"Massive Gravity"
Claudia de Rham 
1 Introduction
2 Massive and Interacting Fields
2.1 Proca field
2.2 Spin-2 field
2.3 From linearized diffeomorphism to full diffeomorphism invariance
2.4 Non-linear Stückelberg decomposition
2.5 Boulware–Deser ghost
I Massive Gravity from Extra Dimensions
3 Higher-Dimensional Scenarios
4 The Dvali–Gabadadze–Porrati Model
4.1 Gravity induced on a brane
4.2 Brane-bending mode
4.3 Phenomenology of DGP
4.4 Self-acceleration branch
4.5 Degravitation
5 Deconstruction
5.1 Formalism
5.2 Ghost-free massive gravity
5.3 Multi-gravity
5.4 Bi-gravity
5.5 Coupling to matter
5.6 No new kinetic interactions
II Ghost-free Massive Gravity
6 Massive, Bi- and Multi-Gravity Formulation: A Summary
7 Evading the BD Ghost in Massive Gravity
7.1 ADM formulation
7.2 Absence of ghost in the Stückelberg language
7.3 Absence of ghost in the vielbein formulation
7.4 Absence of ghosts in multi-gravity
8 Decoupling Limits
8.1 Scaling versus decoupling
8.2 Massive gravity as a decoupling limit of bi-gravity
8.3 Decoupling limit of massive gravity
8.4 Λ3-decoupling limit of bi-gravity
9 Extensions of Ghost-free Massive Gravity
9.1 Mass-varying
9.2 Quasi-dilaton
9.3 Partially massless
10 Massive Gravity Field Theory
10.1 Vainshtein mechanism
10.2 Validity of the EFT
10.3 Non-renormalization
10.4 Quantum corrections beyond the decoupling limit
10.5 Strong coupling scale vs cutoff
10.6 Superluminalities and (a)causality
10.7 Galileon duality
III Phenomenological Aspects of Ghost-free Massive Gravity
11 Phenomenology
11.1 Gravitational waves
11.2 Solar system
11.3 Lensing
11.4 Pulsars
11.5 Black holes
12 Cosmology
12.1 Cosmology in the decoupling limit
12.2 FLRW solutions in the full theory
12.3 Inhomogenous/anisotropic cosmological solutions
12.4 Massive gravity on FLRW and bi-gravity
12.5 Other proposals for cosmological solutions
IV Other Theories of Massive Gravity
13 New Massive Gravity
13.1 Formulation
13.2 Absence of Boulware–Deser ghost
13.3 Decoupling limit of new massive gravity
13.4 Connection with bi-gravity
13.5 3D massive gravity extensions
13.6 Other 3D theories
13.7 Black holes and other exact solutions
13.8 New massive gravity holography
13.9 Zwei-dreibein gravity
14 Lorentz-Violating Massive Gravity
14.1 SO(3)-invariant mass terms
14.2 Phase m1 = 0
14.3 General massive gravity (m0 = 0)
15 Non-local massive gravity
16 Outlook

7 Evading the BD Ghost in Massive Gravity

The deconstruction framework gave an intuitive approach on how to construct a theory of massive gravity or multiple interacting ‘gravitons’. This lead to the ghost-free dRGT theory of massive gravity and its bi- and multi-gravity extensions in a natural way. However, these developments were only possible a posteriori.

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 [347] for the proof of the absence of ghosts in other closely related models).

7.1 ADM formulation

7.1.1 ADM formalism for GR

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),

( )( ) ds2 = − N 2dt2 + γij dxi + N idt dxj + N j dt , (7.1 )
with the lapse N, the shift i N and the 3-dimensional space metric γij. In this section indices are raised and lowered with respect to γij and dots represent derivatives with respect to t. In terms of these variables, the action density for GR is
∫ M-2Pl (√ --- [√ --- ]) ℒGR = 2 dt − gR + ∂t − g[k] (7.2 ) 2 ∫ = M-Pl dtN √ γ-((3)R [γ ] + [k]2 − [k2]), (7.3 ) 2
where (3) R is the three-dimensional scalar curvature built out of γ (no time derivatives in (3) R) and kij is the three-dimensional extrinsic curvature,
-1-( ) kij = 2N γ˙ij − ∇ (iNj) . (7.4 )
The GR action can thus be expressed in a way which has no double or higher time derivatives and only first time-derivatives squared of γij. This means that neither the shift nor the lapse are truly dynamical and they do not have any associated conjugate momenta. The conjugate momentum associated with γ is,
√ --- pij = ∂--−-gR-. (7.5 ) ∂˙γij
We can now construct the Hamiltonian density for GR in terms of the 12 phase space variables (γij and ij p carry 6 component each),
ℋ = N ℛ (γ,p) + N iℛ (γ,p). (7.6 ) GR 0 i
So we see that in GR, both the shift and the lapse play the role of Lagrange multipliers. Thus they propagate a first-class constraint each which removes 2 phase space degrees of freedom per constraint. The counting of the number of degrees of freedom in phase space thus goes as follows:
(2 × 6) − 2 lapse constraints − 2 × 3 shift constraints = 4 = 2 × 2, (7.7 )
corresponding to a total of 4 degrees of freedom in phase space, or 2 independent degrees of freedom in field space. This is the very well-known and established result that in four dimensions GR propagates 2 physical degrees of freedom, or gravitational waves have two polarizations.

