Rather than starting directly with a Kaluza–Klein decomposition (discretization in Fourier space), we perform instead a discretization in real space, known as “deconstruction” of five-dimensional gravity [24*, 25*, 170*, 168*, 28*, 443*, 340*]. The deconstruction framework helps making the connection with massive gravity more explicit. However, we can also obtain multi-gravity out of it which is then completely equivalent to the Kaluza–Klein decomposition (after a non-linear field redefinition).
The idea behind deconstruction is simply to ‘replace’ the continuous fifth dimension by a series of sites separated by a distance . So that the five-dimensional metric is replaced by a set of interacting metrics depending only on .
In what follows, we review the procedure derived in [152*] to recover four-dimensional ghost-free massive gravity as well as bi- and multi-gravity out of five-dimensional GR. The procedure works in any dimensions and we only focus to deconstructing five-dimensional GR for sake of concreteness.
Before going further, let us first describe five-dimensional general relativity in its Einstein–Cartan formulation, where we introduce a set of vielbein , so that the relation between the metric and the vielbein is simply,
Under the torsionless condition, , the antisymmetric spin connection , is uniquely determined in terms of the vielbeins
The counting of the degrees of freedom in both languages is of course equivalent and goes as follows: In -spacetime dimensions, the metric has independent components. Covariance removes of them,10 which leads to independent degrees of freedom. In four-dimensions, we recover the usual independent polarizations for gravitational waves. In five-dimensions, this leads to degrees of freedom which is the same number of degrees of freedom as a massive spin-2 field in four dimensions. This is as expect from the Kaluza–Klein philosophy (massless bosons in dimensions have the same number of degrees of freedom as massive bosons in dimensions – this counting does not directly apply to fermions).
In the Einstein–Cartan formulation, the counting goes as follows: The vielbein has independent components. Covariance removes of them, and the additional global Lorentz invariance removes an additional , leading once again to a total of independent degrees of freedom.
In GR one usually considers the metric and the vielbein formulation as being fully equivalent. However, this perspective is true only in the bosonic sector. The limitations of the metric formulation becomes manifest when coupling gravity to fermions. For such couplings one requires the vielbein formulation of GR. For instance, in four spacetime dimensions, the covariant action for a Dirac fermion at the quadratic order is given by (see Ref. ),
In the bosonic sector, one can convert the covariant action of bosonic fields (e.g., of scalar, vector fields, etc…) between the vielbein and the metric language without much confusion, however this is not possible for the covariant Dirac action, or other half-spin fields. For these types of matter fields, the Einstein–Cartan Formulation of GR is more fundamental than its metric formulation. In doubt, one should always start with the vielbein formulation. This is especially important in the case of deconstruction when a discretization in the metric language is not equivalent to a discretization in the vielbein variables. The same holds for Kaluza–Klein decomposition, a point which might have been under-appreciated in the past.
The discretization process breaks covariance and so before staring this procedure it is wise to fix the gauge (failure to do so leads to spurious degrees of freedom which then become ghost in the four-dimensional description). We thus start in five spacetime dimensions by setting the gauge
In vielbein language, we fix the five-dimensional coordinate system and use four Lorentz transformations to set
One could in principle go ahead and perform the discretization directly at the level of the metric but first this would not lead to a consistent truncated theory of massive gravity.11 As explained previously, the vielbein is more fundamental than the metric itself, and in what follows we discretize the theory keeping the vielbein as the fundamental object.5.9*) then implies  condition appearing in Eq. (5.17*), which corresponds to the symmetric vielbein condition. This is a sufficient condition to allow for a formulation back into the metric language [410*, 314*, 172]. Note, however, that as mentioned in [152*], we have not assumed that this symmetric vielbein condition was true, we simply derived it from the discretization procedure in the five-dimensional gauge choice .
In terms of the extrinsic curvature, this implies
From the metric language, we thus see that the discretization procedure amounts to converting the extrinsic curvature to an interaction between neighboring sites through the building block .
In this subsection we focus on deriving a consistent theory of massive gravity from the discretization procedure (5.19*, 5.20*). For this, we consider a discretization with only two sites and will only be considered in the four-dimensional action induced on one site (say site 1), rather than the sum of both sites. This picture is analogous in spirit to a braneworld picture where we induce the action at one point along the extra dimension. This picture gives the theory of a unique dynamical metric, expressed in terms of a reference metric which corresponds to the fixed metric on the other site. We emphasize that this picture corresponds to a trick to build a consistent theory of massive gravity, and would otherwise be more artificial than its multi-gravity extension. However, as we shall see later, massive gravity can be seen as a perfectly consistent limit of multi (or bi-)gravity where the massless spin-2 field (and other fields in the multi-case) decouple and is thus perfectly acceptable.
