  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.
