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

TL;DR

This paper develops a multipole analysis for linearized $f(R, ext{G})$ gravity, showing that the Gauss-Bonnet term does not affect gravitational wave energy, momentum, or angular momentum in the linear regime, despite its nonlinear significance.

Contribution

It provides a detailed multipole expansion for linearized $f(R, ext{G})$ gravity and clarifies the role of the Gauss-Bonnet term in gravitational wave properties.

Findings

Gauss-Bonnet scalar $ ext{G}$ does not contribute to gravitational wave stress-energy in linearized theory.

Linearized $f(R, ext{G})$ gravity's multipole expansion matches that of $f(R)$ gravity.

Energy, momentum, and angular momentum of gravitational waves are unaffected by $ ext{G}$ in the linear approximation.

Abstract

The field equations of $f(R,\mathcal{G})$ gravity are rewritten in the form of obvious wave equations with the stress-energy pseudotensor of the matter fields and the gravitational field, as their sources, under the de Donder condition. The linearized field equations of $f(R,\mathcal{G})$ gravity are the same as those of linearized $f(R)$ gravity, and thus, their multipole expansions under the de Donder condition are also the same. It is also shown that the Gauss-Bonnet curvature scalar $\mathcal{G}$ does not contribute to the effective stress-energy tensor of gravitational waves in linearized $f(R,\mathcal{G})$ gravity, though $\mathcal{G}$ plays an important role in the nonlinear effects in general. Further, by applying the $1/r$ expansion in the distance to the source to the linearized $f(R,\mathcal{G})$ gravity, the energy, momentum, and angular momentum carried by gravitational waves in linearized $f(R,\mathcal{G})$ gravity are provided, which shows that $\mathcal{G}$, unlike the nonlinear term $R^2$ in the gravitational Lagrangian, does not contribute to them either.