On the 1/c Expansion of f(R) Gravity

Understand

We derive for applications to isolated systems - on the scale of the Solar System - the first relativistic terms in the $1/c$ expansion of the space time metric $g_{\mu\nu}$ for metric $f(R)$ gravity theories, where $f$ is assumed to be analytic at $R=0$.

• For our purpose it suffices to take into account up to quadratic terms in the expansion of $f(R)$, thus we can approximate $f(R) = R + aR^2$ with a positive dimensional parameter $a$.
• In the non-relativistic limit, we get an additional Yukawa correction with coupling strength $G/3$ and Compton wave length $\sqrt{6a}$ to the Newtonian potential, which is a known result in the literature.
• As an application, we derive to the same order the correction to the geodetic precession of a gyroscope in a gravitational field and the precession of binary pulsars.