Multipole expansion of the gravitational field in a general class of fourth-order theories of gravity and the application in gyroscopic precession

Abstract

A viable weak-field and slow-motion approximation method is constructed in $F(R,R_{\mu\nu}R^{\mu\nu}, R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma})$ gravity, a general class of fourth-order theories of gravity. By applying this method, the metric, presented in the form of the multipole expansion, outside a spatially compact source up to $1/c^3$ order is provided, and the closed-form expressions for the source multipole moments are all presented explicitly. The metric consists of the massless tensor part, the massive scalar part, and the massive tensor part, where the former is exactly the metric in General Relativity, and the latter two are the corrections to it. It is shown that the corrections bear the Yukawa-like dependence on the two massive parameters and predict the appearance of six additional sets of source multipole moments, which indicates that up to $1/c^3$ order, there exist six degrees of freedom beyond General Relativity within $F(R,R_{\mu\nu}R^{\mu\nu}, R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma})$ gravity. By means of the metric, for a gyroscope moving around the source without experiencing any torque, the multipole expansions of its spin's angular velocities of the Thomas precession, the geodetic precession, and the Lense-Thirring precession are derived, and from them, the corrections to the angular velocities of the three types of precession in General Relativity can be read off. These results indicate that differently from $f(R)$ or $f(R,\mathcal{G})$ gravity, the most salient feature of the general $F(R,R_{\mu\nu}R^{\mu\nu}, R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma})$ gravity is that it gives the nonvanishing correction to the gyroscopic spin's angular velocity of the Lense-Thirring precession in General Relativity.