## 6 Massive, Bi- and Multi-Gravity Formulation: A Summary

The previous ‘deconstruction’ framework gave a intuitive argument for the emergence of a potential of the form (6.3*) (or (6.1*) in the vielbein language) and its bi- and multi-metric generalizations. In deconstruction or Kaluza–Klein decomposition a certain type of interaction arises naturally and we have seen that the whole spectrum of allowed potentials (or interactions) could be generated by extending the deconstruction procedure to a more general notion of derivative or by involving the mixing of more sites in the definition of the derivative along the extra dimensions. We here summarize the most general formulation for the theories of massive gravity about a generic reference metric, bi-gravity and multi-gravity and provide a dictionary between the different languages used in the literature.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)

with or in the metric language [144*], In what follows we will use the notation for the overall potential of massive gravity so that where is the standard GR Einstein–Hilbert Lagrangian for the dynamical metric and is the reference metric and for bi-gravity, where both and are then dynamical metrics.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

and the Lagrangians are defined as follows in arbitrary dimensions [144*] with and or equivalently in four dimensions [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.

Alternatively, we may express these scalars as follows [144*]

These are easily generalizable to any number of dimensions, and in dimensions we find such independent scalars.The multi-gravity action is a generalization to multiple interacting spin-2 fields with the same form for the interactions, and bi-gravity is the special case of two metrics (), [314*]

or

#### Inverse argument

We could have written this set of interactions in terms of rather than ,

with Interestingly, the absence of tadpole and cosmological constant for say the metric implies which in turn implies the absence of tadpole and cosmological constant for the other metric , , and thus .

#### Alternative variables

Alternatively, another fully equivalent convention has also been used in the literature [292] in terms of defined in (2.76*),

which is equivalent to (6.4*) with and or the inverse relation, so that in order to avoid a tadpole and a cosmological constant we need to set for instance and .

#### Expansion about the reference metric

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 as

where . The mass terms is then expressed as where the have the same expression as the in (6.9*) – (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 [278], it still has the correct structure to avoid any BD ghost, about any background [295*, 294*, 300, 297*].