To simplify the notation for this two-site case, we write the vielbein on both sites as , , and similarly for the metrics and . Out of the five-dimensional action for GR, we obtain the theory of massive gravity in four dimensions, (on site 1),
The theory of massive gravity (5.22*), or equivalently (5.23*) is one special example of a ghost-free theory of massive gravity (i.e., for which the BD ghost is absent). In terms of the ‘Stückelbergized’ tensor introduced in Eq. (2.76*), we see that2.83*) or (2.84*), but as we shall see, a generalization of the Fierz–Pauli mass term which remains free of the BD ghost as is proven in depth in Section 7. We emphasize that the idea of the approach is not to give a proof of the absence of ghost (which is provided later) but rather to provide an intuitive argument of why the mass term takes its very peculiar structure.
This mass term is not the unique acceptable generalization of Fierz–Pauli gravity and by considering more general discretization procedures we can generate the entire 2-parameter family of acceptable potentials for gravity which will also be shown to be free of ghost in Section 7.
Rather than considering the straight-forward discretization , we could consider the average value on one site, pondered with arbitrary weight ,12
In particular for the two-site case, this generates the two-parameter family of mass terms
We see that in the vielbein language, the expression for the mass term is extremely natural and simple. In fact this form was guessed at already for special cases in Ref. [410*] and even earlier in [502*]. However, the crucial analysis on the absence of ghosts and the reason for these terms was incorrect in both of these presentations. Subsequently, after the development of the consistent metric formulation, the generic form of the mass terms was given in Refs. [95*]13 and [314*].5.28*). Their explicit form is given in what follows in the relations (6.11*) – (6.13*) or (6.16*) – (6.18*).
This procedure is easily generalizable to any number of dimensions, and massive gravity in dimensions has -free parameters which are related to the discretization parameters.
In Section 5.2, we showed how to obtain massive gravity from considering the five-dimensional Einstein–Hilbert action on one site.14 Instead in this section, we integrate over the whole of the extra dimension, which corresponds to summing over all the sites after discretization. Following the procedure of , we consider sites to start which leads to multi-gravity [314*], and then focus on the two-site case leading to bi-gravity [293*].[314*] (see also  for other ‘topologies’ in the way the multiple gravitons interact), where each metric only interacts with two other metrics, i.e., with its closest neighbors, leading to -free parameters. For any fixed , one has , and .
To see the mass spectrum of this multi-gravity theory, we perform a Fourier decomposition, which is what one would obtain (after a field redefinition) by performing a KK decomposition rather than a real space discretization. KK decomposition and deconstruction are thus perfectly equivalent (after a non-linear – but benign15 – field redefinition). We define the discrete Fourier transform of the vielbein variables,
The counting of the degrees of freedom in multi-gravity goes as follows: the theory contains massive spin-2 fields with five degrees of freedom each and one massless spin-2 field with two degrees of freedom, corresponding to a total of degrees of freedom. In the continuum limit, we also need to account for the zero mode of the lapse and the shift which have been gauged fixed in five dimensions (see Ref. [443*] for a nice discussion of this point). This leads to three additional degrees of freedom, summing up to a total of degrees of freedom of the four coordinates .
Let us end this section with the special case of bi-gravity. Bi-gravity can also be derived from the deconstruction paradigm, just as massive gravity and multi-gravity, but the idea has been investigated for many years (see for instance [436, 324]). Like massive gravity, bi-gravity was for a long time thought to host a BD ghost parasite, but a ghost-free realization was recently proposed by Hassan and Rosen [293*] and bi-gravity is thus experiencing a revived amount of interested. This extensions is nothing other than the ghost-free massive gravity Lagrangian for a dynamical reference metric with the addition of an Einstein–Hilbert term for the now dynamical reference metric.
Let us consider a two-site discretization with periodic boundary conditions, with quantities at the site being identified with that at the site . Similarly, as in Section 5.2 we denote by and by ƒƒ the metrics and vielbeins at the respective locations and .5.25*) and we use the notation . We can equivalently well write the mass terms in terms of rather than as performed in (6.21*).
Notice that the most naive discretization procedure would lead to , but these can be generalized either ‘by hand’ by changing the weight of each site during the discretization, or by considering a non-trivial configuration along the extra dimension (for instance warping along the extra dimension16), or most simply by performing a conformal rescaling of the metric at each site.
Here, corresponds to a cosmological constant for the metric and the special combination , where the are the binomial coefficients is the cosmological constant for the metric , so only correspond to genuine interactions between the two metrics.
In the deconstruction framework, we naturally obtain and no tadpole nor cosmological constant for either metrics.
In this formulation of bi-gravity, both metrics and carry a superposition of the massless and the massive spin-2 field. As already emphasize the notion of mass (and of spin) only makes sense for a field living in Minkowski, and so to analyze the mass spectrum, we expand both metrics about flat spacetime,[300*]. Here we only focus on a background which preserves Lorentz invariance (in principle we could also include other maximally symmetric backgrounds which hae the same amount of symmetry as Minkowski).
