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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 $(\Lambda f^{\prime }(4\Lambda )-f(4\Lambda )/2)g_{ab}$, giving the displayed necessary and sufficient condition for a fixed cosmological constant. Both function and derivative may vanish, so division by $f^{\prime }$ is unnecessary. Requiring this for all real $\Lambda$ and a smooth $f$ instead yields $f(R)=C{R}^2$.

In four dimensions, a local natural symmetric divergence-free metric tensor involving the metric and its derivatives of at most second order has the Lovelock form $a{G}_{ab}+b{g}_{ab}$ with constant coefficients. Generic nonlinear metric f(R) gravity instead has fourth-order terms and does not meet that derivative-order hypothesis. For affine $f$, the metric field equation is precisely of Lovelock form.

For the canonical scalar action and the stress-tensor convention in the paper, varying the inverse metric and volume element gives the stress-energy tensor of a canonical scalar fieldThe vacuum f(R) gravity equation, divided by $F$, isInsert this in the transformed Einstein tensor from part (a). Since $g_{\mu \nu }=-{\tilde{g}}_{\mu \nu }/F$, its algebraic term becomes ${\tilde{g}}_{\mu \nu }(RF-f)/(2F^2)$. Also $\varphi =\sqrt{3/2}\log (-F)$ gives ${\partial }_{\mu }\varphi =\sqrt{3/2}{\partial }_{\mu }F/F$. The remaining derivative tensor is exactly the kinetic part of $T_{\mu \nu }$: the combination $g_{\mu \nu }(\nabla F{)}^2$ equals ${\tilde{g}}_{\mu \nu }(\tilde{\nabla }F{)}^2$. ThusThis establishes the Einstein-frame representation of metric f(R) gravity in the specified normalization. To express the potential as a function of $\varphi$, choose a branch where $R\mapsto f^{\prime }(R)$ is locally invertible; the conformal metric already requires $f^{\prime }<0$.

In four dimensions, Lovelock's theorem restricts a local natural symmetric divergence-free metric tensor with at most second derivatives of the metric to a constant linear combination of the Einstein tensor and the metric. A generic nonlinear f(R) gravity equation contains ${\nabla }_a{\nabla }_b{f}^{\prime }(R)$: the Ricci scalar already has second metric derivatives, so this term generally introduces fourth metric derivatives. It therefore violates the second-order hypothesis, although it remains covariant, symmetric and divergence-free.

The exception must be stated. For $f(R)=aR+b$, the same tensor reduces towhich is precisely of Lovelock form. In particular, $f(R)=R$ is a counterexample to a blanket claim that the theorem never applies. The intended exclusion concerns generic nonlinear $f$, with $f^{\prime \prime }$ not identically zero. Restricting attention to a special constant-curvature solution of a nonlinear theory does not turn its off-shell equations into a universally second-order metric tensor.

Take compactly supported metric variations, or impose boundary conditions that remove the integration by parts terms. Let $F=f^{\prime }(R)$ and $\square ={\nabla }^a{\nabla }_a$. Since $\delta \sqrt{-g}=\frac{1}{2}\sqrt{-g}{g}^{ab}{h}_{ab}$, the gravitational action varies asApplying integration by parts twice to the derivative terms givesFor the usual stress-energy tensor definition $T_{ab}=-2(-g{)}^{-1/2}\delta S_m/\delta g^{ab}$, varying the covariant metric tensor gives $\delta S_m=\frac{1}{2}\int \sqrt{-g}{T}^{ab}{h}_{ab}{d}^{4}x$. Thus the action's stated normalization impliesThere is no implicit $1/(16\pi )$ in this gravitational action. This is the normalization of the metric f(R) field equation, rather than the commonly normalized version with $8\pi T_{ab}$ on the right.

Finally, the chain rule gives ${\nabla }_a{\nabla }_{b}F=f^{\prime \prime }{\nabla }_a{\nabla }_{b}R+f^{(3)}{\nabla }_{a}R{\nabla }_{b}R$ and $\square F=f^{\prime \prime }\square R+f^{(3)}(\nabla R{)}^2$. Substitution yieldsThis is a metric f(R) gravity variation: the connection is always the Levi-Civita connection of the varied metric tensor, not an independent variable.

Vary the gravitational volume factor and use part (a)(ii):Twice applying integration by parts moves both derivatives in $\delta R$ from the compactly supported $h_{ab}$ onto $f^{\prime }(R)$. Combining the result with the stated matter variation and requiring every coefficient of $h_{ab}$ to vanish gives the f(R) gravity equationTherefore