This result is fully generalizable to any number of dimensions, and in d spacetime dimensions, gravitational waves carry d(d − 3)∕2 polarizations. We now move to the case of massive gravity.

7.1.2 ADM counting in massive gravity

We now amend the GR Lagrangian with a potential 𝒰. 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 pij This translates directly into a potential at the level of the Hamiltonian density,

i 2 ( i ) ℋ = N ℛ0 (γ, p) + N ℛi (γ,p) + m 𝒰 γij,N ,N , (7.8 )
where the overall potential for ghost-free massive gravity is given in (6.4*).

If 𝒰 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 (2 × 6) 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 6 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.

These 6 polarizations correspond to the five healthy massive spin-2 field degrees of freedom in addition to the sixth BD ghost, as explained in Section 2.5 (see also Section 7.2).

This counting is also generalizable to an arbitrary number of dimensions, in d spacetime dimensions, a massive spin-2 field should propagate the same number of degrees of freedom as a massless spin-2 field in d + 1 dimensions, that is (d + 1)(d − 2)∕2 polarizations. However, an arbitrary potential would allow for d(d − 1)∕2 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 γij. This loophole was first presented in [138] 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 1 + 1 dimensions as presented in [144*]).

7.1.3 Eliminating the BD ghost

Linear Fierz–Pauli massive gravity

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 hμν = MPl (gμν − ημν), (where for simplicity and definiteness we take Minkowski as the reference metric fμν = η μν, although most of what follows can be easily generalizable to an arbitrary reference metric fμν). Expanding the lapse as N = 1 + δN, we have h00 = δN + γijN iN j and h0i = γijN j. In the ADM decomposition, the Fierz–Pauli mass term is then (see Eq. (2.45*))

(2) − 2 1-( 2 2) 𝒰 = − m ℒFPmass = 8 hμν − h 1 ( ( )) = -- h2ij − (hii)2 − 2 N 2i − δN hii , (7.9 ) 8
and is linear in the lapse. This is sufficient to deduce that it will keep imposing a constraint on the three-dimensional phase space variables {γij,pij} 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 [75]. 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 hμν, 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 μ i N = {N, N }. 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.

Example on how to evade the BD ghost non-linearly

As an instructive example presented in [137*], consider the following Hamiltonian,

ℋ = N &tidle;𝒞0(γ,p) + N i𝒞 &tidle;i(γ,p) + m2 𝒰 , (7.10 )
with the following example for the potential
i j 𝒰 = V (γ,p)γijN--N--. (7.11 ) 2N
In this example neither the lapse nor the shift enter linearly, and one might worry on the loss of the constraint to project out the BD ghost. However, upon solving for the shift and substituting back into the Hamiltonian (this is possible since the lapse is not dynamical), we get
( ) &tidle; --γij𝒞 &tidle;i𝒞&tidle;j-- ℋ = N 𝒞0(γ, p) − 2m2V (γ, p) , (7.12 )
and the lapse now appears as a Lagrange multiplier generating a constraint, even though it was not linear in (7.10*). This could have been seen more easily, without the need to explicitly integrating out the shift by computing the Hessian
--∂2-ℋ--- 2---∂2𝒰--- L μν = ∂N μ∂N ν = m ∂N μ ∂N ν. (7.13 )
In the example (7.10*), one has
m2V (γ, p)( N 2 − N N ) L μν = -----3---- i 2 i =⇒ det (Lμν) = 0. (7.14 ) N − N Nj N γij
The Hessian cannot be inverted, which means that the equations of motion cannot be solved for all the shift and the lapse. Instead, one of these ought to be solved for the three-dimensional phase space variables which corresponds to the primary second-class constraint. Note that this constraint is not associated with a symmetry in this case and while the Hamiltonian is then pure constraint in this toy example, it will not be in general.

