f(R) gravity

Remarks

Varying $g$ gives $$f'(R) R_{ab} - \frac12 f(R)\, g_{ab} + \left(g_{ab}\square - \nabla_a\nabla_b\right) f'(R) = \kappa T_{ab}.$$ Here $R_{ab}$ is the Ricci tensor, $\nabla$ the covariant derivative, $\square = \nabla^a\nabla_a$ and $T_{ab}$ the stress–energy tensor. These equations are of fourth order in $g$ when $f'' \ne 0$. Example: $f(R) = R - 2\Lambda$ gives $f' = 1$, and the equations become Einstein's equations with cosmological constant $\Lambda$, from the Einstein–Hilbert action. Non-example: a Lagrangian $R_{ab}R^{ab}$ depends on the Ricci tensor and not only on $R$, so it does not define an $f(R)$ theory. When $f'' \ne 0$, setting $\phi = f'(R)$ rewrites the theory as a scalar–tensor theory: Brans–Dicke theory with $\omega = 0$ and the potential $V(\phi) = R\phi - f(R)$, with $R$ expressed through $\phi$. In the Palatini formalism, where the metric and the connection are varied independently, one obtains a different theory.