Twofold Symmetries of the Pure Gravity Action

TL;DR

The paper addresses whether pure gravity can be reformulated with a manifest twofold Lorentz symmetry suggested by the gravity–gauge theory double copy. It develops an all-orders, local action equivalent to the Einstein–Hilbert action by exploring local field redefinitions and gauge fixing, then achieves index factorization with an exponential graviton basis and a doubled spacetime formulation. The main contributions are a simple, local twofold Lorentz invariant action, a twofold invariant propagator and vertices, and a curved-spacetime extension including all-orders Einstein equations in index-factorized form; the framework also connects to double field theory concepts. This work provides a new structural perspective on gravity that may illuminate the double copy at the level of the action and offer practical tools for perturbative and curved-space gravity computations.

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.