f(R) gravity

An f(R) theory replaces the Ricci scalar in the gravitational action by a smooth function $f(R)$. Its metric field equation is

In the mostly-minus convention with vacuum gravitational action proportional to $\int \sqrt{-g}f(R)$ and $f^{\prime }<0$, the displayed conformal transformation turns its vacuum equation into ${\tilde{G}}_{\mu \nu }=T_{\mu \nu }(\varphi )$, with potential $V=(R{f}^{\prime }-f)/(2f^{\prime 2})$ and the stress-energy tensor of a canonical scalar field. The conformal Ricci formula cancels the second derivatives of $f^{\prime }$ against the original field equation. Its remaining derivative tensor is ${\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -{\tilde{g}}_{\mu \nu }(\tilde{\nabla }\varphi {)}^2/2$. Expressing $V$ as a function of $\varphi$ requires a locally invertible $R\mapsto f^{\prime }(R)$. When $f^{\prime \prime }=0$, this scalar-tensor parametrization degenerates and introduces no new propagating scalar degree of freedom.

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 $f(R)=C{R}^2$.

For a metric action $S_g=A\int \sqrt{-g}f(R)d^{4}x$ and the usual stress-energy tensor convention, covariant metric variation gives $A{E}_{ab}=T_{ab}/2$, where $E_{ab}=f^{\prime }{R}_{ab}-f{g}_{ab}/2+(g_{ab}\square -{\nabla }_a{\nabla }_b)f^{\prime }$. With $A=1/(16\pi G)$ this becomes $E_{ab}=8\pi G{T}_{ab}$; with $A=1$ it becomes $E_{ab}=T_{ab}/2$. The chosen normalization does not alter the vacuum equation.

If $f(0)=0$, $f^{\prime }(0)=1$, and $f^{\prime \prime }(0)=0$, then $f(R)=R+O(R^3)$ near zero. The field equations linearized about Minkowski spacetime are therefore the Linearized Einstein equations; deviations first appear beyond linear order.