Finally, one could also have deduce the existence of a constraint by performing the linear change of variable

Ni Ni → ni = --, (7.15 ) N
in terms of which the Hamiltonian is then explicitly linear in the lapse,
( i j) &tidle; i &tidle; 2 γijn-n- ℋ = N 𝒞0(γ,p) + n 𝒞i(γ, p) + m V (γ,p) 2 , (7.16 )
and generates a constraint that can be read for {ni,γij,pij}.

Condition to evade the ghost

To summarize, the condition to eliminate (at least half of) the BD ghost is that the det of the Hessian (7.13*) L μν 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 ni related to the original one as follows N i = f0(γ,n ) + N f1(γ, n), such that the Hamiltonian takes the following factorizable form

ℋ = (𝒜1 (γ,p) + N 𝒞1(γ,p))ℱ (γ,p,n ) + (𝒜2 (γ,p) + N 𝒞2(γ,p)). (7.17 )
In this form, the equation of motion for the shift is manifestly independent of the lapse and integrating over the shift ni manifestly keeps the Hamiltonian linear in the lapse and has the constraint i 𝒞1(γ,p)ℱ (γ,p,n (γ)) + 𝒞2(γ,p) = 0. However, such a field redefinition has not (yet) been found. Instead, the new shift i n found below does the next best thing (which is entirely sufficient) of a. 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.

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 fμν in its own ADM decomposition, while keep the dynamical metric g μν in its original ADM form (since we work in unitary gauge, we may not simplify the metric further),

μ ν 2 2 ( i i ) ( j j ) gμν dx dx = − N dt + γij (dx + N dt) (dx + N dt ) (7.18 ) fμν dx μdx ν = − ¯𝒩 2dt2 + ¯fij dxi + 𝒩¯idt dxj + 𝒩¯ j dt , (7.19 )
and denote again by ij p the conjugate momentum associated with γij. ¯fij is not dynamical in massive gravity so there is no conjugate momenta associated with it. The bars on the reference metric are there to denote that these quantities are parameters of the theory and not dynamical variables, although the proof for a dynamical reference metric and multi-gravity works equally well, this is performed in Section 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 N i in terms of ni[295*, 296*]

i i i i ( i i) j N → n defined as N − 𝒩¯ = 𝒩¯ δj + N D j n , (7.20 )
where the matrix Di j satisfies the following relation
Di Dk = (P− 1)iγk ℓf ¯ , (7.21 ) k j k ℓj
i i i ¯ ℓ k ¯ ℓ i Pj = δj + (n fjℓn − n fkℓn δj). (7.22 )
In what follows we use the definition
&tidle; i i D j = κD j, (7.23 )
∘ ----------- κ = 1 − ninj ¯fij. (7.24 )

The field redefinition naturally involves a square root through the expression of the matrix D 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 𝒦 μν should exist, which in turns imply that the square root in the definition of Di j 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 then

ℋmGR = ℋGR + m2 𝒰 ( ¯ i ( ¯ i i) j) = N ℛ0 (γ, p) + 𝒩 + 𝒩 δj + N D j n ℛi(γ,p ) (7.25 ) +m2 𝒰 (γ,N i(n),N ),
where 𝒰 includes the new contributions from the mass term. i 𝒰 (γ,N ,N ) is neither linear in the lapse N, nor in the shift i N. There is actually no choice of potential 𝒰 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 ni, the non-linearities in the shift absorb all the original non-linearities in the lapse and 𝒰(γ,ni,N ). In itself this is not sufficient to prove the presence of a constraint, as the integration over the shift i n could in turn lead to higher order lapse in the Hamiltonian,
𝒰 (γ,N i(nj),N ) = N 𝒰0(γ,nj) + 𝒩¯𝒰1 (γ, nj), (7.26 )
M 2Pl√--∑3 (4 − n)βn i 𝒰0 = − -4-- γ ----n!----ℒn[ &tidle;D j] (7.27 ) n=0 M 2Pl√--( i j 𝒰1 = − ---- γ 3!β1κ + 2β2D jP i (7.28 ) 4[ ] ) 2 ∘ -- + β3κ 2D [kni] ¯fijDj nℓ + D [iDj ] − M-Plβ4 ¯f, k ℓ i j 4
where the β’s are expressed in terms of the α’s as in (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 N and impose the following relations in terms of ¯ni = njf¯ij,

[ ( ) ] m2 √ γ- 3!β1 ¯ni + 4β2 &tidle;Dj ¯ni] + β3D&tidle;[jj D&tidle;k ]k¯ni − 2 &tidle;Dki]¯nk = κℛi (γ,p), (7.29 ) [j
which entirely fixes the three shifts ni in terms of γij and pij as well as the reference metric ¯fij (note that 𝒩¯i entirely disappears from these equations of motion).

