Multipole analysis for linearizedf(R)gravity with irreducible Cartesian tensors

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 Relativitylike 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.

Multipole analysis for linearized $$f(R,{\mathcal {G}})$$ gravity with irreducible Cartesian tensors

“…This is the first purpose of the present paper. Ifh μν is a perturbation, the resulting field equations of linearized f (R, G) gravity are the same as those of linearized f (R) gravity [35], as expected. Furthermore, under this condition, the effective stress-energy tensor of GWs in linearized f (R, G) gravity, as a typical second-order nonlinear result, is shown to be the same as that of linearized f (R) gravity, which implies that the GB scalar G does not contribute to the effective stressenergy tensor of GWs in linearized f (R, G) gravity, though G usually plays an important role in the nonlinear effects as mentioned before.…”

“…In our preceding papers [35,36], referred to as I and II hereafter, respectively, the multipole analysis for linearized f (R) gravity with irreducible Cartesian tensors has been presented, and the relevant results include:…”

“…In this paper, we will follow the above process to discuss the multipole analysis for linearized f (R, G) gravity with irreducible Cartesian tensors. For f (R, G) gravity, similarly to f (R) gravity [35], one has to introduce the gravitational field amplitude h μν := √ −gg μν − η μν (1.2) and the effective gravitational field amplitudẽ…”

“…With the help of h μν andh μν , by using the same method in Refs. [35,38], the field equations of f (R, G) gravity can be rewritten in the form of obvious wave equations under the de Donder condition, and the source term, composed of all the nonlinear terms under the post-Minkowskian method, is the stress-energy pseudotensor of the matter fields and the gravitational field. This is the first purpose of the present paper.…”

“…3, by using the same method in Refs. [35,38], we show that the field equations of f (R, G) gravity can be rewritten in the form of obvious wave equations with the stress-energy pseudotensor of the matter fields and the gravitational field as the source under de Donder condition. Upon this, we derive the field equations and the effective stress-energy tensor of GWs for linearized f (R, G) gravity.…”