The deconstruction framework was proposed earlier (see Refs. [24, 25, 168*, 28, 443, 168, 170]) directly in the metric language and despite starting from a perfectly healthy five-dimensional theory of GR, the discretization in the metric language leads to the standard BD issue (this also holds in a KK decomposition when truncating the KK tower at some finite energy scale). Knowing that massive gravity (or multi-gravity) can be naturally derived from a healthy five-dimensional theory of GR is thus not a sufficient argument for the absence of the BD ghost, and a great amount of effort was devoted to that proof, which is known by now a multitude of different forms and languages.
Within this review, one cannot make justice to all the independent proofs that have been formulated by now in the literature. We thus focus on a few of them – the Hamiltonian analysis in the ADM language – as well as the analysis in the Stückelberg language. One of the proofs in the vielbein formalism will be used in the multi-gravity case, and thus we do not emphasize that proof in the context of massive gravity, although it is perfectly applicable (and actually very elegant) in that case. Finally, after deriving the decoupling limit in Section 8.3, we also briefly review how it can be used to prove the absence of ghost more generically.
We note that even though the original argument on how the BD ghost could be circumvented in the full nonlinear theory was presented in [137*] and [144*], the absence of BD ghost in “ghost-free massive gravity” or dRGT has been the subject of many discussions [12, 13, 345*, 342*, 95, 341, 344, 96*] (see also [350, 351*, 349*, 348*, 352] for related discussions in bi-gravity). By now the confusion has been clarified, and see for instance [295*, 294*, 400*, 346, 343*, 297*, 15*, 259*] for thorough proofs addressing all the issues raised in the previous literature. (See also  for the proof of the absence of ghosts in other closely related models).
Before going onto the subtleties associated with massive gravity, let us briefly summarize how the counting of the number of degrees of freedom can be performed in the ADM language using the Hamiltonian for GR. Using an ADM decomposition (where this time, we single out the time, rather than the extra dimension as was performed in Part I),
This result is fully generalizable to any number of dimensions, and in spacetime dimensions, gravitational waves carry polarizations. We now move to the case of massive gravity.
We now amend the GR Lagrangian with a potential . As already explained, this can only be performed by breaking covariance (with the exception of a cosmological constant). This potential could be a priori an arbitrary function of the metric, but contains no derivatives and so does not affect the definition of the conjugate momenta This translates directly into a potential at the level of the Hamiltonian density,6.4*).
If depends non-linearly on the shift or the lapse then these are no longer directly Lagrange multipliers (if they are non-linear, they still appear at the level of the equations of motion, and so they do not propagate a constraint for the metric but rather for themselves). As a result for an arbitrary potential one is left with degrees of freedom in the three-dimensional metric and its momentum conjugate and no constraint is present to reduce the phase space. This leads to degrees of freedom in field space: the two usual transverse polarizations for the graviton (as we have in GR), in addition to two ‘vector’ polarizations and two ‘scalar’ polarizations.
This counting is also generalizable to an arbitrary number of dimensions, in spacetime dimensions, a massive spin-2 field should propagate the same number of degrees of freedom as a massless spin-2 field in dimensions, that is polarizations. However, an arbitrary potential would allow for independent degrees of freedom, which is 1 too many excitations, always corresponding to one BD ghost degree of freedom in an arbitrary number of dimensions.
The only way this counting can be wrong is if the constraints for the shift and the lapse cannot be inverted for the shift and the lapse themselves, and thus at least one of the equations of motion from the shift or the lapse imposes a constraint on the three-dimensional metric . This loophole was first presented in  and an example was provided in [137*]. It was then used in [144*] to explain how the ‘no-go’ on the presence of a ghost in massive gravity could be circumvented. Finally, this argument was then carried through fully non-linearly in [295*] (see also [342*] for the analysis in dimensions as presented in [144*]).
Fierz–Pauli massive gravity is special in that at the linear level (quadratic in the Hamiltonian), the lapse remains linear, so it still acts as a Lagrange multiplier generating a primary second-class constraint. Defining the metric as , (where for simplicity and definiteness we take Minkowski as the reference metric , although most of what follows can be easily generalizable to an arbitrary reference metric ). Expanding the lapse as , we have and . In the ADM decomposition, the Fierz–Pauli mass term is then (see Eq. (2.45*))linear in the lapse. This is sufficient to deduce that it will keep imposing a constraint on the three-dimensional phase space variables and remove at least half of the unwanted BD ghost. The shift, on the other hand, is non-linear already in the Fierz–Pauli theory, so their equations of motion impose a relation for themselves rather than a constraint for the three-dimensional metric. As a result the Fierz–Pauli theory (at that order) propagates three additional degrees of freedom than GR, which are the usual five degrees of freedom of a massive spin-2 field. Non-linearly however the Fierz–Pauli mass term involve a non-linear term in the lapse in such a way that the constraint associated with it disappears and Fierz–Pauli massive gravity has a ghost at the non-linear level, as pointed out in . This is in complete agreement with the discussion in Section 2.5, and is a complementary way to see the issue.
