Perturbations of General Relativity to All Orders and the General $n^{\rm th}$ Order Terms

TL;DR

This paper derives comprehensive all-order perturbation expressions for Einstein-Hilbert action and Einstein equations, introducing minimal building blocks and analyzing growth of terms, with applications to graviton scattering and potential extensions.

Contribution

It provides the first complete all-order perturbation formulas for Einstein-Hilbert action and equations, including general $n$-th order terms, using a unified approach.

Findings

Perturbation terms grow linearly with order.

Minimal building blocks generate all perturbations.

Validated results through graviton scattering amplitude calculations.

Abstract

We derive all-order expressions for perturbations of the Einstein-Hilbert action and the Einstein equation with the general $n$-th order terms. To this end, we employ Cheung and Remmen's perturbation conventions both in tensor density and the usual metric tensor formalisms, including the Einstein-dilaton theory. Remarkably, we find minimal building blocks that generate the entire perturbations for each of our formulations. We show that the number of terms of perturbations grows linearly as the order of perturbations increases. We regard our results as the reference and discuss how to derive perturbations in other conventions from the reference. As a consistency check, we compute graviton scattering amplitudes using the perturbiner method based on the perturbative Einstein equation. Finally we discuss how to generalise the results to curved backgrounds and incorporate additional matter.