Multipole analysis for linearized $f(R)$ gravity with irreducible Cartesian tensors

TL;DR

This paper develops a multipole analysis method for linearized $f(R)$ gravity using irreducible Cartesian tensors, revealing monopole and dipole radiation features and providing explicit expansions.

Contribution

It introduces a multipole expansion framework for linearized $f(R)$ gravity with irreducible Cartesian tensors, including scalar and tensor parts, and extends to massive Klein-Gordon fields.

Findings

Tensor part matches General Relativity's symmetric trace-free tensors

Scalar part predicts monopole and dipole radiation in $f(R)$ gravity

Provides explicit multipole expansion formulas

Abstract

The field equations of $f(R)$ gravity are rewritten in the form of obvious wave equations with the stress-energy pseudotensor of the matter fields and the gravitational field as its source under the de Donder condition. The method of multipole analysis in terms of irreducible Cartesian tensors is applied to the linearized $f(R)$ gravity, and its multipole expansion is presented explicitly. In this expansion, the tensor part is symmetric and trace-free and is the same as that in General Relativity, and the scalar part predicts the appearance of monopole and dipole radiation in $f(R)$ gravity, as shown in literature. As a by-product, the multipole expansion for the massive Klein-Gordon field with an external source in terms of irreducible Cartesian tensors and its corresponding stationary results are provided.