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

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. The result of the Gravity Probe B experiment yields the limit $a \lesssim 5 \times 10^{11} \, \mathrm{m}^2$, whereas for the pulsar B in the PSR J0737-3039 system we get a bound which is about $10^4$ times larger. On the other hand the E\"ot-Wash experiment provides the best laboratory bound $a \lesssim 10^{-10} \, \mathrm{m}^2$. Although the former bounds from geodesic precession are much larger than the laboratory ones, they are still meaningful in the case some type of chameleon effect is present and thus the effective values could be different at different length scales.