On the Potential for General Relativity and its Geometry
Gregory Gabadadze, Kurt Hinterbichler, David Pirtskhalava, Yanwen Shang
TL;DR
The paper develops a diffeomorphism- and LLT-invariant vierbein formulation of ghost-free massive gravity, introducing a new two-index Stueckelberg field $\lambda^a_{\bar{a}}$ that enables an extended local $SL(4)$ symmetry for the mass terms and a geometric interpretation via volume forms. The approach yields a simpler decoupling limit and provides closed-form expressions for vector–scalar interactions in the DL through the $k$-vierbein, while reducing to the standard dRGT theory in unitary gauge. It further allows promoting $\lambda^a_{\bar{a}}$ to a dynamical Nambu-Goldstone sector $v^{a}_{\bar{a}}$, which can be eaten by the antisymmetric part of the vierbein, thereby extending ghost-free massive gravity to a dynamical antisymmetric field with an Anderson-like mechanism. The framework generalizes to $GL(4)$ and higher embedding dimensions, potentially yielding extra scalar (Galileon-like) modes and connections to Cuscuton dynamics in specific limits. Overall, the work clarifies the geometric structure of gravity masses, improves the decoupling-limit analysis, and points toward possible UV completions with richer symmetry and field content.
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.