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

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.

39 pages, v2: reorganized the text, published version

References in corpus (9)

• Covariant Color-Kinematics Duality
• Twofold Symmetries of the Pure Gravity Action
• The Off-Shell Recursion for Gravity and the Classical Double Copy for currents
• Graviton particle statistics and coherent states from classical scattering amplitudes
• New recursions for tree-level correlators in (Anti) de Sitter space
• One-loop off-shell amplitudes from classical equations of motion
• Multiparticle solutions to Einstein's equations
• On the `simple' form of the gravitational action and the self-interacting graviton
• Quantum Off-Shell Recursion Relation

Cited by in corpus (3)

• Berends-Giele currents for extended gravity
• One-loop $N$-point correlators in pure gravity
• The off-shell expansion relation of the Yang-Mills scalar theory