c}
{title2=$f(4\Lambda)=2\Lambda f'(4\Lambda)$}
Every four-dimensional Einstein metric with $R_{ab}=\Lambda g_{ab}$ has constant <Ricci scalar> $R=4\Lambda$. The vacuum metric <f(R) gravity> tensor reduces to $(\Lambda f'(4\Lambda)-f(4\Lambda)/2)g_{ab}$, giving the displayed necessary and sufficient condition for a fixed <cosmological constant>. Both function and derivative may vanish, so division by $f'$ is unnecessary. Requiring this for all real $\Lambda$ and a smooth $f$ instead yields $f(R)=CR^2$.
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