f(R) Gravity and Its Scalar Representation

An $f(R)$ action changes Einstein gravity by replacing a term linear in the Ricci scalar with a nonlinear function. In the metric variational problem this introduces one scalar degree of freedom: the trace of the metric equation is a differential equation for $f_R\equiv df/dR$. An auxiliary-field transformation makes that scalar explicit, but only on a branch where the relevant Legendre map is regular and invertible. Varying an independent connection instead gives a different theory whose scalar is algebraically, rather than dynamically, determined by matter.

Required background. Gravitational Actions and Boundary Terms supplies the variational boundary conditions, while Curvature supplies the Ricci identities and curvature contractions used below.

Helpful background. Einstein Equations supplies the trace and Bianchi checks; Formulations of General Relativity clarifies metric and independent-connection variations; Quantum Gravity provides the neighboring EFT interpretation without turning this classical equivalence into a microscopic claim.

Metric f(R) field equations

Use the site signature $(-,+,+,+)$ and curvature convention, and let matter couple minimally to the metric. A useful Dirichlet problem fixes both the induced metric and $f_R$ at a non-null boundary, with the scalar-weighted Gibbons–Hawking–York completion. Fixing the metric alone is generally insufficient for this higher-derivative action. With those data, or with compactly supported bulk variations, take

$$S[g,\psi] =\frac{1}{16\pi G}\int_M d^4x\sqrt{-g}\,f(R) +S_m[g,\psi]+S_{\partial M}.$$

The two identities needed in the bulk variation are

$$\delta\sqrt{-g}=-\frac12\sqrt{-g}\,g_{ab}\delta g^{ab}, \qquad \delta R=R_{ab}\delta g^{ab} +g_{ab}\Box\delta g^{ab}-\nabla_a\nabla_b\delta g^{ab}.$$

Integrating the last two terms by parts gives

$$f_RR_{ab}-\frac12fg_{ab} +\left(g_{ab}\Box-\nabla_a\nabla_b\right)f_R =8\pi G T_{ab}.$$

This derivation and its boundary assumptions are developed in Sotiriou and Faraoni 2010, § II.A, equations (5)–(12). Taking the trace exposes the extra equation,

$$3\Box f_R+f_RR-2f=8\pi G T.$$

For $f(R)=R-2\Lambda$, $f_R=1$ and the derivative term vanishes, so both equations reduce to their Einstein counterparts. For a genuinely nonlinear $f$, the trace evolves the scalar quantity $f_R$. Covariant divergence gives zero by the Bianchi identity and the commutator acting on $f_R$; this is an independent conservation check, not another equation of motion.

Legendre transformation to a scalar

Introduce an auxiliary field $\chi$ before solving any field equation:

$$S[g,\chi,\psi] =\frac{1}{16\pi G}\int d^4x\sqrt{-g} \left[f(\chi)+f'(\chi)(R-\chi)\right]+S_m+S_{\partial M}.$$

Variation with respect to $\chi$ gives

$$f''(\chi)(R-\chi)=0.$$

On a branch with $f''(\chi)\ne0$, this fixes $\chi=R$. If in addition $\Phi=f'(\chi)$ is one-to-one on the branch, define the Legendre potential

$$V(\Phi)=\chi(\Phi)\Phi-f\!\left(\chi(\Phi)\right).$$

The action becomes

$$S[g,\Phi,\psi] =\frac{1}{16\pi G}\int d^4x\sqrt{-g} \left[\Phi R-V(\Phi)\right]+S_m+S_{\partial M},$$

which is the Jordan-frame Brans–Dicke action with $\omega=0$. Here the action contains $-V$; the potential called $V$ on the general scalar–tensor action page appears as $-2V$ and equals half the Legendre potential used here. Its equations are

$$\begin{aligned} \Phi G_{ab} &=8\pi G T_{ab}+\nabla_a\nabla_b\Phi-g_{ab}\Box\Phi -\frac12Vg_{ab},\\ R&=V_{,\Phi}. \end{aligned}$$

Eliminating $R$ with the metric trace yields

$$3\Box\Phi+2V-\Phi V_{,\Phi}=8\pi G T,$$

the same differential content as the $f(R)$ trace. The complete equivalence and the role of $f''\ne0$ are stated in Sotiriou and Faraoni 2010, § III.A, equations (50)–(63).

