"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

5 Deconstruction

As for DGP and its extensions, to get some insight on how to construct a four-dimensional theory of single massive graviton, we can start with five-dimensional general relativity. This time, we consider the extra dimension to be compactified and of finite size R, with periodic boundary conditions. It is then natural to perform a Kaluza–Klein decomposition and to obtain a tower of Kaluza–Klein graviton mode in four dimensions. The zero mode is then massless and the higher modes are all massive with mass separation m = 1βˆ•R. Since the graviton mass is constant in this formalism we omit the subscript 0 in the rest of this review.

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 y by a series of N sites yj separated by a distance β„“ = Rβˆ•N. So that the five-dimensional metric is replaced by a set of N interacting metrics depending only on x.

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.

5.1 Formalism

5.1.1 Metric versus Einstein–Cartan formulation of GR

Before going further, let us first describe five-dimensional general relativity in its Einstein–Cartan formulation, where we introduce a set of vielbein ea A, so that the relation between the metric and the vielbein is simply,

gAB(x,y ) = eaA(x,y)ebB(x, y)ηab, (5.1 )
where, as mentioned previously, the capital Latin letters label five-dimensional spacetime indices, while letters a,b,c ⋅⋅⋅ label five-dimensional Lorentz indices.

Under the torsionless condition, de + ω ∧ e = 0, the antisymmetric spin connection ω, is uniquely determined in terms of the vielbeins

ab 1- c ab ab b a ωA = 2 eA(O c − O c − O c ), (5.2 )
with ab aA bB O c = 2e e ∂ [AeB ]c. In the Einstein–Cartan formulation of GR, we introduce a 2-form Riemann curvature,
ab ab a cb β„› = dω + ω c ∧ ω , (5.3 )
and up to boundary terms, the Einstein–Hilbert action is then given in the respective metric and the vielbein languages by (here in five dimensions for definiteness),
M 3 ∫ √ --- SE(5H)= --5- d4x dy − gR (5)[g] (5.4 ) 2 ∫ M 53 ab c d e = ------ πœ€abcdeβ„› ∧ e ∧ e ∧ e , (5.5 ) 2 × 3!
where R(5)[g] is the scalar curvature built out of the five-dimensional metric g αβ and M5 is the five-dimensional Planck scale.

The counting of the degrees of freedom in both languages is of course equivalent and goes as follows: In d-spacetime dimensions, the metric has d (d + 1)βˆ•2 independent components. Covariance removes 2d of them,10 which leads to 𝒩d = d(d − 3)βˆ•2 independent degrees of freedom. In four-dimensions, we recover the usual 𝒩4 = 2 independent polarizations for gravitational waves. In five-dimensions, this leads to 𝒩5 = 5 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 d + 1 dimensions have the same number of degrees of freedom as massive bosons in d dimensions – this counting does not directly apply to fermions).

In the Einstein–Cartan formulation, the counting goes as follows: The vielbein has d2 independent components. Covariance removes 2d of them, and the additional global Lorentz invariance removes an additional d(d − 1)βˆ•2, leading once again to a total of 𝒩 = d(d − 3)βˆ•2 d 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 ψ at the quadratic order is given by (see Ref. [392]),

∫ 1 [ i ←→ m ] SDirac = --πœ€abcd ea ∧ eb ∧ ec -¯ψ γdD ψ − --edψ¯ψ , (5.6 ) 3! 2 4
where the γa’s are the Dirac matrices and D represents the covariant derivative, D ψ = dψ − 1ωab [γ ,γ ]ψ 8 a b.

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.

5.1.2 Gauge-fixing

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

GAB (x,y) dxA dxB = dy2 + gμν(x,y )dxμ dxν, (5.7 )
meaning that the lapse is set to unity and the shift to zero. Notice that one could in principle only set the lapse to unity and keep the shift present throughout the discretization. From a four-dimensional point of view, the shift will then ‘morally’ play the role of the Stückelberg fields, however they do so only after a cumbersome field redefinition. So for sake of clarity and simplicity, in what follows we first gauge-fix the shift and then once the four-dimensional theory is obtained to restore gauge invariance by use of the Stückelberg trick presented previously.

