On the Potential for General Relativity and its Geometry

TL;DR

This paper introduces a novel vierbein formalism for ghost-free massive gravity, revealing new symmetries and geometric interpretations, and simplifies the decoupling limit analysis, with potential extensions involving Nambu-Goldstone bosons.

Contribution

It presents a new vierbein-based formulation of massive gravity with an additional auxiliary field and symmetry, enabling clearer geometric insights and easier analysis of vector interactions.

Findings

Identifies a new local SL(4) symmetry in the mass and potential terms.

Provides a geometric interpretation of gravity mass terms via volume forms.

Simplifies the derivation of vector mode interactions in the decoupling limit.

Abstract

The unique ghost-free mass and nonlinear potential terms for general relativity are presented in a diffeomorphism and local Lorentz invariant vierbein formalism. This construction requires an additional two-index Stuckelberg field, beyond the four scalar fields used in the metric formulation, and unveils a new local SL(4) symmetry group of the mass and potential terms, not shared by the Einstein-Hilbert term. The new field is auxiliary but transforms as a vector under two different Lorentz groups, one of them the group of local Lorentz transformations, the other an additional global group. This formulation enables a geometric interpretation of the mass and potential terms for gravity in terms of certain volume forms. Furthermore, we find that the decoupling limit is much simpler to extract in this approach; in particular, we are able to derive expressions for the interactions of the vector modes. We also note that it is possible to extend the theory by promoting the two-index auxiliary field into a Nambu-Goldstone boson nonlinearly realizing a certain space-time symmetry, and show how it is "eaten up" by the antisymmetric part of the vierbein.