In Ref. [111*], the most general potential was considered up to quartic order in the , and it was shown that there is no choice of such potential (apart from a pure cosmological constant) which would prevent the lapse from entering non-linearly. While this result is definitely correct, it does not however imply the absence of a constraint generated by the set of shift and lapse . Indeed there is no reason to believe that the lapse should necessarily be the quantity to generates the constraint necessary to remove the BD ghost. Rather it can be any combination of the lapse and the shift.
7.10*). This could have been seen more easily, without the need to explicitly integrating out the shift by computing the Hessian 7.10*), one has
Finally, one could also have deduce the existence of a constraint by performing the linear change of variable
To summarize, the condition to eliminate (at least half of) the BD ghost is that the of the Hessian (7.13*) vanishes as explained in [144*]. This was shown to be the case in the ghost-free theory of massive gravity (6.3*) [(6.1*)] exactly in some cases and up to quartic order, and then fully non-linearly in [295*]. We summarize the derivation in the general case in what follows.
Ultimately, this means that in massive gravity we should be able to find a new shift related to the original one as follows , such that the Hamiltonian takes the following factorizable forma. Keeping the Hamiltonian linear in the lapse and b. Keeping its own equation of motion independent of the lapse, which is sufficient to infer the presence of a primary constraint.
We now proceed by deriving the primary first-class constraint present in ghost-free (dRGT) massive gravity. The proof works equally well for any reference at no extra cost, and so we consider a general reference metric in its own ADM decomposition, while keep the dynamical metric in its original ADM form (since we work in unitary gauge, we may not simplify the metric further),7.4.
Proceeding similarly as in the previous example, we perform a change of variables similar as in (7.15*) (only more complicated, but which remains linear in the lapse when expressing in terms of ) [295*, 296*]
The field redefinition naturally involves a square root through the expression of the matrix in (7.21*), which should come as no surprise from the square root structure of the potential term. For the potential to be writable in the metric language, the square root in the definition of the tensor should exist, which in turns imply that the square root in the definition of in (7.21*) must also exist. While complicated, the important point to notice is that this field redefinition remains linear in the lapse (and so does not spoil the standard constraints of GR).
The Hamiltonian for massive gravity is thenno choice of potential which would keep it linear in the lapse beyond cubic order [111*]. However, as we shall see, when expressed in terms of the redefined shift , the non-linearities in the shift absorb all the original non-linearities in the lapse and . In itself this is not sufficient to prove the presence of a constraint, as the integration over the shift could in turn lead to higher order lapse in the Hamiltonian, 6.28*). For the purpose of this analysis it is easier to work with that notation.
The structure of the potential is so that the equations of motion with respect to the shift are independent of the lapse and impose the following relations in terms of ,
The two requirements defined previously are thus satisfied: a. The Hamiltonian is linear in the lapse and b. the equations of motion with respect to the shift are independent of the lapse, which is sufficient to infer the presence of a primary constraint. This primary constraint is derived by varying with respect to the lapse and evaluating the shift on the constraint surface (7.29*),[294*].
Let us imagine we start with initial conditions that satisfy the constraints of the system, in particular the modified Hamiltonian constraint (7.30*). As the system evolves the constraint (7.30*) needs to remain satisfied. This means that the modified Hamiltonian constraint ought to be independent of time, or in other words it should commute with the Hamiltonian. This requirement generates a secondary constraint,) but was then successfully derived in [294*] (see also [258, 259] and ). Deriving the whole set of Poisson brackets is beyond the scope of this review and we simply give the expression for the secondary constraint,
The important point to notice is that the secondary constraint (7.33*) only depends on the phase space variables and not on the lapse . Thus it constraints the phase space variables rather than the lapse and provides a genuine secondary constraint in addition to the primary one (7.30*) (indeed one can check that .).17 Rather than a constraint on , (7.36*) must be solved for the lapse. This is only possible if both the two following conditions are satisfied [294*], since these conditions do not vanish at the linear level (the constraints reduce to the Fierz–Pauli ones in that case), we can deduce that they cannot vanish non-linearly and thus the condition (7.36*) fixes the expression for the lapse rather than constraining further the phase space dofs. Thus there is no tertiary constraint on the phase space.
To conclude, we have shown in this section that ghost-free (or dRGT) massive gravity is indeed free from the BD ghost and the theory propagates five physical dofs about generic backgrounds. We now present the proof in other languages, but stress that the proof developed in this section is sufficient to infer the absence of BD ghost.
In bi- or multi-gravity where all the metrics are dynamical the Hamiltonian is pure constraint (every term is linear in the one of the lapses as can be seen explicitly already from (7.25*) and (7.26*)).