In vielbein language, we fix the five-dimensional coordinate system and use four Lorentz transformations to set

( a μ) ea = eμdx , (5.8 ) dy
and use the remaining six Lorentz transformations to set
ab μ[a b] ωy = e ∂yeμ = 0. (5.9 )

In this gauge, the five-dimensional Einstein–Hilbert term (5.4*), (5.5*) is given by

∫ (5) M 35 4 √ ---( 2 2) SEH = ---- d x dy − g R [g] + [K ] − [K ] (5.10 ) 23 ∫ ( = M-5- πœ€ Rab ∧ ec ∧ ed − Ka ∧ Kb ∧ ec ∧ ed (5.11 ) 4 abcd a b c d) +2K ∧ ∂ye ∧ e ∧ e ∧ dy,
where R[g], is the four-dimensional curvature built out of the four-dimensional metric gμν, Rab is the 2-form curvature built out of the four-dimensional vielbein eaμ and its associated connection ωab = ωaμbdx μ, Rab = dωab + ωac ∧ ωcb, and Kνμ= gμαK αν is the extrinsic curvature,
1- a b K μν = 2 ∂ygμν = e(μ ∂yeν)ηab (5.12 ) Ka = eνaK μν. (5.13 ) μ

5.1.3 Discretization in the vielbein

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.

y `→ yj (5.14 ) ea(x, y) `→ e a(x) = ea(x, y ) (5.15 ) μ jμ ( μ j) ∂yeaμ(x, y) `→ mN ej+1aμ − ejaμ . (5.16 )
The gauge choice (5.9*) then implies
ωayb= eμ[a∂yebμ]= 0 `→ ej+1μ[aejbμ]= 0, (5.17 )
where the arrow `→ represents the deconstruction of five-dimensional gravity. We have also introduced the ‘truncation scale’, mN = N m = β„“−1 = N R −1, i.e., the scale of the highest mode in the discretized theory. After discretization, we see the Deser–van Nieuwenhuizen [187] 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 ωab = 0 y.

In terms of the extrinsic curvature, this implies

( ) Kaμ `→ mN ej+1aμ − ejaμ . (5.18 )
This can be written back in the metric language as follows
gμν(x,y ) `→ gj,μν(x ) = g μν(x, yj) (5.19 ) ( ( ∘ -------) μ) K μν `→ − mN 𝒦μν[gj,gj+1] ≡ − mN δμν − g−j1gj+1 , (5.20 ) ν
where the square root in the extrinsic curvature appears after converting back to the metric language. The square root exists as long as the metrics g j and g j+1 have the same signature and g− 1g j j+1 has positive eigenvalues so if both metrics were diagonal the ‘time’ direction associated with each metric would be the same, which is a meaningful requirement.

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 𝒦 μ[g ,g ] ν j j+1.

5.2 Ghost-free massive gravity

5.2.1 Simplest discretization

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 j = 1,2 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 e1 = e, e2 = f, and similarly for the metrics g = g 1,μν μν and g = f 2,μν μν. Out of the five-dimensional action for GR, we obtain the theory of massive gravity in four dimensions, (on site 1),

(5) (4) SEH `→ S mGR, (5.21 )
M 2 ∫ √ ---( ( )) S(m4G)R = --Pl d4x − g R [g] + m2 [𝒦 ]2 − [𝒦2 ] (5.22 ) 2 ∫ M P2l ( ab c d 2 abcd ) = --4- πœ€abcd R ∧ e ∧ e + m π’œ (e,f) , (5.23 )
with the mass term in the vielbein language
π’œabcd(e,f ) = (f a − ea) ∧ (fa − ea) ∧ ec ∧ ed, (5.24 )
or the mass term building block in the metric language,
μ μ (∘ -−-1-)μ 𝒦ν = δν − g f ν . (5.25 )
and we introduced the four-dimensional Planck scale, 2 3 ∫ M Pl = M 5 dy, where in this case we limit the integral about one site.

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 𝕏 introduced in Eq. (2.76*), we see that

μ μ (√ --)μ 𝒦 ν = δν − 𝕏 , (5.26 ) ν
or in other words,
𝕏 μν = δμν − 2𝒦μν + 𝒦 μα𝒦 αν, (5.27 )
and the mass term can be written as
m2M 2Pl√ ---( 2 2) β„’mass = − ---2--- − g [𝒦 ] − [𝒦] (5.28 ) 2 2 ---( √ -- √ -- ) = − m--M-Pl√ − g [(𝕀 − 𝕏)2] − [𝕀 − 𝕏]2 . (5.29 ) 2
This also a generalization of the Fierz–Pauli mass term, albeit more complicated on first sight than the ones considered in (2.83*) 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.

