Given $(M,g)$ with $M$ a manifold of dimension $d>2$ and $g$ an Einstein metric (that is $Ric[g]\propto g$) consider a diffeomorphism $\phi: M\rightarrow M$ such that $\phi^*g=g'$. Is $g'$ necessarily Einstein?
I know that if $\phi$ is a conformal diffeomorphism, the conformal factor must satisfy some specific properties for the resulting metric $\phi^*g$ to be Einstein.