In this case, the evolution equation of the primary constraint can always be solved for their respective Lagrange multiplier (lapses) which can always be set to zero. Setting the lapses to zero would be unphysical in a theory of gravity and instead one should take a ‘bifurcation’ of the Dirac constraint analysis as explained in [48*]. Rather than solving for the Lagrange multipliers we can choose to use the evolution equation of some of the primary constraints to provide additional secondary constraints instead of solving them for the lagrange multipliers.
Choosing this bifurcation leads to statements which are then continuous with the massive gravity case and one recovers the correct number of degrees of freedom. See Ref.  for an enlightening discussion.
Another way to see the absence of ghost in massive gravity is to work directly in the Stückelberg language for massive spin-2 fields introduced in Section 2.4. If the four scalar fields were dynamical, the theory would propagate six degrees of freedom (the two usual helicity-2 which dynamics is encoded in the standard Einstein–Hilbert term, and the four Stückelberg fields). To remove the sixth mode, corresponding to the BD ghost, one needs to check that not all four Stückelberg fields are dynamical but only three of them. See also  for a theory of two Stückelberg fields.
Stated more precisely, in the Stückelberg language beyond the DL, if is the equation of motion with respect to the field , the correct requirement for the absence of ghost is that the Hessian defined as[145*]. This condition was successfully shown to arise in a number of situations for the ghost-free theory of massive gravity with potential given in (6.3*) or equivalently in (6.1*) in Ref. [145*] and then more generically in Ref. [297*].18 For illustrative purposes, we start by showing how this constraint arises in simple two-dimensional realization of ghost-free massive gravity before deriving the more general proof.
Consider massive gravity on a two-dimensional space-time, , with the two Stückelberg fields [145*]. In this case the graviton potential can only have one independent non-trivial term, (excluding the tadpole),[145*], and there are no propagating degrees of freedom. This is as expected for a massive spin-two field in two dimensions.
As shown in Refs. [144*, 145] the square root can be traded for an auxiliary non-dynamical variable . In this two-dimensional example, the mass term (7.43*) can be rewritten with the help of an auxiliary non-dynamical variable as
The full proof in the minimal model (corresponding to and and in (6.3*) or in the alternative formulation (6.23*)), was derived in Ref. [297*]. We briefly review the essence of the argument, although the full technical derivation is beyond the scope of this review and refer the reader to Refs. [297*] and  for a fully-fledged derivation.
Using a set of auxiliary variables (with , so these auxiliary variables contain ten elements in four dimensions) as explained previously, we can rewrite the potential term in the minimal model as [79, 342],2.77*) and is equivalent to used previously. Upon integration over the auxiliary variable we recover the square-root structure as mentioned in Ref. [144*]. We now perform an ADM decomposition as in (7.1*) which implies the ADM decomposition on the matrix , [297*]. The canonical momenta conjugate to is given by 19 on both side), 7.52*) is only consistent if we also have . This is the first constraint found in [297*] which is already sufficient to remove (half) the BD ghost, , we get on the constraint surface
When working about different non-Minkowski backgrounds, one can instead generalize the definition of the helicity-0 mode as was performed in [400*]. The essence of the argument is to perform a rotation in field space so that the fluctuations of the Stückelberg fields about a curved background form a vector field in the new basis, and one can then employ the standard treatment for a vector field. See also  for another study of the Stückelberg fields in an FLRW background.
Recently, a covariant Stückelberg analysis valid about any background was performed in Ref. [369*] using the BRST formalism. Interestingly, this method also allows to derive the decoupling limit of massive gravity about any background.
In what follows, we review the approach derived in [400*] which provides yet another independent argument for the absence of ghost in all generalities. The proofs presented in Sections 7.1 and 7.2 work to all orders about a trivial background while in [400*], the proof is performed about a generic (curved) background, and the analysis can thus stop at quadratic order in the fluctuations. Both types of analysis are equivalent so long as the fields are analytic, which is the case if one wishes to remain within the regime of validity of the theory.
Consider a generic background metric, which in unitary gauge (i.e., in the coordinate system where the Stückelberg background fields are given by ), the background metric is given by , and the background Stückelberg fields are given by .
We now add fluctuations about that background,
If the background configuration is not symmetric, then at every point one needs to perform first an internal Lorentz transformation in the Stückelberg field space, so as to align them with the coordinate basis and recover a symmetric configuration for the background Stückelberg fields. In this new Lorentz frame, the Stückelberg fluctuation is . As a result, to quadratic order in the Stückelberg fluctuation the part of the ghost-free potential which is independent of the metric fluctuation and its curvature goes symbolically as (7.60*) with replaced by , (with ). Interestingly, the Lorentz boost now plays the role of a mass term for what looks like a gauge field . This mass term breaks the symmetry, but there is still no kinetic term for , very much as in a Proca theory. This part of the potential is thus manifestly ghost-free (in the sense that it provides a dynamics for only three of the four Stückelberg fields, independently of the background).8.29*), (8.30*) and (8.31*) for . Starting with the symmetric configuration for the Stückelberg fields, then since we are working at the quadratic level in perturbations, one of the in the is taken to be the fluctuation , while the others are taken to be the background field . As a result in the first terms in in (7.61*) cannot come at the same time as or , and we can thus integrate by parts the time derivative acting on any , leading to a harmless first time derivative on , and no time evolution for .