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 ni 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*),

i j 2 𝒞0 = ℛ0 (γ,p) + D jn ℛi(γ,p ) + m 𝒰0(γ, n(γ,p)) ≈ 0, (7.30 )
where the symbol “≈” means on the constraint surface. The existence of this primary constraint is sufficient to infer the absence of BD ghost. If we were dealing with a generic system (which could allow for some spontaneous parity violation), it could still be in principle that there are no secondary constraints associated with 𝒞0 = 0 and the theory propagates 5.5 physical degrees of freedom (11 dofs in phase space). However, physically this never happens in the theory of gravity we are dealing with preserves parity and is Lorentz invariant. Indeed, to have 5.5 physical degrees of freedom, one of the variables should have an equation of motion which is linear in time derivatives. Lorentz invariance then implies that it must also be linear in space derivatives which would then violate parity. However, this is only an intuitive argument and the real proof is presented below. Indeed, it ghost-free massive gravity admits a secondary constraint which was explicitly found in [294*].

Secondary constraint

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,

𝒞 ≡ -d-𝒞 = {𝒞 ,H } ≈ {𝒞 ,H } ≈ 0, (7.31 ) 2 dt 0 0 mGR 0 1
with H = ∫ d3xℋ mGR,1 mGR,1 and
( i i ) 2 ℋ1 = 𝒩¯ + 𝒩¯n (γ,p) ℛi + m 𝒩¯𝒰1(γ, n(γ,p)). (7.32 )
Finding the precise form of this secondary constraint requires a very careful analysis of the Poisson bracket algebra of this system. This formidable task lead to some confusions at first (see Refs. [345]) but was then successfully derived in [294*] (see also [258, 259] and [343]). Deriving the whole set of Poisson brackets is beyond the scope of this review and we simply give the expression for the secondary constraint,
(¯ i ¯ i) 2 ¯ ( ℓ ) ij 𝒞2 ≡ 𝒞0∇i 𝒩 +-𝒩 n + m 𝒩 (γijp ℓ − 2pij 𝒰-1 ) ( ) (7.33 ) +2m2 𝒩¯ √ γ∇i𝒰 i1jDjknk + ℛjDiknk − √γ ¯ℬji ∇i 𝒩¯j + 𝒩¯nj ( j k) ( i i) + ∇iℛ0 + ∇i ℛjD kn ¯𝒩 + 𝒩¯ n ,
where unless specified otherwise, all indices are raised and lowered with respect to the dynamical metric γij, and the covariant derivatives are also taken with respect to the same metric. We also define
ij 1 ∂𝒰1 𝒰 1 = √------- (7.34 ) γ ∂γij ( ) M-P2l[ −1 k β3- ℬ¯ij = − 4 (D&tidle; ) j ¯fik 3 β1ℒ0[ &tidle;D ] + 2β2ℒ1 [ &tidle;D] + 2 ℒ2 [ &tidle;D] (7.35 ) ] − β2f¯ij + 2 β3 ¯fi[k &tidle;Dkj] .

The important point to notice is that the secondary constraint (7.33*) only depends on the phase space variables ij γij,p and not on the lapse N. 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 𝒞2|𝒞0=0 ⁄= 0.).

Finally, we should also check that this secondary constraint is also maintained in time. This was performed [294*], by inspecting the condition

d ---𝒞2 = {𝒞2,HmGR } ≈ 0. (7.36 ) dt
This condition should be satisfied without further constraining the phase space variables, which would otherwise imply that fewer than five degrees of freedom are propagating. Since five fully fledged dofs are propagating at the linearized level, the same must happen non-linearly.17 Rather than a constraint on ij {γij,p }, (7.36*) must be solved for the lapse. This is only possible if both the two following conditions are satisfied
{𝒞2(x),ℋ1 (y)} ≈∕ 0 and {𝒞2(x ),𝒞0(y )} ≈∕ 0. (7.37 )
As shown in [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.

Secondary constraints in bi- and multi-gravity

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. [48] for an enlightening discussion.

7.2 Absence of ghost in the Stückelberg language

7.2.1 Physical degrees of freedom

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 ϕa 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 [14] for a theory of two Stückelberg fields.

Stated more precisely, in the Stückelberg language beyond the DL, if ℰa is the equation of motion with respect to the field ϕa, the correct requirement for the absence of ghost is that the Hessian 𝒜 ab defined as

δℰa- --δ2ℒ-- 𝒜ab = − ¨b = ˙a ˙b (7.38 ) δϕ δϕ δϕ
be not invertible, so that the dynamics of not all four Stückelberg may be derived from it. This is the case if
det (𝒜ab ) = 0, (7.39 )
as first explained in Ref. [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.