5.2.2 Generalized mass term

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 e(x,y) `→ e (x) j, we could consider the average value on one site, pondered with arbitrary weight r,

e(x, y) `→ rej + (1 − r)ej+1. (5.30 )
The mass term at one site is then generalized to
a b c d 2 abcd K ∧ K ∧ e ∧ e `→ m π’œ r,s (ej,ej+1), (5.31 )
and the most general action for massive gravity with reference vielbein f is thus12
∫ M 2Pl ( ab c d 2 abcd ) SmGR = -4-- πœ€abcd R ∧ e ∧ e + m π’œ r,s (e,f ) , (5.32 )
π’œabrc,sd(e,f) = (fa − ea) ∧ (f b − eb) ∧ ((1 − r)ec + rfc) ∧ ((1 − s)ed + sfd),
for any r,s ∈ ℝ.

In particular for the two-site case, this generates the two-parameter family of mass terms

abcd a b c d a b c d π’œ r,s (e,f ) = c0e ∧ e ∧ e ∧ e + c1e ∧ e ∧ e ∧ f (5.33 ) +c ea ∧ eb ∧ fc ∧ fd + c ea ∧ f b ∧ fc ∧ fd + cf a ∧ f b ∧ fc ∧ fd 2 3 4 ≡ π’œ1−r,1− s(f, e), (5.34 )
with c = (1 − s)(1 − r) 0, c = (− 2 + 3s + 3r − 4rs) 1, c = (1 − 3s − 3r + 6rs) 2, c = (r + s − 4rs) 3 and c4 = rs. This corresponds to the most general potential which, by construction, includes no cosmological constant nor tadpole. One can also always include a cosmological constant for such models, which would naturally arise from a cosmological constant in the five-dimensional picture.

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*].

In the metric Language, this corresponds to the following Lagrangian for dRGT massive gravity [144*], or its generalization to arbitrary reference metric [296*]

∫ ( ) M 2Pl 4 √--- m2 β„’mGR = ---- d x − g R + --- (β„’2 [𝒦] + α3β„’3 [𝒦 ] + α4 β„’4[𝒦 ]) , (5.35 ) 2 2
where the two parameters α3,4 are related to the two discretization parameters r,s as
α3 = r + s, and α4 = rs, (5.36 )
and for any tensor Q, we define the scalar β„’n symbolically as
β„’n[Q ] = πœ€πœ€Qn, (5.37 )
for any n = 0,⋅⋅⋅,d, where d is the number of spacetime dimensions. πœ€ is the Levi-Cevita antisymmetric symbol, so for instance in four dimensions, ′ ′ β„’2 [Q ] = πœ€μναβπœ€μ′ν′αβQ μμ Q νν = 2!([Q]2 − [Q2 ]), so we recover the mass term expressed in (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 d dimensions has (d − 2)-free parameters which are related to the (d − 2) discretization parameters.

5.3 Multi-gravity

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 [152], we consider N = 2M + 1 sites to start which leads to multi-gravity [314*], and then focus on the two-site case leading to bi-gravity [293*].

Starting with the five-dimensional action (5.12*) and applying the discretization procedure (5.31*) with π’œabcd r,s given in (5.33*), we get

N ∫ M-42∑ ( ab c d 2 abcd ) SNmGR = 4 πœ€abcd R [ej] ∧ ej ∧ ej + m Nπ’œ rj,sj(ej,ej+1) (5.38 ) j=1 ( ) M 2∑N ∫ ∘ ---- m2 ∑ 4 = --4- d4x − gj R [gj] +--N- α (jn)β„’n(𝒦j,j+1) , 2 j=1 2 n=0
with 2 3 3 M 4 = M 5R = M 5βˆ•m, (j) α2 = − 1βˆ•2, and in this deconstruction framework we obtain no cosmological constant nor tadpole, α(0j)= α (j1)= 0 at any site j, (but we keep them for generality). In the mass Lagrangian, we use the shorthand notation 𝒦 j,j+1 for the tensor 𝒦 μ[g ,g ] ν j j+1. This is a special case of multi-gravity presented in [314*] (see also [417] 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 2N-free parameters. For any fixed j, one has α(3j)= (rj + sj), and (j) α 4 = rjsj.