As for the second type of term in (7.61*), since on the background field , the second type of terms is forced to be proportional to and cannot involve any at all. As a result is not dynamical, which ensures that the theory is free from the BD ghost.
Finally, to complete the argument, we consider the effect from background curvature, then , with . The space-time curvature is another source of ‘misalignment’ between the coordinates and the Stückelberg fields. To rectify for this misalignment, we could go two ways: Either perform a local change of coordinate so as to align the background metric with the flat reference metric (i.e., going to local inertial frame), or the other way around: i.e., express the flat reference metric in terms of the curved background metric, , in terms of the inverse vielbein, . Then the building block of ghost-free massive gravity is the matrix , defined previously as
Finally, we can also prove the absence of ghost for dRGT in the Vielbein formalism, either directly at the level of the Lagrangian in some special cases as shown in [171*] or in full generality in the Hamiltonian formalism, as shown in [314*]. The later proof also works in all generality for a multi-gravity theory and will thus be presented in more depth in what follows, but we first focus on a special case presented in Ref. [171*].
Let us start with massive gravity in the vielbein formalism (6.1*). As was the case in Part II, we work with the symmetric vielbein condition, . For simplicity we specialize further to the case where , so that the symmetric vielbein condition imposes . Under this condition, the vielbein contains as many independent components as the metric. The symmetric veilbein condition ensures that one is able to reformulate the theory in a metric language. In spacetime dimensions, there is a priori independent components in the symmetric vielbein.[171*], is then to use the Bianchi identities to infer an additional constraint of the form, [171*] that special choices of coefficients for the theory lead to remarkably simple analogous relations fully non-linearly. To give an example, we consider all the coefficients to vanish but . In that case the Bianchi identity (7.65*) implies 5.2*), the torsionless connection is given in term of the vielbein as 7.67*) then implies , so that we obtain an extra constraint of the form (7.66*) with . Ref.  derived similar constraints for other parameters of the theory.
We now turn to the proof for the absence of ghost in multi-gravity and follow the vielbein formulation of Ref. [314*]. In this subsection we use the notation that uppercase Latin indices represent -dimensional Lorentz indices, , while lowercase Latin indices represent the -dimensional Lorentz indices along the space directions after ADM decomposition, . Greek indices represent -dimensional spacetime indices , while the ‘middle’ of the Latin alphabet indices represent pure space indices . Finally, capital indices label the metric and span over .
Let us start with non-interacting spin-2 fields. The theory has then copies of coordinate transformation invariance (the coordinate system associated with each metric can be changed separately), as well as copies of Lorentz invariance. At this level may, for each vielbein , we may use part of the Lorentz freedom to work in the upper triangular form for the vielbein,7.6*) 7.6*) (now expressed in the vielbein variables) and are related to diffeomorphism invariance. In the vielbein language there is an addition primary constraints for each vielbein field
Now rather than setting part of the Lorentz frames to be on the upper diagonal form for all the vielbein (7.69*) we only use one Lorentz boost to set one of the vielbein in that form, say , and ‘unboost’ the other frames, so that for any of the other vielbein one has7.77*) lead to a Hamiltonian which is also linear in every shift and lapse, 6.1*) or (7.77*) ensures that there is one and only one vielbein with time-like index for every term .
Notice that for the interactions, the terms can depend on all the vielbeins and all the ‘boosts’ , (as mentioned previously, part of one Lorentz frame is set so that and is in the upper diagonal form). Following the procedure of , we can now solve for the remaining boosts by using of the shift equations of motion20 (i.e., there is no vielbein which does not appear at least once in the interactions (7.77*) which mix different vielbeins), the shift equations (7.79*) will involve all the boosts and can be solved for them without spoiling the linearity in any of the lapses . As a result, the lapses for are Lagrange multiplier for first class constraints. The lapse for the first vielbein combines with the remaining shift to generate the one remaining copy of diffeomorphism invariance.
We now have all the ingredients to count the number of dofs in phase space: We start with components in each of the vielbein and associated conjugate momenta, that is a total of phase space variables. We then have constraints21 associated with the . There is one copy of diffeomorphism removing phase space dofs (with Lagrange multiplier and ) and additional first-class constraints with Lagrange multipliers removing dofs. As a result we end up with