7.2.2 Two-dimensional case

Consider massive gravity on a two-dimensional space-time, 2 ds2 = − N 2dt2 + γ (dx + Nx dt), with the two Stückelberg fields ϕ0,1 [145*]. In this case the graviton potential can only have one independent non-trivial term, (excluding the tadpole),

2 𝒰 = − M-PlN √ γ-(ℒ (𝒦 ) + 1 ). (7.40 ) 4 2
In light-cone coordinates,
ϕ± = ϕ0 ± ϕ1 (7.41 ) 1 1 [ ] 𝒟± = √--∂x ± ---∂t − Nx∂x , (7.42 ) γ N
the potential is thus
M 2Pl √ -∘ ---------------- 𝒰 = − ----N γ (𝒟 − ϕ − )(𝒟+ ϕ+). (7.43 ) 4
The Hessian of this Lagrangian with respect to the two Stückelberg fields ± ϕ is then
δ2ℒmGR 2 δ2𝒰 𝒜ab = --˙a--˙b- = − m -˙a--˙b (δϕ δϕ δϕ δϕ ) (𝒟 − ϕ − )2 − (𝒟 − ϕ − )(𝒟+ ϕ+ ) ∝ − (𝒟− ϕ− )(𝒟+ ϕ+ ) (𝒟+ ϕ+)2 , (7.44 )
and is clearly non-invertible, which shows that not both Stückelberg fields are dynamical. In this special case, the Hamiltonian is actually pure constraint as shown in [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 λ μν. In this two-dimensional example, the mass term (7.43*) can be rewritten with the help of an auxiliary non-dynamical variable λ as

M 2 √ --( 1 ) 𝒰 = − ---PlN γ λ + ---(𝒟 − ϕ− )(𝒟+ ϕ+ ) . (7.45 ) 4 2λ
A similar trick will be used in the full proof.

7.2.3 Full proof

The full proof in the minimal model (corresponding to α2 = 1 and α3 = − 2∕3 and α4 = 1∕6 in (6.3*) or β2 = β3 = 0 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 [15] for a fully-fledged derivation.

Using a set of auxiliary variables λab (with λab = λba, 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],

M 2√ --- ( ) 𝒰 = ---Pl − g [λ ] + [λ−1 ⋅ Y ] , (7.46 ) 4
where the matrix Y has been defined in (2.77*) and is equivalent to X used previously. Upon integration over the auxiliary variable λ 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 Y,
Y ab = gμν∂μϕa ∂νϕcfcb = − 𝒟t ϕa𝒟t ϕcfcb + Vba, (7.47 )
𝒟 = 1-(∂ − N i∂) (7.48 ) t N t i V a = γij∂iϕa∂jϕcfcb. (7.49 ) b
Since the matrix V uses a projection along the 3 spatial directions it is genuinely a rank-3 matrix rather than rank 4. This implies that detV = 0. Notice that we consider an arbitrary reference metric f, as the proof does not depend on it and can be done for any f at no extra cost [297*]. The canonical momenta conjugate to ϕa is given by
1 −1 b pa = -α&tidle;(λ )ab𝒟0ϕ , (7.50 ) 2
√ -- &tidle;α = 2M 2Plm2 γ. (7.51 )
In terms of these conjugate momenta, the equations of motion with respect to ab λ then imposes the relation (after multiplying with the matrix19 α λ on both side),
λacC λbd = V ab, (7.52 ) ab
with the matrix Cab defined as
Cab = &tidle;α2fab + papb. (7.53 )
Since det V = 0, as mentioned previously, the equation of motion (7.52*) is only consistent if we also have det C = 0. This is the first constraint found in [297*] which is already sufficient to remove (half) the BD ghost,
detC-- 2 −1 ab 𝒞1 ≡ det f = α&tidle; + (f ) papb = 0, (7.54 )
which is the primary constraint on a subset of physical phase space variables {γij,pa}, (by construction det f ⁄= 0). The secondary constraint is then derived by commuting 𝒞 1 with the Hamiltonian. Following the derivation of [297], we get on the constraint surface
∫ 𝒞 = -1---d𝒞1 = -1--- dy{𝒞 (y),H (x)} (7.55 ) 2 &tidle;α2N dt &tidle;α2N 1 − 1∕2 ij &tidle;α − 1 a ij (f) b ∝ − γ γijπ − 2 -(λ )ab∂iϕ γ ∇ j p (7.56 ) γ ≡ 0,
where πij is the momentum conjugate associated with γij, and ∇ (f) is the covariant derivative associated with f.

7.2.4 Stückelberg method on arbitrary backgrounds

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 [10] 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 {x } where the Stückelberg background fields are given by ϕa(x) = x μδa μ), the background metric is given by bg a b gμν = eμ(x)eν(x)ηab, and the background Stückelberg fields are given by ϕabg(x ) = xa − Aabg(x).