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,

N a -1--∑ a i2Nπj &tidle;eμ,n = √N--- eμ,je , (5.39 ) j=1
with the inverse map,
a 1 M∑ a −i2πn eμ,j = √---- &tidle;eμ,ne N . (5.40 ) N n=−M
In terms of the Fourier transform variables, the multi-gravity action then reads at the linear level
∑M [ ] β„’ = (∂&tidle;hn )(∂ &tidle;h−n) + m2 &tidle;hn&tidle;h −n + β„’int (5.41 ) n= −M n
with M −P1l &tidle;hμν,n = &tidle;eaμ,n&tidle;ebν,nηab − ημν and MPl represents the four-dimensional Planck scale, √ --- MPl = M4 βˆ• N. The reality condition on the vielbein imposes &tidle;en = &tidle;e∗ −n and similarly for &tidle;hn. The mass spectrum is then
( n ) mn = mN sin --- ≈ nm for n β‰ͺ N. (5.42 ) N

The counting of the degrees of freedom in multi-gravity goes as follows: the theory contains 2M 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 10M + 2 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 5N degrees of freedom of the four coordinates xa.

5.4 Bi-gravity

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.

Bi-gravity from deconstruction

Let us consider a two-site discretization with periodic boundary conditions, j = 1,2,3 with quantities at the site j = 3 being identified with that at the site j = 1. Similarly, as in Section 5.2 we denote by gμν = eaμebνηab and by fμν =ƒaμƒbνηab the metrics and vielbeins at the respective locations y1 and y2.

Then applying the discretization procedure highlighted in Eqs. (5.14*, 5.15*, 5.18*, 5.19* and 5.20*) and summing over the extra dimension, we obtain the bi-gravity action

2 ∫ --- M 2∫ ∘ ---- Sbi−gravity = M-Pl d4x√ − gR [g ] +--f- d4x − fR [f ] (5.43 ) 2 2 2 2 ∫ √ ---∑ 4 + M-Plm-- d4x − g αnβ„’n [𝒦 [g,f ]], 4 n=0
where 𝒦[g,f ] is given in (5.25*) and we use the notation Mg = MPl. We can equivalently well write the mass terms in terms of 𝒦 [f,g] rather than 𝒦[g,f ] as performed in (6.21*).

Notice that the most naive discretization procedure would lead to Mg = MPl = Mf, 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, β„’0[𝒦[g,f ]] corresponds to a cosmological constant for the metric gμν and the special combination ∑4 (− 1)nCn β„’ [𝒦 [g,f]] n=0 4 n, where the Cm n are the binomial coefficients is the cosmological constant for the metric fμν, so only β„’2,3,4 correspond to genuine interactions between the two metrics.

In the deconstruction framework, we naturally obtain α2 = 1 and no tadpole nor cosmological constant for either metrics.

Mass eigenstates

In this formulation of bi-gravity, both metrics g and f 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,

1 gμν = ημν + ----δgμν (5.44 ) MPl -1-- f μν = ημν + Mf δfμν. (5.45 )
The general mass spectrum about different backgrounds is richer and provided in [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 h, the action for bi-gravity reads (for α0 = α1 = 0 and α2 = − 1 βˆ•2),

