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 nonrelativistic limit, we get an additional Yukawa correction with coupling strength G/3 and Compton wave length root 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 less than or similar to 5 x 10(11) 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 Eot-Wash experiment provides the best laboratory bound a less than or similar to 10(-10) 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.