We now add fluctuations about that background,

ϕa = ϕa − aa = xa − Aa (7.57 ) bg gμν = gbμgν + hμν, (7.58 )
with a a a A = A bg + a.

Flat background metric

First, note that if we consider a flat background metric to start with, then at zeroth order in h, the ghost-free potential is of the form [400*], (this can also be seen from [238*, 419*])

1 ℒA = − --F F (1 + ∂A + ⋅⋅⋅), (7.59 ) 4
with Fab = ∂aAb − ∂bAa. This means that for a symmetric Stückelberg background configuration, i.e., if the matrix ∂μ ϕa bg is symmetric, then F bg = 0 ab, and at quadratic order in the fluctuation a, the action has a U (1)-symmetry. This symmetry is lost non-linearly, but is still relevant when looking at quadratic fluctuations about arbitrary backgrounds. Now using the split about the background, Aa = Aa + aa bg, this means that up to quadratic order in the fluctuations aa, the action at zeroth order in the metric fluctuation is of the form [400]
ℒ (2)= ¯B μανβf f , (7.60 ) a μν αβ
with fμν = ∂ μaν − ∂νaμ and B¯μανβ is a set of constant coefficients which depends on Aabg. This quadratic action has an accidental U (1)-symmetry which is responsible for projecting out one of the four dofs naively present in the four Stückelberg fluctuations aa. Had we considered any other potential term, the U (1) symmetry would have been generically lost and all four Stückelberg fields would have been dynamical.

Non-symmetric background Stückelberg

If the background configuration is not symmetric, then at every point one needs to perform first an internal Lorentz transformation Λ(x) 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 &tidle;aμ = Λ μ(x)a ν ν. 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 f replaced by f → &tidle;f + (∂Λ)Λ −1&tidle;a, (with f&tidle;μν = ∂ μ&tidle;aν − ∂ν&tidle;aμ). Interestingly, the Lorentz boost (∂Λ )Λ−1 now plays the role of a mass term for what looks like a gauge field &tidle;a. This mass term breaks the U (1) symmetry, but there is still no kinetic term for &tidle;a0, 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).

Next, we consider the mixing with metric fluctuation h while still assuming zero curvature. At linear order in h, the ghost-free potential, (6.3*) goes as follows

3 (2) μν ∑ (n) ℒ Ah = h cnX μν + hF (∂A + ⋅⋅⋅), (7.61 ) n=1
where the tensors X (μnν) are similar to the ones found in the decoupling limit, but now expressed in terms of the symmetric full four Stückelberg fields rather than just π, i.e., replacing Πμν by ∂ μAν + ∂νA μ in the respective expressions (8.29*), (8.30*) and (8.31*) for (1,2,3) X μν. Starting with the symmetric configuration for the Stückelberg fields, then since we are working at the quadratic level in perturbations, one of the Aμ in the X (μnν) is taken to be the fluctuation a μ, while the others are taken to be the background field Abg μ. As a result in the first terms in hX in (7.61*) ∂0a0 cannot come at the same time as h 00 or h 0i, and we can thus integrate by parts the time derivative acting on any a0, leading to a harmless first time derivative on hij, and no time evolution for a0.

As for the second type of term in (7.61*), since F = 0 on the background field Abμg, the second type of terms is forced to be proportional to f μν and cannot involve any ∂ a 0 0 at all. As a result a0 is not dynamical, which ensures that the theory is free from the BD ghost.

This part of the argument generalizes easily for non symmetric background Stückelberg configurations, and the same replacement f → &tidle;f + (∂Λ)Λ −1&tidle;a still ensures that a&tidle; 0 acquires no dynamics from (7.61*).

Background curvature

Finally, to complete the argument, we consider the effect from background curvature, then gbμgν ⁄= ημν, with gbg = ea (x )eb(x ) μν μ ν. 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 bg gμν with the flat reference metric ημν (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, ηab = eμaeνbgbμgν, in terms of the inverse vielbein, eμ ≡ (e− 1)μ a a. Then the building block of ghost-free massive gravity is the matrix 𝕏, defined previously as

( )μ β bg 𝕏 μν = g−1η ν = g μγ(e αa∂γϕa)(eb∂νϕb)gαβ. (7.62 )
As a result, the whole formalism derived previously is directly applicable with the only subtlety that the Stückelberg fields a ϕ should be replaced by their ‘vielbein-dependent’ counterparts, i.e., ∂μA ν → gbμgν − gbνgαeαa∂μϕa. In terms of the Stückelberg field fluctuation aa, this implies the replacement aa → ¯a μ = gbgeνaa μν a, and symbolically, f → ¯f + (∂Σ )Σ−1¯a, with Σ = g˙e. The situation is thus the same as when we were dealing with a non-symmetric Stückelberg background configuration, after integration by parts (which might involve curvature harmless contributions), the potential can be written in a way which never involves any time derivative on ¯a0. As a result, ¯aμ plays the role of an effective Proca vector field which only propagates three degrees of freedom, and this about any curved background metric. The beauty of this argument lies in the correct identification of the proper degrees of freedom when dealing with a curved background metric.

