Metric Lagrangians with two propagating degrees of freedom

TL;DR

This paper explores a broad class of metric Lagrangians with two propagating degrees of freedom, revealing new local forms related by non-local field redefinitions rooted in topological symmetries, with implications for quantum gravity.

Contribution

It demonstrates that many two-degree-of-freedom Lagrangians can be generated via non-local field redefinitions from Einstein-Hilbert, expanding the understanding of gravitational theories beyond general relativity.

Findings

Identifies a large family of local Lagrangians with two propagating degrees of freedom.

Shows these Lagrangians are connected through non-local field redefinitions.

Links the redefinitions to topological shift symmetry in BF theory and quantum gravity counterterms.

Abstract

There exists a large class of generally covariant metric Lagrangians that contain only local terms and describe two propagating degrees of freedom. Trivial examples can be be obtained by applying a local field redefinition to the Lagrangian of general relativity, but we show that the class of two propagating degrees of freedom Lagrangians is much larger. Thus, we exhibit a large family of non-local field redefinitions that map the Einstein-Hilbert Lagrangian into ones containing only local terms. These redefinitions have origin in the topological shift symmetry of BF theory, to which GR is related in Plebanski formulation, and can be computed order by order as expansions in powers of the Riemann curvature. At its lowest non-trivial order such a field redefinition produces the (Riemann)^3 invariant that arises as the two-loop quantum gravity counterterm. Possible implications for quantum gravity are discussed.