= Einstein-frame representation of metric f(R) gravity
{c}
{title2=$\widetilde g_{\mu\nu}=-f'g_{\mu\nu},\quad\phi=\sqrt{3/2}\log(-f')$}
In the mostly-minus convention with vacuum gravitational action proportional to $\int\sqrt{-g}f(R)$ and $f'<0$, the displayed <conformal transformation> turns its vacuum equation into $\widetilde G_{\mu\nu}=T_{\mu\nu}(\phi)$, with potential $V=(Rf'-f)/(2f'^2)$ and the <stress-energy tensor of a canonical scalar field>. The conformal Ricci formula cancels the second derivatives of $f'$ against the original field equation. Its remaining derivative tensor is $\partial_\mu\phi\partial_\nu\phi-\widetilde g_{\mu\nu}(\widetilde\nabla\phi)^2/2$. Expressing $V$ as a function of $\phi$ requires a locally invertible $R\mapsto f'(R)$. When $f''=0$, this scalar-tensor parametrization degenerates and introduces no new propagating scalar degree of freedom.
Back to article page
