Twofold Symmetries of the Pure Gravity Action

TL;DR

This paper introduces a novel twofold Lorentz symmetry in the pure gravity action, providing a simple, local, and invariant formulation that connects to double copy relations in scattering amplitudes.

Contribution

It constructs a new twofold Lorentz invariant form of the Einstein-Hilbert action using a specific parameterization and field redefinitions, revealing hidden symmetries in gravity.

Findings

The action is local and simple with twofold Lorentz invariance.

All off-shell Feynman diagrams are twofold Lorentz invariant.

On-shell tree amplitudes are automatically twofold gauge invariant.

Abstract

We recast the action of pure gravity into a form that is invariant under a twofold Lorentz symmetry. To derive this representation, we construct a general parameterization of all theories equivalent to the Einstein-Hilbert action up to a local field redefinition and gauge fixing. We then exploit this freedom to eliminate all interactions except those exhibiting two sets of independently contracted Lorentz indices. The resulting action is local, remarkably simple, and naturally expressed in a field basis analogous to the exponential parameterization of the nonlinear sigma model. The space of twofold Lorentz invariant field redefinitions then generates an infinite class of equivalent representations. By construction, all off-shell Feynman diagrams are twofold Lorentz invariant while all on-shell tree amplitudes are automatically twofold gauge invariant. We extend our results to curved spacetime and calculate the analogue of the Einstein equations. While these twofold invariances are hidden in the canonical approach of graviton perturbation theory, they are naturally expected given the double copy relations for scattering amplitudes in gauge theory and gravity.