Multipole analysis on gyroscopic precession in $f(R)$ gravity with irreducible Cartesian tensors

TL;DR

This paper derives the multipole expansion of the metric and gyroscopic precession effects in $f(R)$ gravity up to order $1/c^3$, highlighting the corrections to General Relativity for extended sources.

Contribution

It provides a detailed multipole analysis of gyroscopic precession in $f(R)$ gravity using irreducible Cartesian tensors, including corrections to GR and applications to extended sources.

Findings

Derived multipole expansions for gravitational field in $f(R)$ gravity.

Calculated gyroscopic precession including $f(R)$ corrections.

Showed that for nonzero acceleration, precession matches GR results up to $1/c^3$ order.

Abstract

In $f(R)$ gravity, the metric, presented in the form of the multipole expansion, for the external gravitational field of a spatially compact supported source up to $1/c^3$ order is provided, where $c$ is the velocity of light in vacuum. The metric consists of General Relativity-like part and $f(R)$ part, where the latter is the correction to the former in $f(R)$ gravity. At the leading pole order, the metric can reduce to that for a point-like or ball-like source. For the gyroscope moving around the source without experiencing any torque, the multipole expansions of its spin's angular velocities of gravitoelectric-type precession, gravitomagnetic-type precession, $f(R)$ precession, and Thomas precession are all derived. The first two types of precession are collectively called General Relativity-like precession, and the $f(R)$ precession is the correction in $f(R)$ gravity. At the leading pole order, these expansions can recover the results for the gyroscope moving around a point-like or ball-like source. If the gyroscope has a nonzero four-acceleration, its spin's total angular velocity of precession up to $1/c^3$ order in $f(R)$ gravity is the same as that in General Relativity.