Higher Derivative Gravity and Conformal Gravity From Bimetric and Partially Massless Bimetric Theory

TL;DR

The paper establishes a concrete link between ghost-free HR bimetric theory and higher derivative gravity, showing that eliminating the second metric yields a HD action that, at leading four-derivative order, reproduces HD gravity and, in the PM case, reduces to conformal gravity (CG). It demonstrates that the spin-2 ghost in HD gravity is an artifact of truncation to four derivatives, and that the full bimetric theory remains ghost-free; in particular, the PM bimetric theory corresponds to CG at the four-derivative level and, more generally, provides a ghost-free completion of CG. It also shows that the PM bimetric theory likely possesses an extra gauge symmetry beyond Weyl invariance, consistent with a six-mode spectrum away from de Sitter backgrounds, and that conformal gravity emerges as the linearized low-curvature limit of PM bimetric theory around Bach-flat backgrounds. The results have implications for nonlinear PM realizations, the structure of higher curvature theories, and potential generalizations to higher dimensions or vielbein formulations. Overall, the work provides a principled framework for understanding how ghost-free bimetric dynamics encode and extend higher derivative gravity, with CG serving as a key bridge.

Abstract

In this paper we establish the correspondence between ghost-free bimetric theory and a class of higher derivative gravity actions, including conformal gravity and New Massive Gravity. We also characterize the relation between the respective equations of motion and classical solutions. We illustrate that, in this framework, the spin-2 ghost of higher derivative gravity is an artifact of the truncation to a 4-derivative theory. The analysis also gives a relation between the proposed partially massless (PM) bimetric theory and conformal gravity, showing, in particular, the equivalence of their equations of motion at the 4-derivative level. For the PM bimetric theory this provides further evidence for the existence of an extra gauge symmetry and the associated loss of a propagating mode away from de Sitter backgrounds. The new symmetry is an extension of Weyl symmetry which also suggests the PM bimetric theory as a ghost-free completion of conformal gravity.