Hidden Simplicity of the Gravity Action
Clifford Cheung, Grant N. Remmen
TL;DR
The paper develops new representations of the Einstein-Hilbert action that dramatically simplify graviton perturbation theory by recasting gravity in a cubic form with a single auxiliary field and, separately, in a highly streamlined gauge-fixed form where all interactions derive from the graviton kinetic term. This yields two coupled first-order equations equivalent to the Einstein equations and enables off-shell recursion relations for tree-level graviton amplitudes, along with explicit propagators and a compact, general n-point vertex formula. The authors further generalize these constructions to curved backgrounds and analyze enhanced symmetries, including a twofold Lorentz structure, and discuss connections to BCJ double copy ideas. The framework offers practical avenues for calculating graviton processes and studying classical and semiclassical gravity contexts, with potential extensions to loops, curved spacetimes, and matter couplings.
Abstract
We derive new representations of the Einstein-Hilbert action in which graviton perturbation theory is immensely simplified. To accomplish this, we recast the Einstein-Hilbert action as a theory of purely cubic interactions among gravitons and a single auxiliary field. The corresponding equations of motion are the Einstein field equations rewritten as two coupled first-order differential equations. Since all Feynman diagrams are cubic, we are able to derive new off-shell recursion relations for tree-level graviton scattering amplitudes. With a judicious choice of gauge fixing, we then construct an especially compact form for the Einstein-Hilbert action in which all graviton interactions are simply proportional to the graviton kinetic term. Our results apply to graviton perturbations about an arbitrary curved background spacetime.