The gyroscopic frequency of metric $f(R)$ and generalised Brans-Dicke theories: constraints from Gravity Probe-B

Abstract

We confront the predicted gyroscopic precession (in particular the geodetic precession) from metric $f(R)$ theory with the data provided by the mission, Gravity Probe--B. We find the constraint, $|a_2| < 1.33\times 10^{12} \mathrm{m}^2$, where $a_2$ is the coefficient assessing the strength of the lowest order correction to the Einstein--Hilbert action for a metric $f(R)$ theory with $f$ analytic. This constraint improves over astrophysical bounds provided by massive black holes and planetary precession which are $|a_2|\gtrsim 10^{17} \mathrm{m}^2$ and $|a_2|\lesssim 1.2\times10^{18} \mathrm{m}^2$ respectively and it is complementary to the stringent ones provided by lab based experiments, like the E\"ot--Wash experiment. We also investigate the modification of our result for gyroscopic precession if the oblateness of Earth is taken into account by considering the quadrupole moment of Earth. Finally, we provide a generalisation of our result for the gyroscopic precession in the context of Brans--Dicke theories with a potential (recovering the previously derived results in the appropriate limits).