The general action for ghost-free (or dRGT) massive gravity [144*] in the vielbein language is [95*, 314*] (see however Footnote 13 with respect to Ref. [95*], see also Refs. [502, 410] for earlier work)[144*],
Both massive gravity and bi-gravity break one copy of diff invariance and so the Stückelberg fields can be introduced in exactly the same way in both cases where the Stückelbergized metric was introduced in (2.75*) (or alternatively ). Thus bi-gravity is by no means an alternative to introducing the Stückelberg fields as is sometimes stated.
In these formulations, (or the term proportional to ) correspond to a cosmological constant, to a tadpole, to the mass term and to allowed higher order interactions. The presence of the tadpole would imply a non-zero vev. The presence of the potentials without would lead to infinitely strongly coupled degrees of freedom and would thus be pathological. We recall that is given in terms of the metrics and as[144*] [292*]
We have introduced the constant ( and is nothing other than the cosmological constant) and the tadpole for completeness. Notice however that not all these five Lagrangians are independent and the tadpole can always be re-expressed in terms of a cosmological constant and the other potential terms.
We could have written this set of interactions in terms of rather than ,
6.4*) with and
In the vielbein language the mass term is extremely simple, as can be seen in Eq. (6.1*) with defined in (2.60*). Back to the metric language, this means that the mass term takes a remarkably simple form when writing the dynamical metric in terms of the reference metric and a difference as6.9*) – (6.13*) so is genuinely order in . The expression (6.27*) is thus at most quartic order in but is valid to all orders in , (there is no assumption that be small). In other words, the mass term (6.27*) is not an expansion in truncated to a finite (quartic) order, but rather a fully equivalent way to rewrite the mass Lagrangian in terms of the variable rather than . Of course the kinetic term is intrinsically non-linear and includes a infinite expansion in . A generalization of such parameterizations are provided in [300*].
The relation between the coefficients and is given by
The quadratic expansion about a background different from the reference metric was derived in Ref. [278*]. Notice however that even though the mass term may not appear as having an exact Fierz–Pauli structure as shown in , it still has the correct structure to avoid any BD ghost, about any background [295*, 294*, 300, 297*].