7.3 Absence of ghost in the vielbein formulation

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, eaμfbνηab = eaνfbμηab. For simplicity we specialize further to the case where faμ = δaμ, so that the symmetric vielbein condition imposes eaμ = eμa. 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 d spacetime dimensions, there is a priori d(d + 1)∕2 independent components in the symmetric vielbein.

Varying the action (6.1*) with respect to the vielbein leads to the modified Einstein equation,

m2 ( Ga = ta = − ---𝜀abcd 4c0eb ∧ ec ∧ ed + 3c1eb ∧ ec ∧ fd (7.63 ) 2 ) +2c2eb ∧ f c ∧ fd + c3fb ∧ fc ∧ f d , (7.64 )
with Ga = 𝜀abcdωbc ∧ ed. From the Bianchi identity, 𝒟Ga = dGa − ωbaGb, we infer the d constraints
𝒟t = dt − ωb t = 0, (7.65 ) a a ab
leading to d(d − 1)∕2 independent components in the vielbein. This is still one too many component, unless an additional constraint is found. The idea behind the proof in Ref. [171*], is then to use the Bianchi identities to infer an additional constraint of the form,
a a m ∧ Ga = m ∧ ta, (7.66 )
where ma is an appropriate one-form which depends on the specific coefficients of the theory. Such a constrain is present at the linear level for Fierz–Pauli massive gravity, and it was further shown in Ref. [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 cn to vanish but c1 ⁄= 0. In that case the Bianchi identity (7.65*) implies
b 𝒟ta = 0 =⇒ ω cb = 0, (7.67 )
where similarly as in (5.2*), the torsionless connection is given in term of the vielbein as
1 ωabμ = -ecμ(oabc − ocab− obc a), (7.68 ) 2
with oabc = 2eaμebν∂ [μeν]c. The Bianchi identity (7.67*) then implies eab∂[beaa] = 0, so that we obtain an extra constraint of the form (7.66*) with ma = ea. Ref. [171] derived similar constraints for other parameters of the theory.

7.4 Absence of ghosts in multi-gravity

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 d-dimensional Lorentz indices, A,B, ⋅⋅⋅ = 0,⋅⋅⋅,d − 1, while lowercase Latin indices represent the d − 1-dimensional Lorentz indices along the space directions after ADM decomposition, a,b,⋅⋅⋅ = 1, ⋅⋅⋅,d − 1. Greek indices represent d-dimensional spacetime indices μ, ν = 0,⋅⋅⋅,d − 1, while the ‘middle’ of the Latin alphabet indices i,j ⋅⋅⋅ represent pure space indices i,j ⋅⋅⋅ = 1,⋅⋅⋅,d − 1. Finally, capital indices label the metric and span over I,J, K, ⋅⋅⋅ = 1,⋅⋅⋅,N.

Let us start with N non-interacting spin-2 fields. The theory has then N copies of coordinate transformation invariance (the coordinate system associated with each metric can be changed separately), as well as N copies of Lorentz invariance. At this level may, for each vielbein e(J), J = 1,⋅⋅⋅,N we may use part of the Lorentz freedom to work in the upper triangular form for the vielbein,

( i a ) ( −1 ) e(J)Aμ = N (J) N (J)e(Ja) i , e(J)μA = N (iJ) −1 0 i , (7.69 ) 0 e(J)i − N (J)N e(J) a
leading to the standard ADM decomposition for the metric,
g(J)μν dxμ dxν = e(J)Aμe (J)BνηAB dxμ dxν 2 2 ( i i ) ( j j ) = − N (J) dt + γ(J)ij dx + N (J)dt dx + N (J) dt , (7.70 )
with the three-dimensional metric γ(J)ij = e(J)a e(J)b δab i j. Starting with non-interacting fields, we simply take N copies of the GR action,
∫ ∑N ∘ ------ LNGR = dt − g(J)R (J), (7.71 ) J=1
and the Hamiltonian in terms of the vielbein variables then takes the form (7.6*)
∫ N ( ) ℋ = ddx ∑ π i˙e a + N 𝒞 + N i 𝒞 − 1-λab 𝒫 , (7.72 ) NGR (J)a (J)i (J) (J)0 (J )(J)i 2 (J) (J)ab J=1
where π(J)a i is the conjugate momentum associated with the vielbein e(J)ai and the constraints 𝒞(J)0,i = 𝒞0,i(e(J),π(J)) are the ones mentioned previously in (7.6*) (now expressed in the vielbein variables) and are related to diffeomorphism invariance. In the vielbein language there is an addition d(d − 1)∕2 primary constraints for each vielbein field
i 𝒫(J)ab = e(J)[ai π (J)b], (7.73 )
related to the residual local Lorentz symmetry still present after fixing the upper triangular form for the vielbeins.