Working about Minkowski, then to quadratic order in , the action for bi-gravity reads (for and ),2.37*). We see appearing the Fierz–Pauli mass term combination introduced in (2.44*) for the massive field with the effective mass defined as [293*] [293*], in the case where there is a large Hierarchy between the two Planck scales and , the massive particles is always the one that enters at the lower Planck mass and the massless one the one that has a large Planck scale. For instance if , the massless particle is mainly given by and the massive one mainly by . This means that in the limit while keeping fixed, we recover the theory of a massive gravity and a fully decoupled massless graviton as will be explained in Section 8.2.
In the two-sight derivative formulation, the action for matter is then5.6*) to five dimensions and applying the discretization procedure. Interestingly, in the case of fermions, the fields and would not directly couple to one another, but they would couple to both the vielbein at the same site and the one on the neighboring site.
Notice, however, that the current full proofs for the absence of the BD ghost do not include such couplings between matter fields living on different metrics (or vielbeins), nor matter fields coupling directly to more than one metric (vielbein).
In GR, diffeomorphism invariance uniquely fixes the kinetic term to be the Einstein–Hilbert one[287, 483, 175, 225, 76] for the uniqueness of GR for the theory of a massless spin-2 field).
In more than four dimensions, the GR action can be supplemented by additional Lovelock invariants  which respect diffeomorphism invariance and are expressed in terms of higher powers of the Riemann curvature but lead to second order equations of motion. In four dimensions there is only one non-trivial additional Lovelock invariant corresponding the Gauss–Bonnet term but it is topological and thus does not affect the theory, unless other degrees of freedom such as a scalar field is included.
So, when dealing with the theory of a single massless spin-2 field in four dimensions the only allowed kinetic term is the well-known Einstein–Hilbert one. Now when it comes to the theory of a massive spin-2 field, diffeomorphism invariance is broken and so in addition to the allowed potential terms described in (6.9*) – (6.13*), one could consider other kinetic terms which break diffeomorphism.
This possibility was explored in Refs. [231*, 310*, 230] where it was shown that in four dimensions, the following derivative interaction is ghost-free at leading order (i.e., there is no higher derivatives for the Stückelberg fields when introducing the Stückelberg fields associated with linear diffeomorphism),does not couple to matter. Note that this interaction can only be considered if the spin-2 field is massive in the first place, so this interaction can only be present if the Fierz–Pauli mass term (2.44*) is already present in the theory.
Now let us turn to a theory of gravity. In that case, we have seen that the coupling to matter forces linear diffeomorphisms to be extended to fully non-linear diffeomorphism. So to be viable in a theory of massive gravity, the derivative interaction (5.57*) should enjoy a ghost-free non-linear completion (the absence of ghost non-linearly can be checked for instance by restoring non-linear diffeomorphism using the non-linear Stückelberg decomposition (2.80*) in terms of the helicity-1 and -0 modes given in (2.46*), or by performing an ADM analysis as will be performed for the mass term in Section 7.) It is easy to check that by itself has a ghost at quartic order and so other non-linear interactions should be included for this term to have any chance of being ghost-free.
Within the deconstruction paradigm, the non-linear completion of could have a natural interpretation as arising from the five-dimensional Gauss–Bonnet term after discretization. Exploring the avenue would indeed lead to a new kinetic interaction of the form , where is the dual Riemann tensor [339*, 153*]. However, a simple ADM analysis shows that such a term propagates more than five degrees of freedom and thus has an Ostrogradsky ghost (similarly as the BD ghost). As a result this new kinetic interaction (5.57*) does not have a natural realization from a five-dimensional point of view (at least in its metric formulation, see Ref. [153*] for more details.)
We can push the analysis even further and show that no matter what the higher order interactions are, as soon as is present it will always lead to a ghost and so such an interaction is never acceptable [153*].
As a result, the Einstein–Hilbert kinetic term is the only allowed kinetic term in Lorentz-invariant (massive) gravity.
This result shows how special and unique the Einstein–Hilbert term is. Even without imposing diffeomorphism invariance, the stability of the theory fixes the kinetic term to be nothing else than the Einstein–Hilbert term and thus forces diffeomorphism invariance at the level of the kinetic term. Even without requiring coordinate transformation invariance, the Riemann curvature remains the building block of the kinetic structure of the theory, just as in GR.
Before summarizing the derivation of massive gravity from higher dimensional deconstruction / Kaluza–Klein decomposition, we briefly comment on other ‘apparent’ modifications of the kinetic structure like in – gravity (see for instance Refs. [89*, 354*, 46*] for massive gravity and their implications to cosmology).
Such kinetic terms à la are also possible without a mass term for the graviton. In that case diffeomorphism invariance allows us to perform a change of frame. In the Einstein-frame gravity is seen to correspond to a theory of gravity with a scalar field, and the same result will hold in massive gravity (in that case the scalar field couples non-trivially to the Stückelberg fields). As a result is not a genuine modification of the kinetic term but rather a standard Einstein–Hilbert term and the addition of a new scalar degree of freedom which not a degree of freedom of the graviton but rather an independent scalar degree of freedom which couples non-minimally to matter (see Ref.  for a review on -gravity.)