An Einstein-frame kinetic term can be obtained locally with $g^E_{ab}=\Phi g_{ab}$ and a logarithmic scalar coordinate, but this requires $\Phi\gt0$. Matter then couples to $\Phi^{-1}g^E_{ab}$, so the map is a field redefinition only when that transformed matter action, its clocks, and its sources are retained.

Metric and Palatini theories are not the same

In the Palatini problem, $g_{ab}$ and a torsion-free connection $\hat\Gamma^a{}_{bc}$ are varied independently:

$$S[g,\hat\Gamma,\psi] =\frac{1}{16\pi G}\int d^4x\sqrt{-g}\,f(\mathcal R) +S_m[g,\psi],$$

where $\mathcal R=g^{ab}\mathcal R_{ab}(\hat\Gamma)$ and the displayed matter action does not depend on $\hat\Gamma$. The equations are

$$f_{\mathcal R}\mathcal R_{(ab)}-\frac12fg_{ab}=8\pi G T_{ab}, \qquad \hat\nabla_c\!\left(\sqrt{-g}\,f_{\mathcal R}g^{ab}\right)=0.$$

The second equation makes $\hat\Gamma$ the Levi-Civita connection of $h_{ab}=f_{\mathcal R}g_{ab}$ on a nonsingular branch. Crucially, the trace is algebraic,

$$f_{\mathcal R}\mathcal R-2f=8\pi G T.$$

The scalar representation therefore has Brans–Dicke parameter $\omega=-3/2$ and no independent scalar wave equation. The derivation and the metric-affine qualification appear in Sotiriou and Faraoni 2010, §§ II.B–II.C and III.B–III.C, equations (14)–(22) and (64)–(70). If matter couples to the independent connection, even this Palatini representation changes; it must not be inferred from the metric theory.

A discriminating R plus R-squared check

Let $f(R)=R+aR^2$. Then

$$\Phi=1+2a\chi, \qquad V(\Phi)=\frac{(\Phi-1)^2}{4a}.$$

In metric $f(R)$ gravity, linearizing the trace about Minkowski space with $T=0$ gives

$$(\Box-m_0^2)\,\delta R=0, \qquad m_0^2=\frac{1}{6a}.$$

Thus $a\gt0$ gives a non-tachyonic scalar in this background, and $a\rightarrow0$ sends its mass above any fixed low-energy scale. This is a controlled Einstein limit only for fields and frequencies that remain below that growing mass and on the branch $\Phi\gt0$. At $f''=0$, at a turning point of $f'$, or where $f_R$ changes sign, the scalar coordinate or Einstein metric is singular; the equivalence cannot be continued by declaration.

This chapter analyzes classical actions, modes, and EFT validity. Empirical likelihoods and numerical bounds remain separate in Tests of Gravity, while claims about a microscopic completion remain in Quantum Gravity.

Common pitfalls

Call every f(R) formulation one theory. Metric, Palatini, and metric-affine variations have different equations and matter couplings. The choice of independent fields is part of the definition.

Use the scalar map through a singular point. The auxiliary action is globally equivalent only where $f''\ne0$ and the selected branch of $f'$ is invertible; an Einstein-frame map further requires $f_R\gt0$.

Exercises

For $f(R)=R+aR^2$, perform the auxiliary-field transformation and derive the linearized scalar mass. Then repeat the trace calculation in the Palatini theory and explain why no Klein–Gordon equation results.

Solution

The auxiliary equation gives $\chi=R$ for $a\ne0$, while $\Phi=1+2a\chi$ and $V=(\Phi-1)^2/(4a)$. The metric trace is $6a\Box R-R=0$ in vacuum, hence $(\Box-1/(6a))R=0$. In the Palatini theory the trace instead reads $(1+2a\mathcal R)\mathcal R-2(\mathcal R+a\mathcal R^2) =-\mathcal R=8\pi GT$. It fixes $\mathcal R=-8\pi GT$ algebraically, so there is no independently propagating scalar.

References

• Sotiriou, Thomas P., and Valerio Faraoni. 2010. “$f(R)$ Theories of Gravity.” Reviews of Modern Physics 82: 451–497. DOI.