∫ (2) 4 [ 1- μν ˆαβ 1- μν ˆαβ 1- 2 ( 2 2) ] Sbi−gravity = d x − 4δg β„°μν δgαβ − 4δf β„°μν δfα β − 8m eff h μν − h , (5.46 )
where all indices are raised and lowered with respect to the flat Minkowski metric and the Lichnerowicz operator β„°ˆμανβ was defined in (2.37*). We see appearing the Fierz–Pauli mass term combination h2μν − h2 introduced in (2.44*) for the massive field with the effective mass Meff defined as [293*]
M 2 = (M − 2+ M −2)−1 (5.47 ) eff Pl f 2 2M-P2l- m eff = m M 2 . (5.48 ) eff
The massive field h is given by
( ) h = M --1-δg − -1-δf = M (g − f ), (5.49 ) μν eff MPl μν Mf μν eff μν μν
while the other combination represents the massless field β„“μν,
( ) β„“μν = Me ff -1--δgμν + -1--δfμν , (5.50 ) Mf MPl
so that in terms of the light and heavy spin-2 fields (or more precisely in terms of the two mass eigenstates h and β„“), the quadratic action for bi-gravity reproduces that of a massless spin-2 field β„“ and a Fierz–Pauli massive spin-2 field h with mass meff,
(2) ∫ [ 1 [ 1 ( )] Sbi−gravity = d4x− -hμν β„°ˆμανβ+ -m2eff δαμ δβν − η αβημν h αβ (5.51 ) 4 ] 2 − 1β„“μν ˆβ„°αββ„“ . 4 μν αβ
As explained in [293*], in the case where there is a large Hierarchy between the two Planck scales M Pl and Mf, 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 Mf ≫ MPl, the massless particle is mainly given by δfμν and the massive one mainly by δgμν. This means that in the limit Mf → ∞ while keeping MPl fixed, we recover the theory of a massive gravity and a fully decoupled massless graviton as will be explained in Section 8.2.

5.5 Coupling to matter

So far we have only focus on an empty five-dimensional bulk with no matter. It is natural, though, to consider matter fields living in five dimensions, χ(x,y ) with Lagrangian (in the gauge choice (5.7*))

( ) √ --- 1- 2 1- 2 β„’matter = − g − 2(∂μχ ) − 2 (∂yχ) − V(χ ) , (5.52 )
in addition to arbitrary potentials (we focus on the case of a scalar field for simplicity, but the same philosophy can be applied to higher-spin species be it bosons or fermions). Then applying the same discretization scheme used for gravity, every matter field then comes in N copies
χ(x,y ) `→ χ (j)(x) = χ(x,yj), (5.53 )
for j = 1,⋅⋅⋅,N and each field (j) χ is coupled to the associated vielbein (j) e or metric (j) (j)a (j)b gμν = eμ eν ηab at the same site. In the discretization procedure, the gradient along the extra dimension yields a mixing (interaction) between fields located on neighboring sites,
∫ N 2 ∑ 2 (j+1) (j) 2 dy (∂yχ) `→ R m (χ (x) − χ (x)) , (5.54 ) j=1
(assuming again periodic boundary conditions, χ(N+1) = χ(1)). The discretization procedure could be also performed using a more complicated definition of the derivative along y involving more than two sites, which leads to further interactions between the different fields.

In the two-sight derivative formulation, the action for matter is then

∫ 1- 4 ∑ ∘ ---(j)( 1-(j)μν (j) (j) Smatter `→ m d x − g − 2g ∂μχ ∂νχ (5.55 ) j 1- 2 (j+1) (j) 2 (j) ) − 2m (χ − χ ) − V (χ ) .
The coupling to gauge fields or fermions can be derived in the same way, and the vielbein formalism makes it natural to extend the action (5.6*) to five dimensions and applying the discretization procedure. Interestingly, in the case of fermions, the fields ψ (j) and ψ (j+1) would not directly couple to one another, but they would couple to both the vielbein e(j) at the same site and the one e(j−1) 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).

5.6 No new kinetic interactions

In GR, diffeomorphism invariance uniquely fixes the kinetic term to be the Einstein–Hilbert one

√ --- β„’EH = − gR, (5.56 )
(see, for instance, Refs. [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 [383] 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 (der) β„’ 3 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),

(der) μνρσ μ′ν′ρ′σ′ β„’3 = πœ€ πœ€ hσσ′∂ ρhμμ′∂ ρ′hνν′. (5.57 )
So this new derivative interaction would be allowed for a theory of a massive spin-2 field which 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 β„’ (der) 3 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 β„’ (d3er) 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 √ --- ∗ μναβ − g𝒦 μν𝒦αβ R, where ∗ R 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 β„’ (der) 3 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 f (R ) – gravity (see for instance Refs. [89*, 354*, 46*] for f (R ) massive gravity and their implications to cosmology).

Such kinetic terms à la f(R ) 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 f (R ) gravity is seen to correspond to a theory of gravity with a scalar field, and the same result will hold in f(R ) massive gravity (in that case the scalar field couples non-trivially to the Stückelberg fields). As a result f(R) 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. [128] for a review on f (R )-gravity.)

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