Now rather than setting part of the N Lorentz frames to be on the upper diagonal form for all the N vielbein (7.69*) we only use one Lorentz boost to set one of the vielbein in that form, say e(1), and ‘unboost’ the N − 1 other frames, so that for any of the other vielbein one has

( i a a i b a) e(J)Aμ = N (J)&tidle;γ (J) + N(Ja)e(J)ip(J)a N (J)p(J) + Nb(J)e(Ja) iS (J)b (7.74 ) e(J) i e(J)iS(J)b S a = δa + &tidle;γ− 1pa p (7.75 ) (J)b ∘b ---(J)-(J)-(J)b &tidle;γ(J ) = 1 + p (J)apa (7.76 ) (J)
where p(J)a is the boost that would bring that vielbein in the upper diagonal form.

We now consider arbitrary interactions between the N fields of the form (6.1*),

N ∑ a1 ad LNint = αJ1,⋅⋅⋅,Jd𝜀a1⋅⋅⋅ade(J1) ∧ ⋅⋅⋅ ∧ e(Jd), (7.77 ) J1,⋅⋅⋅,Jd=1
where for concreteness we assume d ≤ N, otherwise the formalism is exactly the same (there is some redundancy in this formulation, i.e., some interactions are repeated in this formulation, but this has no consequence for the argument). Since the vielbeins A e(J) 0 are linear in their respective shifts and lapse N (J),N i(J) and the vielbeins e(J)Ai do not depend any shift nor lapse, it is easy to see that the general set of interactions (7.77*) lead to a Hamiltonian which is also linear in every shift and lapse,
∑N ( ) ℋNint = N (J)𝒞in(Jt)(e,p) + N(iJ)𝒞i(ntJ)i(e,p) . (7.78 ) J=1
Indeed the wedge structure of (6.1*) or (7.77*) ensures that there is one and only one vielbein with time-like index e(J)A0 for every term a 𝜀a1⋅⋅⋅adea(1J1) ∧ ⋅⋅⋅ ∧ e(Jdd).

Notice that for the interactions, the terms 𝒞int0,i (J) can depend on all the N vielbeins e ′ (J) and all the N − 1 ‘boosts’ p(J′), (as mentioned previously, part of one Lorentz frame is set so that p (1) = 0 and e(1) is in the upper diagonal form). Following the procedure of [314], we can now solve for the N − 1 remaining boosts by using (N − 1) of the N shift equations of motion

𝒞 (J)i(e,π ) + 𝒞 (J)i(e, p) = 0 ∀ J = 1,⋅⋅⋅,N . (7.79 )
Now assuming that all N vielbein are interacting,20 (i.e., there is no vielbein e(J) 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 N − 1 boosts and can be solved for them without spoiling the linearity in any of the N lapses N (J). As a result, the N − 1 lapses N (J) for J = 2,⋅⋅⋅,N are Lagrange multiplier for (N − 1) first class constraints. The lapse N (1) for the first vielbein combines with the remaining shift i N(1) 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 d2 components in each of the N vielbein e a (J) i and associated conjugate momenta, that is a total of 2 2 × d × N phase space variables. We then have 2 × d(d − 1)∕2 × N constraints21 associated with the λab (J). There is one copy of diffeomorphism removing 2 × (d + 1) phase space dofs (with Lagrange multiplier N (1) and i N (1)) and (N − 1) additional first-class constraints with Lagrange multipliers N (J≥2) removing 2 × (N − 1) dofs. As a result we end up with

( ) 2 × d(d-−-1) × N − 2 × d(d −-1)× N − 2 × (d + 1) − 2 × (N − 1 ) 2 2 ( 2 ) = d N − 2N + d(N − 2) phase space dofs 1( 2 ) = 2 d N − 2N + d(N − 2) field space dofs (7.80 ) 1( ) = --d2 − d − 2 dofs for a massless spin -2 field 2 + 1-(d2 + d − 2) × (N − 1) dofs for (N − 1) a massive spin-2 fields, (7.81 ) 2
which is the correct counting in (d + 1) spacetime dimensions, and the theory is thus